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import Std.Tactic.Do
import Mathlib

variable [LinearOrder α] (A : Array α)

def BubbleSort := Id.run do
  let N := A.size
  let mut A := A.toVector
  for i in List.range (N - 1) do
    for hj : j in List.range (N - i - 1) do
      have := List.mem_range.mp hj
      if A[j + 1] < A[j] then
        A := A.swap j (j + 1)
  return A.toArray

def ICan'tBelieveItCanSort := Id.run do
  let N := A.size
  let mut A := A.toVector
  for hi : i in [:N] do
    for hj : j in [:N] do
      if A[i] < A[j] then
        A := A.swap i j
  return A.toArray

#guard let A := #[69, 420, 1, 1, 13, 1, 65536]
  ICan'tBelieveItCanSort A = A.qsort

open Std.Do

theorem perm : ICan'tBelieveItCanSort.{0} A |>.Perm A := by
  generalize h : ICan'tBelieveItCanSort A = x
  suffices Multiset.ofList x.toList = A.toList by
    exact { toList := Multiset.coe_eq_coe.mp this }
  apply Id.of_wp_run_eq h
  mvcgen
  case inv1 | inv2 => exact ⇓⟨_, A'⟩ => ⌜Multiset.ofList A.toList = A'.toList⌝
  all_goals try grind
  case vc1.step.isTrue =>
    expose_names
    simp_all only [Multiset.coe_eq_coe]
    exact h_5.trans <| .symm <| Vector.perm_iff_toList_perm.mp <| Vector.swap_perm (by grind) (by grind)

theorem sorted : ICan'tBelieveItCanSort.{0} A |>.Pairwise (· ≤ ·) := by
  generalize h : ICan'tBelieveItCanSort A = x
  apply Id.of_wp_run_eq h
  mvcgen <;> expose_names
  case inv1 => exact ⇓⟨xs, A'⟩ => ⌜A'.take xs.pos |>.toArray.Pairwise (· ≤ ·)⌝
  case inv2 =>
    exact ⇓⟨xs, A'⟩ => ⌜(A'.take cur |>.toArray.Pairwise (· ≤ ·)) ∧ ∀ i (_ : i < xs.pos), A'[i]'(by
      have : xs.prefix.length + xs.suffix.length = N := by simp [← List.length_append, xs.property]
      grind
      ) ≤ A'[cur]'(by grind)⌝
  case vc1.step.isTrue =>
    simp_all
    constructor
    · rw [Array.pairwise_iff_getElem] at h_5 ⊢
      intro i j hi hj hij
      simp
      grind
    · grind
  case vc2.step.isFalse => constructor <;> grind
  case vc3.step.pre => grind
  case vc4.step.post.success =>
    simp_all
    rw [Array.pairwise_iff_getElem] at h_3 ⊢
    grind
  case vc5.a.pre => grind
  case vc6.a.post.success =>
    simp_all
    rwa [show r.toArray = r.toArray.extract 0 N by grind]

theorem ICan'tBelieveICanProveItCanSort : (ICan'tBelieveItCanSort.{0} A |>.Perm A)
    ∧ (ICan'tBelieveItCanSort.{0} A |>.Pairwise (· ≤ ·)) :=
  ⟨perm A, sorted A⟩