constant f (x y : Nat) : Nat
constant g (x : Nat) : Nat

theorem ex1 (x : Nat) (h₁ : f x x = g x) (h₂ : g x = x) : f x (f x x) = x := by
  simp
  simp [*]

theorem ex2 (x : Nat) (h₁ : f x x = g x) (h₂ : g x = x) : f x (f x x) = x := by
  simp [*]

axiom g_ax (x : Nat) : g x = 0

theorem ex3 (x y : Nat) (h₁ : f x x = g x) (h₂ : f x x < 5) : f x x + f x x = 0 := by
  simp [*] at *
  trace_state
  have aux₁ : f x x = g x := h₁
  have aux₂ : g x < 5     := h₂
  simp [g_ax]