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import Mathlib.Order.Lattice

variable [LinearOrder α]

def ins (a : α) : List α → List α
  | [] => [a]
  | x :: xs =>
    if a ≤ x then a :: x :: xs else x :: ins a xs

def List.insSort : List α → List α
  | [] => []
  | x :: xs => ins x xs.insSort

def Sorted : List α → Prop
  | [] | [_] => True
  | x :: x' :: xs => x ≤ x' ∧ Sorted (x' :: xs)

theorem insSorted (a : α) (xs : List α) : Sorted xs → Sorted (ins a xs) := by
  induction xs
  case nil => simp [Sorted, ins]
  case cons x xs ih =>
    cases xs
    case nil =>
      by_cases h₁ : a ≤ x
      · simp [ins, h₁, Sorted]
      · simp only [Sorted, ins, h₁, ↓reduceIte, and_true, forall_const]
        exact le_of_lt <| lt_of_not_ge h₁
    case cons x' t =>
      by_cases h₁ : a ≤ x
      · simp [ins, h₁, Sorted]
      · by_cases h₂ : a ≤ x'
        · simp only [Sorted, ins, h₁, ↓reduceIte, h₂, true_and, and_imp]
          intro h₃ h₄
          constructor
          · exact le_of_lt <| lt_of_not_ge h₁
          · exact h₄
        · simp only [Sorted, ins, h₁, ↓reduceIte, h₂, and_imp]
          intro h₃ h₄
          constructor
          · exact h₃
          · have ih₂ := ih h₄
            simp only [ins, h₂, ↓reduceIte] at ih₂
            exact ih₂

theorem insSortSorted (xs : List α) : Sorted xs.insSort := by
  induction xs
  case nil => simp [List.insSort, Sorted]
  case cons x xs ih =>
    rw [List.insSort]
    exact (insSorted x xs.insSort) ih

theorem insPerm (xs : List α) (a : α) : (a :: xs).Perm (ins a xs) := by
  induction xs
  case nil => rw [ins]
  case _ x xs ih =>
    rw [ins]
    by_cases h₁ : a ≤ x
    · simp [h₁]
    · simp only [h₁]
      exact .trans (.swap x a xs) (.cons x ih)

theorem insSortPerm (xs : List α) : xs.Perm xs.insSort := by
  induction xs
  case nil => rw [List.insSort]
  case cons h t ih =>
    rw [List.insSort]
    exact .trans (.cons h ih) (insPerm t.insSort h)

theorem insSortCorrect (xs : List α) : Sorted xs.insSort ∧ xs.Perm xs.insSort :=
  ⟨insSortSorted xs, insSortPerm xs⟩