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import Std.Tactic.Do
import Mathlib

def BubbleSort [LT α] [DecidableLT α] (A : Array α) := 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 [LT α] [DecidableLT α] (A : Array α) := 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

open Std.Do

theorem perm_ICan'tBelieveItCanSort [LT α] [DecidableLT α] (A : Array α) : ICan'tBelieveItCanSort.{0} A |>.Perm A := by
  suffices Multiset.ofList (ICan'tBelieveItCanSort.{0} A).toList = A.toList by
    rw [Multiset.coe_eq_coe] at this
    exact { toList := this }
  generalize h : ICan'tBelieveItCanSort A = x
  apply Id.of_wp_run_eq h
  mvcgen
  case inv1 => exact ⇓⟨_, A'⟩ => ⌜Multiset.ofList A.toList = A'.toList⌝
  all_goals try grind
  case vc1.step.isTrue =>
    expose_names
    mleave
    simp_all only [Multiset.coe_eq_coe]
    have h₁ := Membership.get_elem_helper (Spec.forIn'_range._proof_4 pref cur suff h_1) rfl
    have h₂ := Membership.get_elem_helper (Spec.forIn'_range._proof_4 pref_1 cur_1 suff_1 h_3) rfl
    exact h_5.trans <| .symm <| Vector.perm_iff_toList_perm.mp <| Vector.swap_perm h₁ h₂

theorem sorted_ICan'tBelieveItCanSort [LT α] [DecidableLT α] (A : Array α) : ICan'tBelieveItCanSort.{0} A |>.Pairwise (¬ · > ·) := by
  generalize h : ICan'tBelieveItCanSort A = x
  apply Id.of_wp_run_eq h
  mvcgen
  case inv1 => exact ⇓⟨xs, A'⟩ => ⌜A'.take xs.prefix.length |>.toArray.Pairwise (¬ · > ·)⌝
  case vc1.step.isTrue =>
    expose_names
    simp_all
    grind
    sorry
  case vc2.step.isFalse =>
    expose_names
    simp_all
    grind
    sorry
  case vc3.step.pre => grind
  case vc4.step.post.success =>
    expose_names
    simp_all
    grind
    sorry
  case vc5.a.pre => grind
  case vc6.a.post.success =>
    expose_names
    simp_all
    have : r.toArray = r.toArray.extract 0 N := by grind
    rwa [this]