feat(algorithms): add counting sort

Cslib.lean

@@ -1,6 +1,8 @@
 module  -- shake: keep-all
 
+public import Cslib.Algorithms.Lean.CountingSort.CountingSort
 public import Cslib.Algorithms.Lean.MergeSort.MergeSort
+public import Cslib.Algorithms.Lean.Sorting
 public import Cslib.Algorithms.Lean.TimeM
 public import Cslib.Computability.Automata.Acceptors.Acceptor
 public import Cslib.Computability.Automata.Acceptors.OmegaAcceptor

Cslib/Algorithms/Lean/CountingSort/CountingSort.lean

@@ -0,0 +1,225 @@
+/-
+Copyright (c) 2026 Robert Joseph George. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Robert Joseph George
+-/
+
+module
+
+public import Cslib.Algorithms.Lean.Sorting
+public import Cslib.Algorithms.Lean.TimeM
+public import Mathlib.Data.Nat.Basic
+public import Mathlib.Order.RelClasses
+public import Mathlib.Tactic.Attr.Core
+public import Mathlib.Tactic.Finiteness.Attr
+public import Batteries.Data.List.Lemmas
+
+/-!
+# Counting sort on natural-number lists
+
+`countingSort` sorts a `List ℕ` using an explicit upper bound `bound`. It scans the input once for
+each key in `0, ..., bound` and returns a `TimeM ℕ (List ℕ)`.
+
+The cost model counts one key comparison while scanning an input element. Constructing output
+blocks, appending lists, arithmetic, and pattern matching are free.
+
+Correctness is proved through counts: each generated block contributes exactly the copies for one
+key, and bounded inputs ensure no out-of-range key is missing from the output.
+-/
+
+@[expose] public section
+
+set_option autoImplicit false
+
+namespace Cslib.Algorithms.Lean.TimeM
+
+/-- `xs` is bounded by `bound` when every entry is at most `bound`. -/
+abbrev BoundedBy (bound : ℕ) (xs : List ℕ) : Prop := ∀ x ∈ xs, x ≤ bound
+
+/-- Counts occurrences of one key, charging one tick for each inspected input element. -/
+def countKey (key : ℕ) : List ℕ → TimeM ℕ ℕ
+  | [] => return 0
+  | x :: xs => do
+    ✓ let hit := x == key
+    let rest ← countKey key xs
+    return rest + if hit then 1 else 0
+
+/-- Builds the sorted blocks for keys `start, ..., start + fuel - 1`. -/
+def buildCountedFrom (start fuel : ℕ) (xs : List ℕ) : TimeM ℕ (List ℕ) :=
+  match fuel with
+  | 0 => return []
+  | fuel' + 1 => do
+    let count ← countKey start xs
+    let rest ← buildCountedFrom (start + 1) fuel' xs
+    return List.replicate count start ++ rest
+
+/-- Sorts natural-number lists by counting all keys from `0` through `bound`. -/
+def countingSort (bound : ℕ) (xs : List ℕ) : TimeM ℕ (List ℕ) :=
+  buildCountedFrom 0 (bound + 1) xs
+
+section Correctness
+
+/-- Timed key counting computes `List.count`. -/
+@[simp, grind =]
+private theorem ret_countKey (key : ℕ) (xs : List ℕ) : ⟪countKey key xs⟫ = xs.count key := by
+  induction xs with
+  | nil => simp [countKey]
+  | cons x xs ih =>
+    by_cases h : x = key <;> simp [countKey, ih, h]
+
+/-- Counting one key performs one comparison per input element. -/
+@[simp, grind =]
+private theorem countKey_time (key : ℕ) (xs : List ℕ) : (countKey key xs).time = xs.length := by
+  induction xs with
+  | nil => simp [countKey]
+  | cons x xs ih => simp [countKey, ih, Nat.add_comm]
+
+@[simp]
+private theorem count_replicate_of_ne {key value count : ℕ} (h : value ≠ key) :
+    (List.replicate count value).count key = 0 := by
+  simp [List.count_replicate, h]
+
+/-- The generated blocks contain no key below the requested start. -/
+private theorem buildCountedFrom_start_le_of_mem {value start fuel : ℕ} {xs : List ℕ} :
+    value ∈ ⟪buildCountedFrom start fuel xs⟫ → start ≤ value := by
+  induction fuel generalizing start value with
+  | zero => simp [buildCountedFrom]
+  | succ fuel ih =>
+    simp only [buildCountedFrom, ret_bind, ret_pure, ret_countKey, List.mem_append,
+      List.mem_replicate]
+    intro h
+    rcases h with h | h
+    · exact le_of_eq h.2.symm
+    · exact le_trans (Nat.le_succ start) (ih h)
+
+/--
+The generated blocks contain the right number of copies for each key. This is the main counting
+invariant: keys outside `start, ..., start + fuel - 1` occur zero times.
+-/
+@[grind =]
+private theorem buildCountedFrom_count (key start fuel : ℕ) (xs : List ℕ) :
+    (⟪buildCountedFrom start fuel xs⟫).count key =
+      if start ≤ key ∧ key < start + fuel then xs.count key else 0 := by
+  induction fuel generalizing start with
+  | zero => simp [buildCountedFrom]
+  | succ fuel ih =>
+    by_cases hstart : start = key
+    · subst key
+      simp [buildCountedFrom, ih]
+    · simp only [buildCountedFrom, ret_bind, ret_pure, ret_countKey, List.count_append]
+      simp only [ne_eq, hstart, not_false_eq_true, count_replicate_of_ne, ih, Nat.zero_add]
+      by_cases htotal : start ≤ key ∧ key < start + (fuel + 1)
+      · have htail : start + 1 ≤ key ∧ key < start + 1 + fuel := by omega
+        simp [htotal, htail]
+      · have htail : ¬(start + 1 ≤ key ∧ key < start + 1 + fuel) := by omega
+        rw [if_neg htail, if_neg htotal]
+
+/-- In-range keys are counted exactly as in the input. -/
+theorem countingSort_count_of_le_bound {bound key : ℕ} (xs : List ℕ) (hkey : key ≤ bound) :
+    (⟪countingSort bound xs⟫).count key = xs.count key := by
+  rw [countingSort, buildCountedFrom_count]
+  split_ifs with h
+  · rfl
+  · omega
+
+/-- Out-of-range keys never appear in the output. -/
+theorem countingSort_count_of_bound_lt {bound key : ℕ} (xs : List ℕ) (hkey : bound < key) :
+    (⟪countingSort bound xs⟫).count key = 0 := by
+  rw [countingSort, buildCountedFrom_count]
+  split_ifs with h
+  · omega
+  · rfl
+
+/-- If the input is bounded, every out-of-range key has count zero in the input too. -/
+private theorem bounded_count_eq_zero_of_bound_lt {bound key : ℕ} {xs : List ℕ}
+    (hxs : BoundedBy bound xs) (hkey : bound < key) : xs.count key = 0 := by
+  rw [List.count_eq_zero]
+  intro hmem
+  have := hxs key hmem
+  omega
+
+/-- Counting sort is a permutation because every key has the same count in input and output. -/
+theorem countingSort_perm (bound : ℕ) (xs : List ℕ) (hxs : BoundedBy bound xs) :
+    (⟪countingSort bound xs⟫).Perm xs := by
+  rw [List.perm_iff_count]
+  intro key
+  by_cases hkey : key ≤ bound
+  · exact countingSort_count_of_le_bound xs hkey
+  · have hlt : bound < key := Nat.lt_of_not_ge hkey
+    rw [countingSort_count_of_bound_lt xs hlt, bounded_count_eq_zero_of_bound_lt hxs hlt]
+
+/-- Every generated block list is sorted because later blocks only contain larger keys. -/
+private theorem buildCountedFrom_sorted (start fuel : ℕ) (xs : List ℕ) :
+    List.Pairwise (· ≤ ·) ⟪buildCountedFrom start fuel xs⟫ := by
+  induction fuel generalizing start with
+  | zero => simp [buildCountedFrom]
+  | succ fuel ih =>
+    simp only [buildCountedFrom, ret_bind, ret_pure, ret_countKey]
+    rw [List.pairwise_append]
+    refine ⟨by simp [List.pairwise_replicate], ih (start + 1), ?_⟩
+    intro a ha b hb
+    have haEq : a = start := (List.mem_replicate.mp ha).2
+    have hbLower : start + 1 ≤ b := buildCountedFrom_start_le_of_mem hb
+    omega
+
+/-- Counting sort returns a sorted list. -/
+theorem countingSort_sorted (bound : ℕ) (xs : List ℕ) :
+    List.Pairwise (· ≤ ·) ⟪countingSort bound xs⟫ := by
+  exact buildCountedFrom_sorted 0 (bound + 1) xs
+
+/-- Filtering a list by one value gives exactly `count` copies of that value. -/
+private theorem filter_eq_replicate_count (key : ℕ) (xs : List ℕ) :
+    xs.filter (fun x => x = key) = List.replicate (xs.count key) key := by
+  induction xs with
+  | nil => simp
+  | cons x xs ih =>
+    by_cases h : x = key
+    · simp [h, ih, List.replicate_succ]
+    · simp [h, ih]
+
+/--
+Counting sort is stable for natural numbers because every equal-value subsequence is just a block of
+identical copies, and the count theorem proves that block has the same length as in the input.
+-/
+theorem countingSort_stable (bound : ℕ) (xs : List ℕ) (hxs : BoundedBy bound xs) :
+    StableByValue xs ⟪countingSort bound xs⟫ := by
+  intro key
+  rw [filter_eq_replicate_count key ⟪countingSort bound xs⟫, filter_eq_replicate_count key xs]
+  by_cases hkey : key ≤ bound
+  · rw [countingSort_count_of_le_bound xs hkey]
+  · have hlt : bound < key := Nat.lt_of_not_ge hkey
+    rw [countingSort_count_of_bound_lt xs hlt, bounded_count_eq_zero_of_bound_lt hxs hlt]
+
+/-- Counting sort is functionally correct for inputs covered by the supplied bound. -/
+theorem countingSort_correct (bound : ℕ) (xs : List ℕ) (hxs : BoundedBy bound xs) :
+    List.Pairwise (· ≤ ·) ⟪countingSort bound xs⟫ ∧ (⟪countingSort bound xs⟫).Perm xs ∧
+      StableByValue xs ⟪countingSort bound xs⟫ := by
+  exact ⟨countingSort_sorted bound xs, countingSort_perm bound xs hxs,
+    countingSort_stable bound xs hxs⟩
+
+end Correctness
+
+section TimeComplexity
+
+/-- Building counted blocks scans the input once for each generated key. -/
+@[simp]
+private theorem buildCountedFrom_time (start fuel : ℕ) (xs : List ℕ) :
+    (buildCountedFrom start fuel xs).time = fuel * xs.length := by
+  induction fuel generalizing start with
+  | zero => simp [buildCountedFrom]
+  | succ fuel ih => simp [buildCountedFrom, ih, Nat.succ_mul, Nat.add_comm]
+
+/-- Time complexity of counting sort under the stated cost model. -/
+theorem countingSort_time (bound : ℕ) (xs : List ℕ) :
+    (countingSort bound xs).time = (bound + 1) * xs.length := by
+  simp [countingSort]
+
+/-- A convenient upper bound form of `countingSort_time`. -/
+theorem countingSort_time_le (bound : ℕ) (xs : List ℕ) :
+    let n := xs.length
+    (countingSort bound xs).time ≤ (bound + 1) * n := by
+  simp [countingSort_time]
+
+end TimeComplexity
+
+end Cslib.Algorithms.Lean.TimeM

Cslib/Algorithms/Lean/Sorting.lean

@@ -0,0 +1,28 @@
+/-
+Copyright (c) 2026 Robert Joseph George. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Robert Joseph George
+-/
+
+module
+
+public import Cslib.Init
+
+/-!
+# Sorting utilities
+
+For stable list sorts, filtering the input and output by any value records the relative order of
+the copies of that value.
+-/
+
+@[expose] public section
+
+set_option autoImplicit false
+
+namespace Cslib.Algorithms.Lean
+
+/-- `ys` preserves the order of equal values from `xs`. -/
+abbrev StableByValue {α : Type*} [DecidableEq α] (xs ys : List α) : Prop :=
+  ∀ value, ys.filter (fun x => x = value) = xs.filter (fun x => x = value)
+
+end Cslib.Algorithms.Lean