x : Nat
h : f (f x) = x
⊢ (have y := x * x;
    if True then 1 else y + 1) =
    1
theorem ex0 : ∀ (x : Nat),
  f (f x) = x →
    (have y := 0 + x * x;
      if f (f x) = x then 1 else y + 1) =
      1 :=
fun x h =>
  Eq.mpr
    (id
      (congrArg (fun x => x = 1)
        (id
          (id
            (have_congr' (Nat.zero_add (x * x)) fun y =>
              ite_congr (Eq.trans (congrArg (fun x_1 => x_1 = x) h) (eq_self x)) (fun a => Eq.refl 1) fun a =>
                Eq.refl (y + 1))))))
    (of_eq_true (Eq.trans (congrArg (fun x => x = 1) (ite_cond_eq_true 1 (x * x + 1) (Eq.refl True))) (eq_self 1)))
x : Nat
h : f (f x) = x
⊢ (have y := x * x;
    if True then 1 else y + 1) =
    1
theorem ex1 : ∀ (x : Nat),
  f (f x) = x →
    (have y := x * x;
      if f (f x) = x then 1 else y + 1) =
      1 :=
fun x h =>
  Eq.mpr
    (id
      (congrArg (fun x => x = 1)
        (id
          (id
            (have_body_congr' (x * x) fun y =>
              ite_congr (Eq.trans (congrArg (fun x_1 => x_1 = x) h) (eq_self x)) (fun a => Eq.refl 1) fun a =>
                Eq.refl (y + 1))))))
    (of_eq_true (Eq.trans (congrArg (fun x => x = 1) (ite_cond_eq_true 1 (x * x + 1) (Eq.refl True))) (eq_self 1)))
x z : Nat
h : f (f x) = x
h' : z = x
⊢ (have y := x;
    y) =
    z
theorem ex2 : ∀ (x z : Nat),
  f (f x) = x →
    z = x →
      (have y := f (f x);
        y) =
        z :=
fun x z h h' =>
  Eq.mpr (id (congrArg (fun x => x = z) (id (id (have_val_congr' h)))))
    (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
⊢ (have n := 10;
    fun x => True) =
    fun z => p
theorem ex4 : ∀ (p : Prop),
  p →
    (have n := 10;
      fun x => x = x) =
      fun z => p :=
fun p h =>
  Eq.mpr
    (id (congrArg (fun x => x = fun z => p) (id (id (have_body_congr_dep' 10 fun n => funext fun x => eq_self x)))))
    (of_eq_true (Eq.trans (congrArg (Eq fun x => True) (funext fun z => eq_true h)) (eq_self fun x => True)))