diff --git a/.devcontainer/Dockerfile b/.devcontainer/Dockerfile
index ae29c8b..5fe1ccf 100644
--- a/.devcontainer/Dockerfile
+++ b/.devcontainer/Dockerfile
@@ -14,5 +14,5 @@ RUN apt-get update && \
 USER vscode
 WORKDIR /home/vscode
 
-RUN curl https://elan.lean-lang.org/elan-init.sh -sSf | sh -s -- -y --default-toolchain leanprover/lean4:4.26.0
+RUN curl https://elan.lean-lang.org/elan-init.sh -sSf | sh -s -- -y --default-toolchain leanprover/lean4:4.27.0-rc1
 ENV PATH="/home/vscode/.elan/bin:${PATH}"
diff --git a/.devcontainer/devcontainer.json b/.devcontainer/devcontainer.json
index 9af9e7f..55cb235 100644
--- a/.devcontainer/devcontainer.json
+++ b/.devcontainer/devcontainer.json
@@ -39,7 +39,11 @@
 				"editor.formatOnSave": true,
 				"github.copilot.chat.agent.thinkingTool": true,
 				"chat.agent.maxRequests": 50,
-				"chat.tools.autoApprove": true,
+				"chat.useAgentSkills": true,
+				"chat.tools.urls.autoApprove": {
+   					"https://lean-lang.org/doc/*": false
+				},
+				"chat.tools.terminal.enableAutoApprove": true,
 				"chat.tools.terminal.autoApprove": {
 					"lean": true,
 					"lake": true,
diff --git a/AGENT.md b/AGENT.md
index f9bb53f..16b0fb4 100644
--- a/AGENT.md
+++ b/AGENT.md
@@ -2,17 +2,12 @@
 
 ## Important Instructions
 
-- Do not add files.
-- Do not remove files.
-- Do not use `git checkout`and `git reset`.
-- Do not use `python` command.
-- Do not use `cat` to overwrite a file.
-- Do not modify `lakefile.toml`.
-- Never use `axiom` and `set_option` for any reason!
-- When `lemma` or `thorem` is too long, devide it into multiple `lemma`s.
-- Never give up simply because it is complicated or time-consuming; instead, try to implement the missing lemmas step by step.
-- Use `$...$` or `$$...$$` for LaTeX math formatting in markdown.
-- Use `conj` as complex conjugate when declaring `open ComplexConjugate`.
-- Use `simp` when referring `try 'simp' instead of 'simpa'`.
+- Never use tabs. Use spaces instead.
+- Never modify `lakefile.toml`.
+- Never use `python`, `cat`, `git checkout`and `git reset`.
+- Never use `sorry`, `admit`, `axiom`, and `set_option` for any reason!
 - Fix indentation when referring `expected '{' or indented tactic sequence`.
-- Do not use tabs. Use spaces instead.
+- Autonomously continue executing the tasks specified in `PLANS.md` until the maximum request limit is reached.
+- Use `lean-lsp` mcp server tools when searching Mathlib, analyzing codes, etc.
+- Write comments in English.
+- Use `$...$` or `$$...$$` for LaTeX math formatting in markdown.
diff --git a/Project.lean b/Project.lean
index ed54e10..570fa40 100644
--- a/Project.lean
+++ b/Project.lean
@@ -1,3 +1 @@
--- This module serves as the root of the `Project` library.
--- Import modules here that should be built as part of the library.
 import Project.Example
diff --git a/Project/Example.lean b/Project/Example.lean
index c167798..e811e87 100644
--- a/Project/Example.lean
+++ b/Project/Example.lean
@@ -12,9 +12,7 @@ namespace Example
 theorem infinitely_many_primes : ∀ n : ℕ, ∃ p : ℕ, n < p ∧ Nat.Prime p := by
   intro n
   let N := Nat.factorial n + 1
-  have N_gt_one : 1 < N := by
-    simp [N]
-    exact Nat.factorial_pos n
+  have N_gt_one : 1 < N := Nat.succ_lt_succ (Nat.factorial_pos n)
   let p := Nat.minFac N
   have N_not_one : N ≠ 1 := Nat.ne_of_gt N_gt_one
   have pp : Nat.Prime p := Nat.minFac_prime N_not_one
diff --git a/README.md b/README.md
index 86c2492..102f2b8 100644
--- a/README.md
+++ b/README.md
@@ -1,8 +1,14 @@
 # lean-project
 
 ```bash
+$ lake exe mk_all
+No update necessary
 $ lake build
-Build completed successfully (579 jobs).
+ℹ [496/498] Replayed Project.Example
+info: Project/Example.lean:29:0: import Mathlib.Data.Nat.Factorial.Basic
+import Mathlib.Data.Nat.Prime.Defs
+info: Project/Example.lean:29:0: `#`-commands, such as '#min_imports', are not allowed in 'Mathlib' [linter.hashCommand]
+Build completed successfully (498 jobs).
 ```
 
 ## How to use
diff --git a/lake-manifest.json b/lake-manifest.json
index 65e89ac..508bb5d 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -5,17 +5,17 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "2df2f0150c275ad53cb3c90f7c98ec15a56a1a67",
+   "rev": "32d24245c7a12ded17325299fd41d412022cd3fe",
    "name": "mathlib",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v4.26.0",
+   "inputRev": "v4.27.0-rc1",
    "inherited": false,
    "configFile": "lakefile.lean"},
   {"url": "https://github.com/leanprover-community/plausible",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "160af9e8e7d4ae448f3c92edcc5b6a8522453f11",
+   "rev": "8d3713f36dda48467eb61f8c1c4db89c49a6251a",
    "name": "plausible",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -25,7 +25,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "3591c3f664ac3719c4c86e4483e21e228707bfa2",
+   "rev": "19e5f5cc9c21199be466ef99489e3acab370f079",
    "name": "LeanSearchClient",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -35,7 +35,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "e9f31324f15ead11048b1443e62c5deaddd055d2",
+   "rev": "4eb26e1a4806b200ddfe5179d0c2a0fae56c54a7",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -45,17 +45,17 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "b4fb2aa5290ebf61bc5f80a5375ba642f0a49192",
+   "rev": "ef8377f31b5535430b6753a974d685b0019d0681",
    "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v0.0.83",
+   "inputRev": "v0.0.84",
    "inherited": true,
    "configFile": "lakefile.lean"},
   {"url": "https://github.com/leanprover-community/aesop",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "2f6d238744c4cb07fdc91240feaf5d4221a27931",
+   "rev": "fb12f5535c80e40119286d9575c9393562252d21",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -65,7 +65,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "9312503909aa8e8bb392530145cc1677a6298574",
+   "rev": "523ec6fc8062d2f470fdc8de6f822fe89552b5e6",
    "name": "Qq",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -75,7 +75,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "24241822ef9d3e7f6a3bcc53ad136e12663db8f3",
+   "rev": "6254bed25866358ce4f841fa5a13b77de04ffbc8",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -85,10 +85,10 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover",
-   "rev": "933fce7e893f65969714c60cdb4bd8376786044e",
+   "rev": "726b98c53e2da249c1de768fbbbb5e67bc9cef60",
    "name": "Cli",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v4.26.0",
+   "inputRev": "v4.27.0-rc1",
    "inherited": true,
    "configFile": "lakefile.toml"}],
  "name": "Project",
diff --git a/lakefile.toml b/lakefile.toml
index 2c8d1c2..70ece93 100644
--- a/lakefile.toml
+++ b/lakefile.toml
@@ -4,9 +4,14 @@ defaultTargets = ["Project"]
 
 [[lean_lib]]
 name = "Project"
-leanOptions = { warningAsError = true }
+leanOptions.warningAsError = true
+leanOptions.pp.unicode.fun = true
+leanOptions.autoImplicit = false
+leanOptions.relaxedAutoImplicit = false
+leanOptions.linter.mathlibStandardSet = true
+leanOptions.linter.style.longLine = false
 
 [[require]]
 name = "mathlib"
 scope = "leanprover-community"
-rev = "v4.26.0"
+rev = "v4.27.0-rc1"
diff --git a/lean-toolchain b/lean-toolchain
index e59446d..bd19bde 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.26.0
+leanprover/lean4:v4.27.0-rc1
diff --git a/package-lock.json b/package-lock.json
index 90feec6..df94e53 100644
--- a/package-lock.json
+++ b/package-lock.json
@@ -10,7 +10,7 @@
         "@commitlint/config-conventional": "20.2.0",
         "commitizen": "4.3.1",
         "cz-conventional-changelog": "3.3.0",
-        "lefthook": "2.0.11"
+        "lefthook": "2.0.12"
       }
     },
     "node_modules/@babel/code-frame": {
@@ -1607,9 +1607,9 @@
       }
     },
     "node_modules/lefthook": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook/-/lefthook-2.0.11.tgz",
-      "integrity": "sha512-/91k4dt9MRNkzeSr1iMjNi/z8dNuh+XvNfXrWA6PV+M1ZxiNY6uN6bGnr13n+j7N89f4h7YWBhCqhzhK33M5cA==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook/-/lefthook-2.0.12.tgz",
+      "integrity": "sha512-I2FdA9cdnq1icwlNz4RADs7exuqe47q1N9+p2LmcP/WfchWh16mvTB82OAD7w7zK9GxblS9GpF7pASaOSl4c7A==",
       "dev": true,
       "hasInstallScript": true,
       "license": "MIT",
@@ -1617,22 +1617,22 @@
         "lefthook": "bin/index.js"
       },
       "optionalDependencies": {
-        "lefthook-darwin-arm64": "2.0.11",
-        "lefthook-darwin-x64": "2.0.11",
-        "lefthook-freebsd-arm64": "2.0.11",
-        "lefthook-freebsd-x64": "2.0.11",
-        "lefthook-linux-arm64": "2.0.11",
-        "lefthook-linux-x64": "2.0.11",
-        "lefthook-openbsd-arm64": "2.0.11",
-        "lefthook-openbsd-x64": "2.0.11",
-        "lefthook-windows-arm64": "2.0.11",
-        "lefthook-windows-x64": "2.0.11"
+        "lefthook-darwin-arm64": "2.0.12",
+        "lefthook-darwin-x64": "2.0.12",
+        "lefthook-freebsd-arm64": "2.0.12",
+        "lefthook-freebsd-x64": "2.0.12",
+        "lefthook-linux-arm64": "2.0.12",
+        "lefthook-linux-x64": "2.0.12",
+        "lefthook-openbsd-arm64": "2.0.12",
+        "lefthook-openbsd-x64": "2.0.12",
+        "lefthook-windows-arm64": "2.0.12",
+        "lefthook-windows-x64": "2.0.12"
       }
     },
     "node_modules/lefthook-darwin-arm64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-darwin-arm64/-/lefthook-darwin-arm64-2.0.11.tgz",
-      "integrity": "sha512-RfpdcJJQXstdgDiIBDRffncayKiXx+0LyMUCunIxDEO2JMXPpYK2hIdpUU0rkitzptAADchG7u1OXJ31rrtIAA==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-darwin-arm64/-/lefthook-darwin-arm64-2.0.12.tgz",
+      "integrity": "sha512-tuBz1sNLien+nKKb8BDopKjS6EnbXU8rQzhMVBY+bnVfsTiYDfbBr4wo/IzA5TcwoTL/b5somCJhljEw6DvSyg==",
       "cpu": [
         "arm64"
       ],
@@ -1644,9 +1644,9 @@
       ]
     },
     "node_modules/lefthook-darwin-x64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-darwin-x64/-/lefthook-darwin-x64-2.0.11.tgz",
-      "integrity": "sha512-D013UNKQa4FKgpxDMqdaU109U2/Pidtrt9CobQoq8te4eGUglcwxMzuYVTgaYnenz0FgKxSfVaCZsZgwqeMWqA==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-darwin-x64/-/lefthook-darwin-x64-2.0.12.tgz",
+      "integrity": "sha512-FnuUMPPRMJyTEPXg6PotSrFJ8qf8FDLhhD1zLh74D+9Cye5j9n3lcrCQEjXubPT8du/GZLxMBjjffRbcZ8eYDA==",
       "cpu": [
         "x64"
       ],
@@ -1658,9 +1658,9 @@
       ]
     },
     "node_modules/lefthook-freebsd-arm64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-freebsd-arm64/-/lefthook-freebsd-arm64-2.0.11.tgz",
-      "integrity": "sha512-mgfNqG1tiJkCuGNwPG0LEfnAHGJA+Qzl6KidOtX/Zhxmj/sM+6hxiP4LOeEAhCnaZF5kuPtQgbFzShFHc2BK6A==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-freebsd-arm64/-/lefthook-freebsd-arm64-2.0.12.tgz",
+      "integrity": "sha512-DXElB0qR5e6a8cXkFNYakhwCieypbfh6Y4QG39pzMnLsG03g/nhe093o6owfiUZ4mUFyDM6+0xmy0steOooF2g==",
       "cpu": [
         "arm64"
       ],
@@ -1672,9 +1672,9 @@
       ]
     },
     "node_modules/lefthook-freebsd-x64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-freebsd-x64/-/lefthook-freebsd-x64-2.0.11.tgz",
-      "integrity": "sha512-rnHOlQbJfLGCibr7yHM44kPNgf/tFpEbj/cWVHRhjRdbgYSCAjJk0uKd/EVo3v/vjfId2na0AhWbLvO/aY3wQQ==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-freebsd-x64/-/lefthook-freebsd-x64-2.0.12.tgz",
+      "integrity": "sha512-iJN1ZxFeaDi4Fi3b9jcW9wgyNl19LOv2NaVOaAi/tG6mlIn196cmSdXkOA3+943ZbqbdfV9I+bBcIKwneXDA3Q==",
       "cpu": [
         "x64"
       ],
@@ -1686,9 +1686,9 @@
       ]
     },
     "node_modules/lefthook-linux-arm64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-linux-arm64/-/lefthook-linux-arm64-2.0.11.tgz",
-      "integrity": "sha512-1XjDo2/4fM0TbJBwxZh8w+WMOFueg9oYHkryM8vc3vp8wTajdWBazg1K37JIS3FUco3tcOs+eWHQg0ekVjpWoA==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-linux-arm64/-/lefthook-linux-arm64-2.0.12.tgz",
+      "integrity": "sha512-byvmO4Iri6P0COwM8c3lGgeCV3Q0hh1XJpRfrcZDr4Wslq9O63t6J3T6i87oOtY+UjC9pXLl6xGk6hlUcHZ3BQ==",
       "cpu": [
         "arm64"
       ],
@@ -1700,9 +1700,9 @@
       ]
     },
     "node_modules/lefthook-linux-x64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-linux-x64/-/lefthook-linux-x64-2.0.11.tgz",
-      "integrity": "sha512-OKOcfEvozXhO7+y2xgUzvc2kkqfhluql/sjQSzd8Ka+iK3hM4KCfbfgYx9q61Pjr34a0+i03cuH5DF2dlq/rrg==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-linux-x64/-/lefthook-linux-x64-2.0.12.tgz",
+      "integrity": "sha512-KBaiinmf336rA+/dmYs7H7TTeAOByB0CyLA7k8IecTCuaiuKr6ez7ktSjht19poa5G+V0mts4GgEGcx6HViR0w==",
       "cpu": [
         "x64"
       ],
@@ -1714,9 +1714,9 @@
       ]
     },
     "node_modules/lefthook-openbsd-arm64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-openbsd-arm64/-/lefthook-openbsd-arm64-2.0.11.tgz",
-      "integrity": "sha512-n1KEx196M3SKaWVNTQXGgxzBsiYAsdAy6Of6I6TAZwPhG7yoRrKGkQrhOlPgMzYl36udG1Lk4D+mfY9T0oOUYQ==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-openbsd-arm64/-/lefthook-openbsd-arm64-2.0.12.tgz",
+      "integrity": "sha512-1QBMXX1UW5rtgC4TB52OKWB7Rz/kCBRB+bKKLT/gDD79aPzLgJANTitQQzgFNIWoa7aM9UvzvIAJzOo6FcFIbg==",
       "cpu": [
         "arm64"
       ],
@@ -1728,9 +1728,9 @@
       ]
     },
     "node_modules/lefthook-openbsd-x64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-openbsd-x64/-/lefthook-openbsd-x64-2.0.11.tgz",
-      "integrity": "sha512-WAEtKpYUVvuJMVLA38IBoaPnTNSiaEzvUYxjTBlYTLHJwn7HC2GG6P1cnvoua8rfxb9/Bfi7C3D3IPa9VmB33Q==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-openbsd-x64/-/lefthook-openbsd-x64-2.0.12.tgz",
+      "integrity": "sha512-zPcvUzs65GexRA37UHmaZqWuEGSU/zpBaPIY98MybXzzcJfCIf+O0oUQe2riMllwYGvNW0B1y3NOYRziDNe/vA==",
       "cpu": [
         "x64"
       ],
@@ -1742,9 +1742,9 @@
       ]
     },
     "node_modules/lefthook-windows-arm64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-windows-arm64/-/lefthook-windows-arm64-2.0.11.tgz",
-      "integrity": "sha512-HBqW1qfAnmmbpet7gSWatB6H5YIFdGxCqzolMCLwY/0o8oPFiMwdNE5RGp5JMmhZdz/h3XlbaUlIhnxoW8dk5g==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-windows-arm64/-/lefthook-windows-arm64-2.0.12.tgz",
+      "integrity": "sha512-kgwxguS2GssoHM4SMTp+ArD/Gjg9q5MinD6iI5vSFpuJygD13ZWiXQQfESMHq9y/v1XkD0BdHTJej49dx8P+Vw==",
       "cpu": [
         "arm64"
       ],
@@ -1756,9 +1756,9 @@
       ]
     },
     "node_modules/lefthook-windows-x64": {
-      "version": "2.0.11",
-      "resolved": "https://registry.npmjs.org/lefthook-windows-x64/-/lefthook-windows-x64-2.0.11.tgz",
-      "integrity": "sha512-e5TYmV5cBZfRrhPVFCqjauegLI5CjdAd8exyAbMzGHkiwp3ZK145Su/pntgEP3d+ayS9mpgYPJmXYOSL7WHlyg==",
+      "version": "2.0.12",
+      "resolved": "https://registry.npmjs.org/lefthook-windows-x64/-/lefthook-windows-x64-2.0.12.tgz",
+      "integrity": "sha512-Tf/VtSOtF3rBTc9dzRWROa+HuhqaiIV+Xp+1gzlx5+uCueLM0m87Rz6yd4IN5mL7TrDaNkiRXI3FvjCp0dUE4Q==",
       "cpu": [
         "x64"
       ],
diff --git a/package.json b/package.json
index 31ab506..7fe35fd 100644
--- a/package.json
+++ b/package.json
@@ -5,7 +5,7 @@
     "@commitlint/config-conventional": "20.2.0",
     "commitizen": "4.3.1",
     "cz-conventional-changelog": "3.3.0",
-    "lefthook": "2.0.11"
+    "lefthook": "2.0.12"
   },
   "config": {
     "commitizen": {