set_option tactic.simp.trace true
set_option trace.Meta.Tactic.simp.rewrite true

def f (x : α) := x

example (a : α) (b : List α) : f (a::b = []) = False :=
  by simp [f]

def length : List α → Nat
  | []    => 0
  | a::as => length as + 1

example (a b c : α) (as : List α) : length (a :: b :: as) > length as := by
  simp [length]
  apply Nat.lt.step
  apply Nat.lt_succ_self

def fact : Nat → Nat
  | 0 => 1
  | x+1 => (x+1) * fact x

theorem ex3 : fact x > 0 := by
  induction x with
  | zero => decide
  | succ x ih =>
    simp [fact]
    apply Nat.mul_pos
    apply Nat.zero_lt_succ
    apply ih

def head [Inhabited α] : List α → α
  | []   => default
  | a::_ => a

example [Inhabited α] (a : α) (as : List α) : head (a::as) = a :=
  by simp [head]

def foo := 10

example (x : Nat) : foo + x = 10 + x := by
  simp [foo]
  done

def g (x : Nat) : Nat := Id.run <| do
  let x := x
  return x

example : g x = x := by
  simp [g, bind, pure]
  rfl

def f1 : StateM Nat Unit := do
  modify fun x => g x

def f2 : StateM Nat Unit := do
  let s ← get
  set <| g s

example : f1 = f2 := by
  simp [f1, f2, bind, StateT.bind, get, getThe, MonadStateOf.get, StateT.get, pure, set, StateT.set, modify, modifyGet, MonadStateOf.modifyGet, StateT.modifyGet]

def h (x : Nat) : Sum (Nat × Nat) Nat := Sum.inl (x, x)

def bla (x : Nat) :=
  match h x with
  | Sum.inl (y, z) => y + z
  | Sum.inr _ => 0

example (x : Nat) : bla x = x + x := by
  simp [bla, h]

example (x : Nat) (h : 1 ≤ x) : x - 1 + 1 + 2 = x + 2 := by
  simp [h, Nat.sub_add_cancel]

example (x : Nat) : (if h : 1 ≤ x then x - 1 + 1 else 0) = (if _h : 1 ≤ x then x else 0) := by
  simp (config := {contextual := true}) [h, Nat.sub_add_cancel]

theorem my_thm : a ∧ a ↔ a := ⟨fun h => h.1, fun h => ⟨h, h⟩⟩

example : a ∧ (b ∧ b) ↔ a ∧ b := by simp [my_thm]
example : (a ∧ (b ∧ b)) = (a ∧ b) := by simp only [my_thm]

example : x - 1 + 1 = x := by simp (discharger := sorry) [Nat.sub_add_cancel]