feat: Adding proofs of correctness and time complexity of insertion sort

Cslib.lean

@@ -1,5 +1,6 @@
 module  -- shake: keep-all
 
+public import Cslib.Algorithms.Lean.InsertionSort.InsertionSort
 public import Cslib.Algorithms.Lean.MergeSort.MergeSort
 public import Cslib.Algorithms.Lean.TimeM
 public import Cslib.Computability.Automata.Acceptors.Acceptor

Cslib/Algorithms/Lean/InsertionSort/InsertionSort.lean

@@ -0,0 +1,127 @@
+/-
+Copyright (c) 2026 Bhargav Kulkarni. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhargav Kulkarni
+-/
+
+module
+
+public import Cslib.Algorithms.Lean.TimeM
+public import Mathlib.Data.Nat.Cast.Order.Ring
+public import Mathlib.Data.Nat.Lattice
+public import Mathlib.Data.Nat.Log
+
+@[expose] public section
+
+/-!
+# Insertion Sort on a list
+
+In this file we introduce `insert` and `insertionSort` algorithms that returns a time monad
+over the list `TimeM ℕ (List α)`. The time complexity of `insertionSort` is the number of comparisons.
+
+--
+## Main results
+
+- `insertionSort_correct`: `insertionSort` permutes the list into a sorted one.
+- `insertionSort_time`:  The number of comparisons of `insertionSort` is at most `n^2`.
+
+-/
+
+set_option autoImplicit false
+
+namespace Cslib.Algorithms.Lean.TimeM
+
+variable {α : Type} [LinearOrder α]
+
+/-- Inserts an element into a list, counting comparisons as time cost.
+Returns a `TimeM ℕ (List α)` where the time represents the number of comparisons performed. -/
+def insert : α → List α → TimeM ℕ (List α)
+  | x, [] => return [x]
+  | x, y :: ys => do
+    ✓ let c := (x ≤ y : Bool)
+    if c then
+      return (x :: y :: ys)
+    else
+      let rest ← insert x ys
+      return (y :: rest)
+
+/-- Sorts a list using the insertion sort algorithm, counting comparisons as time cost.
+Returns a `TimeM ℕ (List α)` where the time represents the total number of comparisons. -/
+def insertionSort (xs : List α) : TimeM ℕ (List α) := do
+  match xs with
+  | [] => return []
+  | x :: xs' => do
+    let sortedTail ← insertionSort xs'
+    insert x sortedTail
+
+section Correctness
+
+open List
+
+@[grind →]
+theorem mem_either_insert (xs : List α) (a b : α) (hz : a ∈ ⟪insert b xs⟫) : a = b ∨ a ∈ xs := by
+  fun_induction insert with
+  | case1 _ => grind
+  | case2 x y ys ih => grind
+
+/-- A list is sorted if it satisfies the `Pairwise (· ≤ ·)` predicate. -/
+abbrev IsSorted (l : List α) : Prop := List.Pairwise (· ≤ ·) l
+
+theorem sorted_insert {x : α} {xs : List α} (hxs : IsSorted xs) : IsSorted ⟪insert x xs⟫ := by
+  fun_induction insert x xs with
+  | case1 _ => simp
+  | case2 x y ys ih => grind [pairwise_cons]
+
+theorem insertionSort_sorted (xs : List α) : IsSorted ⟪insertionSort xs⟫ := by
+  fun_induction insertionSort xs with
+  | case1 => simp
+  | case2 x xs ih => grind [sorted_insert]
+
+lemma insert_perm x (xs : List α) : ⟪insert x xs⟫ ~ x :: xs := by
+  fun_induction insert with
+  | case1 x => simp
+  | case2 x y ys ih => grind
+
+theorem insertionSort_perm (xs : List α) : ⟪insertionSort xs⟫ ~ xs := by
+  fun_induction insertionSort xs with
+  | case1 => simp
+  | case2 x xs ih =>
+    refine (insert_perm x (insertionSort xs).ret).trans ?_
+    grind [List.Perm.cons]
+
+/-- Insertion Sort is functionally correct. -/
+theorem insertionSort_correct (xs : List α) :
+    IsSorted ⟪insertionSort xs⟫ ∧ ⟪insertionSort xs⟫ ~ xs :=
+  ⟨insertionSort_sorted xs, insertionSort_perm xs⟩
+
+end Correctness
+
+section TimeComplexity
+
+/-- Time complexity of `insert`. -/
+theorem insert_time (x : α) (xs : List α) : (insert x xs).time ≤ xs.length := by
+  fun_induction insert with
+  | case1 _ => simp
+  | case2 x y ys ih => grind
+
+theorem insert_length (x : α) (xs : List α) : (insert x xs).ret.length = xs.length + 1 := by
+  fun_induction insert with
+  | case1 _ => simp
+  | case2 x y ys ih => grind
+
+theorem insertionSort_length (xs : List α) : (insertionSort xs).ret.length = xs.length := by
+  fun_induction insertionSort xs with
+  | case1 => simp
+  | case2 x xs ih => grind [insert_length]
+
+/-- Time complexity of `insertionSort`. -/
+theorem insertionSort_time (xs : List α) : (insertionSort xs).time ≤ xs.length * xs.length := by
+  fun_induction insertionSort with
+  | case1 => simp
+  | case2 x xs ih =>
+    simp
+    grind [insertionSort_length, insert_time]
+
+end TimeComplexity
+
+end Cslib.Algorithms.Lean.TimeM