open nat

definition f : nat → nat → nat,
f _ 0 := 0,
f 0 _ := 1,
f _ _ := arbitrary nat

theorem f_zero_right : ∀ a, f a 0 = 0,
f_zero_right 0        := rfl,
f_zero_right (succ _) := rfl

theorem f_zero_succ (a : nat) : f 0 (a+1) = 1 :=
rfl

theorem f_succ_succ (a b : nat) : f (a+1) (b+1) = arbitrary nat :=
rfl