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classPreorder (α) extends LE α where
 trans {x y z : α} : x ≤ y → y ≤ z → x ≤ z
 refl (x : α) : x ≤ x

classTotalPreorder (α) extends Preorder α where
 total (x y : α) : x ≤ y ∨ y ≤ x
 refl x := (total x x).elim id id

theoremle_of_not_le [I : TotalPreorder α] {x y : α} (h : ¬x ≤ y) : y ≤ x :=
 (I.total x y).elim (False.elim ∘ h) id

instance : TotalPreorder Nat where
 trans := Nat.le_trans
 total := Nat.le_total

inductiveList.Chain (R : α → α → Prop) : α → List α → Prop
 | nil : Chain R a []
 | cons : R a b → Chain R b l → Chain R a (b :: l)

defList.Chain' (R : α → α → Prop) : List α → Prop
 | [] => True
 | a :: l => Chain R a l

inductiveList.Perm : List α → List α → Prop
 | nil : Perm [] []
 | cons (x) : Perm l₁ l₂ → Perm (x :: l₁) (x :: l₂)
 | swap (x y l) : Perm (y :: x :: l) (x :: y :: l)
 | trans : Perm l₁ l₂ → Perm l₂ l₃ → Perm l₁ l₃

theoremList.Perm.refl : (l : List α) → Perm l l
 | [] => nil
 | x :: l => cons x (refl l)

theoremList.Perm.symm (h : Perm l₂ l₁) : Perm l₁ l₂ := by
 induction h with
 | nil => exact nil
 | cons x _ ih => exact cons x ih
 | swap x y l => exact swap y x l
 | trans _ _ ih₁ ih₂ => exact trans ih₂ ih₁

theoremList.Perm.swap' (y) (h : Perm l₁ (x :: l₂)) : Perm (y :: l₁) (x :: y :: l₂) :=
 trans (cons y h) (swap x y l₂)

theoremList.Perm.length_eq (h : Perm l₁ l₂) : length l₁ = length l₂ := by
 induction h <;> simp [*]

theoremList.Perm.middle (x : α) : ∀ l₁ l₂, Perm (l₁ ++ x :: l₂) (x :: l₁ ++ l₂)
 | [], _ => refl _
 | y :: l₁, l₂ => swap' y (middle x l₁ l₂)

theoremList.Perm.inv : Perm (l₁ ++ x :: r₁) (l₂ ++ x :: r₂) → Perm (l₁ ++ r₁) (l₂ ++ r₂) := by
 generalize h₁ : l₁ ++ x :: r₁ = s₁
 generalize h₂ : l₂ ++ x :: r₂ = s₂
 intro h
 induction h generalizing l₁ r₁ l₂ r₂ with
 | nil => sorry
 | cons y h ih =>
 cases l₁ <;> cases l₂ <;> cases h₁ <;> cases h₂
 . exact h
 . exact trans h (middle _ _ _)
 . exact trans (symm (middle _ _ _)) h
 . exact cons y (ih rfl rfl)
 | swap _ _ _ => sorry
 | trans _ _ _ _ => sorry

theoremList.Perm.cons_inv (h : Perm (x :: l₁) (x :: l₂)) : Perm l₁ l₂ :=
 inv (l₁ := []) (l₂ := []) h

--theorem List.Perm.swap'_inv (h : Perm (y :: x :: l₁) (x :: y :: l₂)) : Perm l₁ l₂ :=
--cons_inv (inv (l₁ := [y]) (l₂ := []) h)

theoremList.Perm.swap'_inv (h : Perm (y :: l₁) (x :: y :: l₂)) : Perm l₁ (x :: l₂) :=
sorry

classComparisonSort (f : ∀ {α} [I : LE α] [DecidableRel I.le], List α → List α) where
 perm [I : LE α] [DecidableRel I.le] (l : List α) : (f l).Perm l
 sort [I : TotalPreorder α] [DecidableRel I.le] (l : List α) : (f l).Chain' I.le

section

variable [I : LE α] [DecidableRel I.le]

definsert (x : α) : List α → List α
 | [] => [x]
 | y :: l =>
if x ≤ y
then x :: y :: l
else y :: insert x l

definsertionSort : List α → List α
 | [] => []
 | x :: l => insert x (insertionSort l)

theoremperm_insert (x : α) (l) : (insert x l).Perm (x :: l) := by
 induction l
 case nil => simp [insert]
 case cons y l ih =>
 unfold insert
 split
 . simp
 simp
--. exact .trans (.cons _ ih) (.swap x y l)

end

theoremsort_insert [I : TotalPreorder α] [DecidableRel I.le] (x) (h : l.Chain' I.le) : (insert x l).Chain' I.le := by
 induction l
 case nil => exact .nil
 case cons y l ih =>
 unfold insert
 split
 case _ h' => exact .cons h' h
 case _ h' =>
have h' := le_of_not_le h'
 dsimp [List.Chain']
 cases h
 case nil => exact .cons h' .nil
 case cons z l h₁ h₂ =>
 specialize ih h₁
 unfold insert at ih ⊢
 split
 case _ h'' =>
 simp [h''] at ih
 exact .cons h' ih
 case _ h'' =>
 simp [h''] at ih
 exact .cons h₂ ih

instance : ComparisonSort @insertionSort where
 perm l := by
 induction l
 case nil => exact .nil
 case cons x l ih => exact .trans (perm_insert x (insertionSort l)) (.cons x ih)
 sort l := by
 induction l
 case nil => exact trivial
 case cons x l ih => exact sort_insert x ih

section

variable [I : LE α] [DecidableRel I.le]

defselect (x : α) : List α → α × List α
 | [] => (x, [])
 | y :: l =>
if x ≤ y
then ((select x l).1, y :: (select x l).2)
else ((select y l).1, x :: (select y l).2)

theoremperm_select (x : α) (l) : ((select x l).1 :: (select x l).2).Perm (x :: l) := by
 induction l generalizing x
 case nil => exact .cons x .nil
 case cons y l ih =>
 unfold select
 split
 . exact .trans (.trans (.swap y _ _) (.cons y (ih x))) (.swap x y l)
 . exact .trans (.swap x _ _) (.cons _ (ih y))

defselectionSort : List α → List α
 | [] => []
 | x :: l =>
let (y, l') := select x l
 y :: selectionSort l'
termination_by _ l => l.length

end

-- todo: comparison in monad, time complexity