From ca8acac1d6d0103e3952a37c022efd1a9376851c Mon Sep 17 00:00:00 2001
From: Michael Kirsten <kirsten@kit.edu>
Date: Thu, 21 Sep 2023 06:04:31 +0200
Subject: [PATCH 1/4] First investigation results

---
 virage/pom.xml                                |  2 +-
 .../virage/isabelle/IsabelleProofChecker.java | 51 ++++++++++++-------
 .../virage/isabelle/ScalaIsabelleFacade.java  |  2 +-
 .../main/resources/voting_context.template    |  4 +-
 4 files changed, 36 insertions(+), 23 deletions(-)

diff --git a/virage/pom.xml b/virage/pom.xml
index 8011f17..3166fa0 100644
--- a/virage/pom.xml
+++ b/virage/pom.xml
@@ -387,7 +387,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>versions-maven-plugin</artifactId>
-        <version>2.16.0</version>
+        <version>2.16.1</version>
         <configuration>
           <rulesUri>file:///${basedir}/rules.xml</rulesUri>
           <allowSnapshots>false</allowSnapshots>
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
index e25b05f..0365c7a 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
@@ -148,6 +148,16 @@ public final class IsabelleProofChecker {
      */
     private static final String PWD_CMD = "`pwd`";
 
+    /**
+     * The prefix for ad-hoc Isabelle sessions.
+     */
+    private static final String AD_HOC_SESSION_PREFIX = "ad_hoc_session_";
+
+    /**
+     * The pattern for simple date formats, intended for creating quasi unique file names.
+     */
+    private static final String DATE_FORMAT_PATTERN = "yyyyMMddHHmmss";
+
     /**
      * The session name variable.
      */
@@ -321,7 +331,8 @@ private String buildSessionRoot(final String theoryName, final File theory) {
         // rebuilding single sessions without triggering full rebuilds.
         // TODO: Is there a way to do it?
         final String localSessionName =
-                "ad_hoc_session_" + new SimpleDateFormat("yyyyMMddHHmmss").format(new Date());
+                AD_HOC_SESSION_PREFIX
+                + new SimpleDateFormat(DATE_FORMAT_PATTERN).format(new Date());
         String result = this.rootTemplate
                 .replace(SESSION_NAME_VAR, localSessionName).replace(THEORY_NAME_VAR, theoryName);
         result = result.replace(PARENT_NAME_VAR, this.parentName);
@@ -575,24 +586,26 @@ public Pair<Boolean, File> verifyTheoryFile(final File theory,
                     + "by employing different solvers.");
             final String errors = this.lastEvent.getValue("errors");
 
-            command = "purge_theories "  + sessionAndTheoriesCommand;
-            this.sendCommandAndWaitForOk(command);
-            final Pattern pattern = Pattern.compile("line=[0-9]+");
-            final Matcher matcher = pattern.matcher(errors);
-
-            if (matcher.find()) {
-                final String line = errors.substring(matcher.start(), matcher.end());
-                // Isabelle starts counting at 1.
-                final int lineNum = Integer.parseInt(line.split("=")[1]) - 1;
-                final File newFile = this.replaceSolver(theory, lineNum);
-                if (newFile != null) {
-                    // The content of the file has changed, and this can
-                    // only happen IsabelleUtils.SOLVERS.length times,
-                    // so the recursive call is fine.
-                    return this.verifyTheoryFile(newFile, framework);
-                } else {
-                    LOGGER.info("Automatic verification failed. "
-                            + "You might be able to fix the errors manually within Isabelle.");
+            if (errors != null) {
+                command = "purge_theories "  + sessionAndTheoriesCommand;
+                this.sendCommandAndWaitForOk(command);
+                final Pattern pattern = Pattern.compile("line=[0-9]+");
+                final Matcher matcher = pattern.matcher(errors);
+
+                if (matcher.find()) {
+                    final String line = errors.substring(matcher.start(), matcher.end());
+                    // Isabelle starts counting at 1.
+                    final int lineNum = Integer.parseInt(line.split("=")[1]) - 1;
+                    final File newFile = this.replaceSolver(theory, lineNum);
+                    if (newFile != null) {
+                        // The content of the file has changed, and this can
+                        // only happen IsabelleUtils.SOLVERS.length times,
+                        // so the recursive call is fine.
+                        return this.verifyTheoryFile(newFile, framework);
+                    } else {
+                        LOGGER.info("Automatic verification failed. "
+                                + "You might be able to fix the errors manually within Isabelle.");
+                    }
                 }
             }
             return new Pair<Boolean, File>(false, null);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
index 5fb4579..7c74696 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
@@ -155,7 +155,7 @@ public final class ScalaIsabelleFacade {
      * The theory-name function.
      */
     private static final String THY_NAME =
-            "Context" + IsabelleUtils.THEORY_NAME_SEPARATOR + "theory_name";
+            "Context" + IsabelleUtils.THEORY_NAME_SEPARATOR + "theory_base_name";
 
     /**
      * The specifications-of function.
diff --git a/virage/src/main/resources/voting_context.template b/virage/src/main/resources/voting_context.template
index 41f1db2..fbac216 100644
--- a/virage/src/main/resources/voting_context.template
+++ b/virage/src/main/resources/voting_context.template
@@ -4,9 +4,9 @@ import scala.language.postfixOps
 import scala.io.Source
 
 /* ENUM
-class Bounded[A](entities: List[A]) extends Enum.`enum`[A] {
+class Bounded[A](entities: List[A]) extends Enum.`enuma`[A] {
 
-  val `Enum.enum`: List[A] = entities
+  val `Enum.enuma`: List[A] = entities
 
   val `Enum.enum_all`: (A => Boolean) => Boolean = (p => entities.forall(p))
 

From 25e78c1d38a3e6f4c7de457b5cb3028179318299 Mon Sep 17 00:00:00 2001
From: Michael Kirsten <kirsten@kit.edu>
Date: Mon, 25 Sep 2023 20:00:23 +0200
Subject: [PATCH 2/4] Made code and proof generation work for simple voting
 rules, still need to inspect more complex ones

---
 virage/.gitignore                             |    5 +
 .../kit/kastel/formal/util/StringUtils.java   |   43 +
 .../formal/virage/beast/CCodeGenerator.java   |    4 +-
 .../core/VirageCommandLineInterface.java      |    6 +-
 .../isabelle/IsabelleCodeGenerator.java       |   36 +-
 .../isabelle/IsabelleFrameworkExtractor.java  |   20 +-
 .../isabelle/IsabelleTheoryGenerator.java     |   49 +-
 .../formal/virage/isabelle/IsabelleUtils.java |   15 +
 .../virage/isabelle/ScalaIsabelleFacade.java  |    2 +-
 .../jobs/VirageAnalyzeAllPropertiesJob.java   |    6 +-
 .../jobs/VirageIsabelleGenerateScalaJob.java  |    3 +-
 .../formal/virage/types/CompositionProof.java |   18 +-
 .../resources/c_implementations/wrapper.c     |    4 +-
 virage/src/main/resources/code_file.template  |    2 +-
 .../formal/beast/highlevel/javafx/BEAST.fxml  |   16 +-
 .../javafx/bundles/LangBundle_de.properties   |    4 +-
 .../javafx/bundles/LangBundle_en.properties   |    4 +-
 .../main/resources/voting_context.template    |   54 +-
 .../resources/c_implementations/wrapper.c     |    4 +-
 virage/src/test/resources/instances/README.md |    4 +-
 .../output/document/Aggregator.tex            |  159 -
 .../output/document/Blacks_Rule.tex           |  111 -
 .../output/document/Borda_Module.tex          |  131 -
 .../output/document/Borda_Rule.tex            |  116 -
 .../output/document/Classic_Nanson_Rule.tex   |  117 -
 .../output/document/Condorcet_Module.tex      |  275 --
 .../output/document/Copeland_Module.tex       |  917 ----
 .../output/document/Copeland_Rule.tex         |  179 -
 .../output/document/Defer_Equal_Condition.tex |  101 -
 .../output/document/Defer_Module.tex          |  222 -
 .../document/Defer_One_Loop_Composition.tex   |  130 -
 .../document/Drop_And_Pass_Compatibility.tex  |  459 --
 .../output/document/Drop_Module.tex           |  358 --
 .../output/document/Elect_Composition.tex     | 1049 ----
 .../output/document/Elect_Module.tex          |  190 -
 .../output/document/Electoral_Module.tex      | 3358 -------------
 .../output/document/Elimination_Module.tex    | 1358 ------
 .../output/document/Evaluation_Function.tex   |  344 --
 .../output/document/Loop_Composition.tex      | 2502 ----------
 .../output/document/Maximum_Aggregator.tex    |  561 ---
 .../document/Maximum_Parallel_Composition.tex | 1960 --------
 .../output/document/Minimax_Module.tex        |  690 ---
 .../output/document/Minimax_Rule.tex          |  177 -
 .../output/document/Nanson_Baldwin_Rule.tex   |  113 -
 .../document/Pairwise_Majority_Rule.tex       |  197 -
 .../output/document/Parallel_Composition.tex  |  412 --
 .../output/document/Pass_Module.tex           | 1466 ------
 .../output/document/Plurality_Module.tex      | 1263 -----
 .../output/document/Preference_Relation.tex   | 4290 -----------------
 .../old_theories/output/document/Profile.tex  | 1906 --------
 .../old_theories/output/document/Result.tex   |  469 --
 .../output/document/Revision_Composition.tex  |  470 --
 .../output/document/Schwartz_Rule.tex         |  115 -
 .../document/Sequential_Composition.tex       | 3359 -------------
 .../Sequential_Majority_Comparison.tex        | 1331 -----
 .../output/document/Termination_Condition.tex |   99 -
 .../old_theories/output/document/comment.sty  |  278 --
 .../old_theories/output/document/isabelle.sty |  267 -
 .../output/document/isabellesym.sty           |  427 --
 .../output/document/isabelletags.sty          |    1 -
 .../old_theories/output/document/pdfsetup.sty |    7 -
 .../output/document/railsetup.sty             | 1202 -----
 .../old_theories/output/document/root.bib     |   71 -
 .../old_theories/output/document/root.tex     |   56 -
 .../old_theories/output/document/session.tex  |   36 -
 .../old_theories/output/document/settings.tex |   30 -
 .../Basic_Modules/Plurality_Module.thy        |   19 -
 .../#Preference_Relation.thy#                 | 1115 -----
 .../Social_Choice_Types/#Profile.thy#         |  853 ----
 .../theories/output/document/Aggregator.tex   |  133 -
 .../output/document/Aggregator_Facts.tex      |  258 -
 .../output/document/Aggregator_Properties.tex |   74 -
 .../theories/output/document/Blacks_Rule.tex  |  103 -
 .../theories/output/document/Borda_Rule.tex   |  110 -
 .../output/document/Classic_Nanson_Rule.tex   |  107 -
 .../Composite_Elimination_Modules.tex         | 1019 ----
 .../output/document/Composite_Structures.tex  | 1266 -----
 .../output/document/Condorcet_Consistency.tex |   63 -
 .../output/document/Condorcet_Facts.tex       |  795 ---
 .../output/document/Condorcet_Properties.tex  |   93 -
 .../output/document/Condorcet_Rules.tex       | 1954 --------
 .../output/document/Copeland_Rule.tex         |  162 -
 .../output/document/Defer_Equal_Condition.tex |  101 -
 .../theories/output/document/Defer_Module.tex |  150 -
 .../document/Disjoint_Compatibility.tex       |   70 -
 .../document/Disjoint_Compatibility_Facts.tex |  275 --
 .../document/Disjoint_Compatibility_Rules.tex |  521 --
 .../theories/output/document/Drop_Module.tex  |  272 --
 .../theories/output/document/Elect_Module.tex |  147 -
 .../output/document/Electoral_Module.tex      | 1952 --------
 .../output/document/Elimination_Module.tex    |  596 ---
 .../output/document/Evaluation_Function.tex   |   97 -
 .../theories/output/document/Homogeneity.tex  |   69 -
 .../theories/output/document/Indep_Of_Alt.tex |   60 -
 .../output/document/Loop_Composition.tex      |  806 ----
 .../output/document/Maximum_Aggregator.tex    |  343 --
 .../theories/output/document/Minimax_Rule.tex |  160 -
 .../output/document/Monotonicity_Facts.tex    |  923 ----
 .../document/Monotonicity_Properties.tex      |  108 -
 .../output/document/Monotonicity_Rules.tex    | 2710 -----------
 .../output/document/Nanson_Baldwin_Rule.tex   |  103 -
 .../document/Pairwise_Majority_Rule.tex       |  195 -
 .../output/document/Parallel_Composition.tex  |  214 -
 .../theories/output/document/Pass_Module.tex  |  270 --
 .../output/document/Plurality_Module.tex      |  257 -
 .../output/document/Preference_Relation.tex   | 4290 -----------------
 .../theories/output/document/Profile.tex      | 2959 ------------
 .../theories/output/document/Result.tex       |  469 --
 .../theories/output/document/Result_Facts.tex | 2293 ---------
 .../output/document/Result_Properties.tex     |  350 --
 .../theories/output/document/Result_Rules.tex | 2466 ----------
 .../output/document/Revision_Composition.tex  |  233 -
 .../output/document/Schwartz_Rule.tex         |  107 -
 .../document/Sequential_Composition.tex       | 1373 ------
 .../Sequential_Majority_Comparison.tex        | 1341 ------
 .../output/document/Termination_Condition.tex |   99 -
 .../output/document/Weak_Monotonicity.tex     |   62 -
 .../theories/output/document/comment.sty      |  278 --
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 .../theories/output/document/isabelletags.sty |    1 -
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 .../theories/output/document/root.tex         |   56 -
 .../theories/output/document/session.tex      |   48 -
 .../theories/output/document/settings.tex     |   30 -
 127 files changed, 184 insertions(+), 70396 deletions(-)
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 delete mode 100644 virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty
 delete mode 100644 virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty
 delete mode 100644 virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib
 delete mode 100644 virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex
 delete mode 100644 virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex
 delete mode 100644 virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex

diff --git a/virage/.gitignore b/virage/.gitignore
index 5216f15..0f6ca3f 100644
--- a/virage/.gitignore
+++ b/virage/.gitignore
@@ -4,4 +4,9 @@
 /.classpath
 /.project
 /.settings/
+/.bloop/
+/.metals/
+/.vscode/
 /src/main/resources/generated/
+/.apt_generated/
+/.apt_generated_tests/
diff --git a/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java b/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
index 2e4e9fd..c151a63 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
@@ -122,6 +122,16 @@ public final class StringUtils {
      */
     public static final int TEN = 10;
 
+    /**
+     * An opening bracket sign.
+     */
+    private static final String OPENING_BRACKET = "]";
+
+    /**
+     * A closing bracket sign.
+     */
+    private static final String CLOSING_BRACKET = "[";
+
     /**
      * The disjunction (or) sign.
      */
@@ -372,6 +382,39 @@ public static String parenthesize2(final String... args) {
         return OPENING_PARENTHESIS + printCollection2(Arrays.asList(args)) + CLOSING_PARENTHESIS;
     }
 
+    /**
+     * Puts brackets around a comma-separated list of strings.
+     *
+     * @param args the string arguments to bracketize
+     * @return the bracketized string representation
+     */
+    public static String bracketize(final String... args) {
+        return OPENING_BRACKET + printCollection(Arrays.asList(args)) + CLOSING_BRACKET;
+    }
+
+    /**
+     * Puts parentheses around a comma-separated list of strings.
+     *
+     * @param args the string arguments to bracketize
+     * @return the bracketized string representation
+     */
+    public static String bracketize(final Collection<String> args) {
+        if (args == null || args.isEmpty()) {
+            return EMPTY;
+        }
+        return bracketize(args.toArray(new String[args.size()]));
+    }
+
+    /**
+     * Puts brackets around a space-separated list of strings.
+     *
+     * @param args the string arguments to bracketize
+     * @return the bracketized string representation
+     */
+    public static String bracketize2(final String... args) {
+        return OPENING_BRACKET + printCollection2(Arrays.asList(args)) + CLOSING_BRACKET;
+    }
+
     /**
      * Returns function string with a variable amount of argument.
      *
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
index 4c5059c..c0623c0 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
@@ -267,7 +267,7 @@ private List<String> findMissingTemplates(final DecompositionTree composition,
 
     /*public File generateAndCompileCCodeFromComposition(final String compositionString,
                                                          final int numVoters,
-                                                         final int numCandidates)
+                                                         final int numAlternatives)
                 throws IOException, InterruptedException, CompilationFailedException {
         final File votingRuleFile = this.getCCodeFromComposition(compositionString);
         String cFiles = StringUtils.EMPTY;
@@ -293,7 +293,7 @@ private List<String> findMissingTemplates(final DecompositionTree composition,
         final String gccCommand = ConfigReader.getInstance().getGccExecutable()
                 + StringUtils.prefixSpace(tmpVotingRuleFile.getAbsolutePath())
                 + cFiles + " -o " + exeFile.getAbsolutePath() + " -D V=" + numVoters
-                + " -D C=" + numCandidates;
+                + " -D C=" + numAlternatives;
         final Pair<String, String> output = ProcessUtils.runTerminatingProcess(gccCommand);
         if (!output.getFirstValue().isEmpty()) {
             LOGGER.warn(output.getFirstValue());
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
index e5fd323..1882a83 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
@@ -551,12 +551,10 @@ public String requestString(final String message) {
             this.displayInputMarker();
             final String input = this.scanner.nextLine();
             switch (input) {
-            case StringUtils.QUESTION:
-            case "h":
-            case HELP:
+            case StringUtils.QUESTION, "h", "H", HELP:
                 this.displayHelp();
                 break;
-            case EXIT:
+            case "x", "X", "q", "Q", EXIT:
                 this.core.submit(new VirageExitJob(this, 0));
                 break;
             default:
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
index 0b24c14..2dd2f3f 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
@@ -131,11 +131,6 @@ public final class IsabelleCodeGenerator {
      */
     private static final String PARENT_NAME_VAR = "$PARENT_NAME";
 
-    /**
-     * String for enumerable.
-     */
-    private static final String ENUM = "Enum";
-
     /**
      * Comment for enumerable.
      */
@@ -158,9 +153,11 @@ public final class IsabelleCodeGenerator {
     private static final String RELATION =
             StringUtils.parenthesize2(
                     ScalaIsabelleFacade.DEFAULT_VARIABLE + StringUtils.COLON,
-                    IsabelleUtils.SET + IsabelleUtils.THEORY_NAME_SEPARATOR + "set["
-                    + StringUtils.parenthesize(IsabelleUtils.DEFAULT_SET, IsabelleUtils.DEFAULT_SET)
-                    + "]") + StringUtils.COLON;
+                    IsabelleUtils.SET + IsabelleUtils.THEORY_NAME_SEPARATOR + IsabelleUtils.SET_FUNC
+                    + StringUtils.bracketize(
+                            StringUtils.parenthesize(
+                                    IsabelleUtils.DEFAULT_SET, IsabelleUtils.DEFAULT_SET)))
+            + StringUtils.COLON;
 
     /**
      * Option1 comment.
@@ -410,7 +407,7 @@ private File prepareVotingContext(final String theoryName, final String moduleNa
         String result =
                 IsabelleCodeGenerator.votingContextTemplate
                 .replace(THEORY_NAME_VAR, theoryName).replace(MODULE_NAME_VAR, moduleName);
-        final boolean containsEnum = code.contains(ENUM);
+        final boolean containsEnum = code.contains(IsabelleUtils.ENUM);
         final boolean containsEquality = code.contains(EQUALITY);
         final boolean requiresRelation = code.contains(RELATION);
         final List<String> parameters = new LinkedList<String>();
@@ -423,7 +420,7 @@ private File prepareVotingContext(final String theoryName, final String moduleNa
         if (containsEquality) {
             result = result.replace(COMMENT_OPEN + EQUALITY_COMMENT, StringUtils.EMPTY)
                     .replace(EQUALITY_COMMENT + COMMENT_CLOSE, StringUtils.EMPTY);
-            parameters.add(EQ_PARAMETER);
+            parameters.add(StringUtils.prefixSpace(EQ_PARAMETER));
         }
         if (requiresRelation) {
             result = result.replace(COMMENT_OPEN + OPTION2_COMMENT, StringUtils.EMPTY)
@@ -618,8 +615,10 @@ public File generateScalaCodeAndCompile(final File theory)
                         theory.getName().length() - (IsabelleUtils.DOT_THY.length()));
         final String sessionName = this.buildSessionRoot(theoryName, theory);
         final File codeFile = invokeScalaCodeFromIsabelleTheory(theory, sessionName, theoryName);
-        // First, try using implicit values only
-        File votingContext = generateVotingContextFile(theoryName, moduleName, codeFile, false);
+        // First, try using implicit values only.
+        final File cxtImplVals = generateVotingContextFile(theoryName, moduleName, codeFile, false);
+        // If implicit values did not work, try setting them explicitly.
+        final File cxtExplVals = generateVotingContextFile(theoryName, moduleName, codeFile, true);
         final String jarPath = codeFile.getParent() + File.separator + moduleName + DOT_JAR;
         final String isaExec;
         try {
@@ -627,10 +626,15 @@ public File generateScalaCodeAndCompile(final File theory)
         } catch (ExternalSoftwareUnavailableException e) {
             throw new CodeGenerationFailedException(e);
         }
-        int status = runScalaCompileProcess(codeFile, votingContext, isaExec, jarPath);
-        if (status != 0) { // Implicit values did not work, try setting them explicitly.
-            votingContext = generateVotingContextFile(theoryName, moduleName, codeFile, true);
-            status = runScalaCompileProcess(codeFile, votingContext, isaExec, jarPath);
+        int status = 1;
+        try {
+            status = runScalaCompileProcess(codeFile, cxtImplVals, isaExec, jarPath);
+        } catch (CodeGenerationFailedException e) {
+            LOGGER.info("Scala compilation with implicit values did not work,"
+                        + "now trying with explicit values.");
+        }
+        if (status != 0) {
+            status = runScalaCompileProcess(codeFile, cxtExplVals, isaExec, jarPath);
             if (status != 0) {
                 throw new CodeGenerationFailedException(
                         new CompilationFailedException(
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
index ce4eecc..78072c0 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
@@ -334,16 +334,16 @@ public FrameworkRepresentation extract(final String sessionDir, final String ses
             changesMade = false;
             for (final PrologPredicate alias: toReturn.keySet()) {
                 final List<PrologPredicate> aliasExpansions = toReturn.get(alias);
-                for (final PrologPredicate candidate: toReturn.keySet()) {
-                    if (candidate == alias) {
+                for (final PrologPredicate predicate: toReturn.keySet()) {
+                    if (predicate == alias) {
                         continue;
                     }
-                    final List<PrologPredicate> candidateExpansions = toReturn.get(candidate);
+                    final List<PrologPredicate> predicateExpansions = toReturn.get(predicate);
                     final List<PrologPredicate> newValues =
-                        computeAllExpansions(alias, aliasExpansions, candidateExpansions);
+                        computeAllExpansions(alias, aliasExpansions, predicateExpansions);
                     for (final PrologPredicate newValue: newValues) {
-                        if (!candidateExpansions.contains(newValue)) {
-                            candidateExpansions.add(newValue);
+                        if (!predicateExpansions.contains(newValue)) {
+                            predicateExpansions.add(newValue);
                             changesMade = true;
                         }
                     }
@@ -356,15 +356,15 @@ public FrameworkRepresentation extract(final String sessionDir, final String ses
     private static List<PrologPredicate>
                 computeAllExpansions(final PrologPredicate alias,
                                      final List<PrologPredicate> aliasExpansions,
-                                     final List<PrologPredicate> candidateExpansions)
+                                     final List<PrologPredicate> predicateExpansions)
                     throws ExternalSoftwareUnavailableException {
         final List<PrologPredicate> toReturn = new LinkedList<PrologPredicate>();
-        for (final PrologPredicate candidateExpansion: candidateExpansions) {
-            final List<PrologPredicate> children = candidateExpansion.getAllChildren();
+        for (final PrologPredicate predicateExpansion: predicateExpansions) {
+            final List<PrologPredicate> children = predicateExpansion.getAllChildren();
             for (final PrologPredicate child: children) {
                 if (child.getName().equals(alias.getName())) {
                     for (final PrologPredicate aliasExpansion: aliasExpansions) {
-                        toReturn.add(replaceAlias(candidateExpansion, child, aliasExpansion));
+                        toReturn.add(replaceAlias(predicateExpansion, child, aliasExpansion));
                     }
                 }
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
index 5567d87..a66df73 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
@@ -5,9 +5,11 @@
 import java.io.InputStream;
 import java.io.StringWriter;
 import java.nio.charset.StandardCharsets;
+import java.util.ArrayList;
 import java.util.Collections;
 import java.util.HashMap;
 import java.util.HashSet;
+import java.util.LinkedHashSet;
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Map;
@@ -269,26 +271,31 @@ private static String generateModuleParameterTypes(final List<String> modulePara
     private static Pair<String, String> buildModuleDef(final Map<String, Set<String>> moduleDefMap,
                                                        final String importsParam,
                                                        final String sessionName) {
-        String moduleDef = StringUtils.EMPTY;
-        final StringBuilder imports = new StringBuilder(importsParam);
-        // There will be only one iteration, but it is a bit tedious to extract
-        // single elements from sets.
-        for (final Map.Entry<String, Set<String>> entry: moduleDefMap.entrySet()) {
-            moduleDef = entry.getKey();
-            // Additional imports might have occurred, which are only relevant
-            // for the definition of the module, but not used within the proofs.
-            // Add those.
-            for (final String importString: entry.getValue()) {
-                final String importStringWithoutSuffix =
-                        importString.replace(IsabelleUtils.DOT_THY, StringUtils.EMPTY);
-                if (!imports.toString().contains(importStringWithoutSuffix)) {
-                    final String importWithSuffix =
-                            sessionName + StringUtils.PERIOD + importStringWithoutSuffix;
-                    imports.append(StringUtils.surroundWithSpaces(importWithSuffix));
-                }
+        final Set<String> imports = new LinkedHashSet<String>();
+        imports.add(importsParam);
+        // There exists only one element, otherwise there is an exception thrown beforehand.
+        final String moduleDef = new ArrayList<String>(moduleDefMap.keySet()).get(0);
+        // imports.add(sessionName + StringUtils.PERIOD + moduleDef);
+        // Additional imports might have occurred, which are only relevant
+        // for the definition of the module, but not used within the proofs.
+        // Add those.
+        for (final String importString: new ArrayList<Set<String>>(moduleDefMap.values()).get(0)) {
+            final String importStringWithoutSuffix =
+                    importString.replace(IsabelleUtils.DOT_THY, StringUtils.EMPTY);
+            if (!imports.contains(importStringWithoutSuffix)) {
+                final String importWithSuffix =
+                        sessionName + StringUtils.PERIOD + importStringWithoutSuffix;
+                imports.add(importWithSuffix);
             }
         }
-        return new Pair<String, String>(moduleDef, imports.toString());
+        // Build string from chain of imports
+        final StringBuilder importBuild = new StringBuilder(importsParam);
+        for (final String entry: imports) {
+            importBuild.append(StringUtils.surroundWithSpaces(entry));
+        }
+        final String importString =
+                importBuild.toString().isBlank() ? IsabelleUtils.MAIN : importBuild.toString();
+        return new Pair<String, String>(moduleDef, importString);
     }
 
     private static String replaceVariables(final String theoryName, final String imports,
@@ -356,16 +363,18 @@ public File generateTheoryFile(final String passedComposition,
         }
         final Pair<String, String> moduleDefResult =
                 buildModuleDef(moduleDefMap, imports, this.framework.getSessionName());
+        if (moduleDefResult.getSecondValue().isBlank()) {
+            throw new IllegalArgumentException();
+        }
         final StringBuilder proofsString = new StringBuilder();
         for (final CompositionProof proof: proofs) {
             proofsString.append(this.generator.generateIsabelleProof(proof)
                                 + System.lineSeparator() + System.lineSeparator());
         }
-        final String prfString = proofsString.toString();
         final String fileContents =
                 replaceVariables(theoryName, moduleDefResult.getSecondValue(),
                                  moduleParameterTypeString, moduleName, moduleParameterString,
-                                 moduleDefResult.getFirstValue(), prfString);
+                                 moduleDefResult.getFirstValue(), proofsString.toString());
         final SimpleFileWriter writer = new SimpleFileWriter();
         final String outputPath = buildOutputPath(passedOutputPath, theoryName);
         writer.writeToFile(outputPath, fileContents);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
index 52ade5f..d9fffe3 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
@@ -81,6 +81,11 @@ public final class IsabelleUtils {
      */
     public static final String THEORY_NAME_SEPARATOR = StringUtils.PERIOD;
 
+    /**
+     * Main theory.
+     */
+    public static final String MAIN = "Main";
+
     /**
      * Higher-order logic theory.
      */
@@ -91,6 +96,16 @@ public final class IsabelleUtils {
      */
     public static final String SET = "Set";
 
+    /**
+     * Isabelle's enumerable theory.
+     */
+    public static final String ENUM = "Enum";
+
+    /**
+     * Isabelle's set function.
+     */
+    public static final String SET_FUNC = "set";
+
     /**
      * String to represent the HOL type for natural numbers.
      */
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
index 7c74696..f89902b 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
@@ -155,7 +155,7 @@ public final class ScalaIsabelleFacade {
      * The theory-name function.
      */
     private static final String THY_NAME =
-            "Context" + IsabelleUtils.THEORY_NAME_SEPARATOR + "theory_base_name";
+            "Context" + IsabelleUtils.THEORY_NAME_SEPARATOR + "theory_long_name";
 
     /**
      * The specifications-of function.
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
index 377d70a..a2006b9 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
@@ -80,9 +80,9 @@ public final String presentConcreteResult() {
     protected final void concreteExecute() {
         this.unaryProperties = new LinkedList<Property>();
         final VirageCore execCore = this.getExecutingCore();
-        for (final Property candidate: execCore.getFrameworkRepresentation().getProperties()) {
-            if (!candidate.isAtomic() && candidate.getArity() == 1) {
-                this.unaryProperties.add(candidate);
+        for (final Property property: execCore.getFrameworkRepresentation().getProperties()) {
+            if (!property.isAtomic() && property.getArity() == 1) {
+                this.unaryProperties.add(property);
             }
         }
         Collections.sort(this.unaryProperties,
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
index b9e72e7..12608c0 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
@@ -30,9 +30,8 @@ public final class VirageIsabelleGenerateScalaJob extends VirageJobWithExplicitR
      * @param compositionValue the composition
      */
     public VirageIsabelleGenerateScalaJob(final VirageUserInterface issuer,
-            final String compositionValue) {
+                                          final String compositionValue) {
         super(issuer);
-
         this.composition = compositionValue;
     }
 
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
index 9b24ba4..3a0aaae 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
@@ -36,29 +36,27 @@ public final class CompositionProof {
     private String id = StringUtils.EMPTY;
 
     /**
-     * Simple constructor for a terminal proof step.
+     * Simple constructor.
      *
      * @param goalValue the goal
+     * @param subgoalsValue the subgoals
      * @param ruleValue the rule
      */
-    public CompositionProof(final String goalValue, final CompositionRule ruleValue) {
+    public CompositionProof(final String goalValue, final List<CompositionProof> subgoalsValue,
+                            final CompositionRule ruleValue) {
         this.goal = goalValue;
-        this.subgoals = new LinkedList<CompositionProof>();
+        this.subgoals = subgoalsValue;
         this.rule = ruleValue;
     }
 
     /**
-     * Simple constructor.
+     * Simple constructor for a terminal proof step.
      *
      * @param goalValue the goal
-     * @param subgoalsValue the subgoals
      * @param ruleValue the rule
      */
-    public CompositionProof(final String goalValue, final List<CompositionProof> subgoalsValue,
-                            final CompositionRule ruleValue) {
-        this.goal = goalValue;
-        this.subgoals = subgoalsValue;
-        this.rule = ruleValue;
+    public CompositionProof(final String goalValue, final CompositionRule ruleValue) {
+        this(goalValue, new LinkedList<CompositionProof>(), ruleValue);
     }
 
     /**
diff --git a/virage/src/main/resources/c_implementations/wrapper.c b/virage/src/main/resources/c_implementations/wrapper.c
index 869fb71..5662b26 100644
--- a/virage/src/main/resources/c_implementations/wrapper.c
+++ b/virage/src/main/resources/c_implementations/wrapper.c
@@ -25,9 +25,9 @@ char** split_string(char* str) {
   return res;
 }
 
-int find_order_index(char* candidate) {
+int find_order_index(char* alternative) {
   for (int i = 0; i < C; i++) {
-    if (!strcmp(candidate, order[i])) return i;
+    if (!strcmp(alternative, order[i])) return i;
   }
 
   return C;
diff --git a/virage/src/main/resources/code_file.template b/virage/src/main/resources/code_file.template
index 7289df2..c753081 100644
--- a/virage/src/main/resources/code_file.template
+++ b/virage/src/main/resources/code_file.template
@@ -7,7 +7,7 @@
 // via an additional header or via gcc option "-D DEFAULT_X="concrete implementation"",
 // where "concrete implementation" has to be supplied by the user.
 //
-// "V" and "C": These defines specify the number of Voters and Candidates. For SBMC
+// "V" and "C": These defines specify the number of Voters and Alternatives. For SBMC
 // performance, these have to be known at compile time. Default values are 3 and 3,
 // but they have to fit your test instances. If the numbers don't match, segmentation
 // faults and undefined behaviour will occur. Recompile and explicitly set the correct
diff --git a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml
index dd22382..96ea06c 100644
--- a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml
+++ b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml
@@ -251,7 +251,7 @@
                </children>
             </VBox>
             <Separator orientation="VERTICAL" />
-            <Text strokeType="OUTSIDE" strokeWidth="0.0" text="Candidates:" />
+            <Text strokeType="OUTSIDE" strokeWidth="0.0" text="Alternatives:" />
             <VBox>
                <children>
                   <HBox>
@@ -261,7 +261,7 @@
                               <Text strokeType="OUTSIDE" strokeWidth="0.0" text="min" BorderPane.alignment="CENTER" />
                            </left>
                         </BorderPane>
-              <TextField fx:id="minCandidates" alignment="CENTER_RIGHT" onAction="#minCandidates" prefWidth="50.0" text="1" />
+              <TextField fx:id="minAlternatives" alignment="CENTER_RIGHT" onAction="#minAlternatives" prefWidth="50.0" text="1" />
                      </children>
                   </HBox>
                   <HBox>
@@ -271,7 +271,7 @@
                               <Text strokeType="OUTSIDE" strokeWidth="0.0" text="max" BorderPane.alignment="CENTER" />
                            </left>
                         </BorderPane>
-              <TextField fx:id="maxCandidates" alignment="CENTER_RIGHT" onAction="#maxCandidates" prefWidth="50.0" text="1" />
+              <TextField fx:id="maxAlternatives" alignment="CENTER_RIGHT" onAction="#maxAlternatives" prefWidth="50.0" text="1" />
                      </children>
                   </HBox>
                </children>
@@ -354,7 +354,7 @@
                                                                               </MenuItem>
                                                                               <MenuItem mnemonicParsing="false" onAction="#addSymbCand">
                                                                                  <graphic>
-                                                                                    <Label text="%key.addCandidate" />
+                                                                                    <Label text="%key.addAlternative" />
                                                                                  </graphic>
                                                                               </MenuItem>
                                                                               <MenuItem mnemonicParsing="false" onAction="#addSymbSeat">
@@ -458,9 +458,9 @@
                                                       </BorderPane>
                                                    </left>
                                                    <center>
-                                                      <ScrollPane fx:id="candidateScrollPane" fitToHeight="true" hbarPolicy="NEVER" vbarPolicy="NEVER" BorderPane.alignment="CENTER">
+                                                      <ScrollPane fx:id="alternativeScrollPane" fitToHeight="true" hbarPolicy="NEVER" vbarPolicy="NEVER" BorderPane.alignment="CENTER">
                                                          <content>
-                                                            <GridPane fx:id="candidateGridPane">
+                                                            <GridPane fx:id="alternativeGridPane">
                                                                <columnConstraints>
                                                                   <ColumnConstraints hgrow="SOMETIMES" minWidth="10.0" prefWidth="50.0" />
                                                                </columnConstraints>
@@ -480,8 +480,8 @@
                                             <items>
                                                 <Label text="Voters" />
                                                 <TextField fx:id="inputVoterField" maxWidth="50.0" text="0" />
-                                                <Label text="Candidates" />
-                                                <TextField fx:id="inputCandidateField" maxWidth="50.0" text="0" />
+                                                <Label text="Alternatives" />
+                                                <TextField fx:id="inputAlternativeField" maxWidth="50.0" text="0" />
                                                 <Label text="Seats" />
                                                 <TextField fx:id="inputSeatField" maxWidth="50.0" text="0" />
                                                 <Pane HBox.hgrow="ALWAYS" />
diff --git a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties
index e4c6ea8..b424e54 100644
--- a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties
+++ b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties
@@ -10,7 +10,7 @@ key.code=Code
 key.property=Eigenschaft
 key.result=Ergebnis
 key.voters=Wähler
-key.candidates=Kandidaten
+key.alternatives=Alternativen
 key.seats=Sitze
 key.file=Datei
 key.edit=Bearbeiten
@@ -45,7 +45,7 @@ key.newProp=Property hinzufügen
 key.loadProp=Property laden
 key.loadPropList=Propertyliste laden
 key.addVoter=Wählen hinzufügen
-key.addCandidate=Kandidat hinzufügen
+key.addAlternative=Alternative hinzufügen
 key.addSeat=Sitz hinzufügen
 key.deleteItems=Temporäre Dateien löschen
 key.reset=Reset
diff --git a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties
index 50f9b38..5e56e79 100644
--- a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties
+++ b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties
@@ -10,7 +10,7 @@ key.code=Code
 key.property=Property
 key.result=Result
 key.voters=Voters
-key.candidates=Candidates
+key.alternatives=Alternatives
 key.seats=Seats
 key.file=File
 key.edit=Edit
@@ -46,7 +46,7 @@ key.newProp=Create New Property
 key.loadProp=Load Property
 key.loadPropList=Load List of Multiple Properties
 key.addVoter=Add Voter
-key.addCandidate=Add Candidate
+key.addAlternative=Add Alternative
 key.addSeat=Add Seat
 key.deleteItems=Delete Temporary Files
 key.reset=Reset
diff --git a/virage/src/main/resources/voting_context.template b/virage/src/main/resources/voting_context.template
index fbac216..5f7553e 100644
--- a/virage/src/main/resources/voting_context.template
+++ b/virage/src/main/resources/voting_context.template
@@ -1,7 +1,6 @@
-import scala.util.chaining.scalaUtilChainingOps
-import scala.language.postfixOps
-
 import scala.io.Source
+import scala.language.postfixOps
+import scala.util.chaining.scalaUtilChainingOps
 
 /* ENUM
 class Bounded[A](entities: List[A]) extends Enum.`enuma`[A] {
@@ -11,7 +10,7 @@ class Bounded[A](entities: List[A]) extends Enum.`enuma`[A] {
   val `Enum.enum_all`: (A => Boolean) => Boolean = (p => entities.forall(p))
 
   val `Enum.enum_ex`: (A => Boolean) => Boolean = (p => entities.exists(p))
-} 
+}
 ENUM */
 
 /* EQUALITY
@@ -21,46 +20,46 @@ class Equality[A] extends HOL.equal[A] {
 EQUALITY */
 
 object VotingContext {
-  def readBallotsFromFile(file: String): (List[String], List[Set.seta[(String,String)]]) = {
+  def readBallotsFromFile(file: String): (List[String], List[Set.seta[(String, String)]]) = {
     var ballots: List[List[String]] = Nil
-    
-    var candidates: List[String] = Nil
+
+    var alternatives: List[String] = Nil
 
     for (line <- Source.fromFile(file).getLines()) {
       val strings: Array[String] = line split ';'
-      
+
       val stringList = List(strings.toList)
-      if (candidates == Nil) candidates = strings.toList
-      
+      if (alternatives == Nil) alternatives = strings.toList
+
       ballots = ballots ::: stringList
     }
-    
-    (candidates, transformProfileToRelations[String](ballots))
+
+    (alternatives, transformProfileToRelations[String](ballots))
   }
   
-  def transformProfileToRelations[A](profile: List[List[A]]): List[Set.seta[(A,A)]] = profile match {
+  def transformProfileToRelations[A](profile: List[List[A]]): List[Set.seta[(A, A)]] = profile match {
     case Nil => Nil
     case p :: ps =>
-      val list = List(Set.seta[(A,A)](buildLinearOrder(p)))
+      val list = List(Set.seta[(A, A)](buildLinearOrder(p)))
       list ::: transformProfileToRelations(ps)
   }
-  
-  def buildLinearOrder[A](ballot: List[A]): List[(A,A)] = ballot match {
+
+  def buildLinearOrder[A](ballot: List[A]): List[(A, A)] = ballot match {
     case Nil => Nil
-    case x :: xs => 
-      var newList = List((x,x))
+    case x :: xs =>
+      var newList = List((x, x))
       for (next <- xs) {
-        newList = (x,next) :: newList
+        newList = (x, next) :: newList
       }
-    
+
       newList ::: buildLinearOrder(xs)
   }
-  
+
   def prettyPrint(result: (Set.set[String], (Set.set[String], Set.set[String]))): Unit = {
     val elected = result._1
     val rejected = result._2._1
     val deferred = result._2._2
-    
+
     print("elected: "); println(elected)
     print("deferred: "); println(deferred)
     print("rejected: "); println(rejected)
@@ -69,12 +68,12 @@ object VotingContext {
 
 object Main {
   def main(args: Array[String]): Unit = {
-    val (candidates, ballots) = VotingContext.readBallotsFromFile(args(0))
+    val (alternatives, ballots) = VotingContext.readBallotsFromFile(args(0))
 
     println("Reading finished.")
-    
+
     /* ENUM
-    implicit val bounded = new Bounded[String](candidates) 
+    implicit val bounded = new Bounded[String](alternatives)
     ENUM */
     /* EQUALITY
     implicit val eq = new Equality[String]
@@ -83,11 +82,11 @@ object Main {
     val t1 = System.nanoTime
 
     /* OPTION1
-    val module = $THEORY_NAME.$MODULE_NAME[String]$PARAMS(Set.seta[String](candidates))
+    val module = $THEORY_NAME.$MODULE_NAME[String]$PARAMS(Set.seta[String](alternatives))
     OPTION1 */
     /* OPTION2
     val relation = $THEORY_NAME.$MODULE_NAME[String]$PARAMS(ballots(1))
-    val module = relation(Set.seta[String](candidates))
+    val module = relation(Set.seta[String](alternatives))
     OPTION2 */
 
     val result = module(ballots)
@@ -98,4 +97,3 @@ object Main {
     println("time in seconds: " + duration)
   }
 }
-
diff --git a/virage/src/test/resources/c_implementations/wrapper.c b/virage/src/test/resources/c_implementations/wrapper.c
index 9b4342b..be35a1e 100644
--- a/virage/src/test/resources/c_implementations/wrapper.c
+++ b/virage/src/test/resources/c_implementations/wrapper.c
@@ -25,9 +25,9 @@ char** split_string(char* str) {
   return res;
 }
 
-int find_order_index(char* candidate) {
+int find_order_index(char* alternative) {
   for (int i = 0; i < C; i++) {
-    if (!strcmp(candidate, order[i])) return i;
+    if (!strcmp(alternative, order[i])) return i;
   }
 
   return C;
diff --git a/virage/src/test/resources/instances/README.md b/virage/src/test/resources/instances/README.md
index 0cd7234..86be839 100644
--- a/virage/src/test/resources/instances/README.md
+++ b/virage/src/test/resources/instances/README.md
@@ -2,5 +2,5 @@ These are randomly generated test instances for the voting rule implementations
 They can be used for both the Scala and C implementations.
 
 The format is as follows:
-The file name gives both the number of voters and candidates, the file XcYv contains a profile with X candidates and Y voters.
-Each line represents a ballot, with candidates separated by ; and ordered from lowest to highest preference (might be counterintuitive).
\ No newline at end of file
+The file name gives both the number of voters and alternatives, the file XcYv contains a profile with X alternatives and Y voters.
+Each line represents a ballot, with alternatives separated by ; and ordered from lowest to highest preference (might be counterintuitive).
diff --git a/virage/src/test/resources/old_theories/output/document/Aggregator.tex b/virage/src/test/resources/old_theories/output/document/Aggregator.tex
deleted file mode 100644
index a0a5617..0000000
--- a/virage/src/test/resources/old_theories/output/document/Aggregator.tex
+++ /dev/null
@@ -1,159 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An aggregator gets two partitions (results of electoral modules) as input and
-output another partition. They are used to aggregate results of parallel
-composed electoral modules.
-They are commutative, i.e., the order of the aggregated modules does not affect
-the resulting aggregation. Moreover, they are conservative in the sense that
-the resulting decisions are subsets of the two given partitions' decisions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}aggregator\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}conservative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Aggregator.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%31=11%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%82=33%:%
-%:%92=35%:%
-%:%93=35%:%
-%:%94=36%:%
-%:%96=38%:%
-%:%97=39%:%
-%:%98=40%:%
-%:%99=40%:%
-%:%100=41%:%
-%:%106=47%:%
-%:%109=48%:%
-%:%114=49%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Blacks_Rule.tex b/virage/src/test/resources/old_theories/output/document/Blacks_Rule.tex
deleted file mode 100644
index 7833272..0000000
--- a/virage/src/test/resources/old_theories/output/document/Blacks_Rule.tex
+++ /dev/null
@@ -1,111 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Blacks{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Black's Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Blacks{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \ \ \ \ \ \ \ \ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is Black's voting rule. It is composed of a function that determines
-the Condorcet winner, i.e., the Pairwise Majority rule, and the Borda rule.
-Whenever there exists no Condorcet winner, it elects the choice made by the
-Borda rule, otherwise the Condorcet winner is elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ blacks{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}blacks{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isasymtriangleright}\ borda{\isacharunderscore}{\kern0pt}rule{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ blacks{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}blacks{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isasymtriangleright}\ borda{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Blacks_Rule.thy%:%
-%:%6=3%:%
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-%:%13=7%:%
-%:%29=9%:%
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-%:%42=15%:%
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-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Borda_Module.tex b/virage/src/test/resources/old_theories/output/document/Borda_Module.tex
deleted file mode 100644
index 34d3f2e..0000000
--- a/virage/src/test/resources/old_theories/output/document/Borda_Module.tex
+++ /dev/null
@@ -1,131 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Borda{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Borda Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Borda{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Borda module used by the Borda rule. The Borda rule is a voting
-rule, where on each ballot, each alternative is assigned a score that depends
-on how many alternatives are ranked below. The sum of all such scores for an
-alternative is hence called their Borda score. The alternative with the highest
-Borda score is elected. The module implemented herein only rejects the
-alternatives not elected by the voting rule, and defers the alternatives that
-would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Borda_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
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-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%75=30%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%78=33%:%
-%:%79=33%:%
-%:%80=34%:%
-%:%83=35%:%
-%:%88=36%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Borda_Rule.tex b/virage/src/test/resources/old_theories/output/document/Borda_Rule.tex
deleted file mode 100644
index b96ff8a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Borda_Rule.tex
+++ /dev/null
@@ -1,116 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Borda{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupchapter{Voting Rules%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Borda Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Borda rule. On each ballot, each alternative is assigned a score
-that depends on how many alternatives are ranked below. The sum of all such
-scores for an alternative is hence called their Borda score. The alternative
-with the highest Borda score is elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ borda\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ borda{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Borda_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%17=9%:%
-%:%33=11%:%
-%:%34=11%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%46=17%:%
-%:%47=18%:%
-%:%48=19%:%
-%:%49=20%:%
-%:%58=22%:%
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-%:%74=28%:%
-%:%77=29%:%
-%:%82=30%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Classic_Nanson_Rule.tex b/virage/src/test/resources/old_theories/output/document/Classic_Nanson_Rule.tex
deleted file mode 100644
index b84db79..0000000
--- a/virage/src/test/resources/old_theories/output/document/Classic_Nanson_Rule.tex
+++ /dev/null
@@ -1,117 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Classic Nanson Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the classic Nanson's voting rule, i.e., the rule that was originally
-invented by Nanson, but not the Nanson-Baldwin rule. The idea is similar,
-however, as alternatives with a Borda score less or equal than the average
-Borda score are excluded. The Borda scores of the remaining alternatives
-are hence adjusted to the new set of (still) eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Classic_Nanson_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%76=30%:%
-%:%81=31%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Condorcet_Module.tex b/virage/src/test/resources/old_theories/output/document/Condorcet_Module.tex
deleted file mode 100644
index 8b6830e..0000000
--- a/virage/src/test/resources/old_theories/output/document/Condorcet_Module.tex
+++ /dev/null
@@ -1,275 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
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-\isatagdocument
-\isanewline
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-\isamarkupsection{Condorcet Module%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Condorcet module used by the Condorcet (voting) rule. The Condorcet
-rule is a voting rule that implements the Condorcet criterion, i.e., it elects
-the Condorcet winner if it exists, otherwise a tie remains between all
-alternatives. The module implemented herein only rejects the alternatives not
-elected by the voting rule, and defers the alternatives that would be elected
-by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}code\ A\ p\ x{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ f\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymand}\ a\ {\isasymnoteq}\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ a{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ f\ aa\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ condorcet{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}condorcet\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Condorcet_Module.thy%:%
-%:%6=3%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Copeland_Module.tex b/virage/src/test/resources/old_theories/output/document/Copeland_Module.tex
deleted file mode 100644
index 82760ff..0000000
--- a/virage/src/test/resources/old_theories/output/document/Copeland_Module.tex
+++ /dev/null
@@ -1,917 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Copeland{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Copeland Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Copeland{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Copeland module used by the Copeland voting rule. The Copeland
-rule elects the alternatives with the highest difference between the amount
-of simple-majority wins and the amount of simple-majority losses. The module
-implemented herein only rejects the alternatives not elected by the voting
-rule, and defers the alternatives that would be elected by the full voting
-rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ x\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}code\ x\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}code\ y\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ True{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ {\isasymnot}\ wins\ x\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ False{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disjunct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasyminter}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ union{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ finiteness{\isadigit{1}}\ finiteness{\isadigit{2}}\ disjunct\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ union\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ amount{\isadigit{1}}\ amount{\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ w\ A\ p\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ a\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}{\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\ loser\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ M\ f\ {\isachardot}{\kern0pt}\ finite\ M\ {\isasymlongrightarrow}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ f\ x{\isacharbraceright}{\kern0pt}\ {\isasymle}\ card\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ finite{\isacharunderscore}{\kern0pt}Diff\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{3}}\ {\isadigit{4}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ loser{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{5}}\ {\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ copeland{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharless}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}\ {\isacharless}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isacharunderscore}{\kern0pt}Diff\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ one{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}numeral{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ winner\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}copeland\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Copeland_Rule.tex b/virage/src/test/resources/old_theories/output/document/Copeland_Rule.tex
deleted file mode 100644
index 6d860cb..0000000
--- a/virage/src/test/resources/old_theories/output/document/Copeland_Rule.tex
+++ /dev/null
@@ -1,179 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Copeland{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Copeland Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Copeland{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Copeland{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Copeland voting rule. The idea is to elect the alternatives with
-the highest difference between the amount of simple-majority wins and the
-amount of simple-majority losses.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ copeland\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ copeland{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ copeland{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ copeland{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
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-\endisadelimproof
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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diff --git a/virage/src/test/resources/old_theories/output/document/Defer_Equal_Condition.tex b/virage/src/test/resources/old_theories/output/document/Defer_Equal_Condition.tex
deleted file mode 100644
index 055725a..0000000
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-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition}%
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-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Defer Equal Condition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ Termination{\isacharunderscore}{\kern0pt}Condition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\begin{isamarkuptext}%
-This is a family of termination conditions. For a natural number n,
-the according defer-equal condition is true if and only if the given
-result's defer-set contains exactly n elements.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ n\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ result\ in\ card\ d\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\isadelimtheory
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-\endisadelimtheory
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-\isacommand{end}\isamarkupfalse%
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diff --git a/virage/src/test/resources/old_theories/output/document/Defer_Module.tex b/virage/src/test/resources/old_theories/output/document/Defer_Module.tex
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-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Module}%
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-\isadelimdocument
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-\isanewline
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-\isamarkupchapter{Basic Modules%
-}
-\isamarkuptrue%
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-\isamarkupsection{Defer Module%
-}
-\isamarkuptrue%
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-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\begin{isamarkuptext}%
-The defer module is not concerned about the voter's ballots, and
-simply defers all alternatives.
-It is primarily used for defining an empty loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisatagdocument
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
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-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
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-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
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-\isadelimproof
-\isanewline
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-\endisadelimproof
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-\isadelimtheory
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-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\end{isabellebody}%
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diff --git a/virage/src/test/resources/old_theories/output/document/Defer_One_Loop_Composition.tex b/virage/src/test/resources/old_theories/output/document/Defer_One_Loop_Composition.tex
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-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition}%
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-\isamarkupsection{Defer One Loop Composition%
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-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \ \ \ \ \ \ \ \ Elect{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\begin{isamarkuptext}%
-This is a family of loop compositions. It uses the same module in sequence
-until either no new decisions are made or only one alternative is remaining
-in the defer-set. The second family herein uses the above family and
-subsequently elects the remaining alternative.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isadelimdocument
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-\endisadelimdocument
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-\isamarkupsubsection{Definition%
-}
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ iter\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iter\ m\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d\ {\isasymequiv}\ iter\ m{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ iterelect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iterelect\ m\ {\isacharequal}{\kern0pt}\ elector\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\setisabellecontext{Drop{\isacharunderscore}{\kern0pt}And{\isacharunderscore}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Compatibility}%
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-\isamarkupchapter{Compositional Structures%
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-\ Drop{\isacharunderscore}{\kern0pt}And{\isacharunderscore}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Compatibility\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
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-{\isafoldtheory}%
-%
-\isadelimtheory
-%
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-%
-\begin{isamarkuptext}%
-This is a collection of properties about the interplay and compatibility
-of both the drop module and the pass module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
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-\endisatagdocument
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}two{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isadigit{0}}\ x{\isadigit{1}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}v{\isadigit{2}}\ v{\isadigit{3}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ v{\isadigit{2}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isacharparenleft}{\kern0pt}n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{2}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isacartoucheclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defers{\isacharunderscore}{\kern0pt}def\ order\ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}pass{\isacharunderscore}{\kern0pt}disj{\isacharunderscore}{\kern0pt}compat{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}iff\ empty{\isacharunderscore}{\kern0pt}set\ fin\ profile{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Drop_And_Pass_Compatibility.thy%:%
-%:%6=3%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Drop_Module.tex b/virage/src/test/resources/old_theories/output/document/Drop_Module.tex
deleted file mode 100644
index 134f029..0000000
--- a/virage/src/test/resources/old_theories/output/document/Drop_Module.tex
+++ /dev/null
@@ -1,358 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Drop{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Drop Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Drop{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according drop module
-rejects the lexicographically first n alternatives (from A) and
-defers the rest.
-It is primarily used as counterpart to the pass module in a
-parallel composition, in order to segment the alternatives into
-two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Elect_Composition.tex b/virage/src/test/resources/old_theories/output/document/Elect_Composition.tex
deleted file mode 100644
index 997426a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Elect_Composition.tex
+++ /dev/null
@@ -1,1049 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elect{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Elect Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elect{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The elect composition sequences an electoral module and the elect
-module. It finalizes the module's decision as it simply elects all their
-non-rejected alternatives. Thereby, any such elect-composed module induces
-a proper voting rule in the social choice sense, as all alternatives are either
-rejected or elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elector\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elector\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AA\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}block{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Profile\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrsa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAa\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAd\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAb\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isadigit{3}}{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ AAc\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\ Diff{\isacharunderscore}{\kern0pt}empty\ elector{\isachardot}{\kern0pt}simps\ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}block\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\ f{\isadigit{3}}\ f{\isadigit{1}}\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}eq\ finite\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ fst{\isacharunderscore}{\kern0pt}conv\ insert{\isacharunderscore}{\kern0pt}Diff\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ order{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharunderscore}{\kern0pt}Compl{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ empty{\isacharunderscore}{\kern0pt}iff\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ DiffI\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ finite\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}iff\ finite\ prof{\isacharunderscore}{\kern0pt}A\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
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diff --git a/virage/src/test/resources/old_theories/output/document/Elect_Module.tex b/virage/src/test/resources/old_theories/output/document/Elect_Module.tex
deleted file mode 100644
index 34d1c97..0000000
--- a/virage/src/test/resources/old_theories/output/document/Elect_Module.tex
+++ /dev/null
@@ -1,190 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elect{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Elect Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elect{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The elect module is not concerned about the voter's ballots, and
-just elects all alternatives. It is primarily used in sequence after
-an electoral module that only defers alternatives to finalize the decision,
-thereby inducing a proper voting rule in the social choice sense.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Elect_Module.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Electoral_Module.tex b/virage/src/test/resources/old_theories/output/document/Electoral_Module.tex
deleted file mode 100644
index b44bb38..0000000
--- a/virage/src/test/resources/old_theories/output/document/Electoral_Module.tex
+++ /dev/null
@@ -1,3358 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Electoral{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Component Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Electoral Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Electoral{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Electoral modules are the principal component type of the composable modules
-voting framework, as they are a generalization of voting rules in the sense of
-social choice functions.
-These are only the types used for electoral modules. Further restrictions are
-encompassed by the electoral-module predicate.
-
-An electoral module does not need to make final decisions for all alternatives,
-but can instead defer the decision for some or all of them to other modules.
-Hence, electoral modules partition the received (possibly empty) set of
-alternatives into elected, rejected and deferred alternatives. In particular,
-any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives. Just like a voting rule, an electoral module also receives a
-profile which holds the voters’ preferences, which, unlike a voting rule,
-consider only the (sub-)set of alternatives that the module receives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymequiv}\ {\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymequiv}\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymequiv}\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer{\isachardoublequoteclose}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymequiv}\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defers\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defers\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ rejects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rejects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ eliminates\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}eliminates\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ elects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ A\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Equivalence Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def\ ele\ rej\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ set{\isacharunderscore}{\kern0pt}partit\ partit\ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ fst{\isacharunderscore}{\kern0pt}conv\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ snd{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isasymnot}\ disjoint{\isadigit{3}}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D{\isachardot}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharcomma}{\kern0pt}\ C{\isacharcomma}{\kern0pt}\ D{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ B\ {\isasyminter}\ C\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ B\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ C\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disj{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disj{\isadigit{3}}\ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isadigit{3}}\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\ fstI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ module\ profile\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\ prof{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ partit\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ module\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounds{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}elected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ alternative\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}elected\ not{\isacharunderscore}{\kern0pt}rejected\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ m\ A\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ IntI\ assms\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymor}\ x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ alternative\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ not{\isacharunderscore}{\kern0pt}rejected\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ leftover{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubset}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminates{\isacharunderscore}{\kern0pt}def\ eliminating\ eq{\isacharunderscore}{\kern0pt}iff\ leftover{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}zero\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ input{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}collapse\ subsetI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ n\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ n\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ n\ A\ q\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p\ reject{\isacharunderscore}{\kern0pt}q\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ prod{\isacharunderscore}{\kern0pt}eqI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Blocking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ one{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}right\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing\ electing{\isacharunderscore}{\kern0pt}def\ inf{\isachardot}{\kern0pt}orderE\isanewline
-\ \ \ \ \ \ \ \ one{\isacharunderscore}{\kern0pt}alt\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}empty\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ electing\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rejecting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejecting\isanewline
-\ \ \ \ \ \ \ \ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminating\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\ defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ decrementing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decrementing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymforall}\ A\ p\ {\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymnotin}\ reject\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymnotin}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymin}\ reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ p\ {\isasymor}\ elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ {\isasymor}\ defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Inference Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ equals{\isadigit{0}}D\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ insert{\isacharunderscore}{\kern0pt}subset\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymand}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win\ equals{\isadigit{0}}D\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ other{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ one{\isacharunderscore}{\kern0pt}side{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finiteness{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ wins\ d\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ d\ {\isacharless}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ d\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}d\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ n\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ old{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}S{\isacharminus}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ double{\isacharunderscore}{\kern0pt}diff\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ dl{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Social Choice Properties%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Condorcet Consistency%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}e\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ e{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymLongrightarrow}\ electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ cc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ x\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ wins{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\ e{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
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-\endisadelimproof
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsubsection{(Weak) Monotonicity%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsubsection{Homogeneity%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ times\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}times\ n\ l\ {\isacharequal}{\kern0pt}\ concat\ {\isacharparenleft}{\kern0pt}replicate\ n\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ homogeneity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}homogeneity\ m\ {\isasymequiv}\isanewline
-\ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ n\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ {\isacharparenleft}{\kern0pt}times\ n\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Elimination_Module.tex b/virage/src/test/resources/old_theories/output/document/Elimination_Module.tex
deleted file mode 100644
index 2a12805..0000000
--- a/virage/src/test/resources/old_theories/output/document/Elimination_Module.tex
+++ /dev/null
@@ -1,1358 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elimination{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Elimination Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elimination{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ Evaluation{\isacharunderscore}{\kern0pt}Function\isanewline
-\ \ \ \ \ \ \ \ \ \ Electoral{\isacharunderscore}{\kern0pt}Module\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the elimination module. It rejects a set of alternatives only if these
-are not all alternatives. The alternatives potentially to be rejected are put
-in a so-called elimination set. These are all alternatives that score below
-a preset threshold value that depends on the specific voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isacharequal}{\kern0pt}\ nat\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}e\ a\ A\ p{\isacharparenright}{\kern0pt}\ t\ {\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ A\isanewline
-\ \ \ \ \ \ \ \ then\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ else\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Common Eliminators%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}min{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ average\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Threshold{\isacharunderscore}{\kern0pt}Value{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}average\ e\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ {\isacharparenleft}{\kern0pt}card\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ profile\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Inference Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}ccomp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ not{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}\ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}trsh{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ rating\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymor}\ wins\ w\ p\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
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-\endisatagproof
-{\isafoldproof}%
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Evaluation_Function.tex b/virage/src/test/resources/old_theories/output/document/Evaluation_Function.tex
deleted file mode 100644
index fc6ad16..0000000
--- a/virage/src/test/resources/old_theories/output/document/Evaluation_Function.tex
+++ /dev/null
@@ -1,344 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Evaluation{\isacharunderscore}{\kern0pt}Function}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Evaluation Function%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Evaluation{\isacharunderscore}{\kern0pt}Function\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the evaluation function. From a set of currently eligible alternatives,
-the evaluation function computes a numerical value that is then to be used for
-further (s)election, e.g., by the elimination module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ \ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ p\ w\ {\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymlongrightarrow}\ f\ l\ A\ p\ {\isacharless}{\kern0pt}\ f\ w\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Theorems%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eMax\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eW\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ e\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isachardot}{\kern0pt}\ e\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}order\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ a\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{4}}\ Max{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ linA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ l{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ linA\ loser\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\ f{\isacharunderscore}{\kern0pt}prof\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Loop_Composition.tex b/virage/src/test/resources/old_theories/output/document/Loop_Composition.tex
deleted file mode 100644
index 75f4efe..0000000
--- a/virage/src/test/resources/old_theories/output/document/Loop_Composition.tex
+++ /dev/null
@@ -1,2502 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Loop{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Loop Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The loop composition uses the same module in sequence,
-combined with a termination condition, until either
-  (1) the termination condition is met or
-  (2) no new decisions are made (i.e., a fixed point is reached).%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}term{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}inf{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\isanewline
-\ \ \ \ \ \ \ \ measure\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ P\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ bool\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}f\ A\ p\ rs\ fa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fa{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isacharunderscore}{\kern0pt}cases{\isadigit{5}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ P\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ acca\ Aa\ pa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ t\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ta\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acca\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ Aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ pa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ma\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}r{\isachardot}{\kern0pt}\ wf\ r\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ p\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper\ wf{\isacharunderscore}{\kern0pt}measure\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}r\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ Ex\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}ra{\isachardot}{\kern0pt}\ {\isasymforall}f\ A\ rs\ pa\ fa{\isachardot}{\kern0pt}\ {\isasymexists}ra\ pb\ rb\ pc\ pd\ fb\ Aa\ rsa\ fc\ pe{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ wf\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ pa\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wf\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}ra{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}pb{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ wf\ rb\ {\isasymand}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pc{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rb\ {\isasymand}\ wf\ rb\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pd{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}defer\ fb\ {\isacharparenleft}{\kern0pt}Aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharparenright}{\kern0pt}\ Aa\ rsa\ {\isasymsubset}\ defer\ fb\ Aa\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ pe\ {\isacharparenleft}{\kern0pt}fb\ Aa\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ fb{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ rr\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}wf\ rr\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\ p\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rr{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\ wf{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub t\ {\isasymequiv}\ loop{\isacharunderscore}{\kern0pt}composition\ m\ t{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ {\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ profile\isanewline
-\ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ m{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\ m{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ eq{\isacharunderscore}{\kern0pt}iff\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ less{\isacharunderscore}{\kern0pt}irrefl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubset}\ defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ psubsetI\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ q\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ monotone{\isacharunderscore}{\kern0pt}m\ subsetCE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ less{\isachardot}{\kern0pt}hyps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}lifted\ A\ p\ q\ a\ {\isasymand}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}acc\ profile\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\isanewline
-\ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ n\ {\isasymge}\ x\ {\isasymand}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\isanewline
-\ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ term{\isacharunderscore}{\kern0pt}satisfied\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}small{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ i\ \isakeyword{where}\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ f{\isacharunderscore}{\kern0pt}prof{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ enough{\isacharunderscore}{\kern0pt}leftover\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ i\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ step{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ x\ {\isasymor}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}le\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharless}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subset\ step{\isacharunderscore}{\kern0pt}profile\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}new{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymlongrightarrow}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isachardot}{\kern0pt}hyps\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ t\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough\ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ linorder{\isacharunderscore}{\kern0pt}neqE{\isacharunderscore}{\kern0pt}nat\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ less{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}alternatives\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module{\isachardot}{\kern0pt}simps\ monotone{\isacharunderscore}{\kern0pt}m\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}m\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}elect\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ card\ A\ {\isasymge}\ n\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ eliminates{\isacharunderscore}{\kern0pt}def\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ calculation\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Maximum_Aggregator.tex b/virage/src/test/resources/old_theories/output/document/Maximum_Aggregator.tex
deleted file mode 100644
index 7d741d6..0000000
--- a/virage/src/test/resources/old_theories/output/document/Maximum_Aggregator.tex
+++ /dev/null
@@ -1,561 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Maximum{\isacharunderscore}{\kern0pt}Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Maximum Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Maximum{\isacharunderscore}{\kern0pt}Aggregator\isanewline
-\ \ \isakeyword{imports}\ Aggregator\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The max(imum) aggregator takes two partitions of an alternative set A as
-input. It returns a partition where every alternative receives the maximum
-result of the two input partitions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}l\ {\isasymin}\ A{\isachardot}{\kern0pt}\ l\ {\isasymnotin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ set{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Maximum_Aggregator.thy%:%
-%:%6=3%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Maximum_Parallel_Composition.tex b/virage/src/test/resources/old_theories/output/document/Maximum_Parallel_Composition.tex
deleted file mode 100644
index 69a4e07..0000000
--- a/virage/src/test/resources/old_theories/output/document/Maximum_Parallel_Composition.tex
+++ /dev/null
@@ -1,1960 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Maximum{\isacharunderscore}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Maximum Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Maximum{\isacharunderscore}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of parallel compositions. It composes a new electoral module
-from two electoral modules combined with the maximum aggregator. Therein, the
-two modules each make a decision and then a partition is returned where every
-alternative receives the maximum result of the two input partitions. This means
-that, if any alternative is elected by at least one of the modules, then it
-gets elected, if any non-elected alternative is deferred by at least one of the
-modules, then it gets deferred, only alternatives rejected by both modules get
-rejected.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ a\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}aggregator\ in\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymparallel}\isactrlsub {\isasymup}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
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-\endisadelimproof
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x\ {\isasymor}\isanewline
-\ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ x\ {\isasymin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ Un{\isacharunderscore}{\kern0pt}iff\ a{\isadigit{1}}\ empty{\isacharunderscore}{\kern0pt}iff\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}conv\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ f{\isacharunderscore}{\kern0pt}prof\ sndI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}iff\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ a{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}prof\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}unfold\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ surjective{\isacharunderscore}{\kern0pt}pairing\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}iff\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}\ Int{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prodI{\isadigit{2}}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ split{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}iff\ a\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ surjective{\isacharunderscore}{\kern0pt}pairing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymnotin}\ elect\ n\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ prod{\isachardot}{\kern0pt}simps\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ old{\isachardot}{\kern0pt}prod{\isachardot}{\kern0pt}inject\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}collapse\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Set{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}insert\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}insert\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}profs{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n\ lifted{\isacharunderscore}{\kern0pt}x\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{0}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffI\ lifted{\isacharunderscore}{\kern0pt}x\ compatible\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ compatible\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{1}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}x\ defer{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ x\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}sum{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S\ {\isacharplus}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ x\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ x\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ y\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ compatible\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}representation{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ x\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ subsets{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ finite\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ S\ {\isacharplus}{\kern0pt}\ n\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}Int\ reject{\isacharunderscore}{\kern0pt}representation\ reject{\isacharunderscore}{\kern0pt}sum\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subsets\ subset{\isacharunderscore}{\kern0pt}eq\ sup{\isachardot}{\kern0pt}absorb{\isacharunderscore}{\kern0pt}iff{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isachardot}{\kern0pt}cobounded{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ disj{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rejects{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{2}}\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}nat\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ module\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ reject\ m\ A\ p{\isacharcomma}{\kern0pt}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ module\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ case{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ case{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}comp\ prof{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Minimax_Module.tex b/virage/src/test/resources/old_theories/output/document/Minimax_Module.tex
deleted file mode 100644
index 328683a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Minimax_Module.tex
+++ /dev/null
@@ -1,690 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Minimax{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Minimax Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Minimax{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Minimax module used by the Minimax voting rule. The Minimax rule
-elects the alternatives with the highest Minimax score. The module implemented
-herein only rejects the alternatives not elected by the voting rule, and defers
-the alternatives that would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p\ {\isasymle}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ Min\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}lscore\ {\isasymle}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ local{\isachardot}{\kern0pt}finite\ not{\isacharunderscore}{\kern0pt}empty\ Min{\isacharunderscore}{\kern0pt}le\ Min{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ minimax{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ prefer{\isacharunderscore}{\kern0pt}count{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ prof\ winner\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ \ {\isasymge}\ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}\ n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isadigit{6}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\ prof\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ prof\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}minimax\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Minimax_Module.thy%:%
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diff --git a/virage/src/test/resources/old_theories/output/document/Minimax_Rule.tex b/virage/src/test/resources/old_theories/output/document/Minimax_Rule.tex
deleted file mode 100644
index b55f940..0000000
--- a/virage/src/test/resources/old_theories/output/document/Minimax_Rule.tex
+++ /dev/null
@@ -1,177 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Minimax{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Minimax Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Minimax{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Minimax{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Minimax voting rule. It elects the alternatives with the highest
-Minimax score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ minimax\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ minimax{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ minimax{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ minimax{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Minimax_Rule.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Nanson_Baldwin_Rule.tex b/virage/src/test/resources/old_theories/output/document/Nanson_Baldwin_Rule.tex
deleted file mode 100644
index 84a8dab..0000000
--- a/virage/src/test/resources/old_theories/output/document/Nanson_Baldwin_Rule.tex
+++ /dev/null
@@ -1,113 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Nanson-Baldwin Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Nanson-Baldwin voting rule. It excludes alternatives with the
-lowest Borda score from the set of possible winners and then adjusts the
-Borda score to the new (remaining) set of still eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Pairwise_Majority_Rule.tex b/virage/src/test/resources/old_theories/output/document/Pairwise_Majority_Rule.tex
deleted file mode 100644
index 23f1c9a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Pairwise_Majority_Rule.tex
+++ /dev/null
@@ -1,197 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Pairwise Majority Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the pairwise majority rule, a voting rule that implements the
-Condorcet criterion, i.e., it elects the Condorcet winner if it exists,
-otherwise a tie remains between all alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ condorcet\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ condorcet{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}condorcet{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ iterelect\ condorcet{\isacharprime}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ condorcet{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Pairwise_Majority_Rule.thy%:%
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diff --git a/virage/src/test/resources/old_theories/output/document/Parallel_Composition.tex b/virage/src/test/resources/old_theories/output/document/Parallel_Composition.tex
deleted file mode 100644
index 1f55cd6..0000000
--- a/virage/src/test/resources/old_theories/output/document/Parallel_Composition.tex
+++ /dev/null
@@ -1,412 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Parallel{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The parallel composition composes a new electoral module from
-two electoral modules combined with an aggregator.
-Therein, the two modules each make a decision and the aggregator combines
-them to a single (aggregated) result.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ agg\ A\ p\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub a\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ a{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ agg{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ conservative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ agg{\isacharunderscore}{\kern0pt}a\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ c\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ c\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Parallel_Composition.thy%:%
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diff --git a/virage/src/test/resources/old_theories/output/document/Pass_Module.tex b/virage/src/test/resources/old_theories/output/document/Pass_Module.tex
deleted file mode 100644
index 3985b72..0000000
--- a/virage/src/test/resources/old_theories/output/document/Pass_Module.tex
+++ /dev/null
@@ -1,1466 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pass{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Pass Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pass{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according pass module
-defers the lexicographically first n alternatives (from A) and rejects
-the rest. It is primarily used as counterpart to the drop module in a
-parallel composition in order to segment the alternatives into two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ not{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Blocking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ g{\isadigit{0}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ input{\isacharunderscore}{\kern0pt}fine{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ input{\isacharunderscore}{\kern0pt}fine{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ leD\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}dl{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ A{\isacharunderscore}{\kern0pt}valid{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}valid\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ w\ \isakeyword{where}\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit\ above{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}valid\ finite{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ above{\isacharunderscore}{\kern0pt}finite\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}singletonE\ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ singletonD\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}unique\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\ order\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ finA\ profA\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ {\isacharparenleft}{\kern0pt}b\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ Suc{\isacharunderscore}{\kern0pt}leI\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ a{\isacharunderscore}{\kern0pt}above\ card{\isacharunderscore}{\kern0pt}UNIV{\isacharunderscore}{\kern0pt}bool\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ empty{\isacharunderscore}{\kern0pt}iff\ finA\isanewline
-\ \ \ \ \ \ \ \ \ \ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}iff\ limitA{\isacharunderscore}{\kern0pt}order\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \ \ \ \ UNIV{\isacharunderscore}{\kern0pt}bool\ nat{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ finA{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}idemp\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ numeral{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}iff\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\ top{\isacharunderscore}{\kern0pt}greatest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ in\ {\isacharparenleft}{\kern0pt}c\ {\isasympreceq}\isactrlsub q\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b\ Diff{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{2}}\ subset{\isacharunderscore}{\kern0pt}UNIV\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ b{\isacharunderscore}{\kern0pt}best\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ order\ insert{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b\ above{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isachardot}{\kern0pt}\ c\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\ b{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ above{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ finA\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}commute\ numeral{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ finA\ finite{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans\ numeral{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff\ order{\isacharunderscore}{\kern0pt}refl\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Plurality_Module.tex b/virage/src/test/resources/old_theories/output/document/Plurality_Module.tex
deleted file mode 100644
index 95438e8..0000000
--- a/virage/src/test/resources/old_theories/output/document/Plurality_Module.tex
+++ /dev/null
@@ -1,1263 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Plurality{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Plurality Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Plurality{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The plurality module implements the plurality voting rule.
-The plurality rule elects all modules with the maximum amount of top
-preferences among all alternatives, and rejects all the other alternatives.
-It is electing and induces the classical plurality (voting) rule
-from social-choice theory.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ plurality\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}plurality\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}plurality{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasyminter}\ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasymunion}\ reject\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint\ unity\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI\ conjI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ max\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}max\ {\isacharequal}{\kern0pt}\ Max{\isacharparenleft}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ p\ {\isacharbackquote}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ max\ {\isasymand}\ a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Max{\isacharunderscore}{\kern0pt}in\ assm{\isadigit{0}}\ empty{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}image\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}imageI\ imageE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ max\ assm{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem\ assm{\isadigit{1}}\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ sizes{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}size\ p\ {\isacharequal}{\kern0pt}\ size\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}\ insertCI\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ sizes\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}QtoP{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymge}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}pq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ this\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ asm{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}PtoQ{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}pq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ q\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ DiffI\ above{\isacharunderscore}{\kern0pt}pq\ above{\isacharunderscore}{\kern0pt}QtoP\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ sizes\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymnoteq}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ antisym\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ y\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymnotin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ less\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ q\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Preference_Relation.tex b/virage/src/test/resources/old_theories/output/document/Preference_Relation.tex
deleted file mode 100644
index 737d741..0000000
--- a/virage/src/test/resources/old_theories/output/document/Preference_Relation.tex
+++ /dev/null
@@ -1,4290 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Preference{\isacharunderscore}{\kern0pt}Relation}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Social-Choice Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Preference Relation%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The very core of the composable modules voting framework: types and
-functions, derivations, lemmata, operations on preference relations, etc.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ rel{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}preferred{\isacharunderscore}{\kern0pt}than\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasympreceq}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Ranking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ rank\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rank\ r\ x\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rank{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ refl{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rank\ r\ x\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ FieldI{\isadigit{2}}\ refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin\ finite{\isacharunderscore}{\kern0pt}Field{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ FieldI{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ rank{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Preference%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ limited\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limited\ A\ r\ {\isasymequiv}\ r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limitedI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ limited\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}dest{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y{\isacharsemicolon}{\kern0pt}\ limited\ A\ r\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ limit\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ connex\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}connex\ A\ r\ {\isasymequiv}\ limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}s\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub r\ x\ {\isasymor}\ x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex\ connex{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ antisymmetric{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ antisymmetric\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ transitive{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ z\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{5}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}z\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{4}}\ transE\ transitive\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}let\ s\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ C\ r\ {\isacharequal}{\kern0pt}\ limit\ C\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}r\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subrelI\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}singleton{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ singletonI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a\ {\isasymand}\ y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mono\ above{\isacharunderscore}{\kern0pt}def\ assms\ transE\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ above\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ singletonI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b\ {\isasymlongleftrightarrow}\ b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B\ {\isasymand}\ b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ assms\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ \isakeyword{where}\ n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ gr{\isadigit{0}}{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}Suc\ le{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ A\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isadigit{0}}{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ Suc\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ B\ \isakeyword{where}\ B{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ B\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isasymand}\ B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}inject\ add{\isacharunderscore}{\kern0pt}Suc\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\ finite{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}single\ subset{\isacharunderscore}{\kern0pt}insertI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}b{\isasymin}B{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isachardot}{\kern0pt}IH\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ finite{\isacharunderscore}{\kern0pt}subset\ leD\ lessI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ a\ aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ aa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ connex\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\ b\ {\isacharequal}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{6}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ A\ Aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ Aa\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ Aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ aa\ {\isasymin}\ B\ {\isasymand}\ a\ {\isasymin}\ B{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{5}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}B{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ above{\isacharunderscore}{\kern0pt}def\ f{\isadigit{4}}\ ff{\isadigit{1}}\ ff{\isadigit{2}}\ CollectI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Product{\isacharunderscore}{\kern0pt}Type{\isachardot}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}case{\isacharunderscore}{\kern0pt}prodD\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ above{\isacharunderscore}{\kern0pt}def\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subsetI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ DiffE\ a\ asm\ b\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ singletonI\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ Aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ Aa\ {\isasymsubseteq}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ Aa\ {\isacharparenleft}{\kern0pt}limit\ Aa\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ g{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ total{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{1}}\ B\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ total{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa\ {\isasymnotin}\ A\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ B\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ B\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\ B\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ asm\ b\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ singletonI\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ g{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ transE\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\ asm\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}connex\ Diff{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ a\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ CollectD\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ above{\isacharunderscore}{\kern0pt}trans\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ empty{\isacharunderscore}{\kern0pt}Diff\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}if\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ b\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ singletonI\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}dest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}exist{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex\ \isacommand{obtain}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub r\ y\ {\isasymand}\ y\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ transE\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ antisymD\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ DiffD{\isadigit{2}}\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\ singletonI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ ex{\isacharunderscore}{\kern0pt}lifted\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\ rest{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}r\ rest{\isacharunderscore}{\kern0pt}eq\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}r\ trans{\isacharunderscore}{\kern0pt}s\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ refl{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymsubseteq}\ above\ s\ x\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymnotin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ z\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ z{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ z\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ z\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ y\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ member{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ remove{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ {\isasymor}\isanewline
-\ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ subset\ \isacommand{have}\isamarkupfalse%
-\ temp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ subset\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ keep{\isacharunderscore}{\kern0pt}lift{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ u\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ x\ {\isasympreceq}\isactrlsub u\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ temp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}worse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ connex\ A\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymin}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymin}\ A\ {\isasymlongrightarrow}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Ball{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}raw\ connex{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ limit{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ aa\ {\isasymnotin}\ A\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ limit{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ aa\ A\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ limit\ A\ r\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}s{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{2}}\ limit{\isadigit{1}}\ connex{\isadigit{2}}\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ connex{\isadigit{2}}\ limit{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ connex\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ limited{\isacharunderscore}{\kern0pt}dest\ subrelI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ notInA\ subset\ subset{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}insert{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ change\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}above\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymlongrightarrow}\ aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ singletonD\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}connex\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ {\isasymnot}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ a\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ limited\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasympreceq}\isactrlsub r\ a\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ connex\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ limited{\isacharunderscore}{\kern0pt}dest\ insertCI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}x\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mk{\isacharunderscore}{\kern0pt}disjoint{\isacharunderscore}{\kern0pt}insert\ insertE\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ asm{\isadigit{1}}\ assms\ calculation\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ member{\isacharunderscore}{\kern0pt}remove\ remove{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ lifted{\isacharunderscore}{\kern0pt}a\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}others{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ above\ s\ y\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{2}}\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\ singleton{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ y\ \isakeyword{where}\ y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b{\isachardot}{\kern0pt}\ above\ s\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ b\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ y\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Profile.tex b/virage/src/test/resources/old_theories/output/document/Profile.tex
deleted file mode 100644
index b6f1213..0000000
--- a/virage/src/test/resources/old_theories/output/document/Profile.tex
+++ /dev/null
@@ -1,1906 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Profile}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Preference Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Profile\isanewline
-\ \ \isakeyword{imports}\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Preference profiles denote the decisions made by the individual voters on
-the eligible alternatives. They are represented in the form of one preference
-relation (e.g., selected on a ballot) per voter, collectively captured in a
-list of such preference relations.
-Unlike a the common preference profiles in the social-choice sense, the
-profiles described here considers only the (sub-)set of alternatives that are
-received.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Profile\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isacharparenright}{\kern0pt}\ list{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymforall}b\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ all{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}all{\isacharunderscore}{\kern0pt}nth\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymequiv}\ finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Preference Counts and Comparisons%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}above\ p\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ a{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ y\ {\isasympreceq}\isactrlsub p\ x\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}\ f\ x\ {\isacharbar}{\kern0pt}\ x\ {\isachardot}{\kern0pt}\ x\ {\isasymin}\ S\ {\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharbackquote}{\kern0pt}\ S{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}\ y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ neq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ size\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ profile{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ connex\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ antisym\ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisymD\ neq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ size\ p\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{assumes}\ p{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ prof\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ diff{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ neq{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}count\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}p\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}\ x\ y{\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub p\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}constant{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ wins\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}wins\ x\ p\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}wins{\isacharunderscore}{\kern0pt}code\ x\ p\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}wins\ a\ p\ b{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ b\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ w\ p\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Winner%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}code\ A\ p\ w\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}code\ w\ p\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharunderscore}{\kern0pt}c{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ c{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}w\ {\isacharequal}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assumption{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ c{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assumption\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ c{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}against{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ assumption\isanewline
-\ \ \ \ \ \ \ \ \ \ singletonD\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}xa\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}wins\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ C\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ C\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ B\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ length{\isacharunderscore}{\kern0pt}map\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ subset\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}size\ p\ {\isacharequal}{\kern0pt}\ size\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ size\ p\ {\isacharequal}{\kern0pt}\ size\ q\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ size\ p\ {\isacharequal}{\kern0pt}\ size\ q\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}ok{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ size\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ i{\isacharunderscore}{\kern0pt}ok\ lifted\ profile{\isacharunderscore}{\kern0pt}def\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ notInA\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ length{\isacharunderscore}{\kern0pt}map\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ profile\ A\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}size\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ A\ {\isacharquery}{\kern0pt}p\ {\isacharquery}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ inA\ lifted\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Result.tex b/virage/src/test/resources/old_theories/output/document/Result.tex
deleted file mode 100644
index c2899fe..0000000
--- a/virage/src/test/resources/old_theories/output/document/Result.tex
+++ /dev/null
@@ -1,469 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Electoral Result%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An electoral result is the principal result type of the composable modules
-voting framework, as it is a generalization of the set of winning alternatives
-from social choice functions. Electoral results are selections of the received
-(possibly empty) set of alternatives into the three disjoint groups of elected,
-rejected and deferred alternatives.
-Any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Result\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Functions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ disjoint{\isadigit{3}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ result\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ snd\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}elec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}elec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ card\ e\ {\isacharplus}{\kern0pt}\ card\ r\ {\isacharplus}{\kern0pt}\ card\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disj\ set{\isacharunderscore}{\kern0pt}partit\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ finite{\isacharunderscore}{\kern0pt}Un\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}right{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/old_theories/output/document/Revision_Composition.tex b/virage/src/test/resources/old_theories/output/document/Revision_Composition.tex
deleted file mode 100644
index d0a613b..0000000
--- a/virage/src/test/resources/old_theories/output/document/Revision_Composition.tex
+++ /dev/null
@@ -1,470 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Revision{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Revision Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Revision{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-A revised electoral module rejects all originally rejected or deferred
-alternatives, and defers the originally elected alternatives.
-It does not elect any alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ revision{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ rev\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymdown}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m{\isasymdown}\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ revision{\isacharunderscore}{\kern0pt}composition\ m{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ unity\ disjoint\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ electing{\isacharunderscore}{\kern0pt}def\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ no{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ no{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}disjoint\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect\ assms\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Revision_Composition.thy%:%
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diff --git a/virage/src/test/resources/old_theories/output/document/Schwartz_Rule.tex b/virage/src/test/resources/old_theories/output/document/Schwartz_Rule.tex
deleted file mode 100644
index 404ef4d..0000000
--- a/virage/src/test/resources/old_theories/output/document/Schwartz_Rule.tex
+++ /dev/null
@@ -1,115 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Schwartz{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Schwartz Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Schwartz{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Schwartz voting rule. Confusingly, it is sometimes also referred as
-Nanson's rule. The Schwartz rule proceeds as in the classic Nanson's rule, but
-excludes alternatives with a Borda score that is strictly less than the
-average Borda score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ schwartz{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}schwartz{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ schwartz{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}schwartz{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Schwartz_Rule.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Sequential_Composition.tex b/virage/src/test/resources/old_theories/output/document/Sequential_Composition.tex
deleted file mode 100644
index 2619cb1..0000000
--- a/virage/src/test/resources/old_theories/output/document/Sequential_Composition.tex
+++ /dev/null
@@ -1,3359 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The sequential composition creates a new electoral module from
-two electoral modules. In a sequential composition, the second
-electoral module makes decisions over alternatives deferred by
-the first electoral module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sequential{\isacharunderscore}{\kern0pt}composition\ m\ n\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ sequence\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymtriangleright}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymtriangleright}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \ \ \ \ \ \ rev{\isacharunderscore}{\kern0pt}subsetD\ result{\isacharunderscore}{\kern0pt}disj\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}neutr{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ inf{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}iff\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}commute\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ inf{\isachardot}{\kern0pt}commute\ inf{\isachardot}{\kern0pt}orderE\ inf{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}absorb\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ distrib{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ sup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}prof\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ reject\ m\ A\ p\ {\isasymunion}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ module{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ r\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ empty{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ \ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}lower{\isadigit{1}}\ Un{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ empty{\isacharunderscore}{\kern0pt}defer\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assm\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ finite\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ singletonI\ subsetCE\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ defers{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ rej\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ res{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ elect\ n\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card\ def\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\isanewline
-\ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def\ def{\isacharunderscore}{\kern0pt}card\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ res{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ singletonI\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ elect\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}commute\ fst{\isacharunderscore}{\kern0pt}conv\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ in{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymor}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}empty\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ inf{\isachardot}{\kern0pt}absorb{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}m{\isadigit{1}}\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m{\isadigit{1}}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ m{\isadigit{1}}\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ zero{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymlongrightarrow}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ modules\ profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq\ mono{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ modules\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ modules\ profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}lengths{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ mono{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ eql{\isacharunderscore}{\kern0pt}lengths\ mono{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ leI\ not{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}zero\ nth{\isacharunderscore}{\kern0pt}equalityI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m\ mono{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ pos{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ m{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ subsetI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ defers\ n\ f\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymor}\ infinite\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}seq{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}S{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}S\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ p\ q{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ m\ S\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ a\ b\ \isacommand{have}\isamarkupfalse%
-\ profiles{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ DiffD{\isadigit{2}}\ a\ b\ compatible\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}def\ profiles\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ S\ p\ {\isacharequal}{\kern0pt}\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ UnI{\isadigit{1}}\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\ modules\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}w\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p\ lifted{\isacharunderscore}{\kern0pt}w\ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mem{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ empty{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ insertCI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defers{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}subset\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}Diff\ insertCI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}a\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Sequential_Majority_Comparison.tex b/virage/src/test/resources/old_theories/output/document/Sequential_Majority_Comparison.tex
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-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison}%
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-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Sequential Majority Comparison%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}And{\isacharunderscore}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Revision{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\begin{isamarkuptext}%
-Sequential majority comparison compares two alternatives by plurality voting.
-The loser gets rejected, and the winner is compared to the next alternative.
-This process is repeated until only a single alternative is left, which is
-then elected.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ smc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}smc\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isasymup}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ defer{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}plurality{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set\ list\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ reject{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ fin{\isacharunderscore}{\kern0pt}A\ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{(Weak) Monotonicity Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}monotone{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardot}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Termination_Condition.tex b/virage/src/test/resources/old_theories/output/document/Termination_Condition.tex
deleted file mode 100644
index 7ff53ae..0000000
--- a/virage/src/test/resources/old_theories/output/document/Termination_Condition.tex
+++ /dev/null
@@ -1,99 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Termination{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Termination Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Termination{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The termination condition is used in loops. It decides whether or not to
-terminate the loop after each iteration, depending on the current state
-of the loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Termination_Condition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
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-%:%32=11%:%
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-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%67=22%:%
-%:%72=23%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/comment.sty b/virage/src/test/resources/old_theories/output/document/comment.sty
deleted file mode 100644
index 658686f..0000000
--- a/virage/src/test/resources/old_theories/output/document/comment.sty
+++ /dev/null
@@ -1,278 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Comment.sty   version 3.6, October 1999
-%
-% Purpose:
-% selectively in/exclude pieces of text: the user can define new
-% comment versions, and each is controlled separately.
-% Special comments can be defined where the user specifies the
-% action that is to be taken with each comment line.
-%
-% Author
-%    Victor Eijkhout
-%    Department of Computer Science
-%    University of Tennessee
-%    107 Ayres Hall
-%    Knoxville TN 37996
-%    USA
-%
-%    victor@eijkhout.net
-%
-% This program is free software; you can redistribute it and/or
-% modify it under the terms of the GNU General Public License
-% as published by the Free Software Foundation; either version 2
-% of the License, or (at your option) any later version.
-% 
-% This program is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-% GNU General Public License for more details.
-%
-% For a copy of the GNU General Public License, write to the 
-% Free Software Foundation, Inc.,
-% 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA,
-% or find it on the net, for instance at
-% http://www.gnu.org/copyleft/gpl.html
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This style can be used with plain TeX or LaTeX, and probably
-% most other packages too.
-%
-% Usage: all text included between
-%    \comment ... \endcomment
-% or \begin{comment} ... \end{comment}
-% is discarded. 
-%
-% The opening and closing commands should appear on a line
-% of their own. No starting spaces, nothing after it.
-% This environment should work with arbitrary amounts
-% of comment, and the comment can be arbitrary text.
-%
-% Other `comment' environments are defined by
-% and are selected/deselected with
-% \includecomment{versiona}
-% \excludecoment{versionb}
-%
-% These environments are used as
-% \versiona ... \endversiona
-% or \begin{versiona} ... \end{versiona}
-% with the opening and closing commands again on a line of 
-% their own.
-%
-% LaTeX users note: for an included comment, the
-% \begin and \end lines act as if they don't exist.
-% In particular, they don't imply grouping, so assignments 
-% &c are not local.
-%
-% Special comments are defined as
-% \specialcomment{name}{before commands}{after commands}
-% where the second and third arguments are executed before
-% and after each comment block. You can use this for global
-% formatting commands.
-% To keep definitions &c local, you can include \begingroup
-% in the `before commands' and \endgroup in the `after commands'.
-% ex:
-% \specialcomment{smalltt}
-%     {\begingroup\ttfamily\footnotesize}{\endgroup}
-% You do *not* have to do an additional
-% \includecomment{smalltt}
-% To remove 'smalltt' blocks, give \excludecomment{smalltt}
-% after the definition.
-%
-% Processing comments can apply processing to each line.
-% \processcomment{name}{each-line commands}%
-%    {before commands}{after commands}
-% By defining a control sequence 
-% \def\Thiscomment##1{...} in the before commands the user can
-% specify what is to be done with each comment line.
-% BUG this does not work quite yet BUG
-%
-% Trick for short in/exclude macros (such as \maybe{this snippet}):
-%\includecomment{cond}
-%\newcommand{\maybe}[1]{}
-%\begin{cond}
-%\renewcommand{\maybe}[1]{#1}
-%\end{cond}
-%
-% Basic approach of the implementation:
-% to comment something out, scoop up  every line in verbatim mode
-% as macro argument, then throw it away.
-% For inclusions, in LaTeX the block is written out to
-% a file \CommentCutFile (default "comment.cut"), which is
-% then included.
-% In plain TeX (and other formats) both the opening and
-% closing comands are defined as noop.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Changes in version 3.1
-% - updated author's address
-% - cleaned up some code
-% - trailing contents on \begin{env} line is always discarded
-%  even if you've done \includecomment{env}
-% - comments no longer define grouping!! you can even
-%   \includecomment{env}
-%   \begin{env}
-%   \begin{itemize}
-%   \end{env}
-%  Isn't that something ...
-% - included comments are written to file and input again.
-% Changes in 3.2
-% - \specialcomment brought up to date (thanks to Ivo Welch).
-% Changes in 3.3
-% - updated author's address again
-% - parametrised \CommentCutFile
-% Changes in 3.4
-% - added GNU public license
-% - added \processcomment, because Ivo's fix (above) brought an
-%   inconsistency to light.
-% Changes in 3.5
-% - corrected typo in header.
-% - changed author email
-% - corrected \specialcomment yet again.
-% - fixed excludecomment of an earlier defined environment.
-% Changes in 3.6
-% - The 'cut' file is now written more verbatim, using \meaning;
-%   some people reported having trouble with ISO latin 1, or umlaute.sty.
-% - removed some \newif statements.
-%   Has this suddenly become \outer again?
-%
-% Known bugs:
-% - excludecomment leads to one superfluous space
-% - processcomment leads to a superfluous line break
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\makeinnocent#1{\catcode`#1=12 }
-\def\csarg#1#2{\expandafter#1\csname#2\endcsname}
-\def\latexname{lplain}\def\latexename{LaTeX2e}
-\newwrite\CommentStream
-\def\CommentCutFile{comment.cut}
-
-\def\ProcessComment#1% start it all of
-   {\begingroup
-    \def\CurrentComment{#1}%
-    \let\do\makeinnocent \dospecials 
-    \makeinnocent\^^L% and whatever other special cases
-    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-%\def\ProcessCommentWithArg#1#2% to be used in \leveledcomment
-%   {\begingroup
-%    \def\CurrentComment{#1}%
-%    \let\do\makeinnocent \dospecials 
-%    \makeinnocent\^^L% and whatever other special cases
-%    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-{\catcode`\^^M=12 \endlinechar=-1 %
- \gdef\xComment#1^^M{%
-    \expandafter\ProcessCommentLine}
- \gdef\ProcessCommentLine#1^^M{\def\test{#1}
-      \csarg\ifx{End\CurrentComment Test}\test
-          \edef\next{\noexpand\EndOfComment{\CurrentComment}}%
-      \else \ThisComment{#1}\let\next\ProcessCommentLine
-      \fi \next}
-}
-
-\def\CSstringmeaning#1{\expandafter\CSgobblearrow\meaning#1}
-\def\CSstringcsnoescape#1{\expandafter\CSgobbleescape\string#1}
-{\escapechar-1
-\expandafter\expandafter\expandafter\gdef
-  \expandafter\expandafter\expandafter\CSgobblearrow
-    \expandafter\string\csname macro:->\endcsname{}
-}
-\def\CSgobbleescape#1{\ifnum`\\=`#1 \else #1\fi}
-\def\WriteCommentLine#1{\def\CStmp{#1}%
-    \immediate\write\CommentStream{\CSstringmeaning\CStmp}}
-
-% 3.1 change: in LaTeX and LaTeX2e prevent grouping
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-%%%%
-%%%% definitions for LaTeX
-%%%%
-\def\AfterIncludedComment
-   {\immediate\closeout\CommentStream
-    \input{\CommentCutFile}\relax
-    }%
-\def\TossComment{\immediate\closeout\CommentStream}
-\def\BeforeIncludedComment
-   {\immediate\openout\CommentStream=\CommentCutFile
-    \let\ThisComment\WriteCommentLine}
-\def\includecomment
- #1{\message{Include comment '#1'}%
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    % note: \AfterIncludedComment does \input, so #2 goes here!
-    \csarg\def{After#1Comment}{#2\AfterIncludedComment#3}%
-    \csarg\def{#1}{\BeforeIncludedComment\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\processcomment
- #1#2#3#4{\message{Lines-Processing comment '#1'}%
-    \csarg\def{After#1Comment}{#3\AfterIncludedComment#4}%
-    \csarg\def{#1}{\BeforeIncludedComment#2\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\def\leveledcomment
- #1#2{\message{Include comment '#1' up to level '#2'}%
-    %\csname #1IsLeveledCommenttrue\endcsname
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessCommentWithArg{#1}}%
-    \CommentEndDef{#1}}
-\else 
-%%%%
-%%%%plain TeX and other formats
-%%%%
-\def\includecomment
- #1{\message{Including comment '#1'}%
-    \csarg\def{#1}{}%
-    \csarg\def{end#1}{}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    \csarg\def{#1}{\def\ThisComment{}\def\AfterComment{#3}#2%
-           \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\fi
-
-%%%%
-%%%% general definition of skipped comment
-%%%%
-\def\excludecomment
- #1{\message{Excluding comment '#1'}%
-    \csarg\def{#1}{\let\AfterComment\relax
-           \def\ThisComment####1{}\ProcessComment{#1}}%
-    \csarg\let{After#1Comment}\TossComment
-    \CommentEndDef{#1}}
-
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-% latex & latex2e:
-\def\EndOfComment#1{\endgroup\end{#1}%
-    \csname After#1Comment\endcsname}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end\string\{#1\string\}}%
-    }}
-\else
-% plain & other
-\def\EndOfComment#1{\endgroup\AfterComment}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end#1}%
-    }}
-\fi
-
-\excludecomment{comment}
-
-\endinput
diff --git a/virage/src/test/resources/old_theories/output/document/isabelle.sty b/virage/src/test/resources/old_theories/output/document/isabelle.sty
deleted file mode 100644
index b6dcb3e..0000000
--- a/virage/src/test/resources/old_theories/output/document/isabelle.sty
+++ /dev/null
@@ -1,267 +0,0 @@
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-\newcommand{\setisabellecontext}[1]{\def\isabellecontext{#1}}
-
-\newcommand{\isastyle}{\UNDEF}
-\newcommand{\isastylett}{\UNDEF}
-\newcommand{\isastyleminor}{\UNDEF}
-\newcommand{\isastyleminortt}{\UNDEF}
-\newcommand{\isastylescript}{\UNDEF}
-\newcommand{\isastyletext}{\normalsize\normalfont\rmfamily}
-\newcommand{\isastyletxt}{\normalfont\rmfamily}
-\newcommand{\isastylecmt}{\normalfont\rmfamily}
-
-\newcommand{\isaspacing}{%
-  \sfcode 42 1000 % .
-  \sfcode 63 1000 % ?
-  \sfcode 33 1000 % !
-  \sfcode 58 1000 % :
-  \sfcode 59 1000 % ;
-  \sfcode 44 1000 % ,
-}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
-\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
-\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
-\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastyle}{\par}
-
-\newenvironment{isabellebodytt}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastylett}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newenvironment{isabellett}
-{\begin{trivlist}\begin{isabellebodytt}\item\relax}
-{\end{isabellebodytt}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
-\newcommand{\isatt}[1]{\emph{\isaspacing\isastyleminortt #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{}
-\newcommand{\isadigit}[1]{#1}
-
-\newcommand{\isachardefaults}{%
-\def\isacharbell{\isamath{\bigbox}}%requires stmaryrd
-\chardef\isacharbang=`\!%
-\chardef\isachardoublequote=`\"%
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharhash=`\#%
-\chardef\isachardollar=`\$%
-\chardef\isacharpercent=`\%%
-\chardef\isacharampersand=`\&%
-\chardef\isacharprime=`\'%
-\chardef\isacharparenleft=`\(%
-\chardef\isacharparenright=`\)%
-\chardef\isacharasterisk=`\*%
-\chardef\isacharplus=`\+%
-\chardef\isacharcomma=`\,%
-\chardef\isacharminus=`\-%
-\chardef\isachardot=`\.%
-\chardef\isacharslash=`\/%
-\chardef\isacharcolon=`\:%
-\chardef\isacharsemicolon=`\;%
-\chardef\isacharless=`\<%
-\chardef\isacharequal=`\=%
-\chardef\isachargreater=`\>%
-\chardef\isacharquery=`\?%
-\chardef\isacharat=`\@%
-\chardef\isacharbrackleft=`\[%
-\chardef\isacharbackslash=`\\%
-\chardef\isacharbrackright=`\]%
-\chardef\isacharcircum=`\^%
-\chardef\isacharunderscore=`\_%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquote=`\`%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\chardef\isacharbraceleft=`\{%
-\chardef\isacharbar=`\|%
-\chardef\isacharbraceright=`\}%
-\chardef\isachartilde=`\~%
-\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
-\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
-\def\isacartoucheopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacartoucheclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-
-% keyword and section markup
-
-\newcommand{\isakeyword}[1]
-{\emph{\normalfont\bfseries\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isakeywordcontrol}[1]
-{\emph{\normalfont\bfseries\itshape\def\isacharunderscore{\isacharunderscorekeyword}#1\,}}
-
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupparagraph}[1]{\paragraph{#1}}
-\newcommand{\isamarkupsubparagraph}[1]{\subparagraph{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% styles
-
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestyledefault}{%
-\def\isastyle{\small\normalfont\ttfamily\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily\slshape}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily\slshape}%
-\isachardefaults%
-}
-\isabellestyledefault
-
-\newcommand{\isabellestylett}{%
-\def\isastyle{\small\normalfont\ttfamily}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily}%
-\isachardefaults%
-}
-
-\newcommand{\isabellestyleit}{%
-\def\isastyle{\small\normalfont\itshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\itshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\itshape}%
-\isachardefaults%
-\def\isacharunderscorekeyword{\mbox{-}}%
-\def\isacharbang{\isamath{!}}%
-\def\isachardoublequote{}%
-\def\isachardoublequoteopen{}%
-\def\isachardoublequoteclose{}%
-\def\isacharhash{\isamath{\#}}%
-\def\isachardollar{\isamath{\$}}%
-\def\isacharpercent{\isamath{\%}}%
-\def\isacharampersand{\isamath{\&}}%
-\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\def\isacharparenleft{\isamath{(}}%
-\def\isacharparenright{\isamath{)}}%
-\def\isacharasterisk{\isamath{*}}%
-\def\isacharplus{\isamath{+}}%
-\def\isacharcomma{\isamath{\mathord,}}%
-\def\isacharminus{\isamath{-}}%
-\def\isachardot{\isamath{\mathord.}}%
-\def\isacharslash{\isamath{/}}%
-\def\isacharcolon{\isamath{\mathord:}}%
-\def\isacharsemicolon{\isamath{\mathord;}}%
-\def\isacharless{\isamath{<}}%
-\def\isacharequal{\isamath{=}}%
-\def\isachargreater{\isamath{>}}%
-\def\isacharat{\isamath{@}}%
-\def\isacharbrackleft{\isamath{[}}%
-\def\isacharbackslash{\isamath{\backslash}}%
-\def\isacharbrackright{\isamath{]}}%
-\def\isacharunderscore{\mbox{-}}%
-\def\isacharbraceleft{\isamath{\{}}%
-\def\isacharbar{\isamath{\mid}}%
-\def\isacharbraceright{\isamath{\}}}%
-\def\isachartilde{\isamath{{}\sp{\sim}}}%
-\def\isacharbackquoteopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacharbackquoteclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-\newcommand{\isabellestyleliteral}{%
-\isabellestyleit%
-\def\isacharunderscore{\_}%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-}
-
-\newcommand{\isabellestyleliteralunderscore}{%
-\isabellestyleliteral%
-\def\isacharunderscore{\textunderscore}%
-\def\isacharunderscorekeyword{\textunderscore}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\def\isastyle{\small\normalfont\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\slshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\slshape}%
-}
-
-
-% cancel text
-
-\usepackage[normalem]{ulem}
-\newcommand{\isamarkupcancel}[1]{\isa{\xout{#1}}}
-
-
-% tagged regions
-
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isakeeptag{document}
-\isakeeptag{theory}
-\isakeeptag{proof}
-\isakeeptag{ML}
-\isakeeptag{visible}
-\isadroptag{invisible}
-\isakeeptag{important}
-\isakeeptag{unimportant}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
diff --git a/virage/src/test/resources/old_theories/output/document/isabellesym.sty b/virage/src/test/resources/old_theories/output/document/isabellesym.sty
deleted file mode 100644
index b0d076c..0000000
--- a/virage/src/test/resources/old_theories/output/document/isabellesym.sty
+++ /dev/null
@@ -1,427 +0,0 @@
-%%
-%% definitions of standard Isabelle symbols
-%%
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}  %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}  %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}}  %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}  %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}}  %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}}  %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
-\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
-\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
-\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
-\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
-\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLleftarrow}{\isamath{\Lleftarrow}}  %requires amssymb
-\newcommand{\isasymRrightarrow}{\isamath{\Rrightarrow}}  %requires amssymb
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymllangle}{\isamath{\langle\mskip-5mu\langle}}
-\newcommand{\isasymrrangle}{\isamath{\rangle\mskip-5mu\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
-\newcommand{\isasymguillemotleft}{\isatext{\flqq}}  %requires babel
-\newcommand{\isasymguillemotright}{\isatext{\frqq}}  %requires babel
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymcircle}{\isamath{\ocircle}}  %requires wasysym
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymdiamondop}{\isamath{\diamond}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires stmaryrd
-\newcommand{\isasymsetminus}{\isamath{\setminus}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymbbar}{\isamath{[\mskip-1.5mu]}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymsqdot}{\isamath{\sbox\z@{$\centerdot$}\ht\z@=.33333\ht\z@\vcenter{\box\z@}}}  %requires amssymb
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\normalfont\rmfamily\copyright}}
-\newcommand{\isasymregistered}{\isatext{\normalfont\rmfamily\textregistered}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonequarter}{\isatext{\normalfont\rmfamily\textonequarter}}  %requires textcomp
-\newcommand{\isasymonehalf}{\isatext{\normalfont\rmfamily\textonehalf}}  %requires textcomp
-\newcommand{\isasymthreequarters}{\isatext{\normalfont\rmfamily\textthreequarters}}  %requires textcomp
-\newcommand{\isasymordfeminine}{\isatext{\normalfont\rmfamily\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\normalfont\rmfamily\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\normalfont\rmfamily\S}}
-\newcommand{\isasymparagraph}{\isatext{\normalfont\rmfamily\P}}
-\newcommand{\isasymexclamdown}{\isatext{\normalfont\rmfamily\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\normalfont\rmfamily\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\euro}}  %requires eurosym
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\normalfont\rmfamily\textdegree}}  %requires textcomp
-\newcommand{\isasymhyphen}{\isatext{\normalfont\rmfamily-}}
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
-\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
-\newcommand{\isasymopen}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}
-\newcommand{\isasymclose}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}
-\newcommand{\isasymhole}{\isatext{\normalfont\rmfamily\wasylozenge}}  %requires wasysym
-\newcommand{\isasymnewline}{\isatext{\fbox{$\hookleftarrow$}}}
-\newcommand{\isasymcomment}{\isatext{\isastylecmt---}}
-\newcommand{\isasymproof}{\isamath{\,\langle\mathit{proof}\rangle}}
-
-\newcommand{\isactrlmarker}{\isatext{\ding{48}}}  %requires pifont
-\newcommand{\isactrlassert}{\isakeywordcontrol{assert}}
-\newcommand{\isactrlcancel}{\isakeywordcontrol{cancel}}
-\newcommand{\isactrlbinding}{\isakeywordcontrol{binding}}
-\newcommand{\isactrlclass}{\isakeywordcontrol{class}}
-\newcommand{\isactrlclassUNDERSCOREsyntax}{\isakeywordcontrol{class{\isacharunderscore}syntax}}
-\newcommand{\isactrlcommandUNDERSCOREkeyword}{\isakeywordcontrol{command{\isacharunderscore}keyword}}
-\newcommand{\isactrlconst}{\isakeywordcontrol{const}}
-\newcommand{\isactrlconstUNDERSCOREabbrev}{\isakeywordcontrol{const{\isacharunderscore}abbrev}}
-\newcommand{\isactrlconstUNDERSCOREname}{\isakeywordcontrol{const{\isacharunderscore}name}}
-\newcommand{\isactrlconstUNDERSCOREsyntax}{\isakeywordcontrol{const{\isacharunderscore}syntax}}
-\newcommand{\isactrlcontext}{\isakeywordcontrol{context}}
-\newcommand{\isactrlcprop}{\isakeywordcontrol{cprop}}
-\newcommand{\isactrlcterm}{\isakeywordcontrol{cterm}}
-\newcommand{\isactrlctyp}{\isakeywordcontrol{ctyp}}
-\newcommand{\isactrldir}{\isakeywordcontrol{dir}}
-\newcommand{\isactrlfile}{\isakeywordcontrol{file}}
-\newcommand{\isactrlhere}{\isakeywordcontrol{here}}
-\newcommand{\isactrlkeyword}{\isakeywordcontrol{keyword}}
-\newcommand{\isactrllatex}{\isakeywordcontrol{latex}}
-\newcommand{\isactrllocale}{\isakeywordcontrol{locale}}
-\newcommand{\isactrlmakeUNDERSCOREstring}{\isakeywordcontrol{make{\isacharunderscore}string}}
-\newcommand{\isactrlmasterUNDERSCOREdir}{\isakeywordcontrol{master{\isacharunderscore}dir}}
-\newcommand{\isactrlmethod}{\isakeywordcontrol{method}}
-\newcommand{\isactrlnamedUNDERSCOREtheorems}{\isakeywordcontrol{named{\isacharunderscore}theorems}}
-\newcommand{\isactrlnonterminal}{\isakeywordcontrol{nonterminal}}
-\newcommand{\isactrloracleUNDERSCOREname}{\isakeywordcontrol{oracle{\isacharunderscore}name}}
-\newcommand{\isactrlpath}{\isakeywordcontrol{path}}
-\newcommand{\isactrlpathUNDERSCOREbinding}{\isakeywordcontrol{path{\isacharunderscore}binding}}
-\newcommand{\isactrlplugin}{\isakeywordcontrol{plugin}}
-\newcommand{\isactrlprint}{\isakeywordcontrol{print}}
-\newcommand{\isactrlprop}{\isakeywordcontrol{prop}}
-\newcommand{\isactrlscala}{\isakeywordcontrol{scala}}
-\newcommand{\isactrlscalaUNDERSCOREfunction}{\isakeywordcontrol{scala{\isacharunderscore}function}}
-\newcommand{\isactrlscalaUNDERSCOREmethod}{\isakeywordcontrol{scala{\isacharunderscore}method}}
-\newcommand{\isactrlscalaUNDERSCOREobject}{\isakeywordcontrol{scala{\isacharunderscore}object}}
-\newcommand{\isactrlscalaUNDERSCOREtype}{\isakeywordcontrol{scala{\isacharunderscore}type}}
-\newcommand{\isactrlsimproc}{\isakeywordcontrol{simproc}}
-\newcommand{\isactrlsort}{\isakeywordcontrol{sort}}
-\newcommand{\isactrlsyntaxUNDERSCOREconst}{\isakeywordcontrol{syntax{\isacharunderscore}const}}
-\newcommand{\isactrlsystemUNDERSCOREoption}{\isakeywordcontrol{system{\isacharunderscore}option}}
-\newcommand{\isactrlterm}{\isakeywordcontrol{term}}
-\newcommand{\isactrltheory}{\isakeywordcontrol{theory}}
-\newcommand{\isactrltheoryUNDERSCOREcontext}{\isakeywordcontrol{theory{\isacharunderscore}context}}
-\newcommand{\isactrltyp}{\isakeywordcontrol{typ}}
-\newcommand{\isactrltypeUNDERSCOREabbrev}{\isakeywordcontrol{type{\isacharunderscore}abbrev}}
-\newcommand{\isactrltypeUNDERSCOREname}{\isakeywordcontrol{type{\isacharunderscore}name}}
-\newcommand{\isactrltypeUNDERSCOREsyntax}{\isakeywordcontrol{type{\isacharunderscore}syntax}}
-\newcommand{\isactrlundefined}{\isakeywordcontrol{undefined}}
-
-\newcommand{\isactrlcode}{\isakeywordcontrol{code}}
-\newcommand{\isactrlcomputation}{\isakeywordcontrol{computation}}
-\newcommand{\isactrlcomputationUNDERSCOREconv}{\isakeywordcontrol{computation{\isacharunderscore}conv}}
-\newcommand{\isactrlcomputationUNDERSCOREcheck}{\isakeywordcontrol{computation{\isacharunderscore}check}}
diff --git a/virage/src/test/resources/old_theories/output/document/isabelletags.sty b/virage/src/test/resources/old_theories/output/document/isabelletags.sty
deleted file mode 100644
index 8b13789..0000000
--- a/virage/src/test/resources/old_theories/output/document/isabelletags.sty
+++ /dev/null
@@ -1 +0,0 @@
-
diff --git a/virage/src/test/resources/old_theories/output/document/pdfsetup.sty b/virage/src/test/resources/old_theories/output/document/pdfsetup.sty
deleted file mode 100644
index 2bf5ccf..0000000
--- a/virage/src/test/resources/old_theories/output/document/pdfsetup.sty
+++ /dev/null
@@ -1,7 +0,0 @@
-%%
-%% default hyperref setup (both for pdf and dvi output)
-%%
-
-\usepackage{color}
-\definecolor{linkcolor}{rgb}{0,0,0.5}
-\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,urlcolor=linkcolor,pdfpagelabels]{hyperref}
diff --git a/virage/src/test/resources/old_theories/output/document/railsetup.sty b/virage/src/test/resources/old_theories/output/document/railsetup.sty
deleted file mode 100644
index 93cae05..0000000
--- a/virage/src/test/resources/old_theories/output/document/railsetup.sty
+++ /dev/null
@@ -1,1202 +0,0 @@
-% rail.sty - style file to support railroad diagrams
-%
-% 09-Jul-90 L. Rooijakkers
-% 08-Oct-90 L. Rooijakkers	fixed centering bug when \rail@tmpc<0.
-% 07-Feb-91 L. Rooijakkers	added \railoptions command, indexing
-% 08-Feb-91 L. Rooijakkers	minor fixes
-% 28-Jun-94 K. Barthelmann	turned into LaTeX2e package
-% 08-Dec-96 K. Barthelmann	replaced \@writefile
-% 13-Dec-96 K. Barthelmann	cleanup
-% 22-Feb-98 K. Barthelmann	fixed catcodes of special characters
-% 18-Apr-98 K. Barthelmann	fixed \par handling
-% 19-May-98 J. Olsson		Added new macros to support arrow heads.
-% 26-Jul-98 K. Barthelmann	changed \par to output newlines
-% 02-May-11 M. Wenzel           default setup for Isabelle
-%
-% This style file needs to be used in conjunction with the 'rail'
-% program. Running LaTeX as 'latex file' produces file.rai, which should be
-% processed by Rail with 'rail file'. This produces file.rao, which will
-% be picked up by LaTeX on the next 'latex file' run.
-%
-% LaTeX will warn if there is no file.rao or it's out of date.
-%
-% The macros in this file thus consist of two parts: those that read and
-% write the .rai and .rao files, and those that do the actual formatting
-% of the railroad diagrams.
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{rail}[1998/05/19]
-
-% railroad diagram formatting parameters (user level)
-% all of these are copied into their internal versions by \railinit
-%
-% \railunit : \unitlength within railroad diagrams
-% \railextra : extra length at outside of diagram
-% \railboxheight : height of ovals and frames
-% \railboxskip : vertical space between lines
-% \railboxleft : space to the left of a box
-% \railboxright : space to the right of a box
-% \railovalspace : extra space around contents of oval
-% \railframespace : extra space around contents of frame
-% \railtextleft : space to the left of text
-% \railtextright : space to the right of text
-% \railtextup : space to lift text up
-% \railjoinsize : circle size of join/split arcs
-% \railjoinadjust : space to adjust join
-%
-% \railnamesep : separator between name and rule body
-
-\newlength\railunit
-\newlength\railextra
-\newlength\railboxheight
-\newlength\railboxskip
-\newlength\railboxleft
-\newlength\railboxright
-\newlength\railovalspace
-\newlength\railframespace
-\newlength\railtextleft
-\newlength\railtextright
-\newlength\railtextup
-\newlength\railjoinsize
-\newlength\railjoinadjust
-\newlength\railnamesep
-
-% initialize the parameters
-
-\setlength\railunit{1sp}
-\setlength\railextra{4ex}
-\setlength\railboxleft{1ex}
-\setlength\railboxright{1ex}
-\setlength\railovalspace{2ex}
-\setlength\railframespace{2ex}
-\setlength\railtextleft{1ex}
-\setlength\railtextright{1ex}
-\setlength\railjoinadjust{0pt}
-\setlength\railnamesep{1ex}
-
-\DeclareOption{10pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{11pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{12pt}{
-  \setlength\railboxheight{20pt}
-  \setlength\railboxskip{28pt}
-  \setlength\railtextup{6pt}
-  \setlength\railjoinsize{20pt}
-}
-
-\ExecuteOptions{10pt}
-\ProcessOptions
-
-% internal versions of the formatting parameters
-%
-% \rail@extra   : \railextra
-% \rail@boxht   : \railboxheight
-% \rail@boxsp   : \railboxskip
-% \rail@boxlf   : \railboxleft
-% \rail@boxrt   : \railboxright
-% \rail@boxhht  : \railboxheight / 2
-% \rail@ovalsp  : \railovalspace
-% \rail@framesp : \railframespace
-% \rail@textlf  : \railtextleft
-% \rail@textrt  : \railtextright
-% \rail@textup  : \railtextup
-% \rail@joinsz  : \railjoinsize
-% \rail@joinhsz : \railjoinsize / 2
-% \rail@joinadj : \railjoinadjust
-%
-% \railinit : internalize all of the parameters.
-
-\newcount\rail@extra
-\newcount\rail@boxht
-\newcount\rail@boxsp
-\newcount\rail@boxlf
-\newcount\rail@boxrt
-\newcount\rail@boxhht
-\newcount\rail@ovalsp
-\newcount\rail@framesp
-\newcount\rail@textlf
-\newcount\rail@textrt
-\newcount\rail@textup
-\newcount\rail@joinsz
-\newcount\rail@joinhsz
-\newcount\rail@joinadj
-
-\newcommand\railinit{
-\rail@extra=\railextra
-\divide\rail@extra by \railunit
-\rail@boxht=\railboxheight
-\divide\rail@boxht by \railunit
-\rail@boxsp=\railboxskip
-\divide\rail@boxsp by \railunit
-\rail@boxlf=\railboxleft
-\divide\rail@boxlf by \railunit
-\rail@boxrt=\railboxright
-\divide\rail@boxrt by \railunit
-\rail@boxhht=\railboxheight
-\divide\rail@boxhht by \railunit
-\divide\rail@boxhht by 2
-\rail@ovalsp=\railovalspace
-\divide\rail@ovalsp by \railunit
-\rail@framesp=\railframespace
-\divide\rail@framesp by \railunit
-\rail@textlf=\railtextleft
-\divide\rail@textlf by \railunit
-\rail@textrt=\railtextright
-\divide\rail@textrt by \railunit
-\rail@textup=\railtextup
-\divide\rail@textup by \railunit
-\rail@joinsz=\railjoinsize
-\divide\rail@joinsz by \railunit
-\rail@joinhsz=\railjoinsize
-\divide\rail@joinhsz by \railunit
-\divide\rail@joinhsz by 2
-\rail@joinadj=\railjoinadjust
-\divide\rail@joinadj by \railunit
-}
-
-\AtBeginDocument{\railinit}
-
-% \rail@param : declarations for list environment
-%
-% \railparam{TEXT} : sets \rail@param to TEXT
-%
-% \rail@reserved : characters reserved for grammar
-
-\newcommand\railparam[1]{
-\def\rail@param{
-  \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
-  \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
-  \setlength\itemindent{0pt}\setlength\listparindent{0pt}
-  #1
-}
-}
-\railparam{}
-
-\newtoks\rail@reserved
-\rail@reserved={:;|*+?[]()'"}
-
-% \rail@termfont : format setup for terminals
-%
-% \rail@nontfont : format setup for nonterminals
-%
-% \rail@annofont : format setup for annotations
-%
-% \rail@rulefont : format setup for rule names
-%
-% \rail@indexfont : format setup for index entry
-%
-% \railtermfont{TEXT} : set terminal format setup to TEXT
-%
-% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
-%
-% \railannotatefont{TEXT} : set annotation format setup to TEXT
-%
-% \railnamefont{TEXT} : set rule name format setup to TEXT
-%
-% \railindexfont{TEXT} : set index entry format setup to TEXT
-
-\def\rail@termfont{\ttfamily\upshape}
-\def\rail@nontfont{\rmfamily\upshape}
-\def\rail@annofont{\rmfamily\itshape}
-\def\rail@namefont{\rmfamily\itshape}
-\def\rail@indexfont{\rmfamily\itshape}
-
-\newcommand\railtermfont[1]{
-\def\rail@termfont{#1}
-}
-
-\newcommand\railnontermfont[1]{
-\def\rail@nontfont{#1}
-}
-
-\newcommand\railannotatefont[1]{
-\def\rail@annofont{#1}
-}
-
-\newcommand\railnamefont[1]{
-\def\rail@namefont{#1}
-}
-
-\newcommand\railindexfont[1]{
-\def\rail@indexfont{#1}
-}
-
-% railroad read/write macros
-%
-% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
-%                                as \rail@i{NR}{TEXT}. Then the matching
-%                                \rail@o{NR}{FMT} from the .rao file is
-%                                executed (if defined).
-%
-% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
-%                         as \rail@p{OPTIONS}.
-%
-% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
-%                              \rail@t{IDENT} to the .rai file
-%
-% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
-%                           TEXT.
-%
-% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
-%                           (for backward compatibility)
-%
-% \rail@setcodes : guards special characters
-%
-% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
-%                              used inside a loop for \rail@setcodes
-%
-% \rail@nr : railroad diagram counter
-%
-% \ifrail@match : current \rail@i{NR}{TEXT} matches
-%
-% \rail@first : actions to be done first. read in .rao file,
-%               open .rai file if \@filesw true, undefine \rail@first.
-%               executed from \begin{rail}, \railoptions and \railterm.
-%
-% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
-%                     file by \rail, read from the .rao file by
-%                     \rail@first
-%
-% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
-%                  written to the .rai file by \railterm.
-%
-% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
-%                     file by \rail@first.
-%
-% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
-%                    \railoptions
-%
-% \rail@write{TEXT} : write TEXT to the .rai file
-%
-% \rail@warn : warn user for mismatching diagrams
-%
-% \rail@endwarn : either \relax or \rail@warn
-%
-% \ifrail@all : checked at the end of the document
-
-\def\rail@makeother#1{
-  \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
-}
-
-\def\rail@setcodes{
-\let\par=\relax
-\let\\=\relax
-\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
-  \the\rail@reserved
-\do{\expandafter\rail@makeother\rail@symbol}
-}
-
-\newcount\rail@nr
-
-\newif\ifrail@all
-\rail@alltrue
-
-\newif\ifrail@match
-
-\def\rail@first{
-\begingroup
-\makeatletter
-\rail@setcodes
-\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
-\makeatother
-\endgroup
-\if@filesw
-\newwrite\tf@rai
-\immediate\openout\tf@rai=\jobname.rai
-\fi
-\global\let\rail@first=\relax
-}
-
-\long\def\rail@body#1\end{
-{
-\newlinechar=`^^J
-\def\par{\string\par^^J}
-\rail@write{\string\rail@i{\number\rail@nr}{#1}}
-}
-\xdef\rail@i@{#1}
-\end
-}
-
-\newenvironment{rail}{
-\global\advance\rail@nr by 1
-\rail@first
-\begingroup
-\rail@setcodes
-\rail@body
-}{
-\endgroup
-\rail@matchtrue
-\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
-\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
-\else
-\rail@matchfalse
-\fi
-\ifrail@match
-\csname rail@o@\number\rail@nr\endcsname
-\else
-\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
-\global\let\rail@endwarn=\rail@warn
-\begin{list}{}{\rail@param}
-\rail@begin{1}{}
-\rail@setbox{\bfseries ???}
-\rail@oval
-\rail@end
-\end{list}
-\fi
-}
-
-\newcommand\railoptions[1]{
-\rail@first
-\rail@write{\string\rail@p{#1}}
-}
-
-\newcommand\railterm[1]{
-\rail@first
-\@for\rail@@:=#1\do{
-\rail@write{\string\rail@t{\rail@@}}
-}
-}
-
-\newcommand\railalias[2]{
-\expandafter\def\csname rail@t@#1\endcsname{#2}
-}
-
-\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
-
-\long\def\rail@i#1#2{
-\expandafter\gdef\csname rail@i@#1\endcsname{#2}
-}
-
-\def\rail@o#1#2{
-\expandafter\gdef\csname rail@o@#1\endcsname{
-\begin{list}{}{\rail@param}
-#2
-\end{list}
-}
-}
-
-\def\rail@t#1{}
-
-\def\rail@p#1{}
-
-\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
-
-\def\rail@warn{
-\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
-                            Use 'rail' and rerun}
-}
-
-\let\rail@endwarn=\relax
-
-\AtEndDocument{\rail@endwarn}
-
-% index entry macro
-%
-% \rail@index{IDENT} : add index entry for IDENT
-
-\def\rail@index#1{
-\index{\rail@indexfont#1}
-}
-
-% railroad formatting primitives
-%
-% \rail@x : current x
-% \rail@y : current y
-% \rail@ex : current end x
-% \rail@sx : starting x for \rail@cr
-% \rail@rx : rightmost previous x for \rail@cr
-%
-% \rail@tmpa : temporary count
-% \rail@tmpb : temporary count
-% \rail@tmpc : temporary count
-%
-% \rail@put : put at (\rail@x,\rail@y)
-% \rail@vput : put vector at (\rail@x,\rail@y)
-%
-% \rail@eline : end line by drawing from \rail@ex to \rail@x
-%
-% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
-%
-% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
-%
-% \rail@sety{LEVEL} : set \rail@y to level LEVEL
-
-\newcount\rail@x
-\newcount\rail@y
-\newcount\rail@ex
-\newcount\rail@sx
-\newcount\rail@rx
-
-\newcount\rail@tmpa
-\newcount\rail@tmpb
-\newcount\rail@tmpc
-
-\def\rail@put{\put(\number\rail@x,\number\rail@y)}
-
-\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
-
-\def\rail@eline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\line(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vreline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@vput{\vector(1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vleline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\vector(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@sety#1{
-\rail@y=#1
-\multiply\rail@y by -\rail@boxsp
-\advance\rail@y by -\rail@boxht
-}
-
-% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
-%
-% \rail@end : end a railroad diagram
-%
-% \rail@expand{IDENT} : expand IDENT
-
-\def\rail@begin#1#2{
-\item
-\begin{minipage}[t]{\linewidth}
-\ifx\@empty#2\else
-{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
-\fi
-\unitlength=\railunit
-\rail@tmpa=#1
-\multiply\rail@tmpa by \rail@boxsp
-\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
-\rail@ex=0
-\rail@rx=0
-\rail@x=\rail@extra
-\rail@sx=\rail@x
-\rail@sety{0}
-}
-
-\def\rail@end{
-\advance\rail@x by \rail@extra
-\rail@eline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@vend{
-\advance\rail@x by \rail@extra
-\rail@vreline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
-
-% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
-% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
-% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
-%
-% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
-% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
-% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
-%
-% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
-% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
-% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
-%
-% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
-% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow left
-% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
-% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
-% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
-%
-% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
-% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
-% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@annote[TEXT] : format TEXT as annotation
-
-\def\rail@token#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@ltoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rtoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@ctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@nont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@frame
-}
-
-\def\rail@lnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlframe
-}
-
-\def\rail@rnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrframe
-}
-
-\def\rail@cnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@cframe
-}
-
-\def\rail@lcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcframe
-}
-
-\def\rail@rcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcframe
-}
-
-\def\rail@term#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@lterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@cterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@annote[#1]{
-\rail@setbox{\rail@annofont #1}
-\rail@text
-}
-
-% \rail@box : temporary box for \rail@oval and \rail@frame
-%
-% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
-%
-% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
-% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
-% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
-%
-% \rail@coval : same as \rail@oval, but centered between \rail@x and
-%               \rail@mx
-% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector left
-% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector right
-%
-% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
-% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
-% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
-%
-% \rail@cframe : same as \rail@frame, but centered between \rail@x and
-%                \rail@mx
-% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector left
-% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector right
-%
-% \rail@text : format \rail@box of width \rail@tmpa above the line
-
-\newbox\rail@box
-
-\def\rail@setbox#1{
-\setbox\rail@box\hbox{\strut#1}
-\rail@tmpa=\wd\rail@box
-\divide\rail@tmpa by \railunit
-}
-
-\def\rail@oval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vloval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vroval{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@coval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@oval
-}
-
-\def\rail@vlcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vloval
-}
-
-\def\rail@vrcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vroval
-}
-
-\def\rail@frame{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vlframe{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vrframe{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@cframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@frame
-}
-
-\def\rail@vlcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vlframe
-}
-
-\def\rail@vrcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vrframe
-}
-
-\def\rail@text{
-\advance\rail@x by \rail@textlf
-\advance\rail@y by \rail@textup
-\rail@put{\box\rail@box}
-\advance\rail@y by -\rail@textup
-\advance\rail@x by \rail@tmpa
-\advance\rail@x by \rail@textrt
-}
-
-% alternatives
-%
-% \rail@jx \rail@jy : current join point
-%
-% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
-%                                         to pass values over group closings
-%
-% \rail@mx : maximum x so far
-%
-% \rail@sy : starting \rail@y for alternatives
-%
-% \rail@jput : put at (\rail@jx,\rail@jy)
-%
-% \rail@joval[PART] : put \oval[PART] with adjust
-
-\newcount\rail@jx
-\newcount\rail@jy
-
-\newcount\rail@gx
-\newcount\rail@gy
-\newcount\rail@gex
-\newcount\rail@grx
-
-\newcount\rail@sy
-\newcount\rail@mx
-
-\def\rail@jput{
-\put(\number\rail@jx,\number\rail@jy)
-}
-
-\def\rail@joval[#1]{
-\advance\rail@jx by \rail@joinadj
-\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
-\advance\rail@jx by -\rail@joinadj
-}
-
-% \rail@barsplit : incoming split for '|'
-%
-% \rail@plussplit : incoming split for '+'
-%
-
-\def\rail@barsplit{
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@plussplit{
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-}
-
-% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
-%
-
-\def\rail@alt#1{
-\rail@sy=\rail@y
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\rail@mx=0
-\let\rail@list=\@empty
-\let\rail@comma=\@empty
-\let\rail@split=#1
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
-%                         and fix-up FIX
-%
-
-\def\rail@nextalt#1#2{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-#1
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\def\rail@comma{,}
-\rail@split
-\let\rail@split=\@empty
-\rail@sety{#2}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@ex=\rail@x
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@barjoin : outgoing join for first '|' alternative
-%
-% \rail@plusjoin : outgoing join for first '+' alternative
-%
-% \rail@altjoin : join for subsequent alternative
-%
-
-\def\rail@barjoin{
-\ifnum\rail@y<\rail@sy
-\global\rail@gex=\rail@jx
-\else
-\global\rail@gex=\rail@ex
-\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\ifnum\rail@y<\rail@sy
-\rail@altjoin
-\fi
-}
-
-\def\rail@plusjoin{
-\global\rail@gex=\rail@ex
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by -\rail@joinsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@altjoin{
-\rail@eline
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-}
-
-% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
-%
-% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
-
-\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
-
-\def\rail@endalt#1{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\rail@x=\rail@mx
-\rail@jx=\rail@x
-\rail@jy=\rail@sy
-\advance\rail@jx by \rail@joinsz
-\let\rail@join=#1
-\@for\rail@elt:=\rail@list\do{
-\expandafter\rail@eltsplit\rail@elt;
-\rail@join
-\let\rail@join=\rail@altjoin
-}
-\rail@x=\rail@mx
-\rail@y=\rail@sy
-\rail@ex=\rail@gex
-\advance\rail@x by \rail@joinsz
-}
-
-% \rail@bar : start '|' alternatives
-%
-% \rail@nextbar : next '|' alternative
-%
-% \rail@endbar : end '|' alternatives
-%
-
-\def\rail@bar{
-\rail@alt\rail@barsplit
-}
-
-\def\rail@nextbar{
-\rail@nextalt\relax
-}
-
-\def\rail@endbar{
-\rail@endalt\rail@barjoin
-}
-
-% \rail@plus : start '+' alternatives
-%
-% \rail@nextplus: next '+' alternative
-%
-% \rail@endplus : end '+' alternatives
-%
-
-\def\rail@plus{
-\rail@alt\rail@plussplit
-}
-
-\def\rail@nextplus{
-\rail@nextalt\rail@fixplus
-}
-
-\def\rail@fixplus{
-\ifnum\rail@gy<\rail@sy
-\begingroup
-\rail@x=\rail@gx
-\rail@y=\rail@gy
-\rail@ex=\rail@gex
-\rail@rx=\rail@grx
-\ifnum\rail@x<\rail@rx
-\rail@x=\rail@rx
-\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-\rail@tmpa=\rail@sy
-\advance\rail@tmpa by -\rail@joinhsz
-\advance\rail@tmpa by -\rail@jy
-\rail@jput{\line(0,1){\number\rail@tmpa}}
-\rail@jy=\rail@sy
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jy by \rail@joinhsz
-\global\rail@gx=\rail@jx
-\global\rail@gy=\rail@jy
-\global\rail@gex=\rail@gx
-\global\rail@grx=\rail@rx
-\endgroup
-\fi
-}
-
-\def\rail@endplus{
-\rail@endalt\rail@plusjoin
-}
-
-% \rail@cr{Y} : carriage return to vertical position Y
-
-\def\rail@cr#1{
-\rail@tmpa=\rail@sx
-\advance\rail@tmpa by \rail@joinsz
-\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-\rail@sety{#1}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@boxsp
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@boxsp
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jy by -\rail@joinhsz
-\rail@tmpa=\rail@jx
-\advance\rail@tmpa by -\rail@sx
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(-1,0){\number\rail@tmpa}}
-\rail@jx=\rail@sx
-\advance\rail@jx by \rail@joinhsz
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@tmpa=\rail@boxsp
-\advance\rail@tmpa by -\rail@joinsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\advance\rail@jy by -\rail@tmpa
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\rail@x=\rail@jx
-\rail@ex=\rail@x
-}
-
-% default setup for Isabelle
-\newenvironment{railoutput}%
-{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
-
-\def\rail@termfont{\isabellestyle{tt}}
-\def\rail@nontfont{\isabellestyle{it}}
-\def\rail@namefont{\isabellestyle{it}}
diff --git a/virage/src/test/resources/old_theories/output/document/root.bib b/virage/src/test/resources/old_theories/output/document/root.bib
deleted file mode 100644
index 6f429f0..0000000
--- a/virage/src/test/resources/old_theories/output/document/root.bib
+++ /dev/null
@@ -1,71 +0,0 @@
-@InProceedings{adt2019,
-    author    = {Karsten Diekhoff and
-                 Michael Kirsten and
-                 Jonas Kr{\"{a}}mer},
-    editor    = {Sa{\v{s}}a Peke{\v{c}} and
-                 {Kristen Brent} Venable},
-    title     = {Formal Property-Oriented Design of
-                 Voting Rules Using Composable Modules},
-    booktitle = {6th International Conference on Algorithmic
-                 Decision Theory ({ADT} 2019)},
-    series    = {Lecture Notes in Artificial Intelligence},
-    part      = {Short Papers},
-    year      = {2019},
-    abstract  = {Voting rules aggregate multiple individual preferences in order to
-                 make a collective decision. Commonly, these mechanisms are expected
-                 to respect a multitude of different notions of fairness and reliability,
-                 which must be carefully balanced to avoid inconsistencies.
-
-                 We present an approach for the sound and flexible design of voting rules
-                 from composable modules. Formal composition rules guarantee social
-                 choice properties from properties of the individual components. The
-                 approach can be applied to many voting rules from the literature.},
-    publisher = {Springer},
-    volume    = {11834},
-    pages     = {164--166},
-    doi       = {10.1007/978-3-030-31489-7}
-}
-
-@InProceedings{lopstr2019,
-  author        = {Karsten Diekhoff and
-                   Michael Kirsten and
-                   Jonas Kr{\"{a}}mer},
-  title         = {Verified Construction of Fair Voting Rules},
-  booktitle     = {29th International Symposium on Logic-Based Program
-                   Synthesis and Transformation ({LOPSTR} 2019),
-                   Revised Selected Papers},
-  series        = {Lecture Notes in Computer Science},
-  editor        = {Maurizio Gabbrielli},
-  year          = {2020},
-  abstract      = {Voting rules aggregate multiple individual preferences in order to
-                   make collective decisions. Commonly, these mechanisms are expected
-                   to respect a multitude of different fairness and reliability
-                   properties, e.g., to ensure that each voter's ballot accounts for
-                   the same proportion of the elected alternatives, or that a voter
-                   cannot change the election outcome in her favor by insincerely
-                   filling out her ballot. However, no voting rule is fair in all
-                   respects, and trade-off attempts between such properties often
-                   bring out inconsistencies, which makes the construction of arguably
-                   practical and fair voting rules non-trivial and error-prone.
- 
-                   In this paper, we present a formal and systematic approach for the
-                   flexible and verified construction of voting rules from composable
-                   core modules to respect such properties by construction. Formal
-                   composition rules guarantee resulting properties from properties of
-                   the individual components, which are of generic nature to be reused
-                   for various voting rules. We provide a prototypical logic-based
-                   implementation with proofs for a selected set of structures and
-                   composition rules within the theorem prover Isabelle/HOL. The
-                   approach can be readily extended in order to support many voting
-                   rules from the literature by extending the set of basic modules and
-                   composition rules. We exemplarily construct the well-known voting
-                   rule sequential majority comparison (SMC) from simple generic
-                   modules, and automatically produce a formal proof that SMC
-                   satisfies the fairness property monotonicity. Monotonicity is a
-                   well-known social-choice property that is easily violated by voting
-                   rules in practice.},
-  publisher     = {Springer},
-  doi           = {10.1007/978-3-030-45260-5_6},
-  volume        = {12042},
-  pages         = {90--104}
-}
diff --git a/virage/src/test/resources/old_theories/output/document/root.tex b/virage/src/test/resources/old_theories/output/document/root.tex
deleted file mode 100644
index ec9759f..0000000
--- a/virage/src/test/resources/old_theories/output/document/root.tex
+++ /dev/null
@@ -1,56 +0,0 @@
-\documentclass[11pt,a4paper,notitlepage]{report}
-\input{settings}
-\usepackage{isabelle,isabellesym}
-\usepackage{amssymb}
-\usepackage{pdfsetup}
-
-\urlstyle{rm}
-\isabellestyle{it}
-
-% for uniform font size
-%\renewcommand{\isastyle}{\isastyleminor}
-
-% This is required due to some unusual behaviour by Isabelle.
-% It should be included in isabellesym.sty automatically, but it is not.
-\newcommand{\isasymcirclearrowleft}{\isamath{\circlearrowleft}}
-
-\begin{document}
-
-\title{Verified Construction of Fair Voting Rules}
-\author{Michael Kirsten}
-\institute{Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany\\
-\email{kirsten@kit.edu}}
-\maketitle
-
-\begin{abstract}
-Voting rules aggregate multiple individual preferences in order to make a
-collective decision.
-Commonly, these mechanisms are expected to respect a multitude of different
-notions of fairness and reliability, which must be carefully balanced to avoid
-inconsistencies.
-
-This article contains a formalisation of a framework for the construction of
-such fair voting rules using composable modules~\cite{adt2019,lopstr2019}.
-The framework is a formal and systematic approach for the flexible and verified
-construction of voting rules from individual composable modules to respect such
-social-choice properties by construction.
-Formal composition rules guarantee resulting social-choice properties from
-properties of the individual components which are of generic nature to be
-reused for various voting rules.
-We provide proofs for a selected set of structures and composition rules.
-The approach can be readily extended in order to support more voting rules,
-e.g., from the literature by extending the sets of modules and composition
-rules.
-\end{abstract}
-
-\tableofcontents
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-% optional bibliography
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
diff --git a/virage/src/test/resources/old_theories/output/document/session.tex b/virage/src/test/resources/old_theories/output/document/session.tex
deleted file mode 100644
index 43d554b..0000000
--- a/virage/src/test/resources/old_theories/output/document/session.tex
+++ /dev/null
@@ -1,36 +0,0 @@
-\input{Preference_Relation.tex}
-\input{Result.tex}
-\input{Profile.tex}
-\input{Electoral_Module.tex}
-\input{Evaluation_Function.tex}
-\input{Elimination_Module.tex}
-\input{Aggregator.tex}
-\input{Maximum_Aggregator.tex}
-\input{Termination_Condition.tex}
-\input{Defer_Equal_Condition.tex}
-\input{Defer_Module.tex}
-\input{Drop_Module.tex}
-\input{Pass_Module.tex}
-\input{Elect_Module.tex}
-\input{Plurality_Module.tex}
-\input{Borda_Module.tex}
-\input{Condorcet_Module.tex}
-\input{Copeland_Module.tex}
-\input{Minimax_Module.tex}
-\input{Drop_And_Pass_Compatibility.tex}
-\input{Revision_Composition.tex}
-\input{Sequential_Composition.tex}
-\input{Parallel_Composition.tex}
-\input{Loop_Composition.tex}
-\input{Maximum_Parallel_Composition.tex}
-\input{Elect_Composition.tex}
-\input{Defer_One_Loop_Composition.tex}
-\input{Borda_Rule.tex}
-\input{Pairwise_Majority_Rule.tex}
-\input{Copeland_Rule.tex}
-\input{Minimax_Rule.tex}
-\input{Blacks_Rule.tex}
-\input{Nanson_Baldwin_Rule.tex}
-\input{Classic_Nanson_Rule.tex}
-\input{Schwartz_Rule.tex}
-\input{Sequential_Majority_Comparison.tex}
diff --git a/virage/src/test/resources/old_theories/output/document/settings.tex b/virage/src/test/resources/old_theories/output/document/settings.tex
deleted file mode 100644
index 9a1f927..0000000
--- a/virage/src/test/resources/old_theories/output/document/settings.tex
+++ /dev/null
@@ -1,30 +0,0 @@
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{sectsty}
-\usepackage{xpatch}
-\usepackage{upquote}
-
-\makeatletter
-\xpatchcmd{\@maketitle}
-  {\@title}
-{%\vspace{-2.45\baselineskip}
-\bfseries\@title\vspace{.25\baselineskip}}
-  {}{}
-\makeatother
-
-\makeatletter
-\providecommand{\institute}[1]{%
-    \apptocmd{\@author}{\end{tabular}%
-    \vspace{.5\baselineskip}\par
-    \begin{tabular}[t]{c}
-    #1}{}{}
-}
-\makeatother
-
-\newcommand{\email}[1]{%
-    \texttt{\href{mailto:#1}{\color{black}{#1}}}%
-}
-
-\pretocmd{\tableofcontents}{\sectionfont{\Huge}}{}{}
-%\let\clearpage\relax
-\apptocmd{\tableofcontents}{\sectionfont{\Large}}{}{}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy
index 65d7b0a..8cff2e6 100644
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy
+++ b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy
@@ -59,23 +59,4 @@ proof -
     by (simp add: electoral_modI)
 qed
 
-ML \<open>
-    (* val w = (Defs.all_specifications_of (Theory.defs_of @{theory}))
-    val x = filter (fn x => String.isPrefix "Seq" x) (map (fn x => snd (fst x))
-        (Defs.all_specifications_of (Theory.defs_of @{theory}))) *)
-    val y = filter (fn x => (String.isPrefix ("Plurality_Module.plurality") (snd (fst (x))))) 
-        (Defs.all_specifications_of (Theory.defs_of @{theory}))
-    val z = filter (fn x => String.isPrefix ("Plurality_Module.plurality_def") (#description x)) (map (fn x => (hd (snd x))) y)
-
-    val def = hd z
-    val def_rhs = #rhs def
-
-    val def_funs = filter (fn x => String.isPrefix ("fun") (snd (fst x))) (def_rhs)
-    val def_consts = filter (fn x => not (String.isPrefix ("fun") (snd (fst x)))) (def_rhs)
-
-    val fun_type = Defs.pretty_entry (Defs.global_context @{theory}) (List.last def_funs)
-
-   (* val a = Defs.pretty_entry (Defs.global_context @{theory}) (nth (snd (#lhs (hd z), #rhs (hd z)) )2) *)
- \<close>
-
 end
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Preference_Relation.thy# b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Preference_Relation.thy#
deleted file mode 100644
index aae5f7a..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Preference_Relation.thy#
+++ /dev/null
@@ -1,1115 +0,0 @@
-(*  File:       Preference_Relation.thy
-    Copyright   2021  Karlsruhe Institute of Technology (KIT)
-*)
-\<^marker>\<open>creator "Jonas Kraemer, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Karsten Diekhoff, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Michael Kirsten, Karlsruhe Institute of Technology (KIT)"\<close>
-
-chapter \<open>Social-Choice Types\<close>
-
-section \<open>Preference Relation\<close>
-
-theory Preference_Relation
-  imports Main
-begin
-
-text
-\<open>The very core of the composable modules voting framework: types and
-functions, derivations, lemmata, operations on preference relations, etc.\<close>
-
-subsection \<open>Definition\<close>
-
-type_synonym Alternative = Nat.nat
-
-(*
-  Each voter expresses pairwise relations between all alternatives,
-  thereby inducing a linear order.
-*)
-type_synonym Preference_Relation = "Alternative rel"
-
-fun is_less_preferred_than ::
-  "Alternative \<Rightarrow> Preference_Relation \<Rightarrow> Alternative \<Rightarrow> bool" ("_ \<preceq>\<^sub>_ _" [50, 1000, 51] 50) where
-    "x \<preceq>\<^sub>r y = ((x, y) \<in> r)"
-
-lemma lin_imp_antisym:
-  assumes "linear_order_on A r"
-  shows "antisym r"
-  using assms linear_order_on_def partial_order_on_def
-  by auto
-
-lemma lin_imp_trans:
-  assumes "linear_order_on A r"
-  shows "trans r"
-  using assms order_on_defs
-  by blast
-
-subsection \<open>Ranking\<close>
-
-fun rank :: "'a Preference_Relation \<Rightarrow> 'a \<Rightarrow> nat" where
-  "rank r x = card (above r x)"
-
-lemma rank_gt_zero:
-  assumes
-    refl: "x \<preceq>\<^sub>r x" and
-    fin:  "finite r"
-  shows "rank r x \<ge> 1"
-proof -
-  have "x \<in> {y \<in> Field r. (x, y) \<in> r}"
-    using FieldI2 refl
-    by fastforce
-  hence "{y \<in> Field r. (x, y) \<in> r} \<noteq> {}"
-    by blast
-  hence "card {y \<in> Field r. (x, y) \<in> r} \<noteq> 0"
-    by (simp add: fin finite_Field)
-  moreover have "card{y \<in> Field r. (x, y) \<in> r} \<ge> 0"
-    using fin
-    by auto
-  ultimately show ?thesis
-    using Collect_cong FieldI2 above_def
-          less_one not_le_imp_less rank.elims
-    by (metis (no_types, lifting))
-qed
-
-subsection \<open>Limited Preference\<close>
-
-definition limited :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow> bool" where
-  "limited A r \<equiv> r \<subseteq> A \<times> A"
-
-lemma limitedI:
-  "(\<And>x y. \<lbrakk> x \<preceq>\<^sub>r y \<rbrakk> \<Longrightarrow>  x \<in> A \<and> y \<in> A) \<Longrightarrow> limited A r"
-  unfolding limited_def
-  by auto
-
-lemma limited_dest:
-  "(\<And>x y. \<lbrakk> x \<preceq>\<^sub>r y; limited A r \<rbrakk> \<Longrightarrow>  x \<in> A \<and> y \<in> A)"
-  unfolding limited_def
-  by auto
-
-fun limit :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow> 'a Preference_Relation" where
-  "limit A r = {(a, b) \<in> r. a \<in> A \<and> b \<in> A}"
-
-definition connex :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow> bool" where
-  "connex A r \<equiv> limited A r \<and> (\<forall>x \<in> A. \<forall>y \<in> A. x \<preceq>\<^sub>r y \<or> y \<preceq>\<^sub>r x)"
-
-lemma connex_imp_refl:
-  assumes "connex A r"
-  shows "refl_on A r"
-proof
-  show "r \<subseteq> A \<times> A"
-    using assms connex_def limited_def
-    by metis
-next
-  fix
-    x :: "'a"
-  assume
-    x_in_A: "x \<in> A"
-  have "x \<preceq>\<^sub>r x"
-    using assms connex_def x_in_A
-    by metis
-  thus "(x, x) \<in> r"
-    by simp
-qed
-
-lemma lin_ord_imp_connex:
-  assumes "linear_order_on A r"
-  shows "connex A r"
-  unfolding connex_def limited_def
-proof (safe)
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "a \<in> A"
-    using asm1 assms partial_order_onD(1)
-          order_on_defs(3) refl_on_domain
-    by metis
-next
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "b \<in> A"
-    using asm1 assms partial_order_onD(1)
-          order_on_defs(3) refl_on_domain
-    by metis
-next
-  fix
-    x :: "'a" and
-    y :: "'a"
-  assume
-    asm1: "x \<in> A" and
-    asm2: "y \<in> A" and
-    asm3: "\<not> y \<preceq>\<^sub>r x"
-  have "(y, x) \<notin> r"
-    using asm3
-    by simp
-  hence "(x, y) \<in> r"
-    using asm1 asm2 assms partial_order_onD(1)
-          linear_order_on_def refl_onD total_on_def
-    by metis
-  thus "x \<preceq>\<^sub>r y"
-    by simp
-qed
-
-lemma connex_antsym_and_trans_imp_lin_ord:
-  assumes
-    connex_r: "connex A r" and
-    antisym_r: "antisym r" and
-    trans_r: "trans r"
-  shows "linear_order_on A r"
-  unfolding connex_def linear_order_on_def partial_order_on_def
-            preorder_on_def refl_on_def total_on_def
-proof (safe)
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "a \<in> A"
-    using asm1 connex_r refl_on_domain connex_imp_refl
-    by metis
-next
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "b \<in> A"
-    using asm1 connex_r refl_on_domain connex_imp_refl
-    by metis
-next
-  fix
-    x :: "'a"
-  assume
-    asm1: "x \<in> A"
-  show "(x, x) \<in> r"
-    using asm1 connex_r connex_imp_refl refl_onD
-  by metis
-next
-  show "trans r"
-    using trans_r
-    by simp
-next
-  show "antisym r"
-    using antisym_r
-    by simp
-next
-  fix
-    x :: "'a" and
-    y :: "'a"
-  assume
-    asm1: "x \<in> A" and
-    asm2: "y \<in> A" and
-    asm3: "x \<noteq> y" and
-    asm4: "(y, x) \<notin> r"
-  have "x \<preceq>\<^sub>r y \<or> y \<preceq>\<^sub>r x"
-    using asm1 asm2 connex_r connex_def
-    by metis
-  hence "(x, y) \<in> r \<or> (y, x) \<in> r"
-    by simp
-  thus "(x, y) \<in> r"
-    using asm4
-    by metis
-qed
-
-lemma limit_to_limits: "limited A (limit A r)"
-  unfolding limited_def
-  by auto
-
-lemma limit_presv_connex:
-  assumes
-    connex: "connex S r" and
-    subset: "A \<subseteq> S"
-  shows "connex A (limit A r)"
-  unfolding connex_def limited_def
-proof (simp, safe)
-  let ?s = "{(a, b). (a, b) \<in> r \<and> a \<in> A \<and> b \<in> A}"
-  fix
-    x :: "'a" and
-    y :: "'a" and
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "x \<in> A" and
-    asm2: "y \<in> A" and
-    asm3: "(y, x) \<notin> r"
-  have "y \<preceq>\<^sub>r x \<or> x \<preceq>\<^sub>r y"
-    using asm1 asm2 connex connex_def in_mono subset
-    by metis
-  hence
-    "x \<preceq>\<^sub>?s y \<or> y \<preceq>\<^sub>?s x"
-    using asm1 asm2
-    by auto
-  hence "x \<preceq>\<^sub>?s y"
-    using asm3
-    by simp
-  thus "(x, y) \<in> r"
-    by simp
-qed
-
-lemma limit_presv_antisym:
-  assumes
-    antisymmetric: "antisym r" and
-    subset: "A \<subseteq> S"
-  shows "antisym (limit A r)"
-  using antisym_def antisymmetric
-  by auto
-
-lemma limit_presv_trans:
-  assumes
-    transitive: "trans r" and
-    subset:     "A \<subseteq> S"
-  shows "trans (limit A r)"
-  unfolding trans_def
-proof (simp, safe)
-  fix
-    x :: "'a" and
-    y :: "'a" and
-    z :: "'a"
-  assume
-    asm1: "(x, y) \<in> r" and
-    asm2: "x \<in> A" and
-    asm3: "y \<in> A" and
-    asm4: "(y, z) \<in> r" and
-    asm5: "z \<in> A"
-  show  "(x, z) \<in> r"
-    using asm1 asm4 transE transitive
-    by metis
-qed
-
-lemma limit_presv_lin_ord:
-  assumes
-    "linear_order_on S r" and
-      "A \<subseteq> S"
-    shows "linear_order_on A (limit A r)"
-  using assms connex_antsym_and_trans_imp_lin_ord
-            limit_presv_antisym limit_presv_connex
-            limit_presv_trans lin_ord_imp_connex
-            order_on_defs(1) order_on_defs(2)
-            order_on_defs(3)
-  by metis
-
-lemma limit_presv_prefs1:
-  assumes
-    x_less_y: "x \<preceq>\<^sub>r y" and
-    x_in_A: "x \<in> A" and
-    y_in_A: "y \<in> A"
-  shows "let s = limit A r in x \<preceq>\<^sub>s y"
-  using x_in_A x_less_y y_in_A
-  by simp
-
-lemma limit_presv_prefs2:
-  assumes x_less_y: "(x, y) \<in> limit A r"
-  shows "x \<preceq>\<^sub>r y"
-  using mem_Collect_eq x_less_y
-  by auto
-
-lemma limit_trans:
-  assumes
-    "B \<subseteq> A" and
-    "C \<subseteq> B" and
-    "linear_order_on A r"
-  shows "limit C r = limit C (limit B r)"
-  using assms
-  by auto
-
-lemma lin_ord_not_empty:
-  assumes "r \<noteq> {}"
-  shows "\<not> linear_order_on {} r"
-  using assms connex_imp_refl lin_ord_imp_connex
-        refl_on_domain subrelI
-  by fastforce
-
-lemma lin_ord_singleton:
-  "\<forall>r. linear_order_on {a} r \<longrightarrow> r = {(a, a)}"
-proof
-  fix r :: "'a Preference_Relation"
-  show "linear_order_on {a} r \<longrightarrow> r = {(a, a)}"
-  proof
-    assume asm: "linear_order_on {a} r"
-    hence "a \<preceq>\<^sub>r a"
-      using connex_def lin_ord_imp_connex singletonI
-      by metis
-    moreover have "\<forall>(x, y) \<in> r. x = a \<and> y = a"
-      using asm connex_imp_refl lin_ord_imp_connex
-            refl_on_domain split_beta
-      by fastforce
-    ultimately show "r = {(a, a)}"
-      by auto
-  qed
-qed
-
-subsection \<open>Auxiliary Lemmata\<close>
-
-lemma above_trans:
-  assumes
-    "trans r" and
-    "(a, b) \<in> r"
-  shows "above r b \<subseteq> above r a"
-  using Collect_mono above_def assms transE
-  by metis
-
-lemma above_refl:
-  assumes
-    "refl_on A r" and
-     "a \<in> A"
-  shows "a \<in> above r a"
-  using above_def assms refl_onD
-  by fastforce
-
-lemma above_subset_geq_one:
-  assumes
-    "linear_order_on A r \<and> linear_order_on A s" and
-    "above r a \<subseteq> above s a" and
-    "above s a = {a}"
-  shows "above r a = {a}"
-  using above_def assms connex_imp_refl above_refl insert_absorb
-        lin_ord_imp_connex mem_Collect_eq refl_on_domain
-        singletonI subset_singletonD
-  by metis
-
-lemma above_connex:
-  assumes
-    "connex A r" and
-    "a \<in> A"
-  shows "a \<in> above r a"
-  using assms connex_imp_refl above_refl
-  by metis
-
-lemma pref_imp_in_above: "a \<preceq>\<^sub>r b \<longleftrightarrow> b \<in> above r a"
-  by (simp add: above_def)
-
-lemma limit_presv_above:
-  assumes
-    "b \<in> above r a" and
-    (*"linear_order_on A r" and
-    "B \<subseteq> A" and ??? *)
-    "a \<in> B \<and> b \<in> B"
-  shows "b \<in> above (limit B r) a"
-  using pref_imp_in_above assms limit_presv_prefs1
-  by metis
-
-lemma limit_presv_above2:
-  assumes
-    "b \<in> above (limit B r) a" and
-    "linear_order_on A r" and
-    "B \<subseteq> A" and
-    "a \<in> B" and
-    "b \<in> B"
-  shows "b \<in> above r a"
-  unfolding above_def
-  using above_def assms(1) limit_presv_prefs2
-        mem_Collect_eq pref_imp_in_above
-  by metis
-
-lemma above_one:
-  assumes
-    "linear_order_on A r" and
-    "finite A \<and> A \<noteq> {}"
-  shows "\<exists>a\<in>A. above r a = {a} \<and> (\<forall>x\<in>A. above r x = {x} \<longrightarrow> x = a)"
-proof -
-  obtain n::nat where n: "n+1 = card A"
-    using Suc_eq_plus1 antisym_conv2 assms(2) card_eq_0_iff
-          gr0_implies_Suc le0
-    by metis
-  have
-    "(linear_order_on A r \<and> finite A \<and> A \<noteq> {} \<and> n+1 = card A)
-          \<longrightarrow> (\<exists>a. a\<in>A \<and> above r a = {a})"
-  proof (induction n arbitrary: A r)
-    case 0
-    show ?case
-    proof
-      assume asm: "linear_order_on A r \<and> finite A \<and> A \<noteq> {} \<and> 0+1 = card A"
-      then obtain a where "{a} = A"
-        using card_1_singletonE add.left_neutral
-        by metis
-      hence "a \<in> A \<and> above r a = {a}"
-        using above_def asm connex_imp_refl above_refl
-              lin_ord_imp_connex refl_on_domain
-        by fastforce
-      thus "\<exists>a. a\<in>A \<and> above r a = {a}"
-        by auto
-    qed
-  next
-    case (Suc n)
-    show ?case
-    proof
-      assume asm:
-        "linear_order_on A r \<and> finite A \<and> A \<noteq> {} \<and> Suc n+1 = card A"
-      then obtain B where B: "card B = n+1 \<and> B \<subseteq> A"
-        using Suc_inject add_Suc card.insert_remove finite.cases
-              insert_Diff_single subset_insertI
-        by (metis (mono_tags, lifting))
-      then obtain a where a: "{a} = A - B"
-        using Suc_eq_plus1 add_diff_cancel_left' asm card_1_singletonE
-              card_Diff_subset finite_subset
-        by metis
-      have "\<exists>b\<in>B. above (limit B r) b = {b}"
-        using B One_nat_def Suc.IH add_diff_cancel_left' asm
-              card_eq_0_iff diff_le_self finite_subset leD lessI
-              limit_presv_lin_ord
-        by metis
-      then obtain b where b: "above (limit B r) b = {b}"
-        by blast
-      show "\<exists>a. a \<in> A \<and> above r a = {a}"
-      proof cases
-        assume
-          asm1: "a \<preceq>\<^sub>r b"
-        have f1:
-          "\<forall>A r a aa.
-            \<not> refl_on A r \<or> (a::'a, aa) \<notin> r \<or> a \<in> A \<and> aa \<in> A"
-          using refl_on_domain
-          by metis
-        have f2:
-          "\<forall>A r. \<not> connex (A::'a set) r \<or> refl_on A r"
-          using connex_imp_refl
-          by metis
-        have f3:
-          "\<forall>A r. \<not> linear_order_on (A::'a set) r \<or> connex A r"
-          by (simp add: lin_ord_imp_connex)
-        hence "refl_on A r"
-          using f2 asm
-          by metis
-        hence "a \<in> A \<and> b \<in> A"
-          using f1 asm1
-          by simp
-        hence f4:
-          "\<forall>a. a \<notin> A \<or> b = a \<or> (b, a) \<in> r \<or> (a, b) \<in> r"
-          using asm order_on_defs(3) total_on_def
-          by metis
-        have f5:
-          "(b, b) \<in> limit B r"
-          using above_def b mem_Collect_eq singletonI
-          by metis
-        have f6:
-          "\<forall>a A Aa. (a::'a) \<notin> A - Aa \<or> a \<in> A \<and> a \<notin> Aa"
-          by simp
-        have ff1:
-          "{a. (b, a) \<in> limit B r} = {b}"
-          using above_def b
-          by (metis (no_types))
-        have ff2:
-          "(b, b) \<in> {(aa, a). (aa, a) \<in> r \<and> aa \<in> B \<and> a \<in> B}"
-          using f5
-          by simp
-        moreover have b_wins_B:
-          "\<forall>x \<in> B. b \<in> above r x"
-          using B above_def f4 ff1 ff2 CollectI
-                Product_Type.Collect_case_prodD
-          by fastforce
-        moreover have "b \<in> above r a"
-          using asm1 pref_imp_in_above
-          by metis
-        ultimately have b_wins:
-          "\<forall>x \<in> A. b \<in> above r x"
-          using Diff_iff a empty_iff insert_iff
-          by (metis (no_types))
-        hence "\<forall>x \<in> A. x \<in> above r b \<longrightarrow> x = b"
-          using CollectD above_def antisym_def asm lin_imp_antisym
-          by metis
-        hence "\<forall>x \<in> A. x \<in> above r b \<longleftrightarrow> x = b"
-          using b_wins
-          by blast
-        moreover have above_b_in_A: "above r b \<subseteq> A"
-          using above_def asm connex_imp_refl lin_ord_imp_connex
-                mem_Collect_eq refl_on_domain subsetI
-          by metis
-        ultimately have "above r b = {b}"
-          using above_def b
-          by fastforce
-        thus ?thesis
-          using above_b_in_A
-          by blast
-      next
-        assume "\<not>a \<preceq>\<^sub>r b"
-        hence b_smaller_a: "b \<preceq>\<^sub>r a"
-          using B DiffE a asm b limit_to_limits connex_def
-                limited_dest singletonI subset_iff
-                lin_ord_imp_connex pref_imp_in_above
-          by metis
-        hence b_smaller_a_0: "(b, a) \<in> r"
-          by simp
-        have g1:
-          "\<forall>A r Aa.
-            \<not> linear_order_on (A::'a set) r \<or>
-              \<not> Aa \<subseteq> A \<or>
-              linear_order_on Aa (limit Aa r)"
-          using limit_presv_lin_ord
-          by metis
-        have
-          "{a. (b, a) \<in> limit B r} = {b}"
-          using above_def b
-          by metis
-        hence g2: "b \<in> B"
-          by auto
-        have g3:
-          "partial_order_on B (limit B r) \<and> total_on B (limit B r)"
-          using g1 B asm order_on_defs(3)
-          by metis
-        have
-          "\<forall>A r.
-            total_on A r = (\<forall>a. (a::'a) \<notin> A \<or>
-              (\<forall>aa. (aa \<notin> A \<or> a = aa) \<or> (a, aa) \<in> r \<or> (aa, a) \<in> r))"
-          using total_on_def
-          by metis
-        hence
-          "\<forall>a. a \<notin> B \<or>
-            (\<forall>aa. aa \<notin> B \<or> a = aa \<or>
-                (a, aa) \<in> limit B r \<or> (aa, a) \<in> limit B r)"
-          using g3
-          by simp
-        have "\<forall>x \<in> B. b \<in> above r x"
-          using limit_presv_above2 B pref_imp_in_above asm b above_def
-                limit_presv_lin_ord order_on_defs(3) singletonD
-                singletonI total_on_def mem_Collect_eq g2
-          by (smt (verit, ccfv_threshold))
-        hence b_wins2:
-          "\<forall>x \<in> B. x \<preceq>\<^sub>r b"
-          by (simp add: above_def)
-        hence b_wins2_0:
-          "\<forall>x \<in> B. (x, b) \<in> r"
-          by simp
-        have "trans r"
-          using asm lin_imp_trans
-          by metis
-        hence "\<forall>x \<in> B. (x, a) \<in> r"
-          using transE b_smaller_a_0 b_wins2_0
-          by metis
-        hence "\<forall>x \<in> B. x \<preceq>\<^sub>r a"
-          by simp
-        hence nothing_above_a: "\<forall>x \<in> A. x \<preceq>\<^sub>r a"
-          using a asm lin_ord_imp_connex above_connex Diff_iff
-                empty_iff insert_iff pref_imp_in_above
-          by metis
-        have "\<forall>x \<in> A. x \<in> above r a \<longleftrightarrow> x = a"
-          using antisym_def asm lin_imp_antisym
-                nothing_above_a pref_imp_in_above
-                CollectD above_def
-          by metis
-        moreover have above_a_in_A: "above r a \<subseteq> A"
-          using above_def asm connex_imp_refl lin_ord_imp_connex
-                mem_Collect_eq refl_on_domain
-          by fastforce
-        ultimately have "above r a = {a}"
-          using above_def a
-          by auto
-        thus ?thesis
-          using above_a_in_A
-          by blast
-      qed
-    qed
-  qed
-  hence "\<exists>a. a\<in>A \<and> above r a = {a}"
-    using assms n
-    by blast
-  thus ?thesis
-    using Diff_eq_empty_iff above_trans assms(1) empty_Diff insertE
-          insert_Diff_if insert_absorb insert_not_empty order_on_defs(1)
-          order_on_defs(2) order_on_defs(3) total_on_def
-    by (smt (verit, ccfv_SIG))
-qed
-
-lemma above_one2:
-  assumes
-    lin_ord: "linear_order_on A r" and
-    fin_not_emp: "finite A \<and> A \<noteq> {}" and
-    above1: "above r a = {a} \<and> above r b = {b}"
-  shows "a = b"
-proof -
-  have "a \<preceq>\<^sub>r a \<and> b \<preceq>\<^sub>r b"
-    using above1 singletonI pref_imp_in_above
-    by metis
-  also have
-    "\<exists>a\<in>A. above r a = {a} \<and>
-      (\<forall>x\<in>A. above r x = {x} \<longrightarrow> x = a)"
-    using lin_ord fin_not_emp
-    by (simp add: above_one)
-  moreover have "connex A r"
-    using lin_ord
-    by (simp add: lin_ord_imp_connex)
-  ultimately show "a = b"
-    using above1 connex_def limited_dest
-    by metis
-qed
-
-lemma above_presv_limit:
-  assumes "linear_order r"
-  shows "above (limit A r) x \<subseteq> A"
-  unfolding above_def
-  by auto
-
-subsection \<open>Lifting Property\<close>
-
-definition equiv_rel_except_a :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow>
-                                    'a Preference_Relation \<Rightarrow> 'a \<Rightarrow> bool" where
-  "equiv_rel_except_a A r s a \<equiv>
-    linear_order_on A r \<and> linear_order_on A s \<and> a \<in> A \<and>
-    (\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y)"
-
-definition lifted :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow>
-                        'a Preference_Relation \<Rightarrow> 'a \<Rightarrow> bool" where
-  "lifted A r s a \<equiv>
-    equiv_rel_except_a A r s a \<and> (\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a)"
-
-lemma trivial_equiv_rel:
-  assumes order: "linear_order_on A p"
-  shows "\<forall>a \<in> A. equiv_rel_except_a A p p a"
-  by (simp add: equiv_rel_except_a_def order)
-
-lemma lifted_imp_equiv_rel_except_a:
-  assumes lifted: "lifted A r s a"
-  shows "equiv_rel_except_a A r s a"
-proof -
-  from lifted have
-    "linear_order_on A r \<and> linear_order_on A s \<and> a \<in> A \<and>
-      (\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y)"
-    by (simp add: lifted_def equiv_rel_except_a_def)
-  thus ?thesis
-    by (simp add: equiv_rel_except_a_def)
-qed
-
-lemma lifted_mono:
-  assumes lifted: "lifted A r s a"
-  shows "\<forall>x \<in> A - {a}. \<not>(x \<preceq>\<^sub>r a \<and> a \<preceq>\<^sub>s x)"
-proof (safe)
-  fix
-    x :: "'a"
-  assume
-    x_in_A:   "x \<in> A" and
-    x_exist:  "x \<notin> {}" and
-    x_neq_a:  "x \<noteq> a" and
-    x_pref_a: "x \<preceq>\<^sub>r a" and
-    a_pref_x: "a \<preceq>\<^sub>s x"
-  from x_pref_a
-  have x_pref_a_0: "(x, a) \<in> r"
-    by simp
-  from a_pref_x
-  have a_pref_x_0: "(a, x) \<in> s"
-    by simp
-  have "antisym r"
-    using equiv_rel_except_a_def lifted
-          lifted_imp_equiv_rel_except_a
-          lin_imp_antisym
-    by metis
-  hence antisym_r:
-    "(\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
-    using antisym_def
-    by metis
-  hence imp_x_eq_a_0:
-    "\<lbrakk>(x, a) \<in> r; (a, x) \<in> r\<rbrakk> \<Longrightarrow> x = a"
-    by simp
-  have lift_ex: "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a"
-    using lifted lifted_def
-    by metis
-  from lift_ex obtain y :: 'a where
-    f1: "y \<in> A - {a} \<and> a \<preceq>\<^sub>r y \<and> y \<preceq>\<^sub>s a"
-    by metis
-  hence f1_0:
-    "y \<in> A - {a} \<and> (a, y) \<in> r \<and> (y, a) \<in> s"
-    by simp
-  have f2:
-    "equiv_rel_except_a A r s a"
-    using lifted lifted_def
-    by metis
-  hence f2_0:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    using equiv_rel_except_a_def
-    by metis
-  hence f2_1:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. (x, y) \<in> r \<longleftrightarrow> (x, y) \<in> s"
-    by simp
-  have trans: "\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r"
-    using f2 equiv_rel_except_a_def linear_order_on_def
-          partial_order_on_def preorder_on_def trans_def
-    by metis
-  have x_pref_y_0: "(x, y) \<in> s"
-    using equiv_rel_except_a_def f1_0 f2 f2_1 insertE
-          insert_Diff x_in_A x_neq_a x_pref_a_0 trans
-    by metis
-  have a_pref_y_0: "(a, y) \<in> s"
-    using a_pref_x_0 imp_x_eq_a_0 x_neq_a x_pref_a_0
-          equiv_rel_except_a_def f2 lin_imp_trans
-          transE x_pref_y_0
-    by metis
-  show "False"
-    using a_pref_y_0 antisymD equiv_rel_except_a_def
-          DiffD2 f1_0 f2 lin_imp_antisym singletonI
-    by metis
-qed
-
-lemma lifted_mono2:
-  assumes
-    lifted: "lifted A r s a" and
-    x_pref_a: "x \<preceq>\<^sub>r a"
-  shows "x \<preceq>\<^sub>s a"
-proof (simp)
-  have x_pref_a_0: "(x, a) \<in> r"
-    using x_pref_a
-    by simp
-  have x_in_A: "x \<in> A"
-    using connex_imp_refl equiv_rel_except_a_def
-          lifted lifted_def lin_ord_imp_connex
-          refl_on_domain x_pref_a_0
-    by metis
-  have "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    using lifted lifted_def equiv_rel_except_a_def
-    by metis
-  hence rest_eq:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. (x, y) \<in> r \<longleftrightarrow> (x, y) \<in> s"
-    by simp
-  have "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a"
-    using lifted lifted_def
-    by metis
-  hence ex_lifted:
-    "\<exists>x \<in> A - {a}. (a, x) \<in> r \<and> (x, a) \<in> s"
-    by simp
-  show "(x, a) \<in> s"
-  proof (cases "x = a")
-    case True
-    thus ?thesis
-      using connex_imp_refl equiv_rel_except_a_def refl_onD
-            lifted lifted_def lin_ord_imp_connex
-      by metis
-  next
-    case False
-    thus ?thesis
-      using equiv_rel_except_a_def insertE insert_Diff
-            lifted lifted_imp_equiv_rel_except_a x_in_A
-            x_pref_a_0 ex_lifted lin_imp_trans rest_eq
-            trans_def
-      by metis
-  qed
-qed
-
-lemma lifted_above:
-  assumes "lifted A r s a"
-  shows "above s a \<subseteq> above r a"
-  unfolding above_def
-proof (safe)
-  fix
-    x :: "'a"
-  assume
-    a_pref_x: "(a, x) \<in> s"
-  have "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a"
-    using assms lifted_def
-    by metis
-  hence lifted_r:
-    "\<exists>x \<in> A - {a}. (a, x) \<in> r \<and> (x, a) \<in> s"
-    by simp
-  have "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    using assms lifted_def equiv_rel_except_a_def
-    by metis
-  hence rest_eq:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. (x, y) \<in> r \<longleftrightarrow> (x, y) \<in> s"
-    by simp
-  have trans_r:
-    "\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r"
-    using trans_def lifted_def lin_imp_trans
-          equiv_rel_except_a_def assms
-    by metis
-  have trans_s:
-    "\<forall>x y z. (x, y) \<in> s \<longrightarrow> (y, z) \<in> s \<longrightarrow> (x, z) \<in> s"
-    using trans_def lifted_def lin_imp_trans
-          equiv_rel_except_a_def assms
-    by metis
-  have refl_r:
-    "(a, a) \<in> r"
-    using assms connex_imp_refl equiv_rel_except_a_def
-          lifted_def lin_ord_imp_connex refl_onD
-    by metis
-  have x_in_A: "x \<in> A"
-    using a_pref_x assms connex_imp_refl equiv_rel_except_a_def
-          lifted_def lin_ord_imp_connex refl_onD2
-    by metis
-  show "(a, x) \<in> r"
-    using Diff_iff a_pref_x lifted_r rest_eq singletonD
-          trans_r trans_s x_in_A refl_r
-    by (metis (full_types))
-qed
-
-lemma lifted_above2:
-  assumes
-    "lifted A r s a" and
-    "x \<in> A-{a}"
-  shows "above r x \<subseteq> above s x \<union> {a}"
-proof (safe, simp)
-  fix y :: "'a"
-  assume
-    y_in_above_r: "y \<in> above r x" and
-    y_not_in_above_s: "y \<notin> above s x"
-  have "\<forall>z \<in> A-{a}. x \<preceq>\<^sub>r z \<longleftrightarrow> x \<preceq>\<^sub>s z"
-    using assms lifted_def equiv_rel_except_a_def
-    by metis
-  hence "\<forall>z \<in> A-{a}. (x, z) \<in> r \<longleftrightarrow> (x, z) \<in> s"
-    by simp
-  hence "\<forall>z \<in> A-{a}. z \<in> above r x \<longleftrightarrow> z \<in> above s x"
-    by (simp add: above_def)
-  hence "y \<in> above r x \<longleftrightarrow> y \<in> above s x"
-    using y_not_in_above_s assms(1) connex_def
-          equiv_rel_except_a_def lifted_def lifted_mono2
-          limited_dest lin_ord_imp_connex member_remove
-          pref_imp_in_above remove_def
-    by metis
-  thus "y = a"
-    using y_in_above_r y_not_in_above_s
-    by simp
-qed
-
-lemma limit_lifted_imp_eq_or_lifted:
-  assumes
-    lifted: "lifted S r s a" and
-    subset: "A \<subseteq> S"
-  shows
-    "limit A r = limit A s \<or>
-      lifted A (limit A r) (limit A s) a"
-proof -
-  from lifted have
-    "\<forall>x \<in> S - {a}. \<forall>y \<in> S - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    by (simp add: lifted_def equiv_rel_except_a_def)
-  with subset have temp:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    by auto
-  hence eql_rs:
-      "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}.
-      (x, y) \<in> (limit A r) \<longleftrightarrow> (x, y) \<in> (limit A s)"
-    using DiffD1 limit_presv_prefs1 limit_presv_prefs2
-    by auto
-  show ?thesis
-  proof cases
-    assume a1: "a \<in> A"
-    thus ?thesis
-    proof cases
-      (* (\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y) *)
-      assume a1_1: "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a" (* \<and> a1: "a \<in> A" *)
-      from lifted subset have
-        "linear_order_on A (limit A r) \<and> linear_order_on A (limit A s)"
-        using lifted_def equiv_rel_except_a_def limit_presv_lin_ord
-        by metis
-      moreover from a1 a1_1 have keep_lift:
-        "\<exists>x \<in> A - {a}. (let q = limit A r in a \<preceq>\<^sub>q x) \<and>
-            (let u = limit A s in x \<preceq>\<^sub>u a)"
-        using DiffD1 limit_presv_prefs1
-        by simp
-      ultimately show ?thesis
-        using a1 temp
-        by (simp add: lifted_def equiv_rel_except_a_def)
-    next
-      assume
-        "\<not>(\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a)" (* \<and> a1: "a \<in> A" *)
-      hence a1_2:
-        "\<forall>x \<in> A - {a}. \<not>(a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a)"
-        by auto
-      moreover have not_worse:
-        "\<forall>x \<in> A - {a}. \<not>(x \<preceq>\<^sub>r a \<and> a \<preceq>\<^sub>s x)"
-        using lifted subset lifted_mono
-        by fastforce
-      moreover have connex:
-        "connex A (limit A r) \<and> connex A (limit A s)"
-        using lifted subset lifted_def equiv_rel_except_a_def
-              limit_presv_lin_ord lin_ord_imp_connex
-        by metis
-      moreover have connex1:
-        "\<forall>A r. connex A r =
-          (limited A r \<and> (\<forall>a. (a::'a) \<in> A \<longrightarrow>
-            (\<forall>aa. aa \<in> A \<longrightarrow> a \<preceq>\<^sub>r aa \<or> aa \<preceq>\<^sub>r a)))"
-        by (simp add: Ball_def_raw connex_def)
-      hence limit1:
-        "limited A (limit A r) \<and>
-          (\<forall>a. a \<notin> A \<or>
-            (\<forall>aa.
-              aa \<notin> A \<or> (a, aa) \<in> limit A r \<or>
-                (aa, a) \<in> limit A r ))"
-        using connex connex1
-        by simp
-      have limit2:
-        "\<forall>a aa A r. (a::'a, aa) \<notin> limit A r \<or> a \<preceq>\<^sub>r aa"
-        using limit_presv_prefs2
-        by metis
-      have
-        "limited A (limit A s) \<and>
-          (\<forall>a. a \<notin> A \<or>
-            (\<forall>aa. aa \<notin> A \<or>
-              (let q = limit A s in a \<preceq>\<^sub>q aa \<or> aa \<preceq>\<^sub>q a)))"
-        using connex connex_def
-        by metis
-      hence connex2:
-        "limited A (limit A s) \<and>
-          (\<forall>a. a \<notin> A \<or>
-            (\<forall>aa. aa \<notin> A \<or>
-              ((a, aa) \<in> limit A s \<or> (aa, a) \<in> limit A s)))"
-        by simp
-      ultimately have
-          "\<forall>x \<in> A - {a}. (a \<preceq>\<^sub>r x \<and> a \<preceq>\<^sub>s x) \<or> (x \<preceq>\<^sub>r a \<and> x \<preceq>\<^sub>s a)"
-        using DiffD1 limit1 limit_presv_prefs2 a1
-        by metis
-      hence r_eq_s_on_A_0:
-        "\<forall>x \<in> A - {a}. ((a, x) \<in> r \<and> (a, x) \<in> s) \<or> ((x, a) \<in> r \<and> (x, a) \<in> s)"
-        by simp
-      have
-        "\<forall>x \<in> A - {a}. (a, x) \<in> (limit A r) \<longleftrightarrow> (a, x) \<in> (limit A s)"
-        using DiffD1 limit2 limit1 connex2 a1 a1_2 not_worse
-        by metis
-      hence
-        "\<forall>x \<in> A - {a}.
-          (let q = limit A r in a \<preceq>\<^sub>q x) \<longleftrightarrow> (let q = limit A s in a \<preceq>\<^sub>q x)"
-        by simp
-      moreover have
-        "\<forall>x \<in> A - {a}. (x, a) \<in> (limit A r) \<longleftrightarrow> (x, a) \<in> (limit A s)"
-        using a1 a1_2 not_worse DiffD1 limit_presv_prefs2 connex2 limit1
-        by metis
-      moreover have
-        "(a, a) \<in> (limit A r) \<and> (a, a) \<in> (limit A s)"
-        using a1 connex connex_imp_refl refl_onD
-        by metis
-      moreover have
-        "limited A (limit A r) \<and> limited A (limit A s)"
-        using limit_to_limits
-        by metis
-      ultimately have
-        "\<forall>x y. (x, y) \<in> limit A r \<longleftrightarrow> (x, y) \<in> limit A s"
-        using eql_rs
-        by auto
-      thus ?thesis
-        by simp
-    qed
-  next
-    assume a2: "a \<notin> A"
-    with eql_rs have
-      "\<forall>x \<in> A. \<forall>y \<in> A. (x, y) \<in> (limit A r) \<longleftrightarrow> (x, y) \<in> (limit A s)"
-      by simp
-    thus ?thesis
-      using limit_to_limits limited_dest subrelI subset_antisym
-      by auto
-  qed
-qed
-
-lemma negl_diff_imp_eq_limit:
-  assumes
-    change: "equiv_rel_except_a S r s a" and
-    subset: "A \<subseteq> S" and
-    notInA: "a \<notin> A"
-  shows "limit A r = limit A s"
-proof -
-  have "A \<subseteq> S-{a}"
-    by (simp add: notInA subset subset_Diff_insert)
-  hence "\<forall>x \<in> A. \<forall>y \<in> A. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    by (meson change equiv_rel_except_a_def in_mono)
-  thus ?thesis
-    by auto
-qed
-
-theorem lifted_above_winner:
-  assumes
-    lifted_a: "lifted A r s a" and
-    above_x: "above r x = {x}" and
-    fin_A: "finite A"
-  shows "above s x = {x} \<or> above s a = {a}"
-proof cases
-  assume "x = a"
-  thus ?thesis
-    using above_subset_geq_one lifted_a above_x
-          lifted_above lifted_def equiv_rel_except_a_def
-    by metis
-next
-  assume asm1: "x \<noteq> a"
-  thus ?thesis
-  proof cases
-    assume "above s x = {x}"
-    thus ?thesis
-      by simp
-  next
-    assume asm2: "above s x \<noteq> {x}" (* \<and> asm1: x \<noteq> a *)
-    have "\<forall>y \<in> A. y \<preceq>\<^sub>r x"
-    proof -
-      fix aa :: 'a
-      have imp_a: "x \<preceq>\<^sub>r aa \<longrightarrow> aa \<notin> A \<or> aa \<preceq>\<^sub>r x"
-        using singletonD pref_imp_in_above above_x
-        by metis
-      also have f1:
-        "\<forall>A r.
-          (connex A r \<or>
-            (\<exists>a. (\<exists>aa. \<not> (aa::'a) \<preceq>\<^sub>r a \<and> \<not> a \<preceq>\<^sub>r aa \<and> aa \<in> A) \<and> a \<in> A) \<or>
-              \<not> limited A r) \<and>
-            ((\<forall>a. (\<forall>aa. aa \<preceq>\<^sub>r a \<or> a \<preceq>\<^sub>r aa \<or> aa \<notin> A) \<or> a \<notin> A) \<and> limited A r \<or>
-              \<not> connex A r)"
-        using connex_def
-        by metis
-      moreover have eq_exc_a:
-        "equiv_rel_except_a A r s a"
-        using lifted_def lifted_a
-        by metis
-      ultimately have "aa \<notin> A \<or> aa \<preceq>\<^sub>r x"
-        using pref_imp_in_above above_x equiv_rel_except_a_def
-              lin_ord_imp_connex limited_dest insertCI
-        by metis
-      thus ?thesis
-        using f1 eq_exc_a above_one above_one2 above_x fin_A
-              equiv_rel_except_a_def insert_not_empty pref_imp_in_above
-              lin_ord_imp_connex mk_disjoint_insert insertE
-        by metis
-    qed
-    moreover have "equiv_rel_except_a A r s a"
-      using lifted_a lifted_def
-      by metis
-    moreover have "x \<in> A-{a}"
-      using above_one above_one2 asm1 assms calculation
-            equiv_rel_except_a_def insert_not_empty
-            member_remove remove_def insert_absorb
-      by metis
-    ultimately have "\<forall>y \<in> A-{a}. y \<preceq>\<^sub>s x"
-      using DiffD1 lifted_a equiv_rel_except_a_def
-      by metis
-    hence not_others: "\<forall>y \<in> A-{a}. above s y \<noteq> {y}"
-      using asm2 empty_iff insert_iff pref_imp_in_above
-      by metis
-    hence "above s a = {a}"
-      using Diff_iff all_not_in_conv lifted_a fin_A lifted_def
-            equiv_rel_except_a_def above_one singleton_iff
-      by metis
-    thus ?thesis
-      by simp
-  qed
-qed
-
-theorem lifted_above_winner2:
-  assumes
-    "lifted A r s a" and
-    "above r a = {a}" and
-    "finite A"
-  shows "above s a = {a}"
-  using assms lifted_above_winner
-  by metis
-
-theorem lifted_above_winner3:
-  assumes
-    lifted_a: "lifted A r s a" and
-    above_x: "above s x = {x}" and
-    fin_A: "finite A" and
-    x_not_a: "x \<noteq> a"
-  shows "above r x = {x}"
-proof (rule ccontr)
-  assume asm: "above r x \<noteq> {x}"
-  then obtain y where y: "above r y = {y}"
-    using lifted_a fin_A insert_Diff insert_not_empty
-          lifted_def equiv_rel_except_a_def above_one
-    by metis
-  hence "above s y = {y} \<or> above s a = {a}"
-    using lifted_a fin_A lifted_above_winner
-    by metis
-  moreover have "\<forall>b. above s b = {b} \<longrightarrow> b = x"
-    using all_not_in_conv lifted_a above_x lifted_def
-          fin_A equiv_rel_except_a_def above_one2
-    by metis
-  ultimately have "y = x"
-    using x_not_a
-    by presburger
-  moreover have "y \<noteq> x"
-    using asm y
-    by blast
-  ultimately show "False"
-    by simp
-qed
-
-end
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Profile.thy# b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Profile.thy#
deleted file mode 100644
index 507abac..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Profile.thy#
+++ /dev/null
@@ -1,853 +0,0 @@
-(*  File:       Profile.thy
-    Copyright   2021  Karlsruhe Institute of Technology (KIT)
-*)
-\<^marker>\<open>creator "Jonas Kraemer, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Karsten Diekhoff, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Michael Kirsten, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Stephan Bohr, Karlsruhe Institute of Technology (KIT)"\<close>
-
-section \<open>Preference Profile\<close>
-
-theory Profile
-  imports Preference_Relation
-begin
-
-text
-\<open>Preference profiles denote the decisions made by the individual voters on
-the eligible alternatives. They are represented in the form of one preference
-relation (e.g., selected on a ballot) per voter, collectively captured in a
-list of such preference relations.
-Unlike a the common preference profiles in the social-choice sense, the
-profiles described here considers only the (sub-)set of alternatives that are
-received.\<close>
-
-subsection \<open>Definition\<close>
-
-type_synonym Alternative = Nat.nat
-
-(*A profile contains one ballot for each voter.*)
-type_synonym Profile = "(Alternative Preference_Relation) list"
-
-(*
-   A profile on a finite set of alternatives A contains only ballots that are
-   linear orders on A.
-*)
-definition profile :: "'a set \<Rightarrow> Profile \<Rightarrow> bool" where
-  "profile A p \<equiv> \<forall>i::nat. i < length p \<longrightarrow> linear_order_on A (p!i)"
-
-lemma profile_set : "profile A p \<equiv> (\<forall>b \<in> (set p). linear_order_on A b)"
-  by (simp add: all_set_conv_all_nth profile_def)
-
-abbreviation finite_profile :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> bool" where
-  "finite_profile A p \<equiv> finite A \<and> profile A p"
-
-subsection \<open>Preference Counts and Comparisons\<close>
-
-(*
-   The win count for an alternative a in a profile p is
-   the amount of ballots in p that rank alternative a in first position.
-*)
-fun win_count :: "'a Profile \<Rightarrow> 'a \<Rightarrow> nat" where
-  "win_count p a =
-    card {i::nat. i < length p \<and> above (p!i) a = {a}}"
-
-fun win_count_code :: "'a Profile \<Rightarrow> 'a \<Rightarrow> nat" where
-  "win_count_code Nil a = 0" |
-  "win_count_code (p#ps) a =
-      (if (above p a = {a}) then 1 else 0) + win_count_code ps a"
-
-lemma win_count_equiv[code]: "win_count p x = win_count_code p x"
-proof (induction p rule: rev_induct, simp)
-  case (snoc a p)
-  fix
-    a :: "'a Preference_Relation" and
-    p :: "'a Profile"
-  assume
-    base_case:
-    "win_count p x = win_count_code p x"
-  have size_one: "length [a] = 1"
-    by simp
-  have p_pos_in_ps:
-    "\<forall>i<length p. p!i = (p@[a])!i"
-    by (simp add: nth_append)
-  have
-    "win_count [a] x =
-      (let q = [a] in
-        card {i::nat. i < length q \<and>
-              (let r = (q!i) in (above r x = {x}))})"
-    by simp
-  hence one_ballot_equiv:
-    "win_count [a] x = win_count_code [a] x"
-    using size_one
-    by (simp add: nth_Cons')
-  have pref_count_induct:
-    "win_count (p@[a]) x =
-      win_count p x + win_count [a] x"
-  proof (simp)
-    have
-      "{i. i = 0 \<and> (above ([a]!i) x = {x})} =
-        (if (above a x = {x}) then {0} else {})"
-      by (simp add: Collect_conv_if)
-    hence shift_idx_a:
-      "card {i. i = length p \<and> (above ([a]!0) x = {x})} =
-        card {i. i = 0 \<and> (above ([a]!i) x = {x})}"
-      by simp
-    have set_prof_eq:
-      "{i. i < Suc (length p) \<and> (above ((p@[a])!i) x = {x})} =
-        {i. i < length p \<and> (above (p!i) x = {x})} \<union>
-          {i. i = length p \<and> (above ([a]!0) x = {x})}"
-    proof (safe, simp_all)
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xa \<noteq> length p" and
-        "xaa \<in> above (p!xa) x"
-      thus "xaa = x"
-        using less_antisym p_pos_in_ps singletonD
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xa \<noteq> length p"
-      thus "x \<in> above (p!xa) x"
-        using less_antisym insertI1 p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xaa \<in> above a x" and
-        "xaa \<noteq> x"
-      thus "xa < length p"
-        using less_antisym nth_append_length
-              p_pos_in_ps singletonD
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a" and
-        xb :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xaa \<in> above a x" and
-        "xaa \<noteq> x" and
-        "xb \<in> above (p!xa) x"
-      thus "xb = x"
-        using less_antisym p_pos_in_ps
-              nth_append_length singletonD
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xaa \<in> above a x" and
-        "xaa \<noteq> x"
-      thus "x \<in> above (p!xa) x"
-        using insertI1 less_antisym nth_append
-              nth_append_length singletonD
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "x \<notin> above a x"
-      thus "xa < length p"
-        using insertI1 less_antisym nth_append_length
-        by metis
-    next
-      fix
-        xa :: nat and
-        xb :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "x \<notin> above a x" and
-        "xb \<in> above (p!xa) x"
-      thus "xb = x"
-        using insertI1 less_antisym nth_append_length
-              p_pos_in_ps singletonD
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "x \<notin> above a x"
-      thus "x \<in> above (p!xa) x"
-        using insertI1 less_antisym nth_append_length p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < length p" and
-        "above (p!xa) x = {x}" and
-        "xaa \<in> above ((p@[a])!xa) x"
-      thus "xaa = x"
-        by (simp add: nth_append)
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < length p" and
-        "above (p!xa) x = {x}"
-      thus "x \<in> above ((p@[a])!xa) x"
-        by (simp add: nth_append)
-    qed
-    have f1:
-      "finite {n. n < length p \<and> (above (p!n) x = {x})}"
-      by simp
-    have f2:
-      "finite {n. n = length p \<and> (above ([a]!0) x = {x})}"
-      by simp
-    have
-      "card ({i. i < length p \<and> (above (p!i) x = {x})} \<union>
-        {i. i = length p \<and> (above ([a]!0) x = {x})}) =
-          card {i. i < length p \<and> (above (p!i) x = {x})} +
-            card {i. i = length p \<and> (above ([a]!0) x = {x})}"
-      using f1 f2 card_Un_disjoint
-      by blast
-    thus
-      "card {i. i < Suc (length p) \<and> (above ((p@[a])!i) x = {x})} =
-        card {i. i < length p \<and> (above (p!i) x = {x})} +
-          card {i. i = 0 \<and> (above ([a]!i) x = {x})}"
-      using set_prof_eq shift_idx_a
-      by auto
-  qed
-  have pref_count_code_induct:
-    "win_count_code (p@[a]) x =
-      win_count_code p x + win_count_code [a] x"
-  proof (induction p, simp)
-    fix
-      aa :: "'a Preference_Relation" and
-      p :: "'a Profile"
-    assume
-      "win_count_code (p@[a]) x =
-        win_count_code p x + win_count_code [a] x"
-    thus
-      "win_count_code ((aa#p)@[a]) x =
-        win_count_code (aa#p) x + win_count_code [a] x"
-      by simp
-  qed
-  show "win_count (p@[a]) x = win_count_code (p@[a]) x"
-    using pref_count_code_induct pref_count_induct
-          base_case one_ballot_equiv
-    by presburger
-qed
-
-fun prefer_count :: "'a Profile \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> nat" where
-  "prefer_count p x y =
-      card {i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))}"
-
-fun prefer_count_code :: "'a Profile \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> nat" where
-  "prefer_count_code Nil x y = 0" |
-  "prefer_count_code (p#ps) x y =
-      (if y \<preceq>\<^sub>p x then 1 else 0) + prefer_count_code ps x y"
-
-lemma pref_count_equiv[code]: "prefer_count p x y = prefer_count_code p x y"
-proof (induction p rule: rev_induct, simp)
-  case (snoc a p)
-  fix
-    a :: "'a Preference_Relation" and
-    p :: "'a Profile"
-  assume
-    base_case:
-    "prefer_count p x y = prefer_count_code p x y"
-  have size_one: "length [a] = 1"
-    by simp
-  have p_pos_in_ps:
-    "\<forall>i<length p. p!i = (p@[a])!i"
-    by (simp add: nth_append)
-  have
-    "prefer_count [a] x y =
-      (let q = [a] in
-        card {i::nat. i < length q \<and>
-              (let r = (q!i) in (y \<preceq>\<^sub>r x))})"
-    by simp
-  hence one_ballot_equiv:
-    "prefer_count [a] x y = prefer_count_code [a] x y"
-    using size_one
-    by (simp add: nth_Cons')
-  have pref_count_induct:
-    "prefer_count (p@[a]) x y =
-      prefer_count p x y + prefer_count [a] x y"
-  proof (simp)
-    have
-      "{i. i = 0 \<and> (y, x) \<in> [a]!i} =
-        (if ((y, x) \<in> a) then {0} else {})"
-      by (simp add: Collect_conv_if)
-    hence shift_idx_a:
-      "card {i. i = length p \<and> (y, x) \<in> [a]!0} =
-        card {i. i = 0 \<and> (y, x) \<in> [a]!i}"
-      by simp
-    have set_prof_eq:
-      "{i. i < Suc (length p) \<and> (y, x) \<in> (p@[a])!i} =
-        {i. i < length p \<and> (y, x) \<in> p!i} \<union>
-          {i. i = length p \<and> (y, x) \<in> [a]!0}"
-    proof (safe, simp_all)
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "(y, x) \<in> (p@[a])!xa" and
-        "xa \<noteq> length p"
-      thus "(y, x) \<in> p!xa"
-        using less_antisym p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "(y, x) \<in> (p@[a])!xa" and
-        "(y, x) \<notin> a"
-      thus "xa < length p"
-        using less_antisym nth_append_length
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "(y, x) \<in> (p@[a])!xa" and
-        "(y, x) \<notin> a"
-      thus "(y, x) \<in> p!xa"
-        using less_antisym nth_append_length p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < length p" and
-        "(y, x) \<in> p!xa"
-      thus "(y, x) \<in> (p@[a])!xa"
-        using less_antisym p_pos_in_ps
-        by metis
-    qed
-    have f1:
-      "finite {n. n < length p \<and> (y, x) \<in> p!n}"
-      by simp
-    have f2:
-      "finite {n. n = length p \<and> (y, x) \<in> [a]!0}"
-      by simp
-    have
-      "card ({i. i < length p \<and> (y, x) \<in> p!i} \<union>
-        {i. i = length p \<and> (y, x) \<in> [a]!0}) =
-          card {i. i < length p \<and> (y, x) \<in> p!i} +
-            card {i. i = length p \<and> (y, x) \<in> [a]!0}"
-      using f1 f2 card_Un_disjoint
-      by blast
-    thus
-      "card {i. i < Suc (length p) \<and> (y, x) \<in> (p@[a])!i} =
-        card {i. i < length p \<and> (y, x) \<in> p!i} +
-          card {i. i = 0 \<and> (y, x) \<in> [a]!i}"
-      using set_prof_eq shift_idx_a
-      by auto
-  qed
-  have pref_count_code_induct:
-    "prefer_count_code (p@[a]) x y =
-      prefer_count_code p x y + prefer_count_code [a] x y"
-  proof (simp, safe)
-    assume
-      assm: "(y, x) \<in> a"
-    show
-      "prefer_count_code (p@[a]) x y = Suc (prefer_count_code p x y)"
-    proof (induction p, simp_all)
-      show "(y, x) \<in> a"
-        using assm
-        by simp
-    qed
-  next
-    assume
-      assm: "(y, x) \<notin> a"
-    show
-      "prefer_count_code (p@[a]) x y = prefer_count_code p x y"
-    proof (induction p, simp_all, safe)
-      assume
-        "(y, x) \<in> a"
-      thus "False"
-        using assm
-        by simp
-    qed
-  qed
-  show "prefer_count (p@[a]) x y = prefer_count_code (p@[a]) x y"
-    using pref_count_code_induct pref_count_induct
-          base_case one_ballot_equiv
-    by presburger
-qed
-
-lemma set_compr: "{ f x | x . x \<in> S } = f ` S"
-  by auto
-
-lemma pref_count_set_compr: "{prefer_count p x y | y . y \<in> A-{x}} =
-          (prefer_count p x) ` (A-{x})"
-  by auto
-
-lemma pref_count:
-  assumes prof: "profile A p"
-  assumes x_in_A: "x \<in> A"
-  assumes y_in_A: "y \<in> A"
-  assumes neq: "x \<noteq> y"
-  shows "prefer_count p x y = (length p) - (prefer_count p y x)"
-proof -
-  have 00: "card {i::nat. i < length p} = length p"
-    by simp
-  have 10:
-    "{i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))} =
-        {i::nat. i < length p} -
-          {i::nat. i < length p \<and> \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x))}"
-    by auto
-  have 0: "\<forall> i::nat . i < length p \<longrightarrow> linear_order_on A (p!i)"
-    using prof profile_def
-    by metis
-  hence "\<forall> i::nat . i < length p \<longrightarrow> connex A (p!i)"
-    by (simp add: lin_ord_imp_connex)
-  hence 1: "\<forall> i::nat . i < length p \<longrightarrow>
-              \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x)) \<longrightarrow> (let r = (p!i) in (x \<preceq>\<^sub>r y))"
-    using connex_def x_in_A y_in_A
-    by metis
-  from 0 have
-    "\<forall> i::nat . i < length p \<longrightarrow> antisym  (p!i)"
-    using lin_imp_antisym
-    by metis
-  hence "\<forall> i::nat . i < length p \<longrightarrow> ((y, x) \<in> (p!i) \<longrightarrow> (x, y) \<notin> (p!i))"
-    using antisymD neq
-    by metis
-  hence "\<forall> i::nat . i < length p \<longrightarrow>
-          ((let r = (p!i) in (y \<preceq>\<^sub>r x)) \<longrightarrow> \<not> (let r = (p!i) in (x \<preceq>\<^sub>r y)))"
-    by simp
-  with 1 have
-    "\<forall> i::nat . i < length p \<longrightarrow>
-      \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x)) = (let r = (p!i) in (x \<preceq>\<^sub>r y))"
-    by metis
-  hence 2:
-    "{i::nat. i < length p \<and> \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x))} =
-        {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))}"
-    by metis
-  hence 20:
-    "{i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))} =
-        {i::nat. i < length p} -
-          {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))}"
-    using "10" "2"
-    by simp
-  have
-    "{i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))} \<subseteq>
-        {i::nat. i < length p}"
-    by (simp add: Collect_mono)
-  hence 30:
-    "card ({i::nat. i < length p} -
-        {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))}) =
-      (card {i::nat. i < length p}) -
-        card({i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))})"
-    by (simp add: card_Diff_subset)
-  have "prefer_count p x y =
-          card {i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))}"
-    by simp
-  also have
-    "... = card({i::nat. i < length p} -
-            {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))})"
-    using "20"
-    by simp
-  also have
-    "... = (card {i::nat. i < length p}) -
-              card({i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))})"
-    using "30"
-    by metis
-  also have
-    "... = length p - (prefer_count p y x)"
-    by simp
-  finally show ?thesis
-    by (simp add: "20" "30")
-qed
-
-lemma pref_count_sym:
-    assumes p1: "prefer_count p a x \<ge> prefer_count p x b"
-    assumes prof: "profile A p"
-    assumes a_in_A: "a \<in> A"
-    assumes b_in_A: "b \<in> A"
-    assumes x_in_A: "x \<in> A"
-    assumes neq1: "a \<noteq> x"
-    assumes neq2: "x \<noteq> b"
-    shows "prefer_count p b x \<ge> prefer_count p x a"
-proof -
-  from prof a_in_A x_in_A neq1 have 0:
-    "prefer_count p a x = (length p) - (prefer_count p x a)"
-    using pref_count
-    by metis
-  moreover from prof x_in_A b_in_A neq2 have 1:
-    "prefer_count p x b = (length p) - (prefer_count p b x)"
-    using pref_count
-    by (metis (mono_tags, lifting))
-  hence 2: "(length p) - (prefer_count p x a) \<ge>
-              (length p) - (prefer_count p b x)"
-    using calculation p1
-    by auto
-  hence 3: "(prefer_count p x a) - (length p) \<le>
-              (prefer_count p b x) - (length p)"
-    using a_in_A diff_is_0_eq diff_le_self neq1
-          pref_count prof x_in_A
-    by (metis (no_types))
-  hence "(prefer_count p x a) \<le> (prefer_count p b x)"
-    using "1" "3" calculation p1
-    by linarith
-  thus ?thesis
-    by linarith
-qed
-
-lemma empty_prof_imp_zero_pref_count:
-  assumes "p = []"
-  shows "\<forall> x y. prefer_count p x y = 0"
-  using assms
-  by simp
-
-lemma empty_prof_imp_zero_pref_count_code:
-  assumes "p = []"
-  shows "\<forall> x y. prefer_count_code p x y = 0"
-  using assms
-  by simp
-
-lemma pref_count_code_incr:
-  assumes
-    "prefer_count_code ps x y = n" and
-    "y \<preceq>\<^sub>p x"
-  shows "prefer_count_code (p#ps) x y = n+1"
-  using assms
-  by simp
-
-lemma pref_count_code_not_smaller_imp_constant:
-  assumes
-    "prefer_count_code ps x y = n" and
-    "\<not>(y \<preceq>\<^sub>p x)"
-  shows "prefer_count_code (p#ps) x y = n"
-  using assms
-  by simp
-
-fun wins :: "'a \<Rightarrow> 'a Profile \<Rightarrow> 'a \<Rightarrow> bool" where
-  "wins x p y =
-    (prefer_count p x y > prefer_count p y x)"
-
-(* Alternative a wins against b implies that b does not win against a. *)
-lemma wins_antisym:
-  assumes "wins a p b"
-  shows "\<not> wins b p a"
-  using assms
-  by simp
-
-lemma wins_irreflex: "\<not> wins w p w"
-  using wins_antisym
-  by metis
-
-subsection \<open>Condorcet Winner\<close>
-
-fun condorcet_winner :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a \<Rightarrow> bool" where
-  "condorcet_winner A p w =
-      (finite_profile A p \<and>  w \<in> A \<and> (\<forall>x \<in> A - {w} . wins w p x))"
-
-lemma cond_winner_unique:
-  assumes winner_c: "condorcet_winner A p c" and
-          winner_w: "condorcet_winner A p w"
-  shows "w = c"
-proof (rule ccontr)
-  assume
-    assumption: "w \<noteq> c"
-  from winner_w
-  have "wins w p c"
-    using assumption insert_Diff insert_iff winner_c
-    by simp
-  hence "\<not> wins c p w"
-    by (simp add: wins_antisym)
-  moreover from winner_c
-  have
-    c_wins_against_w: "wins c p w"
-    using Diff_iff assumption
-          singletonD winner_w
-    by simp
-  ultimately show False
-    by simp
-qed
-
-lemma cond_winner_unique2:
-  assumes winner: "condorcet_winner A p w" and
-          not_w:  "x \<noteq> w" and
-          in_A:   "x \<in> A"
-        shows "\<not> condorcet_winner A p x"
-  using not_w cond_winner_unique winner
-  by metis
-
-lemma cond_winner_unique3:
-  assumes "condorcet_winner A p w"
-  shows "{a \<in> A. condorcet_winner A p a} = {w}"
-proof (safe, simp_all, safe)
-  fix
-    x :: "'a"
-  assume
-    fin_A: "finite A" and
-    prof_A: "profile A p" and
-    x_in_A: "x \<in> A" and
-    x_wins:
-      "\<forall>xa \<in> A - {x}.
-        card {i. i < length p \<and> (x, xa) \<in> p!i} <
-          card {i. i < length p \<and> (xa, x) \<in> p!i}"
-  from assms have assm:
-    "finite_profile A p \<and>  w \<in> A \<and>
-      (\<forall>x \<in> A - {w}.
-        card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-          card {i::nat. i < length p \<and> (x, w) \<in> p!i})"
-    by simp
-  hence
-    "(\<forall>x \<in> A - {w}.
-      card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-        card {i::nat. i < length p \<and> (x, w) \<in> p!i})"
-    by simp
-  hence w_beats_x:
-    "x \<noteq> w \<Longrightarrow>
-      card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-        card {i::nat. i < length p \<and> (x, w) \<in> p!i}"
-    using x_in_A
-    by simp
-  also from assm have
-    "finite_profile A p"
-    by simp
-  moreover from assm have
-    "w \<in> A"
-    by simp
-  hence x_beats_w:
-    "x \<noteq> w \<Longrightarrow>
-      card {i. i < length p \<and> (x, w) \<in> p!i} <
-        card {i. i < length p \<and> (w, x) \<in> p!i}"
-    using x_wins
-    by simp
-  from w_beats_x x_beats_w show
-    "x = w"
-    by linarith
-next
-  fix
-    x :: "'a"
-  from assms show "w \<in> A"
-    by simp
-next
-  fix
-    x :: "'a"
-  from assms show "finite A"
-    by simp
-next
-  fix
-    x :: "'a"
-  from assms show "profile A p"
-    by simp
-next
-  fix
-    x :: "'a"
-  from assms show "w \<in> A"
-    by simp
-next
-  fix
-    x :: "'a" and
-    xa :: "'a"
-  assume
-    xa_in_A: "xa \<in> A" and
-    w_wins:
-      "\<not> card {i. i < length p \<and> (w, xa) \<in> p!i} <
-        card {i. i < length p \<and> (xa, w) \<in> p!i}"
-  from assms have
-    "finite_profile A p \<and>  w \<in> A \<and>
-      (\<forall>x \<in> A - {w} .
-        card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-          card {i::nat. i < length p \<and> (x, w) \<in> p!i})"
-    by simp
-  thus "xa = w"
-    using xa_in_A w_wins insert_Diff insert_iff
-    by (metis (no_types, lifting))
-qed
-
-subsection \<open>Limited Profile\<close>
-
-(*
-   This function restricts a profile p to a set A and
-   keeps all of A's preferences.
-*)
-fun limit_profile :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a Profile" where
-  "limit_profile A p = map (limit A) p"
-
-lemma limit_prof_trans:
-  assumes
-    "B \<subseteq> A" and
-    "C \<subseteq> B" and
-    "finite_profile A p"
-  shows "limit_profile C p = limit_profile C (limit_profile B p)"
-  using assms
-  by auto
-
-lemma limit_profile_sound:
-  assumes
-    profile: "finite_profile S p" and
-    subset: "A \<subseteq> S"
-  shows "finite_profile A (limit_profile A p)"
-proof (simp)
-  from profile
-  show "finite_profile A (map (limit A) p)"
-    using length_map limit_presv_lin_ord nth_map
-          profile_def subset infinite_super
-    by metis
-qed
-
-lemma limit_prof_presv_size:
-  assumes f_prof: "finite_profile S p" and
-          subset:  "A \<subseteq> S"
-  shows "length p = length (limit_profile A p)"
-  by simp
-
-subsection \<open>Lifting Property\<close>
-
-definition equiv_prof_except_a :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a Profile \<Rightarrow>
-                                        'a \<Rightarrow> bool" where
-  "equiv_prof_except_a A p q a \<equiv>
-    finite_profile A p \<and> finite_profile A q \<and>
-      a \<in> A \<and> length p = length q \<and>
-      (\<forall>i::nat.
-        i < length p \<longrightarrow>
-          equiv_rel_except_a A (p!i) (q!i) a)"
-
-(*
-   An alternative gets lifted from one profile to another iff
-   its ranking increases in at least one ballot, and nothing else changes.
-*)
-definition lifted :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a Profile \<Rightarrow> 'a \<Rightarrow> bool" where
-  "lifted A p q a \<equiv>
-    finite_profile A p \<and> finite_profile A q \<and>
-      a \<in> A \<and> length p = length q \<and>
-      (\<forall>i::nat.
-        (i < length p \<and> \<not>Preference_Relation.lifted A (p!i) (q!i) a) \<longrightarrow>
-          (p!i) = (q!i)) \<and>
-      (\<exists>i::nat. i < length p \<and> Preference_Relation.lifted A (p!i) (q!i) a)"
-
-lemma lifted_imp_equiv_prof_except_a:
-  assumes lifted: "lifted A p q a"
-  shows "equiv_prof_except_a A p q a"
-proof -
-  have
-    "\<forall>i::nat. i < length p \<longrightarrow>
-      equiv_rel_except_a A (p!i) (q!i) a"
-  proof
-    fix i :: nat
-    show
-      "i < length p \<longrightarrow>
-        equiv_rel_except_a A (p!i) (q!i) a"
-    proof
-      assume i_ok: "i < length p"
-      show "equiv_rel_except_a A (p!i) (q!i) a"
-        using lifted_def i_ok lifted profile_def trivial_equiv_rel
-              lifted_imp_equiv_rel_except_a
-        by metis
-    qed
-  qed
-  thus ?thesis
-    using lifted_def lifted equiv_prof_except_a_def
-    by metis
-qed
-
-lemma negl_diff_imp_eq_limit_prof:
-  assumes
-    change: "equiv_prof_except_a S p q a" and
-    subset: "A \<subseteq> S" and
-    notInA: "a \<notin> A"
-  shows "limit_profile A p = limit_profile A q"
-proof -
-  have
-    "\<forall>i::nat. i < length p \<longrightarrow>
-      equiv_rel_except_a S (p!i) (q!i) a"
-    using change equiv_prof_except_a_def
-    by metis
-  hence "\<forall>i::nat. i < length p \<longrightarrow> limit A (p!i) = limit A (q!i)"
-    using notInA negl_diff_imp_eq_limit subset
-    by metis
-  hence "map (limit A) p = map (limit A) q"
-    using change equiv_prof_except_a_def
-          length_map nth_equalityI nth_map
-    by (metis (mono_tags, lifting))
-  thus ?thesis
-    by simp
-qed
-
-lemma limit_prof_eq_or_lifted:
-  assumes
-    lifted: "lifted S p q a" and
-    subset: "A \<subseteq> S"
-  shows
-    "limit_profile A p = limit_profile A q \<or>
-        lifted A (limit_profile A p) (limit_profile A q) a"
-proof cases
-  assume inA: "a \<in> A"
-  have
-    "\<forall>i::nat. i < length p \<longrightarrow>
-        (Preference_Relation.lifted S (p!i) (q!i) a \<or> (p!i) = (q!i))"
-    using lifted_def lifted
-    by metis
-  hence one:
-    "\<forall>i::nat. i < length p \<longrightarrow>
-         (Preference_Relation.lifted A (limit A (p!i)) (limit A (q!i)) a \<or>
-           (limit A (p!i)) = (limit A (q!i)))"
-    using limit_lifted_imp_eq_or_lifted subset
-    by metis
-  thus ?thesis
-  proof cases
-    assume "\<forall>i::nat. i < length p \<longrightarrow> (limit A (p!i)) = (limit A (q!i))"
-    thus ?thesis
-      using lifted_def length_map lifted
-            limit_profile.simps nth_equalityI nth_map
-      by (metis (mono_tags, lifting))
-  next
-    assume assm:
-      "\<not>(\<forall>i::nat. i < length p \<longrightarrow> (limit A (p!i)) = (limit A (q!i)))"
-    let ?p = "limit_profile A p"
-    let ?q = "limit_profile A q"
-    have "profile A ?p \<and> profile A ?q"
-      using lifted_def lifted limit_profile_sound subset
-      by metis
-    moreover have "length ?p = length ?q"
-      using lifted_def lifted
-      by fastforce
-    moreover have
-      "\<exists>i::nat. i < length ?p \<and> Preference_Relation.lifted A (?p!i) (?q!i) a"
-      using assm lifted_def length_map lifted
-            limit_profile.simps nth_map one
-      by (metis (no_types, lifting))
-    moreover have
-      "\<forall>i::nat.
-        (i < length ?p \<and> \<not>Preference_Relation.lifted A (?p!i) (?q!i) a) \<longrightarrow>
-          (?p!i) = (?q!i)"
-      using lifted_def length_map lifted
-            limit_profile.simps nth_map one
-      by metis
-    ultimately have "lifted A ?p ?q a"
-      using lifted_def inA lifted rev_finite_subset subset
-      by (metis (no_types, lifting))
-    thus ?thesis
-      by simp
-  qed
-next
-  assume "a \<notin> A"
-  thus ?thesis
-    using lifted negl_diff_imp_eq_limit_prof subset
-          lifted_imp_equiv_prof_except_a
-    by metis
-qed
-
-end
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator.tex
deleted file mode 100644
index b743c4d..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator.tex
+++ /dev/null
@@ -1,133 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An aggregator gets two partitions (results of electoral modules) as input and
-output another partition. They are used to aggregate results of parallel
-composed electoral modules.
-They are commutative, i.e., the order of the aggregated modules does not affect
-the resulting aggregation. Moreover, they are conservative in the sense that
-the resulting decisions are subsets of the two given partitions' decisions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}aggregator\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Aggregator.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%31=11%:%
-%:%32=11%:%
-%:%33=12%:%
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-%:%43=16%:%
-%:%44=17%:%
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-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
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-%:%70=27%:%
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-%:%72=28%:%
-%:%82=33%:%
-%:%98=36%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Facts.tex
deleted file mode 100644
index bbc60bb..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Facts.tex
+++ /dev/null
@@ -1,258 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Aggregator{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Properties.tex
deleted file mode 100644
index 4ff5f97..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Properties.tex
+++ /dev/null
@@ -1,74 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}conservative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Blacks_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Blacks_Rule.tex
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-\isamarkupsection{Black's Rule%
-}
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-\ Blacks{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \ \ \ \ \ \ \ \ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\isakeyword{begin}%
-\endisatagtheory
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-\begin{isamarkuptext}%
-This is Black's voting rule. It is composed of a function that determines
-the Condorcet winner, i.e., the Pairwise Majority rule, and the Borda rule.
-Whenever there exists no Condorcet winner, it elects the choice made by the
-Borda rule, otherwise the Condorcet winner is elected.%
-\end{isamarkuptext}\isamarkuptrue%
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-\ blacks{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}blacks{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isasymtriangleright}\ borda{\isacharunderscore}{\kern0pt}rule{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Borda_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Borda_Rule.tex
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-\isamarkupchapter{Voting Rules%
-}
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-\isamarkupsection{Borda Rule%
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-\ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isanewline
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-\begin{isamarkuptext}%
-This is the Borda rule. On each ballot, each alternative is assigned a score
-that depends on how many alternatives are ranked below. The sum of all such
-scores for an alternative is hence called their Borda score. The alternative
-with the highest Borda score is elected.%
-\end{isamarkuptext}\isamarkuptrue%
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-\ borda{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ borda\ A\ p{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Classic_Nanson_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Classic_Nanson_Rule.tex
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-\isamarkupsection{Classic Nanson Rule%
-}
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-\ Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\begin{isamarkuptext}%
-This is the classic Nanson's voting rule, i.e., the rule that was originally
-invented by Nanson, but not the Nanson-Baldwin rule. The idea is similar,
-however, as alternatives with a Borda score less or equal than the average
-Borda score are excluded. The Borda scores of the remaining alternatives
-are hence adjusted to the new set of (still) eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
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-\ classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Elimination_Modules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Elimination_Modules.tex
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-\ Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
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-\isamarkupsection{Borda Module%
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-\begin{isamarkuptext}%
-This is the Borda module used by the Borda rule. The Borda rule is a voting
-rule, where on each ballot, each alternative is assigned a score that depends
-on how many alternatives are ranked below. The sum of all such scores for an
-alternative is hence called their Borda score. The alternative with the highest
-Borda score is elected. The module implemented herein only rejects the
-alternatives not elected by the voting rule, and defers the alternatives that
-would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
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-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
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-\ \ {\isachardoublequoteopen}borda\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}%
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-\begin{isamarkuptext}%
-This is the Condorcet module used by the Condorcet (voting) rule. The Condorcet
-rule is a voting rule that implements the Condorcet criterion, i.e., it elects
-the Condorcet winner if it exists, otherwise a tie remains between all
-alternatives. The module implemented herein only rejects the alternatives not
-elected by the voting rule, and defers the alternatives that would be elected
-by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
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-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
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-This is the Copeland module used by the Copeland voting rule. The Copeland
-rule elects the alternatives with the highest difference between the amount
-of simple-majority wins and the amount of simple-majority losses. The module
-implemented herein only rejects the alternatives not elected by the voting
-rule, and defers the alternatives that would be elected by the full voting
-rule.%
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-\ copeland{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ x\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ True{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ {\isasymnot}\ wins\ x\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ False{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disjunct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasyminter}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ union{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ finiteness{\isadigit{1}}\ finiteness{\isadigit{2}}\ disjunct\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ union\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ amount{\isadigit{1}}\ amount{\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ w\ A\ p\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ a\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}{\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\ loser\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ M\ f\ {\isachardot}{\kern0pt}\ finite\ M\ {\isasymlongrightarrow}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ f\ x{\isacharbraceright}{\kern0pt}\ {\isasymle}\ card\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ finite{\isacharunderscore}{\kern0pt}Diff\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{3}}\ {\isadigit{4}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ loser{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{5}}\ {\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Minimax Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is the Minimax module used by the Minimax voting rule. The Minimax rule
-elects the alternatives with the highest Minimax score. The module implemented
-herein only rejects the alternatives not elected by the voting rule, and defers
-the alternatives that would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p\ {\isasymle}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ Min\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}lscore\ {\isasymle}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ local{\isachardot}{\kern0pt}finite\ not{\isacharunderscore}{\kern0pt}empty\ Min{\isacharunderscore}{\kern0pt}le\ Min{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
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-\isadelimproof
-\isanewline
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-\endisadelimproof
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Structures.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Structures.tex
deleted file mode 100644
index fe948a3..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Structures.tex
+++ /dev/null
@@ -1,1266 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Composite{\isacharunderscore}{\kern0pt}Structures}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Composite{\isacharunderscore}{\kern0pt}Structures\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Aggregator{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Aggregator{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Elect Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-The elect composition sequences an electoral module and the elect
-module. It finalizes the module's decision as it simply elects all their
-non-rejected alternatives. Thereby, any such elect-composed module induces
-a proper voting rule in the social choice sense, as all alternatives are either
-rejected or elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elector\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elector\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
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-\endisadelimproof
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsection{Defer One Loop Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
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-\begin{isamarkuptext}%
-This is a family of loop compositions. It uses the same module in sequence
-until either no new decisions are made or only one alternative is remaining
-in the defer-set. The second family herein uses the above family and
-subsequently elects the remaining alternative.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ iter\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iter\ m\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d\ {\isasymequiv}\ iter\ m{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ iterelect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iterelect\ m\ {\isacharequal}{\kern0pt}\ elector\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsection{Maximum Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is a family of parallel compositions. It composes a new electoral module
-from two electoral modules combined with the maximum aggregator. Therein, the
-two modules each make a decision and then a partition is returned where every
-alternative receives the maximum result of the two input partitions. This means
-that, if any alternative is elected by at least one of the modules, then it
-gets elected, if any non-elected alternative is deferred by at least one of the
-modules, then it gets deferred, only alternatives rejected by both modules get
-rejected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ a\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}aggregator\ in\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymparallel}\isactrlsub {\isasymup}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x\ {\isasymor}\isanewline
-\ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ x\ {\isasymin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ Un{\isacharunderscore}{\kern0pt}iff\ a{\isadigit{1}}\ empty{\isacharunderscore}{\kern0pt}iff\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}conv\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ f{\isacharunderscore}{\kern0pt}prof\ sndI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}iff\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ a{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}prof\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}unfold\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ surjective{\isacharunderscore}{\kern0pt}pairing\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}iff\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}\ Int{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prodI{\isadigit{2}}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ split{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}iff\ a\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ surjective{\isacharunderscore}{\kern0pt}pairing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymnotin}\ elect\ n\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ prod{\isachardot}{\kern0pt}simps\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ old{\isachardot}{\kern0pt}prod{\isachardot}{\kern0pt}inject\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}collapse\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Set{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}insert\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}insert\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ x\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}sum{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S\ {\isacharplus}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ x\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ x\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ y\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ compatible\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}representation{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ x\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ subsets{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ finite\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ S\ {\isacharplus}{\kern0pt}\ n\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}Int\ reject{\isacharunderscore}{\kern0pt}representation\ reject{\isacharunderscore}{\kern0pt}sum\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subsets\ subset{\isacharunderscore}{\kern0pt}eq\ sup{\isachardot}{\kern0pt}absorb{\isacharunderscore}{\kern0pt}iff{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isachardot}{\kern0pt}cobounded{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
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-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Consistency.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Consistency.tex
deleted file mode 100644
index 58090d9..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Consistency.tex
+++ /dev/null
@@ -1,63 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Consistency}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Consistency\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}e\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ e{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Properties/Condorcet_Consistency.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Facts.tex
deleted file mode 100644
index c47d28f..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Facts.tex
+++ /dev/null
@@ -1,795 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Consistency{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Condorcet{\isacharunderscore}{\kern0pt}Rules\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ f\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymand}\ a\ {\isasymnoteq}\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ a{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ f\ aa\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ copeland{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharless}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}\ {\isacharless}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isacharunderscore}{\kern0pt}Diff\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ one{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}numeral{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ winner\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ condorcet{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}condorcet\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}copeland\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ minimax{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ prefer{\isacharunderscore}{\kern0pt}count{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ prof\ winner\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ \ {\isasymge}\ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}\ n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isadigit{6}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\ prof\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ prof\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}minimax\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Properties.tex
deleted file mode 100644
index d89a4f0..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Properties.tex
+++ /dev/null
@@ -1,93 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymnotin}\ reject\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymnotin}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymin}\ reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ p\ w\ {\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymlongrightarrow}\ f\ l\ A\ p\ {\isacharless}{\kern0pt}\ f\ w\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Condorcet_Properties.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Rules.tex
deleted file mode 100644
index 47d61c8..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Rules.tex
+++ /dev/null
@@ -1,1954 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Consistency{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eMax\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eW\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ e\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isachardot}{\kern0pt}\ e\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}order\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ a\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{4}}\ Max{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ linA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ l{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ linA\ loser\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\ f{\isacharunderscore}{\kern0pt}prof\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}ccomp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}eq\ finite\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ fst{\isacharunderscore}{\kern0pt}conv\ insert{\isacharunderscore}{\kern0pt}Diff\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ order{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharunderscore}{\kern0pt}Compl{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ empty{\isacharunderscore}{\kern0pt}iff\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ DiffI\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ finite\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}iff\ finite\ prof{\isacharunderscore}{\kern0pt}A\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ equals{\isadigit{0}}D\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ insert{\isacharunderscore}{\kern0pt}subset\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymand}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win\ equals{\isadigit{0}}D\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ other{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ one{\isacharunderscore}{\kern0pt}side{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finiteness{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ wins\ d\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ d\ {\isacharless}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ d\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}d\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ not{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}\ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}trsh{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ rating\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymor}\ wins\ w\ p\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymLongrightarrow}\ electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ cc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ x\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ wins{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\ e{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Copeland_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Copeland_Rule.tex
deleted file mode 100644
index ce2aad3..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Copeland_Rule.tex
+++ /dev/null
@@ -1,162 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Copeland{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Copeland Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Copeland{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Copeland voting rule. The idea is to elect the alternatives with
-the highest difference between the amount of simple-majority wins and the
-amount of simple-majority losses.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ copeland\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ copeland{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ copeland{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Copeland_Rule.thy%:%
-%:%6=3%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Equal_Condition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Equal_Condition.tex
deleted file mode 100644
index f0237b2..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Equal_Condition.tex
+++ /dev/null
@@ -1,101 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Defer Equal Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\begin{isamarkuptext}%
-This is a family of termination conditions. For a natural number n,
-the according defer-equal condition is true if and only if the given
-result's defer-set contains exactly n elements.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isamarkupsubsection{Definition%
-}
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-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ n\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ result\ in\ card\ d\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Module.tex
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-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Module}%
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-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupchapter{Basic Modules%
-}
-\isamarkuptrue%
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-\isamarkupsection{Defer Module%
-}
-\isamarkuptrue%
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-\endisatagdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\ Defer{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\begin{isamarkuptext}%
-The defer module is not concerned about the voter's ballots, and
-simply defers all alternatives.
-It is primarily used for defining an empty loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isamarkupsubsection{Definition%
-}
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-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
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-\isamarkupsubsection{Soundness%
-}
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-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
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-\isadelimproof
-\ \ %
-\endisadelimproof
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-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility.tex
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-%
-\begin{isabellebody}%
-\setisabellecontext{Disjoint{\isacharunderscore}{\kern0pt}Compatibility}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Disjoint{\isacharunderscore}{\kern0pt}Compatibility\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Indep{\isacharunderscore}{\kern0pt}Of{\isacharunderscore}{\kern0pt}Alt\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Facts.tex
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-\begin{isabellebody}%
-\setisabellecontext{Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}pass{\isacharunderscore}{\kern0pt}disj{\isacharunderscore}{\kern0pt}compat{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
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-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}iff\ empty{\isacharunderscore}{\kern0pt}set\ fin\ profile{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Rules.tex
deleted file mode 100644
index 6851942..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Rules.tex
+++ /dev/null
@@ -1,521 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ n\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ old{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}S{\isacharminus}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ double{\isacharunderscore}{\kern0pt}diff\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}seq{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}S{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}S\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ p\ q{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ m\ S\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ a\ b\ \isacommand{have}\isamarkupfalse%
-\ profiles{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ DiffD{\isadigit{2}}\ a\ b\ compatible\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}def\ profiles\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ S\ p\ {\isacharequal}{\kern0pt}\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ UnI{\isadigit{1}}\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\ modules\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Drop_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Drop_Module.tex
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--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Drop_Module.tex
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-%
-\begin{isabellebody}%
-\setisabellecontext{Drop{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Drop Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Drop{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according drop module
-rejects the lexicographically first n alternatives (from A) and
-defers the rest.
-It is primarily used as counterpart to the pass module in a
-parallel composition, in order to segment the alternatives into
-two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elect_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elect_Module.tex
deleted file mode 100644
index a964962..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elect_Module.tex
+++ /dev/null
@@ -1,147 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elect{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Elect Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elect{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The elect module is not concerned about the voter's ballots, and
-just elects all alternatives. It is primarily used in sequence after
-an electoral module that only defers alternatives to finalize the decision,
-thereby inducing a proper voting rule in the social choice sense.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Elect_Module.thy%:%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Electoral_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Electoral_Module.tex
deleted file mode 100644
index ab4be9a..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Electoral_Module.tex
+++ /dev/null
@@ -1,1952 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Electoral{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Component Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Electoral Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Electoral{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Electoral modules are the principal component type of the composable modules
-voting framework, as they are a generalization of voting rules in the sense of
-social choice functions.
-These are only the types used for electoral modules. Further restrictions are
-encompassed by the electoral-module predicate.
-
-An electoral module does not need to make final decisions for all alternatives,
-but can instead defer the decision for some or all of them to other modules.
-Hence, electoral modules partition the received (possibly empty) set of
-alternatives into elected, rejected and deferred alternatives. In particular,
-any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives. Just like a voting rule, an electoral module also receives a
-profile which holds the voters’ preferences, which, unlike a voting rule,
-consider only the (sub-)set of alternatives that the module receives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymequiv}\ {\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymequiv}\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymequiv}\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer{\isachardoublequoteclose}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymequiv}\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Equivalence Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def\ ele\ rej\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ set{\isacharunderscore}{\kern0pt}partit\ partit\ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ fst{\isacharunderscore}{\kern0pt}conv\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ snd{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isasymnot}\ disjoint{\isadigit{3}}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D{\isachardot}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharcomma}{\kern0pt}\ C{\isacharcomma}{\kern0pt}\ D{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ B\ {\isasyminter}\ C\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ B\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ C\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disj{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disj{\isadigit{3}}\ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isadigit{3}}\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\ fstI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ module\ profile\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\ prof{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ partit\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ module\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounds{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}elected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ alternative\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}elected\ not{\isacharunderscore}{\kern0pt}rejected\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ m\ A\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ IntI\ assms\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymor}\ x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ alternative\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ not{\isacharunderscore}{\kern0pt}rejected\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ input{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}collapse\ subsetI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ n\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ n\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ n\ A\ q\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p\ reject{\isacharunderscore}{\kern0pt}q\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ prod{\isacharunderscore}{\kern0pt}eqI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elimination_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elimination_Module.tex
deleted file mode 100644
index df53134..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elimination_Module.tex
+++ /dev/null
@@ -1,596 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elimination{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Elimination Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elimination{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the elimination module. It rejects a set of alternatives only if these
-are not all alternatives. The alternatives potentially to be rejected are put
-in a so-called elimination set. These are all alternatives that score below
-a preset threshold value that depends on the specific voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isacharequal}{\kern0pt}\ nat\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}e\ a\ A\ p{\isacharparenright}{\kern0pt}\ t\ {\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ A\isanewline
-\ \ \ \ \ \ \ \ then\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ else\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Common Eliminators%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}min{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ average\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Threshold{\isacharunderscore}{\kern0pt}Value{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}average\ e\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ {\isacharparenleft}{\kern0pt}card\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
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-\isadelimproof
-\isanewline
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-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\isadelimproof
-\ \ %
-\endisadelimproof
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-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Evaluation_Function.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Evaluation_Function.tex
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-\begin{isabellebody}%
-\setisabellecontext{Evaluation{\isacharunderscore}{\kern0pt}Function}%
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-\isamarkupsection{Evaluation Function%
-}
-\isamarkuptrue%
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Evaluation{\isacharunderscore}{\kern0pt}Function\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\begin{isamarkuptext}%
-This is the evaluation function. From a set of currently eligible alternatives,
-the evaluation function computes a numerical value that is then to be used for
-further (s)election, e.g., by the elimination module.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isadelimdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ \ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Homogeneity.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Homogeneity.tex
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-\begin{isabellebody}%
-\setisabellecontext{Homogeneity}%
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Homogeneity\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
-\ \isanewline
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-\endisadelimtheory
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ times\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}times\ n\ l\ {\isacharequal}{\kern0pt}\ concat\ {\isacharparenleft}{\kern0pt}replicate\ n\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ homogeneity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}homogeneity\ m\ {\isasymequiv}\isanewline
-\ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ n\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ {\isacharparenleft}{\kern0pt}times\ n\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\isadelimtheory
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-\end{isabellebody}%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Indep_Of_Alt.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Indep_Of_Alt.tex
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-\begin{isabellebody}%
-\setisabellecontext{Indep{\isacharunderscore}{\kern0pt}Of{\isacharunderscore}{\kern0pt}Alt}%
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-\isadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Indep{\isacharunderscore}{\kern0pt}Of{\isacharunderscore}{\kern0pt}Alt\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
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-\endisatagtheory
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-\isacommand{definition}\isamarkupfalse%
-\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ A\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
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-\isacommand{end}\isamarkupfalse%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Loop_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Loop_Composition.tex
deleted file mode 100644
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-\setisabellecontext{Loop{\isacharunderscore}{\kern0pt}Composition}%
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-\isamarkupsection{Loop Composition%
-}
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-\ Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\begin{isamarkuptext}%
-The loop composition uses the same module in sequence,
-combined with a termination condition, until either
-  (1) the termination condition is met or
-  (2) no new decisions are made (i.e., a fixed point is reached).%
-\end{isamarkuptext}\isamarkuptrue%
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}term{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}inf{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\isanewline
-\ \ \ \ \ \ \ \ measure\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ P\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ bool\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}f\ A\ p\ rs\ fa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fa{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isacharunderscore}{\kern0pt}cases{\isadigit{5}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ P\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ acca\ Aa\ pa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ t\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ta\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acca\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ Aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ pa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ma\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}r{\isachardot}{\kern0pt}\ wf\ r\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ p\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper\ wf{\isacharunderscore}{\kern0pt}measure\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}r\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ Ex\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}ra{\isachardot}{\kern0pt}\ {\isasymforall}f\ A\ rs\ pa\ fa{\isachardot}{\kern0pt}\ {\isasymexists}ra\ pb\ rb\ pc\ pd\ fb\ Aa\ rsa\ fc\ pe{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ wf\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ pa\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wf\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}ra{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}pb{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ wf\ rb\ {\isasymand}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pc{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rb\ {\isasymand}\ wf\ rb\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pd{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}defer\ fb\ {\isacharparenleft}{\kern0pt}Aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharparenright}{\kern0pt}\ Aa\ rsa\ {\isasymsubset}\ defer\ fb\ Aa\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ pe\ {\isacharparenleft}{\kern0pt}fb\ Aa\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ fb{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ rr\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}wf\ rr\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\ p\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rr{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\ wf{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub t\ {\isasymequiv}\ loop{\isacharunderscore}{\kern0pt}composition\ m\ t{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ {\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ profile\isanewline
-\ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ m{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\ m{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ eq{\isacharunderscore}{\kern0pt}iff\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
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-\endisadelimproof
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Maximum_Aggregator.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Maximum_Aggregator.tex
deleted file mode 100644
index cd1cf1e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Maximum_Aggregator.tex
+++ /dev/null
@@ -1,343 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Maximum{\isacharunderscore}{\kern0pt}Aggregator}%
-%
-\isadelimdocument
-\isanewline
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-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Maximum Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Maximum{\isacharunderscore}{\kern0pt}Aggregator\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The max(imum) aggregator takes two partitions of an alternative set A as
-input. It returns a partition where every alternative receives the maximum
-result of the two input partitions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}l\ {\isasymin}\ A{\isachardot}{\kern0pt}\ l\ {\isasymnotin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ set{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Minimax_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Minimax_Rule.tex
deleted file mode 100644
index 75dc3d2..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Minimax_Rule.tex
+++ /dev/null
@@ -1,160 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Minimax{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Minimax Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Minimax{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Minimax voting rule. It elects the alternatives with the highest
-Minimax score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ minimax\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ minimax{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ minimax{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Facts.tex
deleted file mode 100644
index 7f79b35..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Facts.tex
+++ /dev/null
@@ -1,923 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Monotonicity{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Monotonicity{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Monotonicity{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}dl{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ sizes{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}length\ p\ {\isacharequal}{\kern0pt}\ length\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}\ insertCI\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ sizes\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}QtoP{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymge}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}pq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ this\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ asm{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}PtoQ{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}pq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ q\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ DiffI\ above{\isacharunderscore}{\kern0pt}pq\ above{\isacharunderscore}{\kern0pt}QtoP\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ sizes\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymnoteq}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ antisym\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ y\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymnotin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ less\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ q\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
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-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Properties.tex
deleted file mode 100644
index d3eb734..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Properties.tex
+++ /dev/null
@@ -1,108 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Monotonicity{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Monotonicity{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Result{\isacharunderscore}{\kern0pt}Properties\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ p\ {\isasymor}\ elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ {\isasymor}\ defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Monotonicity_Properties.thy%:%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Rules.tex
deleted file mode 100644
index b5078d4..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Rules.tex
+++ /dev/null
@@ -1,2710 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Monotonicity{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Monotonicity{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Monotonicity{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Properties{\isacharslash}{\kern0pt}Weak{\isacharunderscore}{\kern0pt}Monotonicity{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Result{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \ \ \ \ \ \ \ \ Monotonicity{\isacharunderscore}{\kern0pt}Facts\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mem{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ empty{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ insertCI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defers{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}subset\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}Diff\ insertCI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}a\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}profs{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n\ lifted{\isacharunderscore}{\kern0pt}x\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{0}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffI\ lifted{\isacharunderscore}{\kern0pt}x\ compatible\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ compatible\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{1}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}x\ defer{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymlongrightarrow}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ modules\ profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq\ mono{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ modules\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ modules\ profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}lengths{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ mono{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ eql{\isacharunderscore}{\kern0pt}lengths\ mono{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ leI\ not{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}zero\ nth{\isacharunderscore}{\kern0pt}equalityI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m\ mono{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ less{\isacharunderscore}{\kern0pt}irrefl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubset}\ defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ psubsetI\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ q\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ monotone{\isacharunderscore}{\kern0pt}m\ subsetCE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ less{\isachardot}{\kern0pt}hyps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}lifted\ A\ p\ q\ a\ {\isasymand}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}acc\ profile\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\isanewline
-\ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module{\isachardot}{\kern0pt}simps\ monotone{\isacharunderscore}{\kern0pt}m\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}m\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ dl{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}w\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p\ lifted{\isacharunderscore}{\kern0pt}w\ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Nanson_Baldwin_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Nanson_Baldwin_Rule.tex
deleted file mode 100644
index f13ef1c..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Nanson_Baldwin_Rule.tex
+++ /dev/null
@@ -1,103 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Nanson-Baldwin Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Nanson-Baldwin voting rule. It excludes alternatives with the
-lowest Borda score from the set of possible winners and then adjusts the
-Borda score to the new (remaining) set of still eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pairwise_Majority_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pairwise_Majority_Rule.tex
deleted file mode 100644
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--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pairwise_Majority_Rule.tex
+++ /dev/null
@@ -1,195 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Pairwise Majority Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the pairwise majority rule, a voting rule that implements the
-Condorcet criterion, i.e., it elects the Condorcet winner if it exists,
-otherwise a tie remains between all alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ condorcet\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}condorcet{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ iterelect\ condorcet{\isacharprime}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ condorcet{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Parallel_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Parallel_Composition.tex
deleted file mode 100644
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--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Parallel_Composition.tex
+++ /dev/null
@@ -1,214 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Parallel{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Parallel Composition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
-%
-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The parallel composition composes a new electoral module from
-two electoral modules combined with an aggregator.
-Therein, the two modules each make a decision and the aggregator combines
-them to a single (aggregated) result.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ agg\ A\ p\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub a\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ a{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ agg{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
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-\endisadelimproof
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-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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-%
-\begin{isabellebody}%
-\setisabellecontext{Pass{\isacharunderscore}{\kern0pt}Module}%
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-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Pass Module%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pass{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according pass module
-defers the lexicographically first n alternatives (from A) and rejects
-the rest. It is primarily used as counterpart to the drop module in a
-parallel composition in order to segment the alternatives into two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ not{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
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-\endisatagproof
-{\isafoldproof}%
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-\isadelimproof
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-\end{isabellebody}%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Plurality_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Plurality_Module.tex
deleted file mode 100644
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--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Plurality_Module.tex
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-%
-\begin{isabellebody}%
-\setisabellecontext{Plurality{\isacharunderscore}{\kern0pt}Module}%
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-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Plurality Module%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Plurality{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
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-\begin{isamarkuptext}%
-The plurality module implements the plurality voting rule.
-The plurality rule elects all modules with the maximum amount of top
-preferences among all alternatives, and rejects all the other alternatives.
-It is electing and induces the classical plurality (voting) rule
-from social-choice theory.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ plurality\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}plurality\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasyminter}\ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasymunion}\ reject\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint\ unity\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimML
-\isanewline
-%
-\endisadelimML
-%
-\isatagML
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ w\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ x\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isachardoublequote}{\kern0pt}Seq{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ y\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardot}{\kern0pt}plurality{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ z\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardot}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}def{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}description\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ def\ {\isacharequal}{\kern0pt}\ hd\ z\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}rhs\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ def\isanewline
-\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}funs\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}consts\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ not\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ fun{\isacharunderscore}{\kern0pt}type\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}List{\isachardot}{\kern0pt}last\ def{\isacharunderscore}{\kern0pt}funs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ a\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}lhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ {\isacartoucheclose}%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-\isanewline
-%
-\endisadelimML
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Preference_Relation.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Preference_Relation.tex
deleted file mode 100644
index a6f4bdc..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Preference_Relation.tex
+++ /dev/null
@@ -1,4290 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Preference{\isacharunderscore}{\kern0pt}Relation}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Social-Choice Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Preference Relation%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The very core of the composable modules voting framework: types and
-functions, derivations, lemmata, operations on preference relations, etc.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ rel{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}preferred{\isacharunderscore}{\kern0pt}than\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasympreceq}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Ranking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ rank\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rank\ r\ x\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rank{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ refl{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rank\ r\ x\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ FieldI{\isadigit{2}}\ refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin\ finite{\isacharunderscore}{\kern0pt}Field{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ FieldI{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ rank{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Preference%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ limited\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limited\ A\ r\ {\isasymequiv}\ r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limitedI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ limited\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}dest{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y{\isacharsemicolon}{\kern0pt}\ limited\ A\ r\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ limit\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ connex\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}connex\ A\ r\ {\isasymequiv}\ limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}s\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub r\ x\ {\isasymor}\ x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex\ connex{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ antisymmetric{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ antisymmetric\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ transitive{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ z\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{5}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}z\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{4}}\ transE\ transitive\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}let\ s\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ C\ r\ {\isacharequal}{\kern0pt}\ limit\ C\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}r\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subrelI\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}singleton{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ singletonI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a\ {\isasymand}\ y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mono\ above{\isacharunderscore}{\kern0pt}def\ assms\ transE\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ above\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ singletonI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b\ {\isasymlongleftrightarrow}\ b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B\ {\isasymand}\ b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ assms\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ \isakeyword{where}\ n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ gr{\isadigit{0}}{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}Suc\ le{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ A\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isadigit{0}}{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ Suc\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ B\ \isakeyword{where}\ B{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ B\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isasymand}\ B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}inject\ add{\isacharunderscore}{\kern0pt}Suc\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\ finite{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}single\ subset{\isacharunderscore}{\kern0pt}insertI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}b{\isasymin}B{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isachardot}{\kern0pt}IH\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ finite{\isacharunderscore}{\kern0pt}subset\ leD\ lessI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ a\ aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ aa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ connex\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\ b\ {\isacharequal}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{6}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ A\ Aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ Aa\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ Aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ aa\ {\isasymin}\ B\ {\isasymand}\ a\ {\isasymin}\ B{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{5}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}B{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ above{\isacharunderscore}{\kern0pt}def\ f{\isadigit{4}}\ ff{\isadigit{1}}\ ff{\isadigit{2}}\ CollectI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Product{\isacharunderscore}{\kern0pt}Type{\isachardot}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}case{\isacharunderscore}{\kern0pt}prodD\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ above{\isacharunderscore}{\kern0pt}def\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subsetI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ DiffE\ a\ asm\ b\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ singletonI\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ Aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ Aa\ {\isasymsubseteq}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ Aa\ {\isacharparenleft}{\kern0pt}limit\ Aa\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ g{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ total{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{1}}\ B\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ total{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa\ {\isasymnotin}\ A\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ B\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ B\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\ B\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ asm\ b\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ singletonI\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ g{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ transE\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\ asm\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}connex\ Diff{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ a\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ CollectD\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ above{\isacharunderscore}{\kern0pt}trans\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ empty{\isacharunderscore}{\kern0pt}Diff\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}if\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ b\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ singletonI\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}dest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}exist{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex\ \isacommand{obtain}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub r\ y\ {\isasymand}\ y\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ transE\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ antisymD\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ DiffD{\isadigit{2}}\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\ singletonI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ ex{\isacharunderscore}{\kern0pt}lifted\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\ rest{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}r\ rest{\isacharunderscore}{\kern0pt}eq\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}r\ trans{\isacharunderscore}{\kern0pt}s\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ refl{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymsubseteq}\ above\ s\ x\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymnotin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ z\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ z{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ z\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ z\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ y\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ member{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ remove{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ {\isasymor}\isanewline
-\ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ subset\ \isacommand{have}\isamarkupfalse%
-\ temp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ subset\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ keep{\isacharunderscore}{\kern0pt}lift{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ u\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ x\ {\isasympreceq}\isactrlsub u\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ temp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}worse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ connex\ A\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymin}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymin}\ A\ {\isasymlongrightarrow}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Ball{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}raw\ connex{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ limit{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ aa\ {\isasymnotin}\ A\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ limit{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ aa\ A\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ limit\ A\ r\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}s{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{2}}\ limit{\isadigit{1}}\ connex{\isadigit{2}}\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ connex{\isadigit{2}}\ limit{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ connex\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ limited{\isacharunderscore}{\kern0pt}dest\ subrelI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ notInA\ subset\ subset{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}insert{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ change\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}above\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymlongrightarrow}\ aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ singletonD\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}connex\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ {\isasymnot}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ a\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ limited\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasympreceq}\isactrlsub r\ a\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ connex\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ limited{\isacharunderscore}{\kern0pt}dest\ insertCI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}x\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mk{\isacharunderscore}{\kern0pt}disjoint{\isacharunderscore}{\kern0pt}insert\ insertE\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ asm{\isadigit{1}}\ assms\ calculation\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ member{\isacharunderscore}{\kern0pt}remove\ remove{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ lifted{\isacharunderscore}{\kern0pt}a\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}others{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ above\ s\ y\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{2}}\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\ singleton{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ y\ \isakeyword{where}\ y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b{\isachardot}{\kern0pt}\ above\ s\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ b\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ y\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Profile.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Profile.tex
deleted file mode 100644
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--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Profile.tex
+++ /dev/null
@@ -1,2959 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Profile}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Preference Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Profile\isanewline
-\ \ \isakeyword{imports}\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Preference profiles denote the decisions made by the individual voters on
-the eligible alternatives. They are represented in the form of one preference
-relation (e.g., selected on a ballot) per voter, collectively captured in a
-list of such preference relations.
-Unlike a the common preference profiles in the social-choice sense, the
-profiles described here considers only the (sub-)set of alternatives that are
-received.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Profile\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isacharparenright}{\kern0pt}\ list{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymforall}b\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ all{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}all{\isacharunderscore}{\kern0pt}nth\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymequiv}\ finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Preference Counts and Comparisons%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}above\ p\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ a{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}equiv{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}snoc\ a\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ base{\isacharunderscore}{\kern0pt}case{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}length\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharless}{\kern0pt}length\ p{\isachardot}{\kern0pt}\ p{\isacharbang}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}Cons{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}above\ a\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ then\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}\ else\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}if{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnoteq}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xaa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnoteq}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ insertI{\isadigit{1}}\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymnoteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xb\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xb\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ above\ a\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xb\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xb\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ above\ a\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xaa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}n{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ f{\isadigit{2}}\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}aa{\isacharhash}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}aa{\isacharhash}{\kern0pt}p{\isacharparenright}{\kern0pt}\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct\isanewline
-\ \ \ \ \ \ \ \ \ \ base{\isacharunderscore}{\kern0pt}case\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ y\ {\isasympreceq}\isactrlsub p\ x\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}equiv{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}snoc\ a\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ base{\isacharunderscore}{\kern0pt}case{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}length\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharless}{\kern0pt}length\ p{\isachardot}{\kern0pt}\ p{\isacharbang}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}Cons{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isacharparenright}{\kern0pt}\ then\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}\ else\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}if{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnoteq}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}n{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ f{\isadigit{2}}\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct\isanewline
-\ \ \ \ \ \ \ \ \ \ base{\isacharunderscore}{\kern0pt}case\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}\ f\ x\ {\isacharbar}{\kern0pt}\ x\ {\isachardot}{\kern0pt}\ x\ {\isasymin}\ S\ {\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharbackquote}{\kern0pt}\ S{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}\ y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ neq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ profile{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ connex\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ antisym\ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisymD\ neq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ length\ p\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{assumes}\ p{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ prof\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ diff{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ neq{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}count\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}p\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}\ x\ y{\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}p\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}\ x\ y{\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub p\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}constant{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ wins\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}wins\ x\ p\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}wins\ a\ p\ b{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ b\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ w\ p\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Winner%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharunderscore}{\kern0pt}c{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ c{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}w\ {\isacharequal}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assumption{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ c{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assumption\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ c{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}against{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ assumption\isanewline
-\ \ \ \ \ \ \ \ \ \ singletonD\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}xa\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}wins\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ C\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ C\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ B\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ length{\isacharunderscore}{\kern0pt}map\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ subset\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}length\ p\ {\isacharequal}{\kern0pt}\ length\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ length\ p\ {\isacharequal}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ length\ p\ {\isacharequal}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}ok{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ i{\isacharunderscore}{\kern0pt}ok\ lifted\ profile{\isacharunderscore}{\kern0pt}def\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ notInA\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ length{\isacharunderscore}{\kern0pt}map\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ profile\ A\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}length\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ A\ {\isacharquery}{\kern0pt}p\ {\isacharquery}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ inA\ lifted\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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-\begin{isabellebody}%
-\setisabellecontext{Result}%
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-\isadelimdocument
-\isanewline
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-\endisadelimdocument
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-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Electoral Result%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\begin{isamarkuptext}%
-An electoral result is the principal result type of the composable modules
-voting framework, as it is a generalization of the set of winning alternatives
-from social choice functions. Electoral results are selections of the received
-(possibly empty) set of alternatives into the three disjoint groups of elected,
-rejected and deferred alternatives.
-Any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Result\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}%
-\isadelimdocument
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-\endisadelimdocument
-%
-\isatagdocument
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-\isamarkupsubsection{Auxiliary Functions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ disjoint{\isadigit{3}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ result\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ snd\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}elec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}elec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ card\ e\ {\isacharplus}{\kern0pt}\ card\ r\ {\isacharplus}{\kern0pt}\ card\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disj\ set{\isacharunderscore}{\kern0pt}partit\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ finite{\isacharunderscore}{\kern0pt}Un\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}right{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
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-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Facts.tex
deleted file mode 100644
index 9b427d4..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Facts.tex
+++ /dev/null
@@ -1,2293 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Revision{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI\ conjI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ max\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}max\ {\isacharequal}{\kern0pt}\ Max{\isacharparenleft}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ p\ {\isacharbackquote}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ max\ {\isasymand}\ a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Max{\isacharunderscore}{\kern0pt}in\ assm{\isadigit{0}}\ empty{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}image\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}imageI\ imageE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ max\ assm{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem\ assm{\isadigit{1}}\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ profile\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ g{\isadigit{0}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ n\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\ top{\isacharunderscore}{\kern0pt}greatest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ fin{\isacharunderscore}{\kern0pt}A\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}all{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ leD\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ card{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{0}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}pos{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isasymnot}\ p\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ p\ {\isacharparenleft}{\kern0pt}aa\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}n{\isachardot}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}\ {\isasymor}\ n\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ Aa{\isachardot}{\kern0pt}\ {\isasymnot}\ connex\ Aa\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ Aa\ {\isasymor}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ finite{\isacharunderscore}{\kern0pt}A\ order\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}pos{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}pos\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ w\ \isakeyword{where}\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit\ above{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}A\ finite{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ above{\isacharunderscore}{\kern0pt}finite\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}singletonE\ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ singletonD\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}unique\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ finA\ profA\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ {\isacharparenleft}{\kern0pt}b\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ Suc{\isacharunderscore}{\kern0pt}leI\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ a{\isacharunderscore}{\kern0pt}above\ card{\isacharunderscore}{\kern0pt}UNIV{\isacharunderscore}{\kern0pt}bool\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ empty{\isacharunderscore}{\kern0pt}iff\ finA\isanewline
-\ \ \ \ \ \ \ \ \ \ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}iff\ limitA{\isacharunderscore}{\kern0pt}order\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \ \ \ \ UNIV{\isacharunderscore}{\kern0pt}bool\ nat{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ finA{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}idemp\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ numeral{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}iff\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\ top{\isacharunderscore}{\kern0pt}greatest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ in\ {\isacharparenleft}{\kern0pt}c\ {\isasympreceq}\isactrlsub q\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b\ Diff{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{2}}\ subset{\isacharunderscore}{\kern0pt}UNIV\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ b{\isacharunderscore}{\kern0pt}best\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ order\ insert{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b\ above{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isachardot}{\kern0pt}\ c\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\ b{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ above{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ finA\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}commute\ numeral{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ finA\ finite{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans\ numeral{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff\ order{\isacharunderscore}{\kern0pt}refl\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rejects\ {\isadigit{0}}\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}pos{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ UNIV\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isasymnot}\ p\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ p\ {\isacharparenleft}{\kern0pt}aa\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex\ subset{\isacharunderscore}{\kern0pt}UNIV\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharunderscore}{\kern0pt}drop{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}pass{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ Aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ connex\ Aa\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ Aa\ {\isasymor}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ bot{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ emptyE\ finite{\isacharunderscore}{\kern0pt}A\ order\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ f{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}two{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharunderscore}{\kern0pt}drop{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}pass{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isadigit{0}}\ x{\isadigit{1}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}v{\isadigit{2}}\ v{\isadigit{3}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ v{\isadigit{2}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isacharparenleft}{\kern0pt}n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{2}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rej{\isacharunderscore}{\kern0pt}drop{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}pass\ defers{\isacharunderscore}{\kern0pt}def\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Properties.tex
deleted file mode 100644
index 3f1f97b..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Properties.tex
+++ /dev/null
@@ -1,350 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ one{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}right\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing\ electing{\isacharunderscore}{\kern0pt}def\ inf{\isachardot}{\kern0pt}orderE\isanewline
-\ \ \ \ \ \ \ \ one{\isacharunderscore}{\kern0pt}alt\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ decrementing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decrementing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymforall}\ A\ p\ {\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ elects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defers\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defers\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ rejects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rejects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ eliminates\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}eliminates\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ leftover{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubset}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminates{\isacharunderscore}{\kern0pt}def\ eliminating\ eq{\isacharunderscore}{\kern0pt}iff\ leftover{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}zero\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rejecting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejecting\isanewline
-\ \ \ \ \ \ \ \ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminating\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\ defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Rules.tex
deleted file mode 100644
index 77c129c..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Rules.tex
+++ /dev/null
@@ -1,2466 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Result{\isacharunderscore}{\kern0pt}Facts\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}empty\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ electing\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymor}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}empty\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ inf{\isachardot}{\kern0pt}absorb{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AA\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}block{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Profile\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrsa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAa\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAd\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAb\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isadigit{3}}{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ AAc\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\ Diff{\isacharunderscore}{\kern0pt}empty\ elector{\isachardot}{\kern0pt}simps\ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}block\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\ f{\isadigit{3}}\ f{\isadigit{1}}\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}m{\isadigit{1}}\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m{\isadigit{1}}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ m{\isadigit{1}}\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ zero{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ conservative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ agg{\isacharunderscore}{\kern0pt}a\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ c\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ c\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}elect\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ electing{\isacharunderscore}{\kern0pt}def\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ no{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ no{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}disjoint\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect\ assms\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ pos{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ m{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ subsetI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ defers\ n\ f\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymor}\ infinite\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ n\ {\isasymge}\ x\ {\isasymand}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\isanewline
-\ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ term{\isacharunderscore}{\kern0pt}satisfied\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}small{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ i\ \isakeyword{where}\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ f{\isacharunderscore}{\kern0pt}prof{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ enough{\isacharunderscore}{\kern0pt}leftover\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ i\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ step{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ x\ {\isasymor}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}le\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharless}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subset\ step{\isacharunderscore}{\kern0pt}profile\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}new{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymlongrightarrow}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isachardot}{\kern0pt}hyps\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ t\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough\ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ linorder{\isacharunderscore}{\kern0pt}neqE{\isacharunderscore}{\kern0pt}nat\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ less{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}alternatives\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ card\ A\ {\isasymge}\ n\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ eliminates{\isacharunderscore}{\kern0pt}def\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ calculation\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ disj{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rejects{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{2}}\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}nat\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ module\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ reject\ m\ A\ p{\isacharcomma}{\kern0pt}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ module\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ case{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ case{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}comp\ prof{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Revision_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Revision_Composition.tex
deleted file mode 100644
index 6d18707..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Revision_Composition.tex
+++ /dev/null
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-%
-\begin{isabellebody}%
-\setisabellecontext{Revision{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
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-\isamarkupsection{Revision Composition%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
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-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Revision{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-A revised electoral module rejects all originally rejected or deferred
-alternatives, and defers the originally elected alternatives.
-It does not elect any alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ revision{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ rev\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymdown}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m{\isasymdown}\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ revision{\isacharunderscore}{\kern0pt}composition\ m{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ unity\ disjoint\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
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-\endisadelimproof
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-\isadelimdocument
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-\endisadelimdocument
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-\isatagdocument
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-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
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-\endisatagdocument
-{\isafolddocument}%
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-\isadelimdocument
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-\endisadelimdocument
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
-\isacommand{end}\isamarkupfalse%
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-\endisatagtheory
-{\isafoldtheory}%
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-\isadelimtheory
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-\endisadelimtheory
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-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Compositional_Structures/Revision_Composition.thy%:%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Schwartz_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Schwartz_Rule.tex
deleted file mode 100644
index 638e0c1..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Schwartz_Rule.tex
+++ /dev/null
@@ -1,107 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Schwartz{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Schwartz Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Schwartz{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Schwartz voting rule. Confusingly, it is sometimes also referred as
-Nanson's rule. The Schwartz rule proceeds as in the classic Nanson's rule, but
-excludes alternatives with a Borda score that is strictly less than the
-average Borda score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ schwartz{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}schwartz{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Schwartz_Rule.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Composition.tex
deleted file mode 100644
index 5a67aac..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Composition.tex
+++ /dev/null
@@ -1,1373 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The sequential composition creates a new electoral module from
-two electoral modules. In a sequential composition, the second
-electoral module makes decisions over alternatives deferred by
-the first electoral module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sequential{\isacharunderscore}{\kern0pt}composition\ m\ n\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ sequence\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymtriangleright}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymtriangleright}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \ \ \ \ \ \ rev{\isacharunderscore}{\kern0pt}subsetD\ result{\isacharunderscore}{\kern0pt}disj\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}neutr{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ inf{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}iff\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}commute\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ inf{\isachardot}{\kern0pt}commute\ inf{\isachardot}{\kern0pt}orderE\ inf{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}absorb\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ distrib{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ sup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}prof\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ reject\ m\ A\ p\ {\isasymunion}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ module{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ r\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ empty{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ \ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}lower{\isadigit{1}}\ Un{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ empty{\isacharunderscore}{\kern0pt}defer\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assm\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ finite\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ singletonI\ subsetCE\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ defers{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ rej\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ res{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ elect\ n\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card\ def\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\isanewline
-\ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def\ def{\isacharunderscore}{\kern0pt}card\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ res{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ singletonI\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ elect\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}commute\ fst{\isacharunderscore}{\kern0pt}conv\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ in{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimML
-\isanewline
-%
-\endisadelimML
-%
-\isatagML
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ w\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ x\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isachardoublequote}{\kern0pt}Seq{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ y\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}sequential{\isacharunderscore}{\kern0pt}composition{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ z\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}sequential{\isacharunderscore}{\kern0pt}composition{\isacharunderscore}{\kern0pt}def{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}description\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ def\ {\isacharequal}{\kern0pt}\ hd\ z\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}rhs\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ def\isanewline
-\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}funs\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}consts\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ not\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ fun{\isacharunderscore}{\kern0pt}type\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}List{\isachardot}{\kern0pt}last\ def{\isacharunderscore}{\kern0pt}funs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ a\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}lhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ {\isacartoucheclose}\isanewline
-\isanewline
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\ val\ facts\ {\isacharequal}{\kern0pt}\ Global{\isacharunderscore}{\kern0pt}Theory{\isachardot}{\kern0pt}facts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ \isanewline
-\ \ \ \ \ val\ filtered\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ true{\isacharparenright}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacartoucheclose}\isanewline
-\isanewline
-\isacommand{ML{\isacharunderscore}{\kern0pt}val}\isamarkupfalse%
-\ {\isacartoucheopen}val\ seq{\isacharunderscore}{\kern0pt}consts\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}Context{\isachardot}{\kern0pt}theory{\isacharunderscore}{\kern0pt}name\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ names\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}description\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}snd{\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ seq{\isacharunderscore}{\kern0pt}consts\isanewline
-\ \ \ \ val\ seq{\isacharunderscore}{\kern0pt}defs\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}def{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ names\ \isanewline
-\ \ \ \ val\ strip{\isacharunderscore}{\kern0pt}fun\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}substring\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}size\ x{\isacharparenright}{\kern0pt}{\isacharminus}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ seq{\isacharunderscore}{\kern0pt}defs{\isacharunderscore}{\kern0pt}stripped\ {\isacharequal}{\kern0pt}\ map\ strip{\isacharunderscore}{\kern0pt}fun\ seq{\isacharunderscore}{\kern0pt}defs\isanewline
-\isanewline
-\ \ \ \ fun\ member{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}item{\isacharcomma}{\kern0pt}\ list{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ List{\isachardot}{\kern0pt}exists\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ item{\isacharparenright}{\kern0pt}\ list\isanewline
-\ \ \ \ val\ filter{\isacharunderscore}{\kern0pt}fun\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ member{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ seq{\isacharunderscore}{\kern0pt}defs{\isacharunderscore}{\kern0pt}stripped{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ constants\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ filtered{\isacharunderscore}{\kern0pt}constants\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}filter{\isacharunderscore}{\kern0pt}fun{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ fst\ constants{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ reduce{\isacharunderscore}{\kern0pt}noise\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ not\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}graph{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ orelse\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}sumC{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ orelse\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}rel{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\isanewline
-\ \ \ \ val\ actual{\isacharunderscore}{\kern0pt}constants\ {\isacharequal}{\kern0pt}\ filter\ reduce{\isacharunderscore}{\kern0pt}noise\ filtered{\isacharunderscore}{\kern0pt}constants\isanewline
-\isanewline
-\ \ \ \ val\ constants\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ filter{\isacharunderscore}{\kern0pt}fun{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ member{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharcomma}{\kern0pt}\ actual{\isacharunderscore}{\kern0pt}constants{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ sign\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}filter\ filter{\isacharunderscore}{\kern0pt}fun{\isadigit{2}}\ constants{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\ \ \ \ val\ x\ {\isacharequal}{\kern0pt}\ hd\ sign\isanewline
-\isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}string\ {\isacharequal}{\kern0pt}\ Syntax{\isachardot}{\kern0pt}string{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}typ{\isacharunderscore}{\kern0pt}global\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ x\isanewline
-\isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}pretty\ {\isacharequal}{\kern0pt}\ Syntax{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}typ{\isacharunderscore}{\kern0pt}global\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ x\isanewline
-\isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}string\ {\isacharequal}{\kern0pt}\ Pretty{\isachardot}{\kern0pt}string{\isacharunderscore}{\kern0pt}of\ x{\isacharunderscore}{\kern0pt}pretty\isanewline
-\isanewline
-\ \ \ \ val\ y\ {\isacharequal}{\kern0pt}\ Logic{\isachardot}{\kern0pt}mk{\isacharunderscore}{\kern0pt}type\ x\isanewline
-\isanewline
-\ \ \ \ val\ n\ {\isacharequal}{\kern0pt}\ Term{\isachardot}{\kern0pt}size{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}typ\ x\isanewline
-\isanewline
-\ \ \ \ val\ pm\ {\isacharequal}{\kern0pt}\ fn\ Basic{\isacharunderscore}{\kern0pt}Term{\isachardot}{\kern0pt}Type\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ snd\ x\isanewline
-\isanewline
-\ \ \ \ fun\ collapse\ {\isacharparenleft}{\kern0pt}lst{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ list\ list{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ case\ lst\ of\isanewline
-\ \ \ \ \ \ \ \ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharbar}{\kern0pt}\ hd{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}tl\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ hd\ {\isacharat}{\kern0pt}\ collapse\ tl\isanewline
-\isanewline
-\ \ \ \ val\ a\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\isanewline
-\ \ \ \ \ \ \ \ fun\ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ {\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}\ Basic{\isacharunderscore}{\kern0pt}Term{\isachardot}{\kern0pt}typ{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}\ string\ {\isacharequal}{\kern0pt}\ case\ x\ of\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Type\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ String{\isachardot}{\kern0pt}concat\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbar}{\kern0pt}\ \ {\isacharunderscore}{\kern0pt}\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ in\isanewline
-\ \ \ \ \ \ \ \ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ x\isanewline
-\ \ \ \ \ \ end\isanewline
-\ \ \ \ \ \ \ \isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}string{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ a\ x\isanewline
-\isanewline
-\ \ \ \ val\ z\ {\isacharequal}{\kern0pt}\ Term{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}Bound\ y\isanewline
-\isanewline
-\isanewline
-\isanewline
-val\ a\ {\isacharequal}{\kern0pt}\ fn\ thy\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ thy{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ let\ fun\ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ {\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}\ Basic{\isacharunderscore}{\kern0pt}Term{\isachardot}{\kern0pt}typ{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}\ string\ {\isacharequal}{\kern0pt}\ case\ x\ of\isanewline
-Type\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ String{\isachardot}{\kern0pt}concat\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\isanewline
-{\isacharbar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\ in\ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ x\ end{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ thy{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ val\ b\ {\isacharequal}{\kern0pt}\ rev\ {\isacharparenleft}{\kern0pt}ListPair{\isachardot}{\kern0pt}zip\ {\isacharparenleft}{\kern0pt}a\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ {\isacartoucheclose}\isanewline
-\isanewline
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-\ \ \ \ val\ constants\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ sign{\isadigit{2}}\ {\isacharequal}{\kern0pt}rev\ \ {\isacharparenleft}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ \ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ constants\ {\isacharparenright}{\kern0pt}\ {\isacartoucheclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{ML{\isacharunderscore}{\kern0pt}val}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-val\ thms\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Global{\isacharunderscore}{\kern0pt}Theory{\isachardot}{\kern0pt}dest{\isacharunderscore}{\kern0pt}thm{\isacharunderscore}{\kern0pt}names\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-val\ thm\ {\isacharequal}{\kern0pt}\ Global{\isacharunderscore}{\kern0pt}Theory{\isachardot}{\kern0pt}get{\isacharunderscore}{\kern0pt}thm\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}rev\ thms{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-{\isacharparenleft}{\kern0pt}Syntax{\isachardot}{\kern0pt}string{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}global\ \isactrltheory {\isasymopen}Main{\isasymclose}\ {\isacharparenleft}{\kern0pt}Thm{\isachardot}{\kern0pt}prop{\isacharunderscore}{\kern0pt}of\ thm{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharbar}{\kern0pt}{\isachargreater}{\kern0pt}\ writeln\isanewline
-{\isacartoucheclose}%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-\ \isanewline
-%
-\endisadelimML
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Compositional_Structures/Sequential_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%74=31%:%
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Majority_Comparison.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Majority_Comparison.tex
deleted file mode 100644
index 471c3ab..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Majority_Comparison.tex
+++ /dev/null
@@ -1,1341 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Majority Comparison%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Revision{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Monotonicity{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Sequential majority comparison compares two alternatives by plurality voting.
-The loser gets rejected, and the winner is compared to the next alternative.
-This process is repeated until only a single alternative is left, which is
-then elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ smc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}smc\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isasymup}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ defer{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}plurality{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set\ list\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ reject{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ fin{\isacharunderscore}{\kern0pt}A\ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{(Weak) Monotonicity Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}monotone{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Termination_Condition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Termination_Condition.tex
deleted file mode 100644
index 5434a3b..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Termination_Condition.tex
+++ /dev/null
@@ -1,99 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Termination{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Termination Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Termination{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The termination condition is used in loops. It decides whether or not to
-terminate the loop after each iteration, depending on the current state
-of the loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Termination_Condition.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Weak_Monotonicity.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Weak_Monotonicity.tex
deleted file mode 100644
index 3a14e3e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Weak_Monotonicity.tex
+++ /dev/null
@@ -1,62 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Weak{\isacharunderscore}{\kern0pt}Monotonicity}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Weak{\isacharunderscore}{\kern0pt}Monotonicity\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Properties/Weak_Monotonicity.thy%:%
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\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/comment.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/comment.sty
deleted file mode 100644
index 658686f..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/comment.sty
+++ /dev/null
@@ -1,278 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Comment.sty   version 3.6, October 1999
-%
-% Purpose:
-% selectively in/exclude pieces of text: the user can define new
-% comment versions, and each is controlled separately.
-% Special comments can be defined where the user specifies the
-% action that is to be taken with each comment line.
-%
-% Author
-%    Victor Eijkhout
-%    Department of Computer Science
-%    University of Tennessee
-%    107 Ayres Hall
-%    Knoxville TN 37996
-%    USA
-%
-%    victor@eijkhout.net
-%
-% This program is free software; you can redistribute it and/or
-% modify it under the terms of the GNU General Public License
-% as published by the Free Software Foundation; either version 2
-% of the License, or (at your option) any later version.
-% 
-% This program is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-% GNU General Public License for more details.
-%
-% For a copy of the GNU General Public License, write to the 
-% Free Software Foundation, Inc.,
-% 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA,
-% or find it on the net, for instance at
-% http://www.gnu.org/copyleft/gpl.html
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This style can be used with plain TeX or LaTeX, and probably
-% most other packages too.
-%
-% Usage: all text included between
-%    \comment ... \endcomment
-% or \begin{comment} ... \end{comment}
-% is discarded. 
-%
-% The opening and closing commands should appear on a line
-% of their own. No starting spaces, nothing after it.
-% This environment should work with arbitrary amounts
-% of comment, and the comment can be arbitrary text.
-%
-% Other `comment' environments are defined by
-% and are selected/deselected with
-% \includecomment{versiona}
-% \excludecoment{versionb}
-%
-% These environments are used as
-% \versiona ... \endversiona
-% or \begin{versiona} ... \end{versiona}
-% with the opening and closing commands again on a line of 
-% their own.
-%
-% LaTeX users note: for an included comment, the
-% \begin and \end lines act as if they don't exist.
-% In particular, they don't imply grouping, so assignments 
-% &c are not local.
-%
-% Special comments are defined as
-% \specialcomment{name}{before commands}{after commands}
-% where the second and third arguments are executed before
-% and after each comment block. You can use this for global
-% formatting commands.
-% To keep definitions &c local, you can include \begingroup
-% in the `before commands' and \endgroup in the `after commands'.
-% ex:
-% \specialcomment{smalltt}
-%     {\begingroup\ttfamily\footnotesize}{\endgroup}
-% You do *not* have to do an additional
-% \includecomment{smalltt}
-% To remove 'smalltt' blocks, give \excludecomment{smalltt}
-% after the definition.
-%
-% Processing comments can apply processing to each line.
-% \processcomment{name}{each-line commands}%
-%    {before commands}{after commands}
-% By defining a control sequence 
-% \def\Thiscomment##1{...} in the before commands the user can
-% specify what is to be done with each comment line.
-% BUG this does not work quite yet BUG
-%
-% Trick for short in/exclude macros (such as \maybe{this snippet}):
-%\includecomment{cond}
-%\newcommand{\maybe}[1]{}
-%\begin{cond}
-%\renewcommand{\maybe}[1]{#1}
-%\end{cond}
-%
-% Basic approach of the implementation:
-% to comment something out, scoop up  every line in verbatim mode
-% as macro argument, then throw it away.
-% For inclusions, in LaTeX the block is written out to
-% a file \CommentCutFile (default "comment.cut"), which is
-% then included.
-% In plain TeX (and other formats) both the opening and
-% closing comands are defined as noop.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Changes in version 3.1
-% - updated author's address
-% - cleaned up some code
-% - trailing contents on \begin{env} line is always discarded
-%  even if you've done \includecomment{env}
-% - comments no longer define grouping!! you can even
-%   \includecomment{env}
-%   \begin{env}
-%   \begin{itemize}
-%   \end{env}
-%  Isn't that something ...
-% - included comments are written to file and input again.
-% Changes in 3.2
-% - \specialcomment brought up to date (thanks to Ivo Welch).
-% Changes in 3.3
-% - updated author's address again
-% - parametrised \CommentCutFile
-% Changes in 3.4
-% - added GNU public license
-% - added \processcomment, because Ivo's fix (above) brought an
-%   inconsistency to light.
-% Changes in 3.5
-% - corrected typo in header.
-% - changed author email
-% - corrected \specialcomment yet again.
-% - fixed excludecomment of an earlier defined environment.
-% Changes in 3.6
-% - The 'cut' file is now written more verbatim, using \meaning;
-%   some people reported having trouble with ISO latin 1, or umlaute.sty.
-% - removed some \newif statements.
-%   Has this suddenly become \outer again?
-%
-% Known bugs:
-% - excludecomment leads to one superfluous space
-% - processcomment leads to a superfluous line break
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\makeinnocent#1{\catcode`#1=12 }
-\def\csarg#1#2{\expandafter#1\csname#2\endcsname}
-\def\latexname{lplain}\def\latexename{LaTeX2e}
-\newwrite\CommentStream
-\def\CommentCutFile{comment.cut}
-
-\def\ProcessComment#1% start it all of
-   {\begingroup
-    \def\CurrentComment{#1}%
-    \let\do\makeinnocent \dospecials 
-    \makeinnocent\^^L% and whatever other special cases
-    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-%\def\ProcessCommentWithArg#1#2% to be used in \leveledcomment
-%   {\begingroup
-%    \def\CurrentComment{#1}%
-%    \let\do\makeinnocent \dospecials 
-%    \makeinnocent\^^L% and whatever other special cases
-%    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-{\catcode`\^^M=12 \endlinechar=-1 %
- \gdef\xComment#1^^M{%
-    \expandafter\ProcessCommentLine}
- \gdef\ProcessCommentLine#1^^M{\def\test{#1}
-      \csarg\ifx{End\CurrentComment Test}\test
-          \edef\next{\noexpand\EndOfComment{\CurrentComment}}%
-      \else \ThisComment{#1}\let\next\ProcessCommentLine
-      \fi \next}
-}
-
-\def\CSstringmeaning#1{\expandafter\CSgobblearrow\meaning#1}
-\def\CSstringcsnoescape#1{\expandafter\CSgobbleescape\string#1}
-{\escapechar-1
-\expandafter\expandafter\expandafter\gdef
-  \expandafter\expandafter\expandafter\CSgobblearrow
-    \expandafter\string\csname macro:->\endcsname{}
-}
-\def\CSgobbleescape#1{\ifnum`\\=`#1 \else #1\fi}
-\def\WriteCommentLine#1{\def\CStmp{#1}%
-    \immediate\write\CommentStream{\CSstringmeaning\CStmp}}
-
-% 3.1 change: in LaTeX and LaTeX2e prevent grouping
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-%%%%
-%%%% definitions for LaTeX
-%%%%
-\def\AfterIncludedComment
-   {\immediate\closeout\CommentStream
-    \input{\CommentCutFile}\relax
-    }%
-\def\TossComment{\immediate\closeout\CommentStream}
-\def\BeforeIncludedComment
-   {\immediate\openout\CommentStream=\CommentCutFile
-    \let\ThisComment\WriteCommentLine}
-\def\includecomment
- #1{\message{Include comment '#1'}%
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    % note: \AfterIncludedComment does \input, so #2 goes here!
-    \csarg\def{After#1Comment}{#2\AfterIncludedComment#3}%
-    \csarg\def{#1}{\BeforeIncludedComment\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\processcomment
- #1#2#3#4{\message{Lines-Processing comment '#1'}%
-    \csarg\def{After#1Comment}{#3\AfterIncludedComment#4}%
-    \csarg\def{#1}{\BeforeIncludedComment#2\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\def\leveledcomment
- #1#2{\message{Include comment '#1' up to level '#2'}%
-    %\csname #1IsLeveledCommenttrue\endcsname
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessCommentWithArg{#1}}%
-    \CommentEndDef{#1}}
-\else 
-%%%%
-%%%%plain TeX and other formats
-%%%%
-\def\includecomment
- #1{\message{Including comment '#1'}%
-    \csarg\def{#1}{}%
-    \csarg\def{end#1}{}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    \csarg\def{#1}{\def\ThisComment{}\def\AfterComment{#3}#2%
-           \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\fi
-
-%%%%
-%%%% general definition of skipped comment
-%%%%
-\def\excludecomment
- #1{\message{Excluding comment '#1'}%
-    \csarg\def{#1}{\let\AfterComment\relax
-           \def\ThisComment####1{}\ProcessComment{#1}}%
-    \csarg\let{After#1Comment}\TossComment
-    \CommentEndDef{#1}}
-
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-% latex & latex2e:
-\def\EndOfComment#1{\endgroup\end{#1}%
-    \csname After#1Comment\endcsname}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end\string\{#1\string\}}%
-    }}
-\else
-% plain & other
-\def\EndOfComment#1{\endgroup\AfterComment}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end#1}%
-    }}
-\fi
-
-\excludecomment{comment}
-
-\endinput
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelle.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelle.sty
deleted file mode 100644
index b6dcb3e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelle.sty
+++ /dev/null
@@ -1,267 +0,0 @@
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-\newcommand{\setisabellecontext}[1]{\def\isabellecontext{#1}}
-
-\newcommand{\isastyle}{\UNDEF}
-\newcommand{\isastylett}{\UNDEF}
-\newcommand{\isastyleminor}{\UNDEF}
-\newcommand{\isastyleminortt}{\UNDEF}
-\newcommand{\isastylescript}{\UNDEF}
-\newcommand{\isastyletext}{\normalsize\normalfont\rmfamily}
-\newcommand{\isastyletxt}{\normalfont\rmfamily}
-\newcommand{\isastylecmt}{\normalfont\rmfamily}
-
-\newcommand{\isaspacing}{%
-  \sfcode 42 1000 % .
-  \sfcode 63 1000 % ?
-  \sfcode 33 1000 % !
-  \sfcode 58 1000 % :
-  \sfcode 59 1000 % ;
-  \sfcode 44 1000 % ,
-}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
-\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
-\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
-\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastyle}{\par}
-
-\newenvironment{isabellebodytt}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastylett}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newenvironment{isabellett}
-{\begin{trivlist}\begin{isabellebodytt}\item\relax}
-{\end{isabellebodytt}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
-\newcommand{\isatt}[1]{\emph{\isaspacing\isastyleminortt #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{}
-\newcommand{\isadigit}[1]{#1}
-
-\newcommand{\isachardefaults}{%
-\def\isacharbell{\isamath{\bigbox}}%requires stmaryrd
-\chardef\isacharbang=`\!%
-\chardef\isachardoublequote=`\"%
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharhash=`\#%
-\chardef\isachardollar=`\$%
-\chardef\isacharpercent=`\%%
-\chardef\isacharampersand=`\&%
-\chardef\isacharprime=`\'%
-\chardef\isacharparenleft=`\(%
-\chardef\isacharparenright=`\)%
-\chardef\isacharasterisk=`\*%
-\chardef\isacharplus=`\+%
-\chardef\isacharcomma=`\,%
-\chardef\isacharminus=`\-%
-\chardef\isachardot=`\.%
-\chardef\isacharslash=`\/%
-\chardef\isacharcolon=`\:%
-\chardef\isacharsemicolon=`\;%
-\chardef\isacharless=`\<%
-\chardef\isacharequal=`\=%
-\chardef\isachargreater=`\>%
-\chardef\isacharquery=`\?%
-\chardef\isacharat=`\@%
-\chardef\isacharbrackleft=`\[%
-\chardef\isacharbackslash=`\\%
-\chardef\isacharbrackright=`\]%
-\chardef\isacharcircum=`\^%
-\chardef\isacharunderscore=`\_%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquote=`\`%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\chardef\isacharbraceleft=`\{%
-\chardef\isacharbar=`\|%
-\chardef\isacharbraceright=`\}%
-\chardef\isachartilde=`\~%
-\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
-\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
-\def\isacartoucheopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacartoucheclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-
-% keyword and section markup
-
-\newcommand{\isakeyword}[1]
-{\emph{\normalfont\bfseries\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isakeywordcontrol}[1]
-{\emph{\normalfont\bfseries\itshape\def\isacharunderscore{\isacharunderscorekeyword}#1\,}}
-
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupparagraph}[1]{\paragraph{#1}}
-\newcommand{\isamarkupsubparagraph}[1]{\subparagraph{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% styles
-
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestyledefault}{%
-\def\isastyle{\small\normalfont\ttfamily\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily\slshape}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily\slshape}%
-\isachardefaults%
-}
-\isabellestyledefault
-
-\newcommand{\isabellestylett}{%
-\def\isastyle{\small\normalfont\ttfamily}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily}%
-\isachardefaults%
-}
-
-\newcommand{\isabellestyleit}{%
-\def\isastyle{\small\normalfont\itshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\itshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\itshape}%
-\isachardefaults%
-\def\isacharunderscorekeyword{\mbox{-}}%
-\def\isacharbang{\isamath{!}}%
-\def\isachardoublequote{}%
-\def\isachardoublequoteopen{}%
-\def\isachardoublequoteclose{}%
-\def\isacharhash{\isamath{\#}}%
-\def\isachardollar{\isamath{\$}}%
-\def\isacharpercent{\isamath{\%}}%
-\def\isacharampersand{\isamath{\&}}%
-\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\def\isacharparenleft{\isamath{(}}%
-\def\isacharparenright{\isamath{)}}%
-\def\isacharasterisk{\isamath{*}}%
-\def\isacharplus{\isamath{+}}%
-\def\isacharcomma{\isamath{\mathord,}}%
-\def\isacharminus{\isamath{-}}%
-\def\isachardot{\isamath{\mathord.}}%
-\def\isacharslash{\isamath{/}}%
-\def\isacharcolon{\isamath{\mathord:}}%
-\def\isacharsemicolon{\isamath{\mathord;}}%
-\def\isacharless{\isamath{<}}%
-\def\isacharequal{\isamath{=}}%
-\def\isachargreater{\isamath{>}}%
-\def\isacharat{\isamath{@}}%
-\def\isacharbrackleft{\isamath{[}}%
-\def\isacharbackslash{\isamath{\backslash}}%
-\def\isacharbrackright{\isamath{]}}%
-\def\isacharunderscore{\mbox{-}}%
-\def\isacharbraceleft{\isamath{\{}}%
-\def\isacharbar{\isamath{\mid}}%
-\def\isacharbraceright{\isamath{\}}}%
-\def\isachartilde{\isamath{{}\sp{\sim}}}%
-\def\isacharbackquoteopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacharbackquoteclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-\newcommand{\isabellestyleliteral}{%
-\isabellestyleit%
-\def\isacharunderscore{\_}%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-}
-
-\newcommand{\isabellestyleliteralunderscore}{%
-\isabellestyleliteral%
-\def\isacharunderscore{\textunderscore}%
-\def\isacharunderscorekeyword{\textunderscore}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\def\isastyle{\small\normalfont\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\slshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\slshape}%
-}
-
-
-% cancel text
-
-\usepackage[normalem]{ulem}
-\newcommand{\isamarkupcancel}[1]{\isa{\xout{#1}}}
-
-
-% tagged regions
-
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isakeeptag{document}
-\isakeeptag{theory}
-\isakeeptag{proof}
-\isakeeptag{ML}
-\isakeeptag{visible}
-\isadroptag{invisible}
-\isakeeptag{important}
-\isakeeptag{unimportant}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabellesym.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabellesym.sty
deleted file mode 100644
index b0d076c..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabellesym.sty
+++ /dev/null
@@ -1,427 +0,0 @@
-%%
-%% definitions of standard Isabelle symbols
-%%
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}  %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}  %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}}  %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}  %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}}  %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}}  %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
-\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
-\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
-\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
-\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
-\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLleftarrow}{\isamath{\Lleftarrow}}  %requires amssymb
-\newcommand{\isasymRrightarrow}{\isamath{\Rrightarrow}}  %requires amssymb
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymllangle}{\isamath{\langle\mskip-5mu\langle}}
-\newcommand{\isasymrrangle}{\isamath{\rangle\mskip-5mu\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
-\newcommand{\isasymguillemotleft}{\isatext{\flqq}}  %requires babel
-\newcommand{\isasymguillemotright}{\isatext{\frqq}}  %requires babel
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymcircle}{\isamath{\ocircle}}  %requires wasysym
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymdiamondop}{\isamath{\diamond}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires stmaryrd
-\newcommand{\isasymsetminus}{\isamath{\setminus}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymbbar}{\isamath{[\mskip-1.5mu]}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymsqdot}{\isamath{\sbox\z@{$\centerdot$}\ht\z@=.33333\ht\z@\vcenter{\box\z@}}}  %requires amssymb
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\normalfont\rmfamily\copyright}}
-\newcommand{\isasymregistered}{\isatext{\normalfont\rmfamily\textregistered}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonequarter}{\isatext{\normalfont\rmfamily\textonequarter}}  %requires textcomp
-\newcommand{\isasymonehalf}{\isatext{\normalfont\rmfamily\textonehalf}}  %requires textcomp
-\newcommand{\isasymthreequarters}{\isatext{\normalfont\rmfamily\textthreequarters}}  %requires textcomp
-\newcommand{\isasymordfeminine}{\isatext{\normalfont\rmfamily\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\normalfont\rmfamily\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\normalfont\rmfamily\S}}
-\newcommand{\isasymparagraph}{\isatext{\normalfont\rmfamily\P}}
-\newcommand{\isasymexclamdown}{\isatext{\normalfont\rmfamily\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\normalfont\rmfamily\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\euro}}  %requires eurosym
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\normalfont\rmfamily\textdegree}}  %requires textcomp
-\newcommand{\isasymhyphen}{\isatext{\normalfont\rmfamily-}}
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
-\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
-\newcommand{\isasymopen}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}
-\newcommand{\isasymclose}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}
-\newcommand{\isasymhole}{\isatext{\normalfont\rmfamily\wasylozenge}}  %requires wasysym
-\newcommand{\isasymnewline}{\isatext{\fbox{$\hookleftarrow$}}}
-\newcommand{\isasymcomment}{\isatext{\isastylecmt---}}
-\newcommand{\isasymproof}{\isamath{\,\langle\mathit{proof}\rangle}}
-
-\newcommand{\isactrlmarker}{\isatext{\ding{48}}}  %requires pifont
-\newcommand{\isactrlassert}{\isakeywordcontrol{assert}}
-\newcommand{\isactrlcancel}{\isakeywordcontrol{cancel}}
-\newcommand{\isactrlbinding}{\isakeywordcontrol{binding}}
-\newcommand{\isactrlclass}{\isakeywordcontrol{class}}
-\newcommand{\isactrlclassUNDERSCOREsyntax}{\isakeywordcontrol{class{\isacharunderscore}syntax}}
-\newcommand{\isactrlcommandUNDERSCOREkeyword}{\isakeywordcontrol{command{\isacharunderscore}keyword}}
-\newcommand{\isactrlconst}{\isakeywordcontrol{const}}
-\newcommand{\isactrlconstUNDERSCOREabbrev}{\isakeywordcontrol{const{\isacharunderscore}abbrev}}
-\newcommand{\isactrlconstUNDERSCOREname}{\isakeywordcontrol{const{\isacharunderscore}name}}
-\newcommand{\isactrlconstUNDERSCOREsyntax}{\isakeywordcontrol{const{\isacharunderscore}syntax}}
-\newcommand{\isactrlcontext}{\isakeywordcontrol{context}}
-\newcommand{\isactrlcprop}{\isakeywordcontrol{cprop}}
-\newcommand{\isactrlcterm}{\isakeywordcontrol{cterm}}
-\newcommand{\isactrlctyp}{\isakeywordcontrol{ctyp}}
-\newcommand{\isactrldir}{\isakeywordcontrol{dir}}
-\newcommand{\isactrlfile}{\isakeywordcontrol{file}}
-\newcommand{\isactrlhere}{\isakeywordcontrol{here}}
-\newcommand{\isactrlkeyword}{\isakeywordcontrol{keyword}}
-\newcommand{\isactrllatex}{\isakeywordcontrol{latex}}
-\newcommand{\isactrllocale}{\isakeywordcontrol{locale}}
-\newcommand{\isactrlmakeUNDERSCOREstring}{\isakeywordcontrol{make{\isacharunderscore}string}}
-\newcommand{\isactrlmasterUNDERSCOREdir}{\isakeywordcontrol{master{\isacharunderscore}dir}}
-\newcommand{\isactrlmethod}{\isakeywordcontrol{method}}
-\newcommand{\isactrlnamedUNDERSCOREtheorems}{\isakeywordcontrol{named{\isacharunderscore}theorems}}
-\newcommand{\isactrlnonterminal}{\isakeywordcontrol{nonterminal}}
-\newcommand{\isactrloracleUNDERSCOREname}{\isakeywordcontrol{oracle{\isacharunderscore}name}}
-\newcommand{\isactrlpath}{\isakeywordcontrol{path}}
-\newcommand{\isactrlpathUNDERSCOREbinding}{\isakeywordcontrol{path{\isacharunderscore}binding}}
-\newcommand{\isactrlplugin}{\isakeywordcontrol{plugin}}
-\newcommand{\isactrlprint}{\isakeywordcontrol{print}}
-\newcommand{\isactrlprop}{\isakeywordcontrol{prop}}
-\newcommand{\isactrlscala}{\isakeywordcontrol{scala}}
-\newcommand{\isactrlscalaUNDERSCOREfunction}{\isakeywordcontrol{scala{\isacharunderscore}function}}
-\newcommand{\isactrlscalaUNDERSCOREmethod}{\isakeywordcontrol{scala{\isacharunderscore}method}}
-\newcommand{\isactrlscalaUNDERSCOREobject}{\isakeywordcontrol{scala{\isacharunderscore}object}}
-\newcommand{\isactrlscalaUNDERSCOREtype}{\isakeywordcontrol{scala{\isacharunderscore}type}}
-\newcommand{\isactrlsimproc}{\isakeywordcontrol{simproc}}
-\newcommand{\isactrlsort}{\isakeywordcontrol{sort}}
-\newcommand{\isactrlsyntaxUNDERSCOREconst}{\isakeywordcontrol{syntax{\isacharunderscore}const}}
-\newcommand{\isactrlsystemUNDERSCOREoption}{\isakeywordcontrol{system{\isacharunderscore}option}}
-\newcommand{\isactrlterm}{\isakeywordcontrol{term}}
-\newcommand{\isactrltheory}{\isakeywordcontrol{theory}}
-\newcommand{\isactrltheoryUNDERSCOREcontext}{\isakeywordcontrol{theory{\isacharunderscore}context}}
-\newcommand{\isactrltyp}{\isakeywordcontrol{typ}}
-\newcommand{\isactrltypeUNDERSCOREabbrev}{\isakeywordcontrol{type{\isacharunderscore}abbrev}}
-\newcommand{\isactrltypeUNDERSCOREname}{\isakeywordcontrol{type{\isacharunderscore}name}}
-\newcommand{\isactrltypeUNDERSCOREsyntax}{\isakeywordcontrol{type{\isacharunderscore}syntax}}
-\newcommand{\isactrlundefined}{\isakeywordcontrol{undefined}}
-
-\newcommand{\isactrlcode}{\isakeywordcontrol{code}}
-\newcommand{\isactrlcomputation}{\isakeywordcontrol{computation}}
-\newcommand{\isactrlcomputationUNDERSCOREconv}{\isakeywordcontrol{computation{\isacharunderscore}conv}}
-\newcommand{\isactrlcomputationUNDERSCOREcheck}{\isakeywordcontrol{computation{\isacharunderscore}check}}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelletags.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelletags.sty
deleted file mode 100644
index 8b13789..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelletags.sty
+++ /dev/null
@@ -1 +0,0 @@
-
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty
deleted file mode 100644
index 2bf5ccf..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty
+++ /dev/null
@@ -1,7 +0,0 @@
-%%
-%% default hyperref setup (both for pdf and dvi output)
-%%
-
-\usepackage{color}
-\definecolor{linkcolor}{rgb}{0,0,0.5}
-\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,urlcolor=linkcolor,pdfpagelabels]{hyperref}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty
deleted file mode 100644
index 93cae05..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty
+++ /dev/null
@@ -1,1202 +0,0 @@
-% rail.sty - style file to support railroad diagrams
-%
-% 09-Jul-90 L. Rooijakkers
-% 08-Oct-90 L. Rooijakkers	fixed centering bug when \rail@tmpc<0.
-% 07-Feb-91 L. Rooijakkers	added \railoptions command, indexing
-% 08-Feb-91 L. Rooijakkers	minor fixes
-% 28-Jun-94 K. Barthelmann	turned into LaTeX2e package
-% 08-Dec-96 K. Barthelmann	replaced \@writefile
-% 13-Dec-96 K. Barthelmann	cleanup
-% 22-Feb-98 K. Barthelmann	fixed catcodes of special characters
-% 18-Apr-98 K. Barthelmann	fixed \par handling
-% 19-May-98 J. Olsson		Added new macros to support arrow heads.
-% 26-Jul-98 K. Barthelmann	changed \par to output newlines
-% 02-May-11 M. Wenzel           default setup for Isabelle
-%
-% This style file needs to be used in conjunction with the 'rail'
-% program. Running LaTeX as 'latex file' produces file.rai, which should be
-% processed by Rail with 'rail file'. This produces file.rao, which will
-% be picked up by LaTeX on the next 'latex file' run.
-%
-% LaTeX will warn if there is no file.rao or it's out of date.
-%
-% The macros in this file thus consist of two parts: those that read and
-% write the .rai and .rao files, and those that do the actual formatting
-% of the railroad diagrams.
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{rail}[1998/05/19]
-
-% railroad diagram formatting parameters (user level)
-% all of these are copied into their internal versions by \railinit
-%
-% \railunit : \unitlength within railroad diagrams
-% \railextra : extra length at outside of diagram
-% \railboxheight : height of ovals and frames
-% \railboxskip : vertical space between lines
-% \railboxleft : space to the left of a box
-% \railboxright : space to the right of a box
-% \railovalspace : extra space around contents of oval
-% \railframespace : extra space around contents of frame
-% \railtextleft : space to the left of text
-% \railtextright : space to the right of text
-% \railtextup : space to lift text up
-% \railjoinsize : circle size of join/split arcs
-% \railjoinadjust : space to adjust join
-%
-% \railnamesep : separator between name and rule body
-
-\newlength\railunit
-\newlength\railextra
-\newlength\railboxheight
-\newlength\railboxskip
-\newlength\railboxleft
-\newlength\railboxright
-\newlength\railovalspace
-\newlength\railframespace
-\newlength\railtextleft
-\newlength\railtextright
-\newlength\railtextup
-\newlength\railjoinsize
-\newlength\railjoinadjust
-\newlength\railnamesep
-
-% initialize the parameters
-
-\setlength\railunit{1sp}
-\setlength\railextra{4ex}
-\setlength\railboxleft{1ex}
-\setlength\railboxright{1ex}
-\setlength\railovalspace{2ex}
-\setlength\railframespace{2ex}
-\setlength\railtextleft{1ex}
-\setlength\railtextright{1ex}
-\setlength\railjoinadjust{0pt}
-\setlength\railnamesep{1ex}
-
-\DeclareOption{10pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{11pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{12pt}{
-  \setlength\railboxheight{20pt}
-  \setlength\railboxskip{28pt}
-  \setlength\railtextup{6pt}
-  \setlength\railjoinsize{20pt}
-}
-
-\ExecuteOptions{10pt}
-\ProcessOptions
-
-% internal versions of the formatting parameters
-%
-% \rail@extra   : \railextra
-% \rail@boxht   : \railboxheight
-% \rail@boxsp   : \railboxskip
-% \rail@boxlf   : \railboxleft
-% \rail@boxrt   : \railboxright
-% \rail@boxhht  : \railboxheight / 2
-% \rail@ovalsp  : \railovalspace
-% \rail@framesp : \railframespace
-% \rail@textlf  : \railtextleft
-% \rail@textrt  : \railtextright
-% \rail@textup  : \railtextup
-% \rail@joinsz  : \railjoinsize
-% \rail@joinhsz : \railjoinsize / 2
-% \rail@joinadj : \railjoinadjust
-%
-% \railinit : internalize all of the parameters.
-
-\newcount\rail@extra
-\newcount\rail@boxht
-\newcount\rail@boxsp
-\newcount\rail@boxlf
-\newcount\rail@boxrt
-\newcount\rail@boxhht
-\newcount\rail@ovalsp
-\newcount\rail@framesp
-\newcount\rail@textlf
-\newcount\rail@textrt
-\newcount\rail@textup
-\newcount\rail@joinsz
-\newcount\rail@joinhsz
-\newcount\rail@joinadj
-
-\newcommand\railinit{
-\rail@extra=\railextra
-\divide\rail@extra by \railunit
-\rail@boxht=\railboxheight
-\divide\rail@boxht by \railunit
-\rail@boxsp=\railboxskip
-\divide\rail@boxsp by \railunit
-\rail@boxlf=\railboxleft
-\divide\rail@boxlf by \railunit
-\rail@boxrt=\railboxright
-\divide\rail@boxrt by \railunit
-\rail@boxhht=\railboxheight
-\divide\rail@boxhht by \railunit
-\divide\rail@boxhht by 2
-\rail@ovalsp=\railovalspace
-\divide\rail@ovalsp by \railunit
-\rail@framesp=\railframespace
-\divide\rail@framesp by \railunit
-\rail@textlf=\railtextleft
-\divide\rail@textlf by \railunit
-\rail@textrt=\railtextright
-\divide\rail@textrt by \railunit
-\rail@textup=\railtextup
-\divide\rail@textup by \railunit
-\rail@joinsz=\railjoinsize
-\divide\rail@joinsz by \railunit
-\rail@joinhsz=\railjoinsize
-\divide\rail@joinhsz by \railunit
-\divide\rail@joinhsz by 2
-\rail@joinadj=\railjoinadjust
-\divide\rail@joinadj by \railunit
-}
-
-\AtBeginDocument{\railinit}
-
-% \rail@param : declarations for list environment
-%
-% \railparam{TEXT} : sets \rail@param to TEXT
-%
-% \rail@reserved : characters reserved for grammar
-
-\newcommand\railparam[1]{
-\def\rail@param{
-  \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
-  \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
-  \setlength\itemindent{0pt}\setlength\listparindent{0pt}
-  #1
-}
-}
-\railparam{}
-
-\newtoks\rail@reserved
-\rail@reserved={:;|*+?[]()'"}
-
-% \rail@termfont : format setup for terminals
-%
-% \rail@nontfont : format setup for nonterminals
-%
-% \rail@annofont : format setup for annotations
-%
-% \rail@rulefont : format setup for rule names
-%
-% \rail@indexfont : format setup for index entry
-%
-% \railtermfont{TEXT} : set terminal format setup to TEXT
-%
-% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
-%
-% \railannotatefont{TEXT} : set annotation format setup to TEXT
-%
-% \railnamefont{TEXT} : set rule name format setup to TEXT
-%
-% \railindexfont{TEXT} : set index entry format setup to TEXT
-
-\def\rail@termfont{\ttfamily\upshape}
-\def\rail@nontfont{\rmfamily\upshape}
-\def\rail@annofont{\rmfamily\itshape}
-\def\rail@namefont{\rmfamily\itshape}
-\def\rail@indexfont{\rmfamily\itshape}
-
-\newcommand\railtermfont[1]{
-\def\rail@termfont{#1}
-}
-
-\newcommand\railnontermfont[1]{
-\def\rail@nontfont{#1}
-}
-
-\newcommand\railannotatefont[1]{
-\def\rail@annofont{#1}
-}
-
-\newcommand\railnamefont[1]{
-\def\rail@namefont{#1}
-}
-
-\newcommand\railindexfont[1]{
-\def\rail@indexfont{#1}
-}
-
-% railroad read/write macros
-%
-% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
-%                                as \rail@i{NR}{TEXT}. Then the matching
-%                                \rail@o{NR}{FMT} from the .rao file is
-%                                executed (if defined).
-%
-% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
-%                         as \rail@p{OPTIONS}.
-%
-% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
-%                              \rail@t{IDENT} to the .rai file
-%
-% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
-%                           TEXT.
-%
-% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
-%                           (for backward compatibility)
-%
-% \rail@setcodes : guards special characters
-%
-% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
-%                              used inside a loop for \rail@setcodes
-%
-% \rail@nr : railroad diagram counter
-%
-% \ifrail@match : current \rail@i{NR}{TEXT} matches
-%
-% \rail@first : actions to be done first. read in .rao file,
-%               open .rai file if \@filesw true, undefine \rail@first.
-%               executed from \begin{rail}, \railoptions and \railterm.
-%
-% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
-%                     file by \rail, read from the .rao file by
-%                     \rail@first
-%
-% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
-%                  written to the .rai file by \railterm.
-%
-% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
-%                     file by \rail@first.
-%
-% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
-%                    \railoptions
-%
-% \rail@write{TEXT} : write TEXT to the .rai file
-%
-% \rail@warn : warn user for mismatching diagrams
-%
-% \rail@endwarn : either \relax or \rail@warn
-%
-% \ifrail@all : checked at the end of the document
-
-\def\rail@makeother#1{
-  \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
-}
-
-\def\rail@setcodes{
-\let\par=\relax
-\let\\=\relax
-\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
-  \the\rail@reserved
-\do{\expandafter\rail@makeother\rail@symbol}
-}
-
-\newcount\rail@nr
-
-\newif\ifrail@all
-\rail@alltrue
-
-\newif\ifrail@match
-
-\def\rail@first{
-\begingroup
-\makeatletter
-\rail@setcodes
-\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
-\makeatother
-\endgroup
-\if@filesw
-\newwrite\tf@rai
-\immediate\openout\tf@rai=\jobname.rai
-\fi
-\global\let\rail@first=\relax
-}
-
-\long\def\rail@body#1\end{
-{
-\newlinechar=`^^J
-\def\par{\string\par^^J}
-\rail@write{\string\rail@i{\number\rail@nr}{#1}}
-}
-\xdef\rail@i@{#1}
-\end
-}
-
-\newenvironment{rail}{
-\global\advance\rail@nr by 1
-\rail@first
-\begingroup
-\rail@setcodes
-\rail@body
-}{
-\endgroup
-\rail@matchtrue
-\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
-\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
-\else
-\rail@matchfalse
-\fi
-\ifrail@match
-\csname rail@o@\number\rail@nr\endcsname
-\else
-\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
-\global\let\rail@endwarn=\rail@warn
-\begin{list}{}{\rail@param}
-\rail@begin{1}{}
-\rail@setbox{\bfseries ???}
-\rail@oval
-\rail@end
-\end{list}
-\fi
-}
-
-\newcommand\railoptions[1]{
-\rail@first
-\rail@write{\string\rail@p{#1}}
-}
-
-\newcommand\railterm[1]{
-\rail@first
-\@for\rail@@:=#1\do{
-\rail@write{\string\rail@t{\rail@@}}
-}
-}
-
-\newcommand\railalias[2]{
-\expandafter\def\csname rail@t@#1\endcsname{#2}
-}
-
-\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
-
-\long\def\rail@i#1#2{
-\expandafter\gdef\csname rail@i@#1\endcsname{#2}
-}
-
-\def\rail@o#1#2{
-\expandafter\gdef\csname rail@o@#1\endcsname{
-\begin{list}{}{\rail@param}
-#2
-\end{list}
-}
-}
-
-\def\rail@t#1{}
-
-\def\rail@p#1{}
-
-\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
-
-\def\rail@warn{
-\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
-                            Use 'rail' and rerun}
-}
-
-\let\rail@endwarn=\relax
-
-\AtEndDocument{\rail@endwarn}
-
-% index entry macro
-%
-% \rail@index{IDENT} : add index entry for IDENT
-
-\def\rail@index#1{
-\index{\rail@indexfont#1}
-}
-
-% railroad formatting primitives
-%
-% \rail@x : current x
-% \rail@y : current y
-% \rail@ex : current end x
-% \rail@sx : starting x for \rail@cr
-% \rail@rx : rightmost previous x for \rail@cr
-%
-% \rail@tmpa : temporary count
-% \rail@tmpb : temporary count
-% \rail@tmpc : temporary count
-%
-% \rail@put : put at (\rail@x,\rail@y)
-% \rail@vput : put vector at (\rail@x,\rail@y)
-%
-% \rail@eline : end line by drawing from \rail@ex to \rail@x
-%
-% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
-%
-% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
-%
-% \rail@sety{LEVEL} : set \rail@y to level LEVEL
-
-\newcount\rail@x
-\newcount\rail@y
-\newcount\rail@ex
-\newcount\rail@sx
-\newcount\rail@rx
-
-\newcount\rail@tmpa
-\newcount\rail@tmpb
-\newcount\rail@tmpc
-
-\def\rail@put{\put(\number\rail@x,\number\rail@y)}
-
-\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
-
-\def\rail@eline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\line(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vreline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@vput{\vector(1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vleline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\vector(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@sety#1{
-\rail@y=#1
-\multiply\rail@y by -\rail@boxsp
-\advance\rail@y by -\rail@boxht
-}
-
-% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
-%
-% \rail@end : end a railroad diagram
-%
-% \rail@expand{IDENT} : expand IDENT
-
-\def\rail@begin#1#2{
-\item
-\begin{minipage}[t]{\linewidth}
-\ifx\@empty#2\else
-{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
-\fi
-\unitlength=\railunit
-\rail@tmpa=#1
-\multiply\rail@tmpa by \rail@boxsp
-\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
-\rail@ex=0
-\rail@rx=0
-\rail@x=\rail@extra
-\rail@sx=\rail@x
-\rail@sety{0}
-}
-
-\def\rail@end{
-\advance\rail@x by \rail@extra
-\rail@eline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@vend{
-\advance\rail@x by \rail@extra
-\rail@vreline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
-
-% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
-% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
-% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
-%
-% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
-% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
-% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
-%
-% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
-% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
-% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
-%
-% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
-% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow left
-% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
-% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
-% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
-%
-% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
-% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
-% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@annote[TEXT] : format TEXT as annotation
-
-\def\rail@token#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@ltoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rtoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@ctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@nont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@frame
-}
-
-\def\rail@lnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlframe
-}
-
-\def\rail@rnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrframe
-}
-
-\def\rail@cnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@cframe
-}
-
-\def\rail@lcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcframe
-}
-
-\def\rail@rcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcframe
-}
-
-\def\rail@term#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@lterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@cterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@annote[#1]{
-\rail@setbox{\rail@annofont #1}
-\rail@text
-}
-
-% \rail@box : temporary box for \rail@oval and \rail@frame
-%
-% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
-%
-% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
-% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
-% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
-%
-% \rail@coval : same as \rail@oval, but centered between \rail@x and
-%               \rail@mx
-% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector left
-% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector right
-%
-% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
-% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
-% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
-%
-% \rail@cframe : same as \rail@frame, but centered between \rail@x and
-%                \rail@mx
-% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector left
-% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector right
-%
-% \rail@text : format \rail@box of width \rail@tmpa above the line
-
-\newbox\rail@box
-
-\def\rail@setbox#1{
-\setbox\rail@box\hbox{\strut#1}
-\rail@tmpa=\wd\rail@box
-\divide\rail@tmpa by \railunit
-}
-
-\def\rail@oval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vloval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vroval{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@coval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@oval
-}
-
-\def\rail@vlcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vloval
-}
-
-\def\rail@vrcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vroval
-}
-
-\def\rail@frame{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vlframe{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vrframe{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@cframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@frame
-}
-
-\def\rail@vlcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vlframe
-}
-
-\def\rail@vrcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vrframe
-}
-
-\def\rail@text{
-\advance\rail@x by \rail@textlf
-\advance\rail@y by \rail@textup
-\rail@put{\box\rail@box}
-\advance\rail@y by -\rail@textup
-\advance\rail@x by \rail@tmpa
-\advance\rail@x by \rail@textrt
-}
-
-% alternatives
-%
-% \rail@jx \rail@jy : current join point
-%
-% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
-%                                         to pass values over group closings
-%
-% \rail@mx : maximum x so far
-%
-% \rail@sy : starting \rail@y for alternatives
-%
-% \rail@jput : put at (\rail@jx,\rail@jy)
-%
-% \rail@joval[PART] : put \oval[PART] with adjust
-
-\newcount\rail@jx
-\newcount\rail@jy
-
-\newcount\rail@gx
-\newcount\rail@gy
-\newcount\rail@gex
-\newcount\rail@grx
-
-\newcount\rail@sy
-\newcount\rail@mx
-
-\def\rail@jput{
-\put(\number\rail@jx,\number\rail@jy)
-}
-
-\def\rail@joval[#1]{
-\advance\rail@jx by \rail@joinadj
-\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
-\advance\rail@jx by -\rail@joinadj
-}
-
-% \rail@barsplit : incoming split for '|'
-%
-% \rail@plussplit : incoming split for '+'
-%
-
-\def\rail@barsplit{
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@plussplit{
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-}
-
-% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
-%
-
-\def\rail@alt#1{
-\rail@sy=\rail@y
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\rail@mx=0
-\let\rail@list=\@empty
-\let\rail@comma=\@empty
-\let\rail@split=#1
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
-%                         and fix-up FIX
-%
-
-\def\rail@nextalt#1#2{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-#1
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\def\rail@comma{,}
-\rail@split
-\let\rail@split=\@empty
-\rail@sety{#2}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@ex=\rail@x
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@barjoin : outgoing join for first '|' alternative
-%
-% \rail@plusjoin : outgoing join for first '+' alternative
-%
-% \rail@altjoin : join for subsequent alternative
-%
-
-\def\rail@barjoin{
-\ifnum\rail@y<\rail@sy
-\global\rail@gex=\rail@jx
-\else
-\global\rail@gex=\rail@ex
-\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\ifnum\rail@y<\rail@sy
-\rail@altjoin
-\fi
-}
-
-\def\rail@plusjoin{
-\global\rail@gex=\rail@ex
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by -\rail@joinsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@altjoin{
-\rail@eline
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-}
-
-% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
-%
-% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
-
-\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
-
-\def\rail@endalt#1{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\rail@x=\rail@mx
-\rail@jx=\rail@x
-\rail@jy=\rail@sy
-\advance\rail@jx by \rail@joinsz
-\let\rail@join=#1
-\@for\rail@elt:=\rail@list\do{
-\expandafter\rail@eltsplit\rail@elt;
-\rail@join
-\let\rail@join=\rail@altjoin
-}
-\rail@x=\rail@mx
-\rail@y=\rail@sy
-\rail@ex=\rail@gex
-\advance\rail@x by \rail@joinsz
-}
-
-% \rail@bar : start '|' alternatives
-%
-% \rail@nextbar : next '|' alternative
-%
-% \rail@endbar : end '|' alternatives
-%
-
-\def\rail@bar{
-\rail@alt\rail@barsplit
-}
-
-\def\rail@nextbar{
-\rail@nextalt\relax
-}
-
-\def\rail@endbar{
-\rail@endalt\rail@barjoin
-}
-
-% \rail@plus : start '+' alternatives
-%
-% \rail@nextplus: next '+' alternative
-%
-% \rail@endplus : end '+' alternatives
-%
-
-\def\rail@plus{
-\rail@alt\rail@plussplit
-}
-
-\def\rail@nextplus{
-\rail@nextalt\rail@fixplus
-}
-
-\def\rail@fixplus{
-\ifnum\rail@gy<\rail@sy
-\begingroup
-\rail@x=\rail@gx
-\rail@y=\rail@gy
-\rail@ex=\rail@gex
-\rail@rx=\rail@grx
-\ifnum\rail@x<\rail@rx
-\rail@x=\rail@rx
-\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-\rail@tmpa=\rail@sy
-\advance\rail@tmpa by -\rail@joinhsz
-\advance\rail@tmpa by -\rail@jy
-\rail@jput{\line(0,1){\number\rail@tmpa}}
-\rail@jy=\rail@sy
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jy by \rail@joinhsz
-\global\rail@gx=\rail@jx
-\global\rail@gy=\rail@jy
-\global\rail@gex=\rail@gx
-\global\rail@grx=\rail@rx
-\endgroup
-\fi
-}
-
-\def\rail@endplus{
-\rail@endalt\rail@plusjoin
-}
-
-% \rail@cr{Y} : carriage return to vertical position Y
-
-\def\rail@cr#1{
-\rail@tmpa=\rail@sx
-\advance\rail@tmpa by \rail@joinsz
-\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-\rail@sety{#1}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@boxsp
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@boxsp
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jy by -\rail@joinhsz
-\rail@tmpa=\rail@jx
-\advance\rail@tmpa by -\rail@sx
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(-1,0){\number\rail@tmpa}}
-\rail@jx=\rail@sx
-\advance\rail@jx by \rail@joinhsz
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@tmpa=\rail@boxsp
-\advance\rail@tmpa by -\rail@joinsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\advance\rail@jy by -\rail@tmpa
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\rail@x=\rail@jx
-\rail@ex=\rail@x
-}
-
-% default setup for Isabelle
-\newenvironment{railoutput}%
-{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
-
-\def\rail@termfont{\isabellestyle{tt}}
-\def\rail@nontfont{\isabellestyle{it}}
-\def\rail@namefont{\isabellestyle{it}}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib
deleted file mode 100644
index 6f429f0..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib
+++ /dev/null
@@ -1,71 +0,0 @@
-@InProceedings{adt2019,
-    author    = {Karsten Diekhoff and
-                 Michael Kirsten and
-                 Jonas Kr{\"{a}}mer},
-    editor    = {Sa{\v{s}}a Peke{\v{c}} and
-                 {Kristen Brent} Venable},
-    title     = {Formal Property-Oriented Design of
-                 Voting Rules Using Composable Modules},
-    booktitle = {6th International Conference on Algorithmic
-                 Decision Theory ({ADT} 2019)},
-    series    = {Lecture Notes in Artificial Intelligence},
-    part      = {Short Papers},
-    year      = {2019},
-    abstract  = {Voting rules aggregate multiple individual preferences in order to
-                 make a collective decision. Commonly, these mechanisms are expected
-                 to respect a multitude of different notions of fairness and reliability,
-                 which must be carefully balanced to avoid inconsistencies.
-
-                 We present an approach for the sound and flexible design of voting rules
-                 from composable modules. Formal composition rules guarantee social
-                 choice properties from properties of the individual components. The
-                 approach can be applied to many voting rules from the literature.},
-    publisher = {Springer},
-    volume    = {11834},
-    pages     = {164--166},
-    doi       = {10.1007/978-3-030-31489-7}
-}
-
-@InProceedings{lopstr2019,
-  author        = {Karsten Diekhoff and
-                   Michael Kirsten and
-                   Jonas Kr{\"{a}}mer},
-  title         = {Verified Construction of Fair Voting Rules},
-  booktitle     = {29th International Symposium on Logic-Based Program
-                   Synthesis and Transformation ({LOPSTR} 2019),
-                   Revised Selected Papers},
-  series        = {Lecture Notes in Computer Science},
-  editor        = {Maurizio Gabbrielli},
-  year          = {2020},
-  abstract      = {Voting rules aggregate multiple individual preferences in order to
-                   make collective decisions. Commonly, these mechanisms are expected
-                   to respect a multitude of different fairness and reliability
-                   properties, e.g., to ensure that each voter's ballot accounts for
-                   the same proportion of the elected alternatives, or that a voter
-                   cannot change the election outcome in her favor by insincerely
-                   filling out her ballot. However, no voting rule is fair in all
-                   respects, and trade-off attempts between such properties often
-                   bring out inconsistencies, which makes the construction of arguably
-                   practical and fair voting rules non-trivial and error-prone.
- 
-                   In this paper, we present a formal and systematic approach for the
-                   flexible and verified construction of voting rules from composable
-                   core modules to respect such properties by construction. Formal
-                   composition rules guarantee resulting properties from properties of
-                   the individual components, which are of generic nature to be reused
-                   for various voting rules. We provide a prototypical logic-based
-                   implementation with proofs for a selected set of structures and
-                   composition rules within the theorem prover Isabelle/HOL. The
-                   approach can be readily extended in order to support many voting
-                   rules from the literature by extending the set of basic modules and
-                   composition rules. We exemplarily construct the well-known voting
-                   rule sequential majority comparison (SMC) from simple generic
-                   modules, and automatically produce a formal proof that SMC
-                   satisfies the fairness property monotonicity. Monotonicity is a
-                   well-known social-choice property that is easily violated by voting
-                   rules in practice.},
-  publisher     = {Springer},
-  doi           = {10.1007/978-3-030-45260-5_6},
-  volume        = {12042},
-  pages         = {90--104}
-}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex
deleted file mode 100644
index 2bf1d6e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex
+++ /dev/null
@@ -1,56 +0,0 @@
-\documentclass[11pt,a4paper,notitlepage]{report}
-\input{settings}
-\usepackage{isabelle,isabellesym}
-\usepackage{amssymb} % amsfonts, amssymb
-\usepackage{pdfsetup}
-
-\urlstyle{rm}
-\isabellestyle{it}
-
-% for uniform font size
-%\renewcommand{\isastyle}{\isastyleminor}
-
-% This is required due to some unusual behaviour by Isabelle.
-% It should be included in isabellesym.sty automatically, but it is not.
-\newcommand{\isasymcirclearrowleft}{\isamath{\circlearrowleft}}
-
-\begin{document}
-
-\title{Verified Construction of Fair Voting Rules}
-\author{Michael Kirsten}
-\institute{Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany\\
-\email{kirsten@kit.edu}}
-\maketitle
-
-\begin{abstract}
-Voting rules aggregate multiple individual preferences in order to make a
-collective decision.
-Commonly, these mechanisms are expected to respect a multitude of different
-notions of fairness and reliability, which must be carefully balanced to avoid
-inconsistencies.
-
-This article contains a formalisation of a framework for the construction of
-such fair voting rules using composable modules~\cite{adt2019,lopstr2019}.
-The framework is a formal and systematic approach for the flexible and verified
-construction of voting rules from individual composable modules to respect such
-social-choice properties by construction.
-Formal composition rules guarantee resulting social-choice properties from
-properties of the individual components which are of generic nature to be
-reused for various voting rules.
-We provide proofs for a selected set of structures and composition rules.
-The approach can be readily extended in order to support more voting rules,
-e.g., from the literature by extending the sets of modules and composition
-rules.
-\end{abstract}
-
-\tableofcontents
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-% optional bibliography
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex
deleted file mode 100644
index 00be90a..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex
+++ /dev/null
@@ -1,48 +0,0 @@
-\input{Preference_Relation.tex}
-\input{Result.tex}
-\input{Profile.tex}
-\input{Electoral_Module.tex}
-\input{Evaluation_Function.tex}
-\input{Aggregator.tex}
-\input{Termination_Condition.tex}
-\input{Defer_Equal_Condition.tex}
-\input{Defer_Module.tex}
-\input{Drop_Module.tex}
-\input{Pass_Module.tex}
-\input{Elect_Module.tex}
-\input{Elimination_Module.tex}
-\input{Maximum_Aggregator.tex}
-\input{Plurality_Module.tex}
-\input{Composite_Elimination_Modules.tex}
-\input{Result_Properties.tex}
-\input{Sequential_Composition.tex}
-\input{Parallel_Composition.tex}
-\input{Loop_Composition.tex}
-\input{Aggregator_Properties.tex}
-\input{Indep_Of_Alt.tex}
-\input{Disjoint_Compatibility.tex}
-\input{Aggregator_Facts.tex}
-\input{Composite_Structures.tex}
-\input{Revision_Composition.tex}
-\input{Borda_Rule.tex}
-\input{Condorcet_Properties.tex}
-\input{Condorcet_Consistency.tex}
-\input{Condorcet_Rules.tex}
-\input{Condorcet_Facts.tex}
-\input{Pairwise_Majority_Rule.tex}
-\input{Copeland_Rule.tex}
-\input{Minimax_Rule.tex}
-\input{Blacks_Rule.tex}
-\input{Nanson_Baldwin_Rule.tex}
-\input{Classic_Nanson_Rule.tex}
-\input{Schwartz_Rule.tex}
-\input{Monotonicity_Properties.tex}
-\input{Weak_Monotonicity.tex}
-\input{Result_Facts.tex}
-\input{Result_Rules.tex}
-\input{Monotonicity_Facts.tex}
-\input{Monotonicity_Rules.tex}
-\input{Disjoint_Compatibility_Facts.tex}
-\input{Disjoint_Compatibility_Rules.tex}
-\input{Sequential_Majority_Comparison.tex}
-\input{Homogeneity.tex}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex
deleted file mode 100644
index 9a1f927..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex
+++ /dev/null
@@ -1,30 +0,0 @@
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{sectsty}
-\usepackage{xpatch}
-\usepackage{upquote}
-
-\makeatletter
-\xpatchcmd{\@maketitle}
-  {\@title}
-{%\vspace{-2.45\baselineskip}
-\bfseries\@title\vspace{.25\baselineskip}}
-  {}{}
-\makeatother
-
-\makeatletter
-\providecommand{\institute}[1]{%
-    \apptocmd{\@author}{\end{tabular}%
-    \vspace{.5\baselineskip}\par
-    \begin{tabular}[t]{c}
-    #1}{}{}
-}
-\makeatother
-
-\newcommand{\email}[1]{%
-    \texttt{\href{mailto:#1}{\color{black}{#1}}}%
-}
-
-\pretocmd{\tableofcontents}{\sectionfont{\Huge}}{}{}
-%\let\clearpage\relax
-\apptocmd{\tableofcontents}{\sectionfont{\Large}}{}{}

From c75172c0783d46b749adb5704b64bbaf2c5fe84f Mon Sep 17 00:00:00 2001
From: Michael Kirsten <kirsten@kit.edu>
Date: Thu, 13 Mar 2025 04:09:31 +0100
Subject: [PATCH 3/4] Updates

---
 virage/pom.xml                                | 101 +--
 .../kit/kastel/formal/util/ProcessUtils.java  |  22 +-
 .../kit/kastel/formal/util/StringUtils.java   |  77 ++-
 .../SimplePrologCompositionAnalyzer.java      |   8 +-
 .../formal/virage/beast/CCodeGenerator.java   |  14 +-
 .../formal/virage/core/ConfigReader.java      |  38 +-
 .../core/VirageCommandLineInterface.java      |  50 +-
 .../virage/core/VirageSearchManager.java      |  18 +-
 .../isabelle/IsabelleClientObserver.java      |  10 +-
 .../isabelle/IsabelleCodeGenerator.java       |   4 +-
 .../isabelle/IsabelleFrameworkExtractor.java  |  35 +-
 .../virage/isabelle/IsabelleProofChecker.java |  18 +-
 .../isabelle/IsabelleProofGenerator.java      |  45 +-
 .../isabelle/IsabelleProofStepGenerator.java  |  21 +-
 .../isabelle/IsabelleTheoryGenerator.java     |  44 +-
 .../virage/isabelle/IsabelleTheoryParser.java |   2 +-
 .../formal/virage/isabelle/IsabelleUtils.java |  27 +-
 .../virage/isabelle/ScalaIsabelleFacade.java  |  48 +-
 .../jobs/VirageAnalyzeAllPropertiesJob.java   |   2 +-
 .../formal/virage/jobs/VirageAnalyzeJob.java  |  19 +-
 .../virage/jobs/VirageGenerateCCodeJob.java   |   9 +-
 .../formal/virage/jobs/VirageGenerateJob.java |   9 +-
 .../jobs/VirageIsabelleGenerateJob.java       |  10 +-
 .../jobs/VirageIsabelleGenerateScalaJob.java  |   9 +-
 .../virage/jobs/VirageIsabelleVerifyJob.java  |  15 +-
 .../kastel/formal/virage/jobs/VirageJob.java  |   6 +-
 .../formal/virage/jobs/VirageProveJob.java    |  10 +-
 .../formal/virage/prolog/JplFacade.java       |  30 +-
 .../formal/virage/prolog/PrologClause.java    |   3 +-
 .../formal/virage/prolog/PrologPredicate.java |  26 +-
 .../prolog/SimpleExtendedPrologParser.java    |  21 +-
 .../formal/virage/types/ComponentType.java    |   5 +-
 .../formal/virage/types/CompositionProof.java |   4 +-
 .../formal/virage/types/CompositionRule.java  |   6 +-
 .../virage/types/DecompositionTree.java       |   8 +-
 .../virage/types/FrameworkRepresentation.java |  10 +-
 .../kastel/formal/virage/types/Property.java  |   3 +-
 .../{settings => settings.properties}         |   7 -
 .../kastel/formal/util/JIsabelleWrapper.scala |   9 -
 .../theories/framework.pl                     | 631 +++++++++++++++++-
 .../theories/output/document.pdf              | Bin 684332 -> 648347 bytes
 .../output/document/session_graph.pdf         | Bin 15635 -> 15838 bytes
 42 files changed, 998 insertions(+), 436 deletions(-)
 rename virage/src/main/resources/{settings => settings.properties} (99%)

diff --git a/virage/pom.xml b/virage/pom.xml
index 3166fa0..8d49197 100644
--- a/virage/pom.xml
+++ b/virage/pom.xml
@@ -40,19 +40,24 @@
     <skipTests>false</skipTests>
     <project.build.sourceEncoding>UTF-8</project.build.sourceEncoding>
     <project.reporting.outputEncoding>UTF-8</project.reporting.outputEncoding>
-    <maven.version>3.8.1</maven.version>
-    <maven.clean.version>3.3.1</maven.clean.version>
-    <maven.install.version>3.1.1</maven.install.version>
-    <maven.jar.version>3.3.0</maven.jar.version>
-    <maven.compiler.release>17</maven.compiler.release>
-    <maven.compiler.source>17</maven.compiler.source>
-    <maven.compiler.target>17</maven.compiler.target>
-    <jdk.version>17</jdk.version>
-    <scala.version>2.13.12</scala.version>
-    <log4j.version>2.20.0</log4j.version>
-    <spotbugs.version>4.7.3</spotbugs.version>
-    <junit.jupiter.version>5.10.0</junit.jupiter.version>
-    <surefire.version>3.1.2</surefire.version>
+    <maven.version>3.6.3</maven.version>
+    <jdk.version>21</jdk.version>
+    <maven.clean.version>3.4.1</maven.clean.version>
+    <maven.install.version>3.1.4</maven.install.version>
+    <maven.exec.version>3.5.0</maven.exec.version>
+    <maven.jar.version>3.4.2</maven.jar.version>
+    <maven.resources.version>3.3.1</maven.resources.version>
+    <maven.checkstyle.version>3.6.0</maven.checkstyle.version>
+    <maven.buildhelper.version>3.6.0</maven.buildhelper.version>
+    <maven.reports.version>3.9.0</maven.reports.version>
+    <maven.compiler.release>${jdk.version}</maven.compiler.release>
+    <maven.compiler.source>${jdk.version}</maven.compiler.source>
+    <maven.compiler.target>${jdk.version}</maven.compiler.target>
+    <scala.version>2.13.16</scala.version>
+    <log4j.version>2.24.3</log4j.version>
+    <spotbugs.version>4.9.2.0</spotbugs.version>
+    <junit.jupiter.version>5.12.0</junit.jupiter.version>
+    <surefire.version>3.5.2</surefire.version>
     <swi.home.dir>/usr/lib/swi-prolog</swi.home.dir>
     <main.class>edu.kit.kastel.formal.virage.core.VirageMain</main.class>
   </properties>
@@ -108,7 +113,7 @@
     <dependency>
       <groupId>org.junit.platform</groupId>
       <artifactId>junit-platform-suite-engine</artifactId>
-      <version>1.10.0</version>
+      <version>1.12.0</version>
       <scope>test</scope>
     </dependency>
     <dependency>
@@ -135,22 +140,22 @@
     <dependency>
       <groupId>com.github.SWI-Prolog</groupId>
       <artifactId>packages-jpl</artifactId>
-      <version>V9.1.15</version>
+      <version>V9.3.21</version>
     </dependency>
     <dependency>
       <groupId>commons-cli</groupId>
       <artifactId>commons-cli</artifactId>
-      <version>1.5.0</version>
+      <version>1.9.0</version>
     </dependency>
     <dependency>
       <groupId>commons-io</groupId>
       <artifactId>commons-io</artifactId>
-      <version>2.13.0</version>
+      <version>2.18.0</version>
     </dependency>
     <dependency>
       <groupId>com.fasterxml.jackson.core</groupId>
       <artifactId>jackson-databind</artifactId>
-      <version>2.15.2</version>
+      <version>2.18.3</version>
     </dependency>
     <!-- <dependency> <groupId>com.github.VeriVote</groupId> <artifactId>ViRAGe</artifactId>
       <version>master-SNAPSHOT</version> </dependency> -->
@@ -158,22 +163,22 @@
     <dependency>
       <groupId>de.unruh</groupId>
       <artifactId>scala-isabelle_2.13</artifactId>
-      <version>0.4.2-RC1</version>
+      <version>0.4.2</version>
     </dependency>
     <dependency>
       <groupId>org.apache.commons</groupId>
       <artifactId>commons-configuration2</artifactId>
-      <version>2.9.0</version>
+      <version>2.11.0</version>
     </dependency>
     <dependency>
       <groupId>commons-beanutils</groupId>
       <artifactId>commons-beanutils</artifactId>
-      <version>1.9.4</version>
+      <version>1.10.1</version>
     </dependency>
     <dependency>
       <groupId>com.github.ferstl</groupId>
       <artifactId>depgraph-maven-plugin</artifactId>
-      <version>4.0.2</version>
+      <version>4.0.3</version>
     </dependency>
   </dependencies>
 
@@ -190,6 +195,7 @@
       <build>
         <plugins>
           <plugin>
+            <groupId>org.apache.maven.plugins</groupId>
             <artifactId>maven-compiler-plugin</artifactId>
             <executions>
               <execution>
@@ -207,6 +213,14 @@
             </executions>
           </plugin>
         </plugins>
+        <resources>
+          <resource>
+            <directory>${user.home}/.virage/</directory>
+            <includes>
+              <include>settings.properties</include>
+            </includes>
+          </resource>
+        </resources>
       </build>
     </profile>
   </profiles>
@@ -222,12 +236,12 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-resources-plugin</artifactId>
-        <version>${maven.clean.version}</version>
+        <version>${maven.resources.version}</version>
       </plugin>
       <plugin>
         <groupId>net.alchim31.maven</groupId>
         <artifactId>scala-maven-plugin</artifactId>
-        <version>4.8.1</version>
+        <version>4.9.2</version>
         <configuration>
           <scalaVersion>${scala.version}</scalaVersion>
         </configuration>
@@ -251,7 +265,12 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-compiler-plugin</artifactId>
-        <version>3.11.0</version>
+        <version>3.14.0</version>
+        <configuration>
+          <source>${jdk.version}</source>
+          <target>${jdk.version}</target>
+          <compilerArgument>-Xlint:deprecation</compilerArgument>
+        </configuration>
       </plugin>
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
@@ -278,7 +297,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-site-plugin</artifactId>
-        <version>3.12.1</version>
+        <version>3.21.0</version>
         <configuration>
           <locales>en</locales>
         </configuration>
@@ -286,7 +305,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-checkstyle-plugin</artifactId>
-        <version>${maven.jar.version}</version>
+        <version>${maven.checkstyle.version}</version>
         <configuration>
           <configLocation>checks.xml</configLocation>
           <encoding>UTF-8</encoding>
@@ -309,14 +328,14 @@
           <dependency>
             <groupId>com.puppycrawl.tools</groupId>
             <artifactId>checkstyle</artifactId>
-            <version>10.12.3</version>
+            <version>10.21.4</version>
           </dependency>
         </dependencies>
       </plugin>
       <plugin>
         <groupId>com.github.spotbugs</groupId>
         <artifactId>spotbugs-maven-plugin</artifactId>
-        <version>${spotbugs.version}.5</version>
+        <version>${spotbugs.version}</version>
         <configuration>
           <xmlOutput>true</xmlOutput>
           <failOnError>false</failOnError>
@@ -336,7 +355,7 @@
         <dependencies>
           <dependency>
             <groupId>com.github.spotbugs</groupId>
-            <artifactId>spotbugs</artifactId>
+            <artifactId>spotbugs-maven-plugin</artifactId>
             <version>${spotbugs.version}</version>
           </dependency>
         </dependencies>
@@ -344,7 +363,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>build-helper-maven-plugin</artifactId>
-        <version>3.4.0</version>
+        <version>${maven.buildhelper.version}</version>
         <executions>
           <execution>
             <id>add-source</id>
@@ -364,7 +383,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-enforcer-plugin</artifactId>
-        <version>3.4.1</version>
+        <version>3.5.0</version>
         <executions>
           <execution>
             <id>enforce-versions</id>
@@ -374,7 +393,7 @@
             <configuration>
               <rules>
                 <requireMavenVersion>
-                  <version>3.3.9</version>
+                  <version>3.6.3</version>
                 </requireMavenVersion>
                 <requireJavaVersion>
                   <version>${jdk.version}</version>
@@ -387,7 +406,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>versions-maven-plugin</artifactId>
-        <version>2.16.1</version>
+        <version>2.18.0</version>
         <configuration>
           <rulesUri>file:///${basedir}/rules.xml</rulesUri>
           <allowSnapshots>false</allowSnapshots>
@@ -407,7 +426,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>properties-maven-plugin</artifactId>
-        <version>1.2.0</version>
+        <version>1.2.1</version>
         <executions>
           <execution>
             <id>read-settings</id>
@@ -417,7 +436,7 @@
             </goals>
             <configuration>
               <files>
-                <file>${user.home}/.virage/settings</file>
+                <file>${user.home}/.virage/settings.properties</file>
               </files>
               <!-- settings will not be there on first startup. -->
               <!-- This is fine, as ViRAGe will copy it and enforce a restart if 
@@ -455,7 +474,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>exec-maven-plugin</artifactId>
-        <version>3.1.0</version>
+        <version>${maven.exec.version}</version>
         <executions>
           <execution>
             <id>execute</id>
@@ -487,7 +506,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-jxr-plugin</artifactId>
-        <version>${maven.jar.version}</version>
+        <version>${maven.checkstyle.version}</version>
       </plugin>
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
@@ -497,12 +516,12 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-javadoc-plugin</artifactId>
-        <version>3.6.0</version>
+        <version>3.11.2</version>
       </plugin>
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-project-info-reports-plugin</artifactId>
-        <version>3.4.5</version>
+        <version>${maven.reports.version}</version>
         <reportSets>
           <reportSet>
             <reports>
@@ -525,7 +544,7 @@
       <plugin>
         <groupId>com.github.spotbugs</groupId>
         <artifactId>spotbugs-maven-plugin</artifactId>
-        <version>4.7.2.1</version>
+        <version>${spotbugs.version}</version>
         <reportSets>
           <reportSet>
             <reports>
diff --git a/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java b/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java
index f43549a..748432d 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java
@@ -3,6 +3,7 @@
 import java.io.File;
 import java.io.IOException;
 import java.nio.charset.StandardCharsets;
+import java.util.StringTokenizer;
 
 import org.apache.logging.log4j.LogManager;
 import org.apache.logging.log4j.Logger;
@@ -40,6 +41,24 @@ public static void start(final String editorExecutable, final File path) {
         }
     }
 
+    /**
+     * Wrapper method for process execution, either strip and escape or simply format.
+     *
+     * @param cmd the command string, multiple ones separated by blanks
+     * @param escape flag for determining whether to strip and escape or not
+     * @return the resulting process
+     * @throws IOException if an I/O error occurs
+     */
+    public static Process exec(final String cmd, final boolean escape) throws IOException {
+        final String command = escape ? StringUtils.stripAndEscape(cmd) : String.format(cmd);
+        final StringTokenizer st = new StringTokenizer(command);
+        final String[] cmdarray = new String[st.countTokens()];
+        for (int i = 0; st.hasMoreTokens(); i++) {
+            cmdarray[i] = st.nextToken();
+        }
+        return new ProcessBuilder(cmdarray).start();
+    }
+
     /**
      * Executes a terminating command and prints its output to System.out/System.err, respectively.
      * <b>Does not return if the command is non-terminating!</b>
@@ -52,8 +71,7 @@ public static void start(final String editorExecutable, final File path) {
      */
     public static Output runTerminatingProcess(final String command)
             throws IOException, InterruptedException {
-        final Runtime rt = Runtime.getRuntime();
-        final Process p = rt.exec(StringUtils.stripAndEscape(command));
+        final Process p = exec(command, true);
         final int status = p.waitFor();
         final String stdErr = new String(p.getErrorStream().readAllBytes(), StandardCharsets.UTF_8);
         final String stdOut = new String(p.getInputStream().readAllBytes(), StandardCharsets.UTF_8);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java b/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
index c151a63..b017afe 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
@@ -37,6 +37,11 @@ public final class StringUtils {
      */
     public static final String HASH = "#";
 
+    /**
+     * A star sign.
+     */
+    public static final String STAR = "*";
+
     /**
      * An opening parenthesis sign.
      */
@@ -205,9 +210,9 @@ public static String printCollection(final Collection<?> c) {
         if (c.isEmpty()) {
             return EMPTY;
         }
-        final StringBuilder res = new StringBuilder(EMPTY);
+        final StringBuilder res = new StringBuilder();
         for (final Object obj: c) {
-            res.append(obj).append(COMMA);
+            res.append(obj).append(COMMA.charAt(0));
         }
         return res.deleteCharAt(res.length() - 1).toString();
     }
@@ -232,9 +237,9 @@ public static String printCollection2(final Collection<?> c) {
         if (c.isEmpty()) {
             return EMPTY;
         }
-        final StringBuilder res = new StringBuilder(EMPTY);
+        final StringBuilder res = new StringBuilder();
         for (final Object obj: c) {
-            res.append(obj).append(SPACE);
+            res.append(obj).append(SPACE.charAt(0));
         }
         return res.deleteCharAt(res.length() - 1).toString();
     }
@@ -289,6 +294,62 @@ public static String surroundWithSpaces(final String s) {
         return SPACE + s + SPACE;
     }
 
+    /**
+     * Adds the separator and a space before and a space and the separator after the given String.
+     *
+     * @param s the given String.
+     * @param separator the separator.
+     * @return new String with added separator and spaces before and after
+     */
+    public static String surroundWithSpacedSeparators(final String s, final String separator) {
+        return separator + surroundWithSpaces(s) + separator;
+    }
+
+    /**
+     * Put the given String in a raw Isabelle comment.
+     *
+     * @param s the given String.
+     * @return the resulting Isabelle comment containing the given String
+     */
+    public static String isabelleComment(final String s) {
+        return parenthesize(surroundWithSpacedSeparators(s, STAR));
+    }
+
+    /**
+     * Put the given String in a raw Isabelle comment line.
+     *
+     * @param s the given String.
+     * @return the resulting Isabelle comment line containing the given String
+     */
+    public static String isabelleCommentLine(final String s) {
+        return isabelleComment(s) + System.lineSeparator();
+    }
+
+    /**
+     * Put the given Strings in raw Isabelle comment lines.
+     *
+     * @param s the given Strings.
+     * @return the resulting Isabelle comment lines containing the given Strings
+     */
+    public static String isabelleCommentLines(final String... s) {
+        final StringBuilder sb = new StringBuilder();
+        for (final String s1: s) {
+            sb.append(isabelleCommentLine(s1));
+        }
+        return sb.toString();
+    }
+
+    /**
+     * Put the given Strings in raw Isabelle comment block.
+     *
+     * @param s the given Strings.
+     * @return the resulting Isabelle comment block containing the given Strings
+     */
+    public static String isabelleCommentBlock(final String... s) {
+        final String starLine = parenthesize(repeat(22, addSpace(STAR)) + STAR);
+        return starLine + System.lineSeparator() + isabelleCommentLines(s) + starLine;
+    }
+
     /**
      * Returns a disjunction of the two string arguments.
      *
@@ -453,12 +514,12 @@ public static String map(final String fun, final String arg1, final String arg2)
     /**
      * Returns map function string with two separately parenthesized arguments.
      *
-     * @param arg1 the function which is to be applied by the map
-     * @param arg2 the collection argument to which the function is applied
+     * @param fun the function which is to be applied by the map
+     * @param app the collection argument to which the function is applied
      * @return the function string representation
      */
-    public static String map(final String arg1, final String arg2) {
-        return map(MAP, arg1, arg2);
+    public static String map(final String fun, final String app) {
+        return map(MAP, fun, app);
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java b/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java
index 4804a83..10381ee 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java
@@ -194,10 +194,10 @@ public List<CompositionProof> proveClaims(final DecompositionTree composition,
             this.facade.setTimeout(oldTimeout);
             if (result.hasValue()) {
                 try {
-                    final Map<String, String> map = result.getValue();
-                    if (map.containsKey(proofVariable)) {
-                        LOGGER.debug(map.get(proofVariable));
-                        proofs.add(PrologProof.createProofFromString(map.get(proofVariable)));
+                    final String val = result.getValue().get(proofVariable);
+                    if (val != null) {
+                        LOGGER.debug(val);
+                        proofs.add(PrologProof.createProofFromString(val));
                     } else {
                         throw new IllegalArgumentException();
                     }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
index c0623c0..def743d 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
@@ -99,11 +99,6 @@ public final class CCodeGenerator {
      */
     private final String codeFileTemplate;
 
-    /**
-     * Templates for the components of the framework.
-     */
-    private final String compositionsTemplate;
-
     /**
      * Templates for each component.
      */
@@ -144,7 +139,6 @@ public CCodeGenerator(final FrameworkRepresentation framework) {
         } catch (final IOException e) {
             e.printStackTrace();
         }
-        this.compositionsTemplate = writer.toString();
         this.templates = new HashMap<String, String>();
         this.signatures = new HashMap<String, List<String>>();
         final String componentNameGroup = "componentName";
@@ -155,7 +149,7 @@ public CCodeGenerator(final FrameworkRepresentation framework) {
                 + signatureGroup + ">[^\\)]*\\)).*)\\/\\/[ ]*\\k<" + componentNameGroup + ">"
                 + System.lineSeparator(),
                 Pattern.DOTALL);
-        final Matcher matcher = pattern.matcher(this.compositionsTemplate);
+        final Matcher matcher = pattern.matcher(writer.toString());
         while (matcher.find()) {
             this.templates.put(matcher.group(componentNameGroup),
                     matcher.group(implementationGroup));
@@ -210,8 +204,8 @@ private Pair<Pair<String, String>, Integer> getCCodeFromComposition(
                 this.frameworkField.getComponent(composition.getLabel());
         if (currentComponent != null) {
             final String componentName = composition.getLabel();
-            if (this.templates.containsKey(componentName)) {
-                String componentTemplate = this.templates.get(componentName);
+            String componentTemplate = this.templates.get(componentName);
+            if (componentTemplate != null) {
                 componentTemplate =
                         componentTemplate.replace(COMPONENT_INDEX_VARIABLE, String.valueOf(ctr));
                 int moduleCounter = 1;
@@ -256,7 +250,7 @@ private List<String> findMissingTemplates(final DecompositionTree composition) {
 
     private List<String> findMissingTemplates(final DecompositionTree composition,
                                               final List<String> found) {
-        if (!this.templates.keySet().contains(composition.getLabel())) {
+        if (!this.templates.containsKey(composition.getLabel())) {
             found.add(composition.getLabel());
         }
         for (final DecompositionTree child: composition.getChildren()) {
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java
index 5f23779..8833481 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java
@@ -1,13 +1,11 @@
 package edu.kit.kastel.formal.virage.core;
 
 import java.io.File;
-import java.io.FileInputStream;
 import java.io.IOException;
 import java.io.InputStream;
 import java.io.StringWriter;
 import java.nio.charset.StandardCharsets;
 import java.nio.file.Files;
-import java.util.HashMap;
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Map;
@@ -24,6 +22,8 @@
 import org.apache.logging.log4j.Logger;
 import org.jpl7.JPL;
 
+import com.google.common.collect.Maps;
+
 import edu.kit.kastel.formal.util.Pair;
 import edu.kit.kastel.formal.util.ProcessUtils;
 import edu.kit.kastel.formal.util.SimpleFileReader;
@@ -68,7 +68,7 @@ public final class ConfigReader {
     /**
      * Name of the settings file.
      */
-    private static final String SETTINGS = "settings";
+    private static final String SETTINGS = "settings.properties";
 
     /**
      * Prefix for the old settings file.
@@ -391,13 +391,12 @@ public String checkAvailabilityAndGetVersions() {
             this.jplAvailable = false;
             this.swiplAvailable = false;
         }
-        res += getVersionString("JPL", null, JPL.version_string(), StringUtils.THREE);
-        return res;
+        return res + getVersionString("JPL", null, JPL.version_string(), StringUtils.THREE);
     }
 
     private Pair<Boolean, List<String>> checkConfigCompatibility() throws IOException {
         final Properties oldProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             oldProperties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -450,7 +449,7 @@ public void copyConfigToExpectedPath() throws IOException {
         }
         writer.writeToFile(oldPropertiesFile.getAbsolutePath(), reader.readFile(this.configFile));
         final Properties oldProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(oldPropertiesFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(oldPropertiesFile.toPath())) {
             oldProperties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -461,15 +460,16 @@ public void copyConfigToExpectedPath() throws IOException {
         IOUtils.copy(configFromResourcesStream, stringWriter, StandardCharsets.UTF_8);
         writer.writeToFile(this.configFile.getAbsolutePath(), stringWriter.toString());
         final Properties newProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             newProperties.load(fis);
         }
         for (final Object o: oldProperties.keySet()) {
-            if (o.toString().equals(VIRAGE_CONFIG_VERSION)) {
+            final String propName = o.toString();
+            if (VIRAGE_CONFIG_VERSION.equals(propName)) {
                 continue;
             }
             if (newProperties.keySet().contains(o)) {
-                this.updateValue(o.toString(), oldProperties.getProperty(o.toString()));
+                this.updateValue(propName, oldProperties.getProperty(propName));
             }
         }
     }
@@ -489,9 +489,11 @@ public List<String> getAdditionalProperties() {
      * @return the properties
      */
     public Map<String, String> getAllProperties() {
-        final Map<String, String> result = new HashMap<String, String>();
-        for (final Object prop: this.properties.keySet()) {
-            result.put(prop.toString(), this.properties.getProperty(prop.toString()));
+        final Properties props = this.properties;
+        final Map<String, String> result = Maps.newHashMapWithExpectedSize(props.size());
+        for (final Object prop: props.keySet()) {
+            final String propString = prop.toString();
+            result.put(propString, props.getProperty(propString));
         }
         return result;
     }
@@ -612,9 +614,9 @@ public String getPathToRootFile() {
      * @return the component aliases
      */
     public Map<String, String> getComponentAliases() {
-        final Map<String, String> toReturn = new HashMap<String, String>();
         final List<String> pairStrings =
                 readAndSplitList(this.properties.getProperty(COMPONENT_ALIASES));
+        final Map<String, String> toReturn = Maps.newHashMapWithExpectedSize(pairStrings.size());
         for (final String pairString: pairStrings) {
             final String[] elements = pairString.split(PAIR_SEPARATOR);
             toReturn.put(elements[0], elements[1]);
@@ -777,7 +779,7 @@ public void openConfigFileForEditing() throws ExternalSoftwareUnavailableExcepti
         ProcessUtils.start(XDG_OPEN, this.configFile);
         final String editorExecutable;
         if (this.properties.containsKey(SYSTEM_EDITOR)
-                && !this.properties.getProperty(SYSTEM_EDITOR).toString().isEmpty()) {
+                && !this.properties.getProperty(SYSTEM_EDITOR).isEmpty()) {
             editorExecutable = this.properties.getProperty(SYSTEM_EDITOR);
         } else if (System.getenv().containsKey(EDITOR)
                 && !System.getenv().get(EDITOR).isEmpty()) {
@@ -818,7 +820,7 @@ public void readConfigFile(final boolean overwriteIfNecessary) throws IOExceptio
             this.copyConfigToExpectedPath();
         }
         final Properties proposedProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             proposedProperties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -835,7 +837,7 @@ public void readConfigFile(final boolean overwriteIfNecessary) throws IOExceptio
                         + StringUtils.printCollection(configCompatibility.getSecondValue()));
             }
         }
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             this.properties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -849,7 +851,7 @@ private String replaceTypeAliases(final String s) {
         }
         final List<String> replacements =
                 readAndSplitList(this.properties.getProperty(SESSION_SPECIFIC_TYPE_ALIASES));
-        final Map<String, String> replMap = new HashMap<String, String>();
+        final Map<String, String> replMap = Maps.newHashMapWithExpectedSize(replacements.size());
         for (final String repl: replacements) {
             final String[] parts = repl.split(PAIR_SEPARATOR);
             if (parts.length != 2) {
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
index 1882a83..f699cb2 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
@@ -13,6 +13,7 @@
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Scanner;
+import java.util.Set;
 
 import org.apache.commons.io.IOUtils;
 import org.apache.logging.log4j.LogManager;
@@ -37,15 +38,13 @@
 import edu.kit.kastel.formal.virage.jobs.VirageProveJob;
 import edu.kit.kastel.formal.virage.prolog.JplFacade;
 import edu.kit.kastel.formal.virage.prolog.PrologParser;
-import edu.kit.kastel.formal.virage.types.Component;
-import edu.kit.kastel.formal.virage.types.ComposableModule;
 import edu.kit.kastel.formal.virage.types.CompositionProof;
-import edu.kit.kastel.formal.virage.types.CompositionalStructure;
 import edu.kit.kastel.formal.virage.types.DecompositionTree;
 import edu.kit.kastel.formal.virage.types.ExternalSoftwareUnavailableException;
 import edu.kit.kastel.formal.virage.types.FrameworkRepresentation;
 import edu.kit.kastel.formal.virage.types.InvalidConfigVersionException;
 import edu.kit.kastel.formal.virage.types.Property;
+import edu.kit.kastel.formal.virage.types.TypedAndParameterized;
 
 /**
  * A simple command line interface for ViRAGe.
@@ -239,11 +238,6 @@ public final class VirageCommandLineInterface implements VirageUserInterface {
      */
     private final Writer outputWriter;
 
-    /**
-     * This UI's thread.
-     */
-    private Thread thread;
-
     /**
      * Simple constructor.
      *
@@ -639,6 +633,14 @@ private String requestPropertyString() {
         return propertiesString;
     }
 
+    private static List<String> convertNames(final Set<? extends TypedAndParameterized> cs) {
+        final List<String> names = new ArrayList<String>(cs.size());
+        for (final TypedAndParameterized c: cs) {
+            names.add(StringUtils.indentWithTab(c.toString()));
+        }
+        return names;
+    }
+
     private String requestCompositionString() {
         boolean invalid = false;
         String compositionString = this
@@ -659,18 +661,16 @@ private String requestCompositionString() {
             }
         }
         if (compositionString.isEmpty() || invalid) {
-            final List<String> sortedStrings = new ArrayList<String>();
-            for (final Component c: this.core.getFrameworkRepresentation().getComponents()) {
-                sortedStrings.add(StringUtils.indentWithTab(c.toString()));
-            }
-            for (final ComposableModule c: this.core.getFrameworkRepresentation()
-                    .getComposableModules()) {
-                sortedStrings.add(StringUtils.indentWithTab(c.toString()));
-            }
-            for (final CompositionalStructure c: this.core.getFrameworkRepresentation()
-                    .getCompositionalStructures()) {
-                sortedStrings.add(StringUtils.indentWithTab(c.toString()));
-            }
+            final FrameworkRepresentation fr = this.core.getFrameworkRepresentation();
+            final List<String> compNames = convertNames(fr.getComponents());
+            final List<String> modNames = convertNames(fr.getComposableModules());
+            final List<String> structNames = convertNames(fr.getCompositionalStructures());
+            final List<String> sortedStrings =
+                    new ArrayList<String>(compNames.size() + modNames.size() + structNames.size());
+
+            sortedStrings.addAll(compNames);
+            sortedStrings.addAll(modNames);
+            sortedStrings.addAll(structNames);
             Collections.sort(sortedStrings);
             for (final String s: sortedStrings) {
                 this.displayMessage(s);
@@ -705,8 +705,7 @@ private VirageGenerateJob createGenerationQuery() {
 
     private VirageProveJob createProofQuery(final String composition, final String propertyString) {
         final List<String> properties = StringUtils.separate(StringUtils.COMMA, propertyString);
-        final VirageProveJob res = new VirageProveJob(this, composition, properties);
-        return res;
+        return new VirageProveJob(this, composition, properties);
     }
 
     private VirageJob<?> createIsabelleQuery() {
@@ -751,9 +750,7 @@ private VirageJob<?> createIsabelleQuery() {
         }
         this.core.submit(generateJob);
         generateJob.waitFor();
-        final VirageIsabelleVerifyJob verifyJob =
-                new VirageIsabelleVerifyJob(this, generateJob.getResult());
-        return verifyJob;
+        return new VirageIsabelleVerifyJob(this, generateJob.getResult());
     }
 
     /**
@@ -935,8 +932,7 @@ private void extractOrParseInputs() {
      */
     @Override
     public void launch() {
-        this.thread = new Thread(this, "vcli");
-        this.thread.start();
+        new Thread(this, "vcli").start();
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java
index 03a5fa5..3fd6c3c 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java
@@ -59,10 +59,8 @@ public List<List<SearchResult<BooleanWithUncertainty>>> analyzeComposition(
         // TODO Parallelize.
         final List<List<SearchResult<BooleanWithUncertainty>>> results =
                 new LinkedList<List<SearchResult<BooleanWithUncertainty>>>();
-        for (int i = 0; i < this.analyzers.size(); i++) {
-            final List<SearchResult<BooleanWithUncertainty>> result =
-                    this.analyzers.get(i).analyzeComposition(composition, properties);
-            results.add(result);
+        for (final CompositionAnalyzer analyzer: this.analyzers) {
+            results.add(analyzer.analyzeComposition(composition, properties));
         }
         return results;
     }
@@ -82,10 +80,8 @@ public List<SearchResult<DecompositionTree>> generateComposition(
         // TODO Parallelize.
         final List<SearchResult<DecompositionTree>> results =
                 new LinkedList<SearchResult<DecompositionTree>>();
-        for (int i = 0; i < this.analyzers.size(); i++) {
-            final SearchResult<DecompositionTree> result =
-                    this.analyzers.get(i).generateComposition(properties);
-            results.add(result);
+        for (final CompositionAnalyzer analyzer: this.analyzers) {
+            results.add(analyzer.generateComposition(properties));
         }
         return results;
     }
@@ -106,10 +102,8 @@ public List<List<CompositionProof>> proveClaims(final DecompositionTree composit
         }
         // TODO Parallelize.
         final List<List<CompositionProof>> results = new LinkedList<List<CompositionProof>>();
-        for (int i = 0; i < this.analyzers.size(); i++) {
-            final List<CompositionProof> result =
-                    this.analyzers.get(i).proveClaims(composition, properties);
-            results.add(result);
+        for (final CompositionAnalyzer analyzer: this.analyzers) {
+            results.add(analyzer.proveClaims(composition, properties));
         }
         return results;
     }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java
index 98249e0..cb7b498 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java
@@ -26,11 +26,6 @@ public final class IsabelleClientObserver implements Runnable {
      */
     private final IsabelleProofChecker listener;
 
-    /**
-     * The Isabelle client process.
-     */
-    private final Process isabelleClient;
-
     /**
      * The Isabelle event factory.
      */
@@ -49,13 +44,12 @@ public final class IsabelleClientObserver implements Runnable {
     private IsabelleClientObserver(final IsabelleProofChecker listenerValue,
             final Process isabelleClientValue) {
         this.listener = listenerValue;
-        this.isabelleClient = isabelleClientValue;
         this.factory = new IsabelleEventFactory();
         this.stdoutReader = new BufferedReader(
-                new InputStreamReader(this.isabelleClient.getInputStream(),
+                new InputStreamReader(isabelleClientValue.getInputStream(),
                                       StandardCharsets.UTF_8));
         this.stderrReader = new BufferedReader(
-                new InputStreamReader(this.isabelleClient.getErrorStream(),
+                new InputStreamReader(isabelleClientValue.getErrorStream(),
                                       StandardCharsets.UTF_8));
         final Thread thread = new Thread(this, "Isabelle Console Output");
         thread.start();
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
index 2dd2f3f..dfa09a4 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
@@ -335,7 +335,7 @@ private String prepareTheoryFile(final File theory, final String language) throw
         String newName = StringUtils.EMPTY;
         final Map<String, String> map =
                 IsabelleTheoryParser.getAllFunctionsAndDefinitions(theory.getCanonicalPath());
-        if (map.keySet().size() != 1) {
+        if (map.size() != 1) {
             throw new IllegalArgumentException();
         }
         for (final String definition: map.keySet()) {
@@ -359,7 +359,7 @@ private String prepareTheoryFile(final File theory, final String language) throw
         final List<String> lines = this.reader.readFileByLine(theory);
         for (int i = 0; i < lines.size(); i++) {
             final String line = lines.get(i);
-            if (StringUtils.removeWhitespace(line).equals(END)) {
+            if (END.equals(StringUtils.removeWhitespace(line))) {
                 lines.add(i, result);
                 break;
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
index 78072c0..3ed7462 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
@@ -12,6 +12,8 @@
 import org.apache.logging.log4j.LogManager;
 import org.apache.logging.log4j.Logger;
 
+import com.google.common.collect.Maps;
+
 import edu.kit.kastel.formal.util.Pair;
 import edu.kit.kastel.formal.util.StringUtils;
 import edu.kit.kastel.formal.util.SystemUtils;
@@ -80,17 +82,11 @@ public final class IsabelleFrameworkExtractor {
      */
     private final PrologParser parser;
 
-    /**
-     * List of Prolog Strings.
-     */
-    private final List<String> prologStrings;
-
     /**
      * Simple constructor.
      */
     public IsabelleFrameworkExtractor() {
         this.parser = new SimplePrologParser();
-        this.prologStrings = new LinkedList<String>();
     }
 
     private static String buildPrologClauseString(final String succedent,
@@ -185,18 +181,13 @@ private void convertCompositionRules(final FrameworkRepresentation framework,
                         continue thmLoop;
                     }
                 }
-                final List<String> prologStringList = new LinkedList<String>();
-                prologStringList.add(buildPrologClauseString(succedent, antecedents));
+                final String prologString = buildPrologClauseString(succedent, antecedents);
                 try {
-                    for (final String prologString: prologStringList) {
-                        final CompositionRule rule =
-                                new CompositionRule(thm.getKey(),
-                                                    thy.getKey().split(DOT_REGEX)[1]
-                                                            + IsabelleUtils.DOT_THY,
-                                                    this.parser.parseSingleClause(prologString));
-                        framework.add(rule);
-                        this.prologStrings.add(prologString);
-                    }
+                    framework.add(
+                            new CompositionRule(thm.getKey(),
+                                                thy.getKey().split(DOT_REGEX)[1]
+                                                        + IsabelleUtils.DOT_THY,
+                                                this.parser.parseSingleClause(prologString)));
                 } catch (final IllegalArgumentException e) {
                     this.logger.warn(e);
                 }
@@ -244,8 +235,7 @@ private static String convertIsabelleToProlog(final String s) {
                 break;
             }
         }
-        res += StringUtils.RIGHT_PAREN;
-        return res;
+        return res + StringUtils.RIGHT_PAREN;
     }
 
     /**
@@ -308,7 +298,7 @@ public FrameworkRepresentation extract(final String sessionDir, final String ses
                 computeTransitiveClosureOfComponentAliases() {
         final Map<String, String> input = ConfigReader.getInstance().getComponentAliases();
         final Map<PrologPredicate, List<PrologPredicate>> res =
-                    new HashMap<PrologPredicate, List<PrologPredicate>>();
+                Maps.newHashMapWithExpectedSize(input.size());
         for (final Map.Entry<String, String> in: input.entrySet()) {
             res.put(this.parser.parsePredicate(in.getKey()),
                     List.of(this.parser.parsePredicate(in.getValue())));
@@ -384,7 +374,7 @@ private static PrologPredicate replaceAlias(final PrologPredicate original,
         } catch (final IllegalArgumentException e) {
             unifiable = false;
         }
-        if (replacements.keySet().contains(PrologPredicate.ANONYMOUS)) {
+        if (replacements.containsKey(PrologPredicate.ANONYMOUS)) {
             replacements.remove(PrologPredicate.ANONYMOUS);
         }
         final boolean onlyOneVariable =
@@ -527,7 +517,6 @@ private static String replaceVariables(final String isaString) {
                                                    varName.toUpperCase());
             matcher = pattern.matcher(prologString);
         }
-        prologString = prologString.replace("?", StringUtils.EMPTY);
-        return prologString;
+        return prologString.replace("?", StringUtils.EMPTY);
     }
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
index 0365c7a..42363b7 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
@@ -173,11 +173,6 @@ public final class IsabelleProofChecker {
      */
     private static final String PARENT_NAME_VAR = "$PARENT_NAME";
 
-    /**
-     * The runtime.
-     */
-    private final Runtime runtime;
-
     /**
      * The server process.
      */
@@ -241,7 +236,6 @@ public final class IsabelleProofChecker {
     private IsabelleProofChecker(final String sessionNameValue, final String theoryPathValue)
             throws ExternalSoftwareUnavailableException, IsabelleBuildFailedException {
         this.theoryPath = theoryPathValue;
-        this.runtime = Runtime.getRuntime();
         this.solvers = ConfigReader.getInstance().getIsabelleTactics();
 
         // Use scala-isabelle to build the session, as my own solution
@@ -254,7 +248,7 @@ private IsabelleProofChecker(final String sessionNameValue, final String theoryP
                             ConfigReader.getInstance().getIsabelleExecutable(),
                             BUILD_TOOL, SYS_OPT, BROWSER_INFO,
                             BUILD_OPT, INCL_SEL_SESS_DIR, PWD_CMD);
-            final Process process = Runtime.getRuntime().exec(String.format(clString));
+            final Process process = ProcessUtils.exec(clString, false);
             process.waitFor();
 
             this.initServer();
@@ -391,7 +385,7 @@ private void initClient(final String localSessionName, final String localTheoryP
                         ConfigReader.getInstance().getIsabelleExecutable(),
                         CLIENT_TOOL, NAME_OPT, SERVER_NAME);
         this.sessionName = localSessionName;
-        this.client = this.runtime.exec(String.format(clString));
+        this.client = ProcessUtils.exec(clString, false);
         this.clientInput = this.client.getOutputStream();
 
         IsabelleClientObserver.start(this, this.client);
@@ -408,7 +402,7 @@ private void initServer() throws IOException, ExternalSoftwareUnavailableExcepti
                 StringUtils.printCollection2(
                         ConfigReader.getInstance().getIsabelleExecutable(),
                         SERVER_TOOL, NAME_OPT, SERVER_NAME);
-        this.server = this.runtime.exec(String.format(clString));
+        this.server = ProcessUtils.exec(clString, false);
         // The server will send a message when startup is finished.
         // Contents are irrelevant, just wait for it to appear.
         try (BufferedReader br = new BufferedReader(new InputStreamReader(
@@ -569,7 +563,7 @@ public Pair<Boolean, File> verifyTheoryFile(final File theory,
         String command = "use_theories " + sessionAndTheoriesCommand;
         this.sendCommandAndWaitForTermination(command);
 
-        final String result = this.lastEvent.getValue(IsabelleUtils.SUCCESS_STRING.toLowerCase());
+        final String result = this.lastEvent.getValue(IsabelleUtils.SUCCESS_STRING);
         if ("true".equals(result)) {
             LOGGER.info("Verification successful.");
             final String adHocSessionName = this.buildSessionRoot(
@@ -580,7 +574,7 @@ public Pair<Boolean, File> verifyTheoryFile(final File theory,
                     | ExternalSoftwareUnavailableException e) {
                 LOGGER.warn("No documentation could be generated.");
             }
-            return new Pair<Boolean, File>(true, theory);
+            return new Pair<Boolean, File>(Boolean.TRUE, theory);
         } else {
             LOGGER.info("Verification failed. Attempting to solve automatically "
                     + "by employing different solvers.");
@@ -608,7 +602,7 @@ public Pair<Boolean, File> verifyTheoryFile(final File theory,
                     }
                 }
             }
-            return new Pair<Boolean, File>(false, null);
+            return new Pair<Boolean, File>(Boolean.FALSE, null);
         }
     }
 
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java
index 95c83f0..b8737e2 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java
@@ -80,11 +80,6 @@ public class IsabelleProofGenerator {
      */
     private final IsabelleProofStepGenerator generator;
 
-    /**
-     * Functions and definitions from an Isabelle session.
-     */
-    private final Map<String, String> functionsAndDefinitions;
-
     /**
      * The parent theory generator.
      */
@@ -111,8 +106,7 @@ public IsabelleProofGenerator(final IsabelleTheoryGenerator parentValue,
             }
             setProofTemplate(writer.toString());
         }
-        this.functionsAndDefinitions = functionsAndDefinitionsValue;
-        this.generator = new IsabelleProofStepGenerator(this, this.functionsAndDefinitions);
+        this.generator = new IsabelleProofStepGenerator(this, functionsAndDefinitionsValue);
         this.parent = parentValue;
     }
 
@@ -123,13 +117,12 @@ private static synchronized void setProofTemplate(final String newTemplate) {
     private static String replaceVariables(final String theoremName, final String goal,
                                            final String proofSteps, final String subgoalIds,
                                            final String assumptions) {
-        String res = IsabelleProofGenerator.proofTemplate;
-        res = res.replace(VAR_THEOREM_NAME, theoremName);
-        res = res.replace(VAR_GOAL, goal);
-        res = res.replace(VAR_PROOF_STEPS, proofSteps);
-        res = res.replace(VAR_SUBGOAL_ID, subgoalIds);
-        res = res.replace(VAR_ASSUMPTIONS, assumptions);
-        return res;
+        return IsabelleProofGenerator.proofTemplate
+                .replace(VAR_THEOREM_NAME, theoremName)
+                .replace(VAR_GOAL, goal)
+                .replace(VAR_PROOF_STEPS, proofSteps)
+                .replace(VAR_SUBGOAL_ID, subgoalIds)
+                .replace(VAR_ASSUMPTIONS, assumptions);
     }
 
     /**
@@ -165,26 +158,24 @@ public String generateIsabelleProof(final CompositionProof proof) {
         final Set<String> assumptions = new HashSet<String>();
         final StringBuilder proofSteps = new StringBuilder();
         for (final CompositionProof subgoal: proof.getAllStepsDepthFirst()) {
-            if (subgoal.getAllCompositionRules().size() == 1) {
-                final CompositionRule rule = subgoal.getAllCompositionRules().iterator().next();
-                if (rule.getOrigin().equals(ExtendedPrologStrings.ASSUMPTION)) {
-                    final Property p =
-                            this.parent.getFramework().getProperty(rule.getSuccedent().getName());
-                    String newAssumptions =
-                            StringUtils.addSpace(StringUtils.QUOTATION
-                                                    + rule.getSuccedent().getName());
+            final Set<CompositionRule> rules = subgoal.getAllCompositionRules();
+            if (rules.size() == 1) {
+                final CompositionRule rule = rules.iterator().next();
+                if (ExtendedPrologStrings.ASSUMPTION.equals(rule.getOrigin())) {
+                    final String succName = rule.getSuccedent().getName();
+                    final Property p = this.parent.getFramework().getProperty(succName);
+                    String newAssumptions = StringUtils.addSpace(StringUtils.QUOTATION + succName);
                     for (final ComponentType child: p.getParameters()) {
+                        final String childVar =
+                                this.getParent().getTypedVariables().get(child.getName());
                         // Only parameters defined within the module definition can be referenced,
                         // so if
                         // any others
                         // occur, we skip.
-                        if (!this.getParent().getTypedVariables().containsKey(child.getName())) {
+                        if (childVar == null) {
                             break;
                         }
-                        newAssumptions +=
-                                StringUtils.addSpace(
-                                        this.getParent().getTypedVariables().get(child.getName()))
-                                    + StringUtils.QUOTATION;
+                        newAssumptions += StringUtils.addSpace(childVar) + StringUtils.QUOTATION;
                         assumptions.add(newAssumptions);
                     }
                     // This is needed to not add subgoal identities for these "virtual" goals.
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java
index 3d5cf58..332e912 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java
@@ -118,13 +118,12 @@ private static synchronized void setProotStepTemplate(final String newTemplate)
     private static String replaceVariables(final String goalId, final String goal,
                                            final String subgoalIds, final String rule,
                                            final String solver) {
-        String res = IsabelleProofStepGenerator.proofStepTemplate;
-        res = res.replace(VAR_GOAL_ID, goalId);
-        res = res.replace(VAR_GOAL, goal);
-        res = res.replace(VAR_SUBGOAL_IDS, subgoalIds);
-        res = res.replace(VAR_RULE, rule);
-        res = res.replace(VAR_SOLVER, solver);
-        return res;
+        return IsabelleProofStepGenerator.proofStepTemplate
+                .replace(VAR_GOAL_ID, goalId)
+                .replace(VAR_GOAL, goal)
+                .replace(VAR_SUBGOAL_IDS, subgoalIds)
+                .replace(VAR_RULE, rule)
+                .replace(VAR_SOLVER, solver);
     }
 
     /**
@@ -141,7 +140,7 @@ public String generateIsabelleProofStep(final CompositionProof step) {
         final Map<String, Set<String>> goalMap =
                 IsabelleUtils
                 .translatePrologToIsabelle(this.functionsAndDefinitions, predicate.toString());
-        if (goalMap.keySet().size() != 1) {
+        if (goalMap.size() != 1) {
             throw new IllegalArgumentException();
         }
         String goal = StringUtils.EMPTY;
@@ -151,9 +150,9 @@ public String generateIsabelleProofStep(final CompositionProof step) {
 
         final StringBuilder subgoalIds = new StringBuilder();
         for (final CompositionProof subgoal: step.getSubgoals()) {
-            if (subgoal.getAllCompositionRules().size() == 1) {
-                final CompositionRule rule = subgoal.getAllCompositionRules().iterator().next();
-                if (rule.getOrigin().equals(ExtendedPrologStrings.ASSUMPTION)) {
+            final Set<CompositionRule> rules = subgoal.getAllCompositionRules();
+            if (rules.size() == 1) {
+                if (ExtendedPrologStrings.ASSUMPTION.equals(rules.iterator().next().getOrigin())) {
                     continue;
                 }
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
index a66df73..6ac866c 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
@@ -196,7 +196,7 @@ private static String findImportsFromCompositionRules(final List<CompositionProo
                 // Isabelle expects imports without suffix.
                 originStrings.add(origin.replace(IsabelleUtils.DOT_THY, StringUtils.EMPTY));
             } else {
-                if (origin.equals(ExtendedPrologStrings.UNPROVEN)) {
+                if (ExtendedPrologStrings.UNPROVEN.equals(origin)) {
                     // Proof relies on unproven facts, add a comment explaining this.
                     usingUnprovenFacts = true;
                 }
@@ -209,23 +209,11 @@ private static String findImportsFromCompositionRules(final List<CompositionProo
         }
         String res = r.toString();
         if (usingUnprovenFacts) {
-            final String star = "*";
-            final String separatorStart = StringUtils.addSpace(star);
-            final String separatorEnd = StringUtils.prefixSpace(star);
-            final String separatorLine =
-                    StringUtils.parenthesize(
-                            StringUtils.repeat(22, StringUtils.addSpace(star)) + star);
             res += System.lineSeparator() + System.lineSeparator()
-                    + separatorLine + System.lineSeparator()
-                    + StringUtils.parenthesize(separatorStart
-                            + "Some proofs appear to rely on facts not yet" + separatorEnd)
-                     + System.lineSeparator()
-                    + StringUtils.parenthesize(separatorStart
-                            + " proven within Isabelle/HOL. Check Isabelle" + separatorEnd)
-                     + System.lineSeparator()
-                    + StringUtils.parenthesize(separatorStart
-                            + "    error messages for more information.   " + separatorEnd)
-                     + System.lineSeparator() + separatorLine;
+                    + StringUtils.isabelleCommentBlock(
+                            "Some proofs appear to rely on facts not yet",
+                            " proven within Isabelle/HOL. Check Isabelle",
+                            "    error messages for more information.   ");
         }
         return res;
     }
@@ -264,8 +252,7 @@ private static String generateModuleParameterTypes(final List<String> modulePara
         for (final String type: moduleParamTypesList) {
             moduleParamTypes.append(type);
         }
-        final String moduleParameterTypeString = moduleParamTypes.toString();
-        return moduleParameterTypeString;
+        return moduleParamTypes.toString();
     }
 
     private static Pair<String, String> buildModuleDef(final Map<String, Set<String>> moduleDefMap,
@@ -302,15 +289,14 @@ private static String replaceVariables(final String theoryName, final String imp
                                            final String moduleParamTypes, final String moduleName,
                                            final String moduleParameters, final String moduleDef,
                                            final String proofs) {
-        String res = IsabelleTheoryGenerator.theoryTemplate;
-        res = res.replace(VAR_PROOFS, proofs);
-        res = res.replace(VAR_THEORY_NAME, theoryName);
-        res = res.replace(VAR_IMPORTS, imports);
-        res = res.replace(VAR_MODULE_PARAM_TYPES, moduleParamTypes);
-        res = res.replace(VAR_MODULE_DEF, moduleDef);
-        res = res.replace(VAR_MODULE_NAME, moduleName);
-        res = res.replace(VAR_MODULE_PARAMETERS, moduleParameters);
-        return res;
+        return IsabelleTheoryGenerator.theoryTemplate
+                .replace(VAR_PROOFS, proofs)
+                .replace(VAR_THEORY_NAME, theoryName)
+                .replace(VAR_IMPORTS, imports)
+                .replace(VAR_MODULE_PARAM_TYPES, moduleParamTypes)
+                .replace(VAR_MODULE_DEF, moduleDef)
+                .replace(VAR_MODULE_NAME, moduleName)
+                .replace(VAR_MODULE_PARAMETERS, moduleParameters);
     }
 
     /**
@@ -358,7 +344,7 @@ public File generateTheoryFile(final String passedComposition,
         final Map<String, Set<String>> moduleDefMap =
                 IsabelleUtils.translatePrologToIsabelle(this.functionsAndDefinitions,
                                                         proofPredicate);
-        if (moduleDefMap.keySet().size() != 1) {
+        if (moduleDefMap.size() != 1) {
             throw new IllegalArgumentException();
         }
         final Pair<String, String> moduleDefResult =
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java
index f3b48b9..2d4e0e6 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java
@@ -112,7 +112,7 @@ public static List<String> getImportsFromTheoryFile(final File theory) throws IO
             if (line.contains(IsabelleUtils.IMPORTS)) {
                 final String[] splits = line.split(StringUtils.SPACE);
                 for (final String split: splits) {
-                    if (split.isBlank() || split.equals(IsabelleUtils.IMPORTS)) {
+                    if (split.isBlank() || IsabelleUtils.IMPORTS.equals(split)) {
                         continue;
                     }
                     res.add(split);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
index d9fffe3..78c94cc 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
@@ -39,7 +39,7 @@ public final class IsabelleUtils {
     /**
      * String used by Isabelle to denote successful termination.
      */
-    public static final String SUCCESS_STRING = "OK ";
+    public static final String SUCCESS_STRING = "ok ";
 
     /**
      * String used by Isabelle to denote the line stating imported theories.
@@ -81,6 +81,31 @@ public final class IsabelleUtils {
      */
     public static final String THEORY_NAME_SEPARATOR = StringUtils.PERIOD;
 
+    /**
+     * Suffix for definition-unfolding rules.
+     */
+    public static final String DEFINITION_SUFFIX = "_def";
+
+    /**
+     * Suffix for relation rules.
+     */
+    public static final String RELATION_SUFFIX = "_rel";
+
+    /**
+     * Suffix for raw rules.
+     */
+    public static final String RAW_SUFFIX = "_raw";
+
+    /**
+     * Suffix for graph rules.
+     */
+    public static final String GRAPH_SUFFIX = "_graph";
+
+    /**
+     * Suffix for sum rules.
+     */
+    public static final String SUM_SUFFIX = "_sumC";
+
     /**
      * Main theory.
      */
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
index f89902b..eb5899e 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
@@ -18,6 +18,8 @@
 import org.apache.logging.log4j.LogManager;
 import org.apache.logging.log4j.Logger;
 
+import com.google.common.collect.Maps;
+
 import de.unruh.isabelle.control.Isabelle;
 import de.unruh.isabelle.control.Isabelle.Setup;
 import de.unruh.isabelle.control.IsabelleBuildException;
@@ -342,13 +344,12 @@ private static Setup setup(final String isabelleHome, final boolean verbose,
                                final String sessionDir,
                                final Iterable<String> sessionDirs) {
         Setup s = JIsabelle.setup(Path.of(isabelleHome));
-        s = JIsabelle.setupSetVerbose(true, s);
+        s = JIsabelle.setupSetVerbose(verbose, s);
         s = JIsabelle.setupSetUserDir(Path.of(isabelleDirectory), s);
         s = JIsabelle.setupSetWorkingDirectory(Path.of(sessionDir), s);
         s = JIsabelle.setupSetSessionRoots(toPaths(sessionDirs), s);
-        s = JIsabelleWrapper.setupSetLogic(sessionName, s);
-        s = JIsabelle.setupSetBuild(true, s);
-        return s;
+        s = JIsabelle.setupSetLogic(sessionName, s);
+        return JIsabelle.setupSetBuild(true, s);
     }
 
     private static synchronized void setIsabelleInstance(final Isabelle instance) {
@@ -458,7 +459,7 @@ public static void buildSession(final String sessionDir, final String sessionNam
      * Extracts functions and definitions from a given framework in Isabelle.
      */
     public void extractFunctionsAndDefinitions() {
-        this.functionsAndDefinitions = new HashMap<String, Map<String, String>>();
+        this.functionsAndDefinitions = Maps.newHashMapWithExpectedSize(this.theoryNames.size());
         final MLFunction<Theory, scala.collection.immutable.List<String>> mlFunToExtractAllNames =
                 MLValue.compileFunction(funToExtractAllNames(THY), ScalaIsabelleFacade.isabelle,
                                         Implicits.theoryConverter(), LIST_CONVERTER);
@@ -468,8 +469,6 @@ public void extractFunctionsAndDefinitions() {
                                     Implicits.theoryConverter(), PAIR_TO_LIST_CONVERTER);
         for (final String thy: this.theoryNames) {
             final String thyWithoutSession = thy.split("\\.")[1];
-            final Map<String, String> toBeFilled = new HashMap<String, String>();
-            this.functionsAndDefinitions.put(thy, toBeFilled);
             final Context ctxt = Context.apply(thy, ScalaIsabelleFacade.isabelle);
             final Theory theory = ctxt.theoryOf(ScalaIsabelleFacade.isabelle);
             final List<String> names =
@@ -480,41 +479,42 @@ public void extractFunctionsAndDefinitions() {
                     JavaConverters.asJava(mlFunToExtractSigns
                             .apply(theory.mlValue(), ScalaIsabelleFacade.isabelle)
                             .retrieveNow(PAIR_TO_LIST_CONVERTER, ScalaIsabelleFacade.isabelle));
-            for (int i = 0; i < names.size(); i++) {
-                String name = names.get(i);
+            final Map<String, String> toBeFilled = Maps.newHashMapWithExpectedSize(names.size());
+            this.functionsAndDefinitions.put(thy, toBeFilled);
+            for (final String name: names) {
                 if (!name.startsWith(thyWithoutSession + IsabelleUtils.THEORY_NAME_SEPARATOR)) {
                     continue;
                 }
-                final String defSfx = "_def";
-                final String relSfx = "_rel";
-                final String rawSfx = "_raw";
-                final String graphSfx = "_graph";
-                final String sumSfx = "_sumC";
-                final String defRawSfx = defSfx + rawSfx;
-                if (name.endsWith(defSfx) || name.endsWith(defRawSfx)) {
-                    name = name.replace(rawSfx, StringUtils.EMPTY);
-                    name = name.replace(defSfx, StringUtils.EMPTY);
-                    if (name.endsWith(relSfx) || name.endsWith(graphSfx) || name.endsWith(sumSfx)) {
+                if (name.endsWith(IsabelleUtils.DEFINITION_SUFFIX)
+                        || name.endsWith(IsabelleUtils.DEFINITION_SUFFIX
+                                            + IsabelleUtils.RAW_SUFFIX)) {
+                    final String def =
+                            name.replace(IsabelleUtils.RAW_SUFFIX, StringUtils.EMPTY)
+                                .replace(IsabelleUtils.DEFINITION_SUFFIX, StringUtils.EMPTY);
+                    if (def.endsWith(IsabelleUtils.RELATION_SUFFIX)
+                            || def.endsWith(IsabelleUtils.GRAPH_SUFFIX)
+                            || def.endsWith(IsabelleUtils.SUM_SUFFIX)) {
                         continue;
                     }
                     String type = StringUtils.EMPTY;
                     for (final Tuple2<String, String> tup: types) {
-                        if (tup._1.equals(name)) {
+                        if (tup._1.equals(def)) {
                             type = tup._2;
+                            break;
                         }
                     }
                     type = type.replace(
                             IsabelleUtils.SET + IsabelleUtils.THEORY_NAME_SEPARATOR
                             + StringUtils.func(SET_FUN, ALTERNATIVE),
                             ALTERNATIVES_NAME);
-                    toBeFilled.put(name, type);
+                    toBeFilled.put(def, type);
                 }
             }
         }
     }
 
     private void extractTheorems() {
-        this.theorems = new HashMap<String, Map<String, String>>();
+        this.theorems = Maps.newHashMapWithExpectedSize(this.theoryNames.size());
         final MLFunction<Theory, scala.collection.immutable.List<String>> mlFun =
                 MLValue.compileFunction(
                         addSpace(fnTo(THY,
@@ -531,13 +531,13 @@ private void extractTheorems() {
                         ScalaIsabelleFacade.isabelle,
                         Implicits.thmConverter(), STRING_CONVERTER);
         for (final String thyName: this.theoryNames) {
-            final Map<String, String> toBeFilled = new HashMap<String, String>();
-            this.theorems.put(thyName, toBeFilled);
             final Context ctxt = Context.apply(thyName, ScalaIsabelleFacade.isabelle);
             final Theory theory = ctxt.theoryOf(ScalaIsabelleFacade.isabelle);
             final scala.collection.immutable.List<String> defs =
                     mlFun.apply(theory, ScalaIsabelleFacade.isabelle, Implicits.theoryConverter())
                     .retrieveNow(LIST_CONVERTER, ScalaIsabelleFacade.isabelle);
+            final Map<String, String> toBeFilled = new HashMap<String, String>(defs.length());
+            this.theorems.put(thyName, toBeFilled);
             for (int i = 0; i < defs.length(); i++) {
                 final String thmName = defs.apply(i);
                 // This filters out all theorems generated by Isabelle for internal usage.
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
index a2006b9..4d948e6 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
@@ -55,7 +55,7 @@ public final String getDescription() {
     public final String presentConcreteResult() {
         final List<Boolean> hasProperties = new LinkedList<Boolean>();
         for (int i = 0; i < this.unaryProperties.size(); i++) {
-            hasProperties.add(false);
+            hasProperties.add(Boolean.FALSE);
         }
         for (final List<SearchResult<BooleanWithUncertainty>> resultList: this.getResult()) {
             for (int i = 0; i < resultList.size(); i++) {
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java
index fbffd5d..6be49dd 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java
@@ -35,16 +35,6 @@ public final class VirageAnalyzeJob
      */
     private final DecompositionTree tree;
 
-    /**
-     * The compositional framework.
-     */
-    private FrameworkRepresentation framework;
-
-    /**
-     * The search manager.
-     */
-    private VirageSearchManager manager;
-
     /**
      * Simple constructor.
      *
@@ -61,13 +51,14 @@ public VirageAnalyzeJob(final VirageUserInterface issuer, final String treeValue
 
     @Override
     public void concreteExecute() {
-        this.framework = this.getExecutingCore().getFrameworkRepresentation();
-        this.manager = this.getExecutingCore().getSearchManager();
+        final FrameworkRepresentation framework =
+                this.getExecutingCore().getFrameworkRepresentation();
+        final VirageSearchManager manager = this.getExecutingCore().getSearchManager();
         this.properties = new LinkedList<Property>();
         for (final String s: this.propertyStrings) {
-            this.properties.add(this.framework.getProperty(s));
+            this.properties.add(framework.getProperty(s));
         }
-        this.setResult(this.manager.analyzeComposition(this.tree, this.properties));
+        this.setResult(manager.analyzeComposition(this.tree, this.properties));
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java
index d218d95..61fd06b 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java
@@ -11,11 +11,6 @@
  * @author VeriVote
  */
 public final class VirageGenerateCCodeJob extends VirageJobWithExplicitResult<File> {
-    /**
-     * The C code generator.
-     */
-    private CCodeGenerator generator;
-
     /**
      * The generation from which code shall be generated.
      */
@@ -34,8 +29,8 @@ public VirageGenerateCCodeJob(final VirageUserInterface issuer, final String com
 
     @Override
     protected void concreteExecute() {
-        this.generator = this.getExecutingCore().getCCodeGenerator();
-        this.setResult(this.generator.getCCodeFromComposition(this.composition));
+        final CCodeGenerator generator = this.getExecutingCore().getCCodeGenerator();
+        this.setResult(generator.getCCodeFromComposition(this.composition));
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java
index ba96d9d..2235aec 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java
@@ -29,11 +29,6 @@ public final class VirageGenerateJob
      */
     private List<Property> properties;
 
-    /**
-     * The search manager to be used.
-     */
-    private VirageSearchManager manager;
-
     /**
      * Simple constructor.
      *
@@ -48,12 +43,12 @@ public VirageGenerateJob(final VirageUserInterface issuer, final List<String> pr
     @Override
     public void concreteExecute() {
         final VirageCore execCore = this.getExecutingCore();
-        this.manager = execCore.getSearchManager();
+        final VirageSearchManager manager = execCore.getSearchManager();
         this.properties = new LinkedList<Property>();
         for (final String s: this.propertyStrings) {
             this.properties.add(execCore.getFrameworkRepresentation().getProperty(s));
         }
-        this.setResult(this.manager.generateComposition(this.properties));
+        this.setResult(manager.generateComposition(this.properties));
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java
index 3169411..9d795f9 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java
@@ -26,11 +26,6 @@ public final class VirageIsabelleGenerateJob extends VirageJobWithExplicitResult
      */
     private final String outputPath;
 
-    /**
-     * The Isabelle theory generator to be used.
-     */
-    private IsabelleTheoryGenerator generator;
-
     /**
      * Simple constructor.
      *
@@ -65,8 +60,9 @@ public String presentConcreteResult() {
 
     @Override
     protected void concreteExecute() {
-        this.generator = this.getExecutingCore().getIsabelleTheoryGenerator();
-        this.setResult(this.generator
+        final IsabelleTheoryGenerator generator =
+                this.getExecutingCore().getIsabelleTheoryGenerator();
+        this.setResult(generator
                 .generateTheoryFile(this.composition, this.proofs, this.outputPath));
     }
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
index 12608c0..c777bd4 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
@@ -13,11 +13,6 @@
  * @author VeriVote
  */
 public final class VirageIsabelleGenerateScalaJob extends VirageJobWithExplicitResult<File> {
-    /**
-     * The Isabelle Scala generator.
-     */
-    private IsabelleCodeGenerator generator;
-
     /**
      * The composition from which an implementation shall be generated.
      */
@@ -52,7 +47,7 @@ public String presentConcreteResult() {
 
     @Override
     protected void concreteExecute() throws CodeGenerationFailedException {
-        this.generator = this.getExecutingCore().getIsabelleCodeGenerator();
-        this.setResult(this.generator.generateScalaCodeAndCompile(this.composition));
+        final IsabelleCodeGenerator generator = this.getExecutingCore().getIsabelleCodeGenerator();
+        this.setResult(generator.generateScalaCodeAndCompile(this.composition));
     }
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java
index 3caef28..c7c8254 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java
@@ -5,8 +5,8 @@
 
 import edu.kit.kastel.formal.util.Pair;
 import edu.kit.kastel.formal.virage.core.ConfigReader;
+import edu.kit.kastel.formal.virage.core.VirageCore;
 import edu.kit.kastel.formal.virage.core.VirageUserInterface;
-import edu.kit.kastel.formal.virage.isabelle.IsabelleProofChecker;
 
 /**
  * A {@link VirageJob} that invokes Isabelle to automatically attempt proof verification.
@@ -15,11 +15,6 @@
  */
 public final class VirageIsabelleVerifyJob
         extends VirageJobWithExplicitResult<Pair<Boolean, File>> {
-    /**
-     * The checker used to verify the proof.
-     */
-    private IsabelleProofChecker checker;
-
     /**
      * The theory file to be checked.
      */
@@ -59,10 +54,8 @@ public String presentConcreteResult() {
 
     @Override
     protected void concreteExecute() throws IOException, InterruptedException {
-        this.checker = this.getExecutingCore().getIsabelleProofChecker();
-
-        this.setResult(this.checker.verifyTheoryFile(this.file,
-                this.getExecutingCore().getFrameworkRepresentation()));
+        final VirageCore core = this.getExecutingCore();
+        this.setResult(core.getIsabelleProofChecker()
+                        .verifyTheoryFile(this.file, core.getFrameworkRepresentation()));
     }
-
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java
index bf011ae..08f6f02 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java
@@ -155,8 +155,7 @@ public String presentResult() {
         } else {
             res += StringUtils.sentence("Something went wrong while executing this job");
         }
-        res += StringUtils.repeat(StringUtils.TEN, StringUtils.DASH);
-        return res;
+        return res + StringUtils.repeat(StringUtils.TEN, StringUtils.DASH);
     }
 
     /**
@@ -198,8 +197,7 @@ public String toString() {
         res += printPropertyLine(finishedString, SystemUtils.getTime(lastMeasurement));
         res += printPropertyLine("Elapsed time:", elapsedTime);
         res += printPropertyLine(StringUtils.repeat(StringUtils.FIVE, StringUtils.DASH));
-        res += printPropertyLine("State:", this.state.toString());
-        return res;
+        return res + printPropertyLine("State:", this.state.toString());
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java
index a156ac9..0632b7b 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java
@@ -33,11 +33,6 @@ public final class VirageProveJob
      */
     private final DecompositionTree tree;
 
-    /**
-     * The framework representation.
-     */
-    private FrameworkRepresentation framework;
-
     /**
      * Simple constructor.
      *
@@ -54,10 +49,11 @@ public VirageProveJob(final VirageUserInterface issuer, final String treeValue,
 
     @Override
     public void concreteExecute() {
-        this.framework = this.getExecutingCore().getFrameworkRepresentation();
+        final FrameworkRepresentation framework =
+                this.getExecutingCore().getFrameworkRepresentation();
         this.properties = new LinkedList<Property>();
         for (final String s: this.propertyStrings) {
-            this.properties.add(this.framework.getProperty(s));
+            this.properties.add(framework.getProperty(s));
         }
         this.setResult(this.getExecutingCore().getSearchManager().proveClaims(this.tree,
                                                                               this.properties));
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java
index 1b200d7..75adaa1 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java
@@ -20,6 +20,8 @@
 import org.jpl7.Term;
 import org.jpl7.Variable;
 
+import com.google.common.collect.Maps;
+
 import edu.kit.kastel.formal.util.StringUtils;
 import edu.kit.kastel.formal.util.SystemUtils;
 import edu.kit.kastel.formal.virage.core.ConfigReader;
@@ -163,19 +165,22 @@ public static String findUnusedVariable(final String queryString) {
      */
     public static Map<String, String> parseReplacementMap(final String variable,
                                                           final Map<String, String> map) {
-        final Map<String, String> result = new HashMap<String, String>();
-        // Looks like this: ['='(_108, 1)]
-        // _NUMBER is an alias for a variable.
         String replacementString = map.get(variable);
         replacementString = StringUtils.removeWhitespace(replacementString);
         replacementString = replacementString.substring(1, replacementString.length() - 1);
+
+        final String[] replacements = replacementString.isEmpty()
+                ? new String[] {} : replacementString.split("\\'=\\'");
+        // replacements[0] is always empty, as the String starts with the splitting
+        // regular expression.
+
+        final Map<String, String> result = new HashMap<String, String>(replacements.length);
+        // Looks like this: ['='(_108, 1)]
+        // _NUMBER is an alias for a variable.
         if (replacementString.isEmpty()) {
             // List of replacements is empty.
             return result;
         }
-        final String[] replacements = replacementString.split("\\'=\\'");
-        // replacements[0] is always empty, as the String starts with the splitting
-        // regular expression.
         for (int i = 1; i < replacements.length; i++) {
             String replacement = replacements[i];
             if (replacement.endsWith(StringUtils.COMMA)) {
@@ -316,18 +321,18 @@ public SearchResult<Boolean> factQuery(final String queryString, final long time
             SearchResult<Boolean> toReturn = null;
             if (result == null) {
                 // No solution, query failed.
-                toReturn = new SearchResult<Boolean>(QueryState.FAILED, false);
+                toReturn = new SearchResult<Boolean>(QueryState.FAILED, Boolean.FALSE);
             } else if (!result.containsKey(unusedVariable)) {
                 // Empty Map was received, timeout.
                 toReturn = new SearchResult<Boolean>(QueryState.TIMEOUT, null);
-            } else if (result.get(unusedVariable).equals(DEPTH_LIMIT_EXCEEDED)) {
+            } else if (DEPTH_LIMIT_EXCEEDED.equals(result.get(unusedVariable))) {
                 // Depth limit exceeded, increase and try again.
                 maxDepth++;
                 continue;
             } else if (StringUtils.isNumeric(result.get(unusedVariable))) {
                 // Query succeeded.
                 result.remove(unusedVariable);
-                toReturn = new SearchResult<Boolean>(QueryState.SUCCESS, true);
+                toReturn = new SearchResult<Boolean>(QueryState.SUCCESS, Boolean.TRUE);
             }
             if (toReturn != null) {
                 return toReturn;
@@ -387,7 +392,7 @@ public SearchResult<Map<String, String>> iterativeDeepeningQuery(final String qu
                 // Empty Map was received, timeout.
                 toReturn = new SearchResult<Map<String, String>>(QueryState.TIMEOUT, null);
             } else {
-                if (result.get(unusedVariable).equals(DEPTH_LIMIT_EXCEEDED)) {
+                if (DEPTH_LIMIT_EXCEEDED.equals(result.get(unusedVariable))) {
                     // Depth limit exceeded, increase and try again.
                     maxDepth++;
                     continue;
@@ -452,7 +457,8 @@ public Map<String, String> simpleQueryWithTimeout(final String queryString,
             if (query.hasMoreSolutions()) {
                 try {
                     final Map<String, Term> solution = query.nextSolution();
-                    final Map<String, String> result = new HashMap<String, String>();
+                    final Map<String, String> result =
+                            Maps.newHashMapWithExpectedSize(solution.size());
                     for (final Map.Entry<String, Term> entry: solution.entrySet()) {
                         final String termString = entry.getValue().toString();
                         result.put(entry.getKey(), termString);
@@ -473,7 +479,7 @@ public Map<String, String> simpleQueryWithTimeout(final String queryString,
                 return null;
             }
         } catch (final PrologException e) {
-            if (!e.getMessage().equals("PrologException: time_limit_exceeded")) {
+            if (!"PrologException: time_limit_exceeded".equals(e.getMessage())) {
                 LOGGER.error("A Prolog error occured.");
                 throw e;
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java
index ff5d891..b5dd558 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java
@@ -143,8 +143,7 @@ public int hashCode() {
         final int prime = 31;
         int result = 1;
         result = prime * result + ((this.antecedents == null) ? 0 : this.antecedents.hashCode());
-        result = prime * result + ((this.succedent == null) ? 0 : this.succedent.hashCode());
-        return result;
+        return prime * result + ((this.succedent == null) ? 0 : this.succedent.hashCode());
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java
index ef368ee..3b899f9 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java
@@ -116,15 +116,6 @@ public List<PrologPredicate> getParameters() {
         return this.parameters;
     }
 
-    /**
-     * Checks whether a PrologPredicate is a variable.
-     *
-     * @return true if this is a variable, false otherwise
-     */
-    public boolean isVariable() {
-        return PrologPredicate.isVariable(this.name);
-    }
-
     /**
      * Checks whether a string represents a Prolog variable.
      *
@@ -135,6 +126,15 @@ public static boolean isVariable(final String s) {
         return s.matches("[A-Z_][a-zA-Z_0-9]*");
     }
 
+    /**
+     * Checks whether a PrologPredicate is a variable.
+     *
+     * @return true if this is a variable, false otherwise
+     */
+    public boolean isVariable() {
+        return isVariable(this.name);
+    }
+
     /**
      * Sets a new name for the Prolog predicate.
      *
@@ -164,8 +164,9 @@ public static PrologPredicate copy(final PrologPredicate pred) {
      * @param replacements the replacements
      */
     public void replaceVariables(final Map<String, String> replacements) {
-        if (replacements.keySet().contains(this.name)) {
-            this.name = replacements.get(this.name);
+        final String s = replacements.get(this.name);
+        if (s != null) {
+            this.name = s;
         } else {
             for (final PrologPredicate parameter: this.parameters) {
                 parameter.replaceVariables(replacements);
@@ -179,8 +180,7 @@ public int hashCode() {
         int result = 1;
         result = prime * result + this.arity;
         result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        result = prime * result + ((this.parameters == null) ? 0 : this.parameters.hashCode());
-        return result;
+        return prime * result + ((this.parameters == null) ? 0 : this.parameters.hashCode());
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java
index 2ca86e9..b399f31 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java
@@ -91,8 +91,7 @@ private FrameworkRepresentation parseFramework(final List<String> representation
         final List<String> propertySection = new LinkedList<String>();
         final List<String> compositionRuleSection = new LinkedList<String>();
 
-        for (int lineNumber = 0; lineNumber < representation.size(); lineNumber++) {
-            String currentLine = representation.get(lineNumber);
+        for (final String currentLine: representation) {
             // Skip comments
             if (currentLine.startsWith("%%")) {
                 continue;
@@ -101,29 +100,29 @@ private FrameworkRepresentation parseFramework(final List<String> representation
                 this.findTheoryPath(currentLine, framework);
             }
             // Remove Prolog comment markings, they are not necessary any more.
-            currentLine = currentLine.replace("% ", StringUtils.EMPTY);
-            currentLine = currentLine.replace("%", StringUtils.EMPTY);
+            final String line =
+                    currentLine.replace("% ", StringUtils.EMPTY).replace("%", StringUtils.EMPTY);
             // Skip empty lines. (Careful: currentLine is not actually sanitized after
             // this!)
-            if (this.sanitizeLine(currentLine).isEmpty()) {
+            if (this.sanitizeLine(line).isEmpty()) {
                 continue;
             }
-            state = this.newState(currentLine, state);
+            state = this.newState(line, state);
             switch (state) {
             case COMPOSITION_TYPE:
-                compositionTypeSection.add(currentLine);
+                compositionTypeSection.add(line);
                 break;
             case COMPOSABLE_MODULE:
-                composableModuleSection.add(currentLine);
+                composableModuleSection.add(line);
                 break;
             case COMPOSITIONAL_STRUCTURE:
-                compositionalStructureSection.add(currentLine);
+                compositionalStructureSection.add(line);
                 break;
             case PROPERTY:
-                propertySection.add(currentLine);
+                propertySection.add(line);
                 break;
             case COMPOSITION_RULE:
-                compositionRuleSection.add(currentLine);
+                compositionRuleSection.add(line);
                 break;
             default: // skip operation, invalid call.
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java
index 0c626f2..1b8192d 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java
@@ -60,9 +60,8 @@ public String getName() {
     @Override
     public int hashCode() {
         final int prime = 31;
-        int result = 1;
-        result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        return result;
+        final int result = 1;
+        return prime * result + ((this.name == null) ? 0 : this.name.hashCode());
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
index 3a0aaae..63a6719 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
@@ -5,6 +5,8 @@
 import java.util.List;
 import java.util.Set;
 
+import com.google.common.collect.Sets;
+
 import edu.kit.kastel.formal.util.StringUtils;
 import edu.kit.kastel.formal.virage.isabelle.IsabelleUtils;
 
@@ -79,8 +81,8 @@ public Set<CompositionRule> getAllCompositionRules() {
      * @return all origins
      */
     public Set<String> getAllOrigins() {
-        final Set<String> origins = new HashSet<String>();
         final Set<CompositionRule> allRules = this.getAllCompositionRules();
+        final Set<String> origins = Sets.newHashSetWithExpectedSize(allRules.size());
         for (final CompositionRule localRule: allRules) {
             origins.add(localRule.getOrigin());
         }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java
index cb2d037..7cc73c3 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java
@@ -149,8 +149,7 @@ public int hashCode() {
         int result = 1;
         result = prime * result + ((this.clause == null) ? 0 : this.clause.hashCode());
         result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        result = prime * result + ((this.origin == null) ? 0 : this.origin.hashCode());
-        return result;
+        return prime * result + ((this.origin == null) ? 0 : this.origin.hashCode());
     }
 
     /**
@@ -162,8 +161,7 @@ public String toEplString() {
         String res = StringUtils.EMPTY;
         res += StringUtils.printCollection2("% =", this.origin) + System.lineSeparator();
         res += StringUtils.printCollection2("%", this.name) + System.lineSeparator();
-        res += this.clause.toString() + System.lineSeparator();
-        return res;
+        return res + this.clause.toString() + System.lineSeparator();
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java
index 8a529a0..17b9951 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java
@@ -208,8 +208,7 @@ public int hashCode() {
         int result = 1;
         result = prime * result + this.arity;
         result = prime * result + ((this.children == null) ? 0 : this.children.hashCode());
-        result = prime * result + ((this.label == null) ? 0 : this.label.hashCode());
-        return result;
+        return prime * result + ((this.label == null) ? 0 : this.label.hashCode());
     }
 
     @Override
@@ -217,9 +216,8 @@ public String toString() {
         if (this.arity == 0) {
             return this.label;
         } else {
-            String res = this.label;
-            res += StringUtils.parenthesize(StringUtils.printCollection(this.children));
-            return res;
+            return this.label
+                    + StringUtils.parenthesize(StringUtils.printCollection(this.children));
         }
     }
 
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java
index 2dcdf23..c946aee 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java
@@ -5,6 +5,7 @@
 import java.time.format.DateTimeFormatter;
 import java.util.Collections;
 import java.util.HashSet;
+import java.util.LinkedHashSet;
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Map;
@@ -93,9 +94,9 @@ public final class FrameworkRepresentation {
     private List<Pair<String, String>> typeSynonyms;
 
     /**
-     * List of atomic types.
+     * Set of atomic types.
      */
-    private List<ComponentType> atomicTypes;
+    private Set<ComponentType> atomicTypes;
 
     /**
      * Alias for modules that can be composed.
@@ -263,7 +264,7 @@ private void addAliasRules() {
      */
     private void addDummyRules() {
         final List<String> atomicTypeStrings = ConfigReader.getInstance().getAtomicTypes();
-        this.atomicTypes = new LinkedList<ComponentType>();
+        this.atomicTypes = new LinkedHashSet<ComponentType>(atomicTypeStrings.size());
         for (final String s: atomicTypeStrings) {
             final ComponentType type = new ComponentType(s);
             this.add(type);
@@ -340,8 +341,7 @@ private String createHeader() {
                 + StringUtils.sentence(
                         "%% % example_rule_without_meaning" + System.lineSeparator()
                                 + "%% property_a(X) :- property_b(X)");
-        res += SEPARATOR;
-        return res;
+        return res + SEPARATOR;
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java
index d56f696..0852934 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java
@@ -173,8 +173,7 @@ public int hashCode() {
         final int prime = 31;
         int result = 1;
         result = prime * result + this.arity;
-        result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        return result;
+        return prime * result + ((this.name == null) ? 0 : this.name.hashCode());
     }
 
     @Override
diff --git a/virage/src/main/resources/settings b/virage/src/main/resources/settings.properties
similarity index 99%
rename from virage/src/main/resources/settings
rename to virage/src/main/resources/settings.properties
index ba54cb3..7f4b3b8 100644
--- a/virage/src/main/resources/settings
+++ b/virage/src/main/resources/settings.properties
@@ -3,7 +3,6 @@ TIMESTAMP=2023-09-01 00:00:00 GMT+02:00
 # This value is used for internal consistency checks, please do not alter it!
 VIRAGE_CONFIG_VERSION=0.1.0
 
-
 ## External executables
 
 ISABELLE_EXECUTABLE=isabelle
@@ -19,7 +18,6 @@ SWI_PROLOG_LIBSWIPL_PATH=/usr/lib/swi-prolog/lib/x86_64-linux/libswipl.so
 # should contain libjpl.so, otherwise install swi-prolog-java)
 SWI_PROLOG_LIBRARIES_PATH=/usr/lib/swi-prolog/lib/x86_64-linux/
 
-
 ## System settings
 #verbosity_level
 
@@ -30,7 +28,6 @@ SYSTEM_DEFAULT_OUTPUT_PATH=./target/generated-sources/
 # variable $EDITOR is used).
 SYSTEM_TEXT_EDITOR=/usr/bin/kate
 
-
 ## Isabelle settings
 
 # Isabelle tactics to be used by the brute force tactics choice
@@ -41,7 +38,6 @@ ISABELLE_TACTICS=simp;blast;metis;fastforce
 # A compositional framework will be extracted and written to an (E)PL file.
 ISABELLE_PATH_TO_ROOT_FILE=src/test/resources/verifiedVotingRuleConstruction/theories/ROOT
 
-
 ## Voting-specific settings
 
 # Names for generic type variables. These can be used in all session-specific
@@ -84,6 +80,3 @@ SESSION_SPECIFIC_COMPONENT_ALIASES=elector(X)->sequential_composition(X,elect_mo
   maximum_parallel_composition(X,Y)->parallel_composition(X,Y,max_aggregator);\
   smc(X)->iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),\
   sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))
-
-
-
diff --git a/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala b/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala
index 6b3ebae..9178e1f 100644
--- a/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala
+++ b/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala
@@ -8,15 +8,6 @@ import de.unruh.isabelle.control.IsabelleBuildException
 import scala.jdk.CollectionConverters._
 
 object JIsabelleWrapper {
-  /** Sets the [[de.unruh.isabelle.control.Isabelle.Setup.logic logic]] directory in the
-   *  setup `setup`.
-   *
-   * @return `setup` with [[de.unruh.isabelle.control.Isabelle.Setup.logic logic]] set to `logic`
-   * @param logic the new value for `setup.`[[de.unruh.isabelle.control.Isabelle.Setup.logic logic]]
-   * */
-  def setupSetLogic(logic: String, setup: Isabelle.Setup): Isabelle.Setup =
-    setup.copy(logic = logic)
-
   /**
    * Invokes the constructor [[de.unruh.isabelle.control.Isabelle]]`(setup: SetupGeneral)` and
    * explicitly declares that exceptions of [[de.unruh.isabelle.control.IsabelleBuildException]]
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl
index f61df48..127f7c3 100644
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl
+++ b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl
@@ -1,5 +1,5 @@
 %%%%%%%%%%%%%%%%%%%%
-%% Generated by ViRAGe 0.1.0 at 2021-07-12 15:25:59.
+%% Generated by ViRAGe 0.1.0 at 2023-09-26 04:35:52 MESZ.
 %%%%%%%%%%%%%%%%%%%%
 %% To add new composition rules, go to the bottom of this file.
 %% A rule definition consists of three parts. The first one represents its origin,
@@ -12,7 +12,7 @@
 %% % example_rule_without_meaning
 %% property_a(X) :- property_b(X).
 %%%%%%%%%%%%%%%%%%%%
-% ==== src/test/resources/verifiedVotingRuleConstruction/theories/ - Verified_Voting_Rule_Construction
+% ==== src/test/resources/verifiedVotingRuleConstruction/theories/ROOT - Verified_Voting_Rule_Construction
 %
 % === COMPONENT_TYPE
 % == Rel.rel(?'a)
@@ -20,54 +20,54 @@
 % elimination_set(Evaluation_Function,Nat.nat,Nat.nat,Nat.nat,HOL.bool,Set.set(?'a),Profile)
 % == HOL.bool
 % == Electoral_Module
-% elimination_module(Evaluation_Function,Nat.nat,Nat.nat,Nat.nat,HOL.bool)
-% leq_average_eliminator(Evaluation_Function)
-% loop_composition(Electoral_Module,Termination_Condition)
-% minimax()
-% revision_composition(Electoral_Module)
-% leq_eliminator(Evaluation_Function,Nat.nat)
-% copeland()
-% nanson_baldwin_rule()
-% condorcet()
-% min_eliminator(Evaluation_Function)
-% sequential_composition(Electoral_Module,Electoral_Module)
-% blacks_rule()
-% minimax_rule()
-% maximum_parallel_composition(Electoral_Module,Electoral_Module)
 % iterelect(Electoral_Module)
-% max_eliminator(Evaluation_Function)
-% drop_module(Nat.nat,Preference_Relation)
-% condorcet'()
-% schwartz_rule()
-% borda()
+% pairwise_majority_rule'()
+% loop_comp_helper(Electoral_Module,Electoral_Module,Termination_Condition)
 % pass_module(Nat.nat,Preference_Relation)
-% less_eliminator(Evaluation_Function,Nat.nat)
+% schwartz_rule()
+% drop_module(Nat.nat,Preference_Relation)
+% sequential_composition(Electoral_Module,Electoral_Module)
+% elimination_module(Evaluation_Function,Nat.nat,Nat.nat,Nat.nat,HOL.bool)
 % borda_rule()
+% borda()
+% copeland_rule()
 % pairwise_majority_rule()
+% loop_composition(Electoral_Module,Termination_Condition)
 % plurality()
-% pairwise_majority_rule'()
+% leq_eliminator(Evaluation_Function,Nat.nat)
 % less_average_eliminator(Evaluation_Function)
-% smc(Preference_Relation)
-% classic_nanson_rule()
-% defer_module()
-% copeland_rule()
-% elector(Electoral_Module)
-% loop_comp_helper(Electoral_Module,Electoral_Module,Termination_Condition)
+% leq_average_eliminator(Evaluation_Function)
+% condorcet()
 % elect_module()
-% parallel_composition(Electoral_Module,Electoral_Module,Aggregator)
+% max_eliminator(Evaluation_Function)
+% maximum_parallel_composition(Electoral_Module,Electoral_Module)
 % iter(Electoral_Module)
+% parallel_composition(Electoral_Module,Electoral_Module,Aggregator)
+% minimax_rule()
+% elector(Electoral_Module)
+% defer_module()
+% minimax()
+% less_eliminator(Evaluation_Function,Nat.nat)
+% revision_composition(Electoral_Module)
+% blacks_rule()
+% classic_nanson_rule()
+% smc(Preference_Relation)
+% min_eliminator(Evaluation_Function)
+% condorcet'()
+% nanson_baldwin_rule()
+% copeland()
 % == Nat.nat
-% prefer_count_code(Profile,?'a,?'a)
+% average(Evaluation_Function,Set.set(?'a),Profile)
 % win_count_code(Profile,?'a)
-% win_count(Profile,?'a)
 % prefer_count(Profile,?'a,?'a)
 % rank(Preference_Relation,?'a)
-% average(Evaluation_Function,Set.set(?'a),Profile)
+% prefer_count_code(Profile,?'a,?'a)
+% win_count(Profile,?'a)
 % == Termination_Condition
-% set_equals_partition(Set.set(?'a))
-% disjoint3()
 % defer_equal_condition(Nat.nat)
+% set_equals_partition(Set.set(?'a))
 % well_formed(Set.set(?'a))
+% disjoint3()
 % == Aggregator
 % max_aggregator()
 % == ?'a
@@ -78,16 +78,17 @@
 % == Profile
 % limit_profile(Set.set(?'a),Profile)
 % == Evaluation_Function
-% condorcet_score()
-% borda_score()
 % copeland_score()
+% condorcet_score()
 % minimax_score()
+% borda_score()
 %
 % === PROPERTY
 % defer_condorcet_consistency(Electoral_Module)
 % wins(?'a,Profile,?'a)
 % is_less_preferred_than(?'a,Preference_Relation,?'a)
 % aggregator(Aggregator)
+% linear_order(Preference_Relation)
 % condorcet_compatibility(Electoral_Module)
 % defer_invariant_monotonicity(Electoral_Module)
 % indep_of_alt(Electoral_Module,Set.set(?'a),?'a)
@@ -99,6 +100,7 @@
 % condorcet_rating(Evaluation_Function)
 % invariant_monotonicity(Electoral_Module)
 % agg_conservative(Aggregator)
+% trans(Preference_Relation)
 % electoral_module(Electoral_Module)
 % connex(Set.set(?'a),Preference_Relation)
 % defers(Nat.nat,Electoral_Module)
@@ -110,6 +112,7 @@
 % defer_lift_invariance(Electoral_Module)
 % monotonicity(Electoral_Module)
 % prof_geq_result(Electoral_Module,Set.set(?'a),Profile,Profile,?'a)
+% custom_greater(Nat.nat,Nat.nat)
 % equiv_prof_except_a(Set.set(?'a),Profile,Profile,?'a)
 % homogeneity(Electoral_Module)
 % prof_leq_result(Electoral_Module,Set.set(?'a),Profile,Profile,?'a)
@@ -120,11 +123,13 @@
 % disjoint_compatibility(Electoral_Module,Electoral_Module)
 % elects(Nat.nat,Electoral_Module)
 % condorcet_consistency(Electoral_Module)
+% finite_profile(Set.set(?'a),Profile)
 % equiv_rel_except_a(Set.set(?'a),Preference_Relation,Preference_Relation,?'a)
 % condorcet_winner(Set.set(?'a),Profile,?'a)
 % linear_order(Preference_Relation)
 % trans(Preference_Relation)
 % finite_profile(Set.set(?'a),Profile)
+% custom_greater(Nat.nat,Nat.nat)
 %
 % === COMPOSITION_RULE
 % = Aggregator_Facts.thy
@@ -143,6 +148,26 @@
 % Preference_Relation.lin_imp_antisym
 antisym(R) :- linear_order_on(_,R).
 
+% = ASSUMPTION
+% condorcet_compatibility_alias_0
+condorcet_compatibility(iter(X)) :- condorcet_compatibility(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_1
+condorcet_compatibility(iterelect(X)) :- condorcet_compatibility(elector(iter(X))).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_2
+condorcet_compatibility(maximum_parallel_composition(X,Y)) :- condorcet_compatibility(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_3
+condorcet_compatibility(smc(X)) :- condorcet_compatibility(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_4
+condorcet_compatibility(elector(X)) :- condorcet_compatibility(sequential_composition(X,elect_module)).
+
 % = Condorcet_Rules.thy
 % Condorcet_Rules.cr_eval_imp_ccomp_max_elim
 condorcet_compatibility(max_eliminator(E)) :- finite_profile(_,_),condorcet_rating(E).
@@ -163,6 +188,26 @@
 % Condorcet_Rules.dcc_imp_cc_elector
 condorcet_consistency(elector(M)) :- defer_condorcet_consistency(M).
 
+% = ASSUMPTION
+% condorcet_consistency_alias_0
+condorcet_consistency(iter(X)) :- condorcet_consistency(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_1
+condorcet_consistency(iterelect(X)) :- condorcet_consistency(elector(iter(X))).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_2
+condorcet_consistency(maximum_parallel_composition(X,Y)) :- condorcet_consistency(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_3
+condorcet_consistency(smc(X)) :- condorcet_consistency(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_4
+condorcet_consistency(elector(X)) :- condorcet_consistency(sequential_composition(X,elect_module)).
+
 % = Condorcet_Facts.thy
 % Condorcet_Facts.minimax_score_cond_rating
 condorcet_rating(minimax_score).
@@ -187,6 +232,26 @@
 % Preference_Relation.lin_ord_imp_connex
 connex(A,R) :- linear_order_on(A,R).
 
+% = ASSUMPTION
+% decrementing_alias_0
+decrementing(iter(X)) :- decrementing(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% decrementing_alias_1
+decrementing(iterelect(X)) :- decrementing(elector(iter(X))).
+
+% = ASSUMPTION
+% decrementing_alias_2
+decrementing(maximum_parallel_composition(X,Y)) :- decrementing(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% decrementing_alias_3
+decrementing(smc(X)) :- decrementing(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% decrementing_alias_4
+decrementing(elector(X)) :- decrementing(sequential_composition(X,elect_module)).
+
 % = Condorcet_Facts.thy
 % Condorcet_Facts.copeland_is_dcc
 defer_condorcet_consistency(copeland).
@@ -203,14 +268,74 @@
 % Condorcet_Rules.cr_eval_imp_dcc_max_elim
 defer_condorcet_consistency(max_eliminator(E)) :- condorcet_rating(E).
 
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_0
+defer_condorcet_consistency(iter(X)) :- defer_condorcet_consistency(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_1
+defer_condorcet_consistency(iterelect(X)) :- defer_condorcet_consistency(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_2
+defer_condorcet_consistency(maximum_parallel_composition(X,Y)) :- defer_condorcet_consistency(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_3
+defer_condorcet_consistency(smc(X)) :- defer_condorcet_consistency(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_4
+defer_condorcet_consistency(elector(X)) :- defer_condorcet_consistency(sequential_composition(X,elect_module)).
+
 % = Condorcet_Rules.thy
 % Condorcet_Rules.ccomp_and_dd_imp_dcc
 defer_condorcet_consistency(M) :- condorcet_compatibility(M),defer_deciding(M).
 
+% = ASSUMPTION
+% defer_deciding_alias_0
+defer_deciding(iter(X)) :- defer_deciding(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_deciding_alias_1
+defer_deciding(iterelect(X)) :- defer_deciding(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_deciding_alias_2
+defer_deciding(maximum_parallel_composition(X,Y)) :- defer_deciding(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_deciding_alias_3
+defer_deciding(smc(X)) :- defer_deciding(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_deciding_alias_4
+defer_deciding(elector(X)) :- defer_deciding(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.rev_comp_def_inv_mono
 defer_invariant_monotonicity(revision_composition(M)) :- invariant_monotonicity(M).
 
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_0
+defer_invariant_monotonicity(iter(X)) :- defer_invariant_monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_1
+defer_invariant_monotonicity(iterelect(X)) :- defer_invariant_monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_2
+defer_invariant_monotonicity(maximum_parallel_composition(X,Y)) :- defer_invariant_monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_3
+defer_invariant_monotonicity(smc(X)) :- defer_invariant_monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_4
+defer_invariant_monotonicity(elector(X)) :- defer_invariant_monotonicity(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Facts.thy
 % Monotonicity_Facts.def_mod_def_lift_inv
 defer_lift_invariance(defer_module).
@@ -227,6 +352,26 @@
 % Monotonicity_Facts.drop_mod_def_lift_inv
 defer_lift_invariance(drop_module(_,R)) :- linear_order(R).
 
+% = ASSUMPTION
+% defer_lift_invariance_alias_0
+defer_lift_invariance(iter(X)) :- defer_lift_invariance(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_1
+defer_lift_invariance(iterelect(X)) :- defer_lift_invariance(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_2
+defer_lift_invariance(maximum_parallel_composition(X,Y)) :- defer_lift_invariance(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_3
+defer_lift_invariance(smc(X)) :- defer_lift_invariance(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_4
+defer_lift_invariance(elector(X)) :- defer_lift_invariance(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.seq_comp_presv_def_lift_inv
 defer_lift_invariance(sequential_composition(M,N)) :- defer_lift_invariance(M),defer_lift_invariance(N).
@@ -247,6 +392,26 @@
 % Monotonicity_Rules.dl_inv_imp_def_mono
 defer_monotonicity(M) :- defer_lift_invariance(M).
 
+% = ASSUMPTION
+% defer_monotonicity_alias_0
+defer_monotonicity(iter(X)) :- defer_monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_1
+defer_monotonicity(iterelect(X)) :- defer_monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_2
+defer_monotonicity(maximum_parallel_composition(X,Y)) :- defer_monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_3
+defer_monotonicity(smc(X)) :- defer_monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_4
+defer_monotonicity(elector(X)) :- defer_monotonicity(sequential_composition(X,elect_module)).
+
 % = Result_Facts.thy
 % Result_Facts.pass_zero_mod_def_zero
 defers(0,pass_module(0,R)) :- linear_order(R).
@@ -259,6 +424,26 @@
 % Result_Facts.pass_two_mod_def_two
 defers(2,pass_module(2,R)) :- linear_order(R).
 
+% = ASSUMPTION
+% defers_alias_0
+defers(ALIAS_VAR_0,iter(X)) :- defers(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defers_alias_1
+defers(ALIAS_VAR_0,iterelect(X)) :- defers(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% defers_alias_2
+defers(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- defers(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defers_alias_3
+defers(ALIAS_VAR_0,smc(X)) :- defers(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defers_alias_4
+defers(ALIAS_VAR_0,elector(X)) :- defers(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.seq_comp_def_one
 defers(1,sequential_composition(M,N)) :- non_blocking(M),non_electing(M),defers(1,N).
@@ -275,6 +460,46 @@
 % Disjoint_Compatibility_Rules.disj_compat_comm
 disjoint_compatibility(N,M) :- disjoint_compatibility(M,N).
 
+% = ASSUMPTION
+% disjoint_compatibility_alias_0
+disjoint_compatibility(iter(X),ALIAS_VAR_1) :- disjoint_compatibility(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_1
+disjoint_compatibility(iterelect(X),ALIAS_VAR_1) :- disjoint_compatibility(elector(iter(X)),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_2
+disjoint_compatibility(maximum_parallel_composition(X,Y),ALIAS_VAR_1) :- disjoint_compatibility(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_3
+disjoint_compatibility(smc(X),ALIAS_VAR_1) :- disjoint_compatibility(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_4
+disjoint_compatibility(elector(X),ALIAS_VAR_1) :- disjoint_compatibility(sequential_composition(X,elect_module),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_0
+disjoint_compatibility(ALIAS_VAR_0,iter(X)) :- disjoint_compatibility(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_1
+disjoint_compatibility(ALIAS_VAR_0,iterelect(X)) :- disjoint_compatibility(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_2
+disjoint_compatibility(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- disjoint_compatibility(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_3
+disjoint_compatibility(ALIAS_VAR_0,smc(X)) :- disjoint_compatibility(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_4
+disjoint_compatibility(ALIAS_VAR_0,elector(X)) :- disjoint_compatibility(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Result_Facts.thy
 % Result_Facts.plurality_electing
 electing(plurality).
@@ -287,6 +512,26 @@
 % Sequential_Majority_Comparison.smc_electing
 electing(smc(X)) :- linear_order(X).
 
+% = ASSUMPTION
+% electing_alias_0
+electing(iter(X)) :- electing(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% electing_alias_1
+electing(iterelect(X)) :- electing(elector(iter(X))).
+
+% = ASSUMPTION
+% electing_alias_2
+electing(maximum_parallel_composition(X,Y)) :- electing(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% electing_alias_3
+electing(smc(X)) :- electing(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% electing_alias_4
+electing(elector(X)) :- electing(sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.elector_electing
 electing(elector(M)) :- electoral_module(M),non_blocking(M).
@@ -359,6 +604,26 @@
 % Revision_Composition.rev_comp_sound
 electoral_module(revision_composition(M)) :- electoral_module(M).
 
+% = ASSUMPTION
+% electoral_module_alias_0
+electoral_module(iter(X)) :- electoral_module(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% electoral_module_alias_1
+electoral_module(iterelect(X)) :- electoral_module(elector(iter(X))).
+
+% = ASSUMPTION
+% electoral_module_alias_2
+electoral_module(maximum_parallel_composition(X,Y)) :- electoral_module(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% electoral_module_alias_3
+electoral_module(smc(X)) :- electoral_module(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% electoral_module_alias_4
+electoral_module(elector(X)) :- electoral_module(sequential_composition(X,elect_module)).
+
 % = Composite_Structures.thy
 % Composite_Structures.max_par_comp_sound
 electoral_module(maximum_parallel_composition(M,N)) :- electoral_module(M),electoral_module(N).
@@ -371,6 +636,46 @@
 % Parallel_Composition.par_comp_sound
 electoral_module(parallel_composition(M,N,A)) :- electoral_module(M),electoral_module(N),aggregator(A).
 
+% = ASSUMPTION
+% elects_alias_0
+elects(ALIAS_VAR_0,iter(X)) :- elects(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% elects_alias_1
+elects(ALIAS_VAR_0,iterelect(X)) :- elects(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% elects_alias_2
+elects(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- elects(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% elects_alias_3
+elects(ALIAS_VAR_0,smc(X)) :- elects(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% elects_alias_4
+elects(ALIAS_VAR_0,elector(X)) :- elects(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
+% = ASSUMPTION
+% eliminates_alias_0
+eliminates(ALIAS_VAR_0,iter(X)) :- eliminates(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% eliminates_alias_1
+eliminates(ALIAS_VAR_0,iterelect(X)) :- eliminates(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% eliminates_alias_2
+eliminates(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- eliminates(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% eliminates_alias_3
+eliminates(ALIAS_VAR_0,smc(X)) :- eliminates(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% eliminates_alias_4
+eliminates(ALIAS_VAR_0,elector(X)) :- eliminates(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.par_comp_elim_one
 eliminates(1,maximum_parallel_composition(M,N)) :- defers(1,M),non_electing(M),rejects(2,N),disjoint_compatibility(M,N).
@@ -391,14 +696,78 @@
 % Preference_Relation.lifted_imp_equiv_rel_except_a
 equiv_rel_except_a(A,R,S,A) :- lifted(A,R,S,A).
 
+% = ASSUMPTION
+% finite_profile_intro
+finite_profile(_,_).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.lifted_imp_fin_prof
 finite_profile(A,P) :- lifted(A,P,_,A).
 
+% = ASSUMPTION
+% homogeneity_alias_0
+homogeneity(iter(X)) :- homogeneity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% homogeneity_alias_1
+homogeneity(iterelect(X)) :- homogeneity(elector(iter(X))).
+
+% = ASSUMPTION
+% homogeneity_alias_2
+homogeneity(maximum_parallel_composition(X,Y)) :- homogeneity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% homogeneity_alias_3
+homogeneity(smc(X)) :- homogeneity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% homogeneity_alias_4
+homogeneity(elector(X)) :- homogeneity(sequential_composition(X,elect_module)).
+
+% = ASSUMPTION
+% indep_of_alt_alias_0
+indep_of_alt(iter(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_1
+indep_of_alt(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_2
+indep_of_alt(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_3
+indep_of_alt(smc(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_4
+indep_of_alt(elector(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2).
+
 % = Monotonicity_Facts.thy
 % Monotonicity_Facts.plurality_inv_mono
 invariant_monotonicity(plurality).
 
+% = ASSUMPTION
+% invariant_monotonicity_alias_0
+invariant_monotonicity(iter(X)) :- invariant_monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_1
+invariant_monotonicity(iterelect(X)) :- invariant_monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_2
+invariant_monotonicity(maximum_parallel_composition(X,Y)) :- invariant_monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_3
+invariant_monotonicity(smc(X)) :- invariant_monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_4
+invariant_monotonicity(elector(X)) :- invariant_monotonicity(sequential_composition(X,elect_module)).
+
 % = ASSUMPTION
 % is_less_preferred_than_intro
 is_less_preferred_than(_,_,_).
@@ -419,6 +788,10 @@
 % limited_intro
 limited(_,_).
 
+% = ASSUMPTION
+% linear_order_intro
+linear_order(_).
+
 % = Preference_Relation.thy
 % Preference_Relation.connex_antsym_and_trans_imp_lin_ord
 linear_order_on(A,R) :- connex(A,R),antisym(R),trans(R).
@@ -427,10 +800,70 @@
 % Electoral_Module.mod_contains_result_comm
 mod_contains_result(N,M,A,P,A) :- mod_contains_result(M,N,A,P,A).
 
+% = ASSUMPTION
+% mod_contains_result_alias_0
+mod_contains_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_1
+mod_contains_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_2
+mod_contains_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_3
+mod_contains_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_4
+mod_contains_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_0
+mod_contains_result(ALIAS_VAR_0,iter(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_1
+mod_contains_result(ALIAS_VAR_0,iterelect(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,elector(iter(X)),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_2
+mod_contains_result(ALIAS_VAR_0,maximum_parallel_composition(X,Y),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_3
+mod_contains_result(ALIAS_VAR_0,smc(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_4
+mod_contains_result(ALIAS_VAR_0,elector(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,sequential_composition(X,elect_module),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
 % = Sequential_Majority_Comparison.thy
 % Sequential_Majority_Comparison.smc_monotone
 monotonicity(smc(X)) :- linear_order(X).
 
+% = ASSUMPTION
+% monotonicity_alias_0
+monotonicity(iter(X)) :- monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% monotonicity_alias_1
+monotonicity(iterelect(X)) :- monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% monotonicity_alias_2
+monotonicity(maximum_parallel_composition(X,Y)) :- monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% monotonicity_alias_3
+monotonicity(smc(X)) :- monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% monotonicity_alias_4
+monotonicity(elector(X)) :- monotonicity(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.seq_comp_mono
 monotonicity(sequential_composition(M,N)) :- defer_lift_invariance(M),non_electing(M),defers(1,M),electing(N).
@@ -439,6 +872,26 @@
 % Result_Rules.electing_imp_non_blocking
 non_blocking(M) :- electing(M).
 
+% = ASSUMPTION
+% non_blocking_alias_0
+non_blocking(iter(X)) :- non_blocking(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% non_blocking_alias_1
+non_blocking(iterelect(X)) :- non_blocking(elector(iter(X))).
+
+% = ASSUMPTION
+% non_blocking_alias_2
+non_blocking(maximum_parallel_composition(X,Y)) :- non_blocking(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% non_blocking_alias_3
+non_blocking(smc(X)) :- non_blocking(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% non_blocking_alias_4
+non_blocking(elector(X)) :- non_blocking(sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.seq_comp_presv_non_blocking
 non_blocking(sequential_composition(M,N)) :- non_blocking(M),non_blocking(N).
@@ -491,6 +944,26 @@
 % Result_Facts.leq_avg_elim_non_electing
 non_electing(leq_average_eliminator(_)) :- finite_profile(_,_).
 
+% = ASSUMPTION
+% non_electing_alias_0
+non_electing(iter(X)) :- non_electing(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% non_electing_alias_1
+non_electing(iterelect(X)) :- non_electing(elector(iter(X))).
+
+% = ASSUMPTION
+% non_electing_alias_2
+non_electing(maximum_parallel_composition(X,Y)) :- non_electing(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% non_electing_alias_3
+non_electing(smc(X)) :- non_electing(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% non_electing_alias_4
+non_electing(elector(X)) :- non_electing(sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.seq_comp_presv_non_electing
 non_electing(sequential_composition(M,N)) :- non_electing(M),non_electing(N).
@@ -503,6 +976,66 @@
 % Result_Rules.conserv_agg_presv_non_electing
 non_electing(parallel_composition(M,N,A)) :- non_electing(M),non_electing(N),agg_conservative(A).
 
+% = ASSUMPTION
+% prof_contains_result_alias_0
+prof_contains_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_1
+prof_contains_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_2
+prof_contains_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_3
+prof_contains_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_4
+prof_contains_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_0
+prof_geq_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_1
+prof_geq_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_2
+prof_geq_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_3
+prof_geq_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_4
+prof_geq_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_0
+prof_leq_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_1
+prof_leq_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_2
+prof_leq_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_3
+prof_leq_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_4
+prof_leq_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
 % = ASSUMPTION
 % profile_intro
 profile(_,_).
@@ -519,10 +1052,34 @@
 % Result_Facts.drop_zero_mod_rej_zero
 rejects(0,drop_module(0,R)) :- linear_order(R).
 
+% = ASSUMPTION
+% rejects_alias_0
+rejects(ALIAS_VAR_0,iter(X)) :- rejects(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% rejects_alias_1
+rejects(ALIAS_VAR_0,iterelect(X)) :- rejects(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% rejects_alias_2
+rejects(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- rejects(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% rejects_alias_3
+rejects(ALIAS_VAR_0,smc(X)) :- rejects(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% rejects_alias_4
+rejects(ALIAS_VAR_0,elector(X)) :- rejects(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Sequential_Composition.thy
 % Sequential_Composition.seq_comp_presv_alts
 set_equals_partition(A,sequential_composition(M,N,A,P)) :- electoral_module(M),electoral_module(N),finite_profile(A,P).
 
+% = ASSUMPTION
+% trans_intro
+trans(_).
+
 % = Preference_Relation.thy
 % Preference_Relation.lin_imp_trans
 trans(R) :- linear_order_on(_,R).
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document.pdf b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document.pdf
index 118aab44f51d6859aa98d13ac16d9b36d1af2dce..4ab743172a8fe72a47dc12b5c31061a017495146 100644
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diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session_graph.pdf b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session_graph.pdf
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From 1040042e050b386787e66125d58952fd496c9bf4 Mon Sep 17 00:00:00 2001
From: "mi.ki" <kirsten@kit.edu>
Date: Thu, 13 Mar 2025 04:14:51 +0100
Subject: [PATCH 4/4] Update maven.yml to JDK 21

---
 .github/workflows/maven.yml | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/.github/workflows/maven.yml b/.github/workflows/maven.yml
index 902077c..f4c8600 100644
--- a/.github/workflows/maven.yml
+++ b/.github/workflows/maven.yml
@@ -27,10 +27,10 @@ jobs:
 
     steps:
     - uses: actions/checkout@v4
-    - name: Set up JDK 17
+    - name: Set up JDK 21
       uses: actions/setup-java@v3
       with:
-        java-version: '17'
+        java-version: '21'
         distribution: 'temurin'
         cache: maven
     - name: Build with Maven