Local Open Scope nat_scope.

Implicit Types m n x y : nat.

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 : forall n, eq_nat n n.
Hint Resolve eq_nat_refl: arith v62.

Lemma eq_eq_nat : forall n m, n = m -> eq_nat n m.
Hint Immediate eq_eq_nat: arith v62.

Lemma eq_nat_eq : forall n m, eq_nat n m -> n = m.
Hint Immediate eq_nat_eq: arith v62.

Theorem eq_nat_is_eq : forall 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}.

Fixpoint beq_nat n m : bool :=
  match n, m with
    | O, O => true
    | O, S _ => false
    | S _, O => false
    | S n1, S m1 => beq_nat n1 m1
  end.

Lemma beq_nat_refl : forall n, true = beq_nat n n.

Definition beq_nat_eq : forall x y, true = beq_nat x y -> x = y.

Lemma beq_nat_true : forall x y, beq_nat x y = true -> x=y.

Lemma beq_nat_false : forall x y, beq_nat x y = false -> x<>y.

Lemma beq_nat_true_iff : forall x y, beq_nat x y = true <-> x=y.

Lemma beq_nat_false_iff : forall x y, beq_nat x y = false <-> x<>y.