x : Nat
h : f (f x) = x
⊢ (let y := x * x;
    if True then 1 else y + 1) =
    1
theorem ex1 : ∀ (x : Nat),
  f (f x) = x →
    (let y := x * x;
      if f (f x) = x then 1 else y + 1) =
      1 :=
fun x h =>
  Eq.mpr
    (congrFun
      (congrArg Eq
        (let_congr (Eq.refl (x * x)) fun y =>
          ite_congr (Eq.trans (congrFun (congrArg Eq h) x) (eq_self x)) (fun a => Eq.refl 1) fun a => Eq.refl (y + 1)))
      1)
    (of_eq_true (eq_self 1))
x z : Nat
h : f (f x) = x
h' : z = x
⊢ (let y := x;
    y) =
    z
theorem ex2 : ∀ (x z : Nat),
  f (f x) = x →
    z = x →
      (let y := f (f x);
        y) =
        z :=
fun x z h h' =>
  Eq.mpr (congrFun (congrArg Eq (let_val_congr (fun y => y) h)) z)
    (of_eq_true (Eq.trans (congrArg (Eq x) h') (eq_self x)))
x z : Nat
⊢ (let α := Nat;
    fun x => 0 + x) =
    id
p : Prop
h : p
⊢ (let n := 10;
    fun x => True) =
    fun z => p
theorem ex4 : ∀ (p : Prop),
  p →
    (let n := 10;
      fun x => x = x) =
      fun z => p :=
fun p h =>
  Eq.mpr (congrFun (congrArg Eq (let_body_congr 10 fun n => funext fun x => eq_self x)) fun z => p)
    (of_eq_true (Eq.trans (congrArg (Eq fun x => True) (funext fun z => eq_true h)) (eq_self fun x => True)))