(** Equality on natural numbers *)

(** * Propositional equality  *)

Fixpoint eq_nat n m : Prop :=
  match n, m with
    | O, O => True
    | O, S _ => False
    | S _, O => False
    | S n1, S m1 => eq_nat n1 m1
  end.

(* Specifications: *)
Theorem eq_nat_refl n : eq_nat n n.
Theorem eq_nat_is_eq n m : eq_nat n m <-> n = m.
Lemma eq_eq_nat n m : n = m -> eq_nat n m.
Lemma eq_nat_eq n m : eq_nat n m -> n = m.
Theorem eq_nat_elim : forall n (P:nat -> Prop), P n -> forall m, eq_nat n m -> P m.
Theorem eq_nat_decide : forall n m, {eq_nat n m} + {~ eq_nat n m}.