import logic
open decidable
open eq

inductive nat : Type :=
zero : nat,
succ : nat → nat
definition refl := @eq.refl
namespace nat
theorem induction_on {P : nat → Prop} (a : nat) (H1 : P zero) (H2 : ∀ (n : nat) (IH : P n), P (succ n)) : P a
:= nat.rec H1 H2 a

definition pred (n : nat) := nat.rec zero (fun m x, m) n
theorem pred_zero : pred zero = zero := refl _
theorem pred_succ (n : nat) : pred (succ n) = n := refl _

theorem zero_or_succ (n : nat) : n = zero ∨ n = succ (pred n)
:= induction_on n
    (or.intro_left _ (refl zero))
    (take m IH, or.intro_right _
      (show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))

theorem zero_or_succ2 (n : nat) : n = zero ∨ n = succ (pred n)
:= @induction_on _ n
    (or.intro_left _ (refl zero))
    (take m IH, or.intro_right _
      (show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
end nat
