Require Import PeanoNat.
Local Open Scope nat_scope.

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.

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}.