# Library Coq.Arith.EqNat

Equality on natural numbers
Open Local Scope nat_scope.
Implicit Types m n x y : nat.

# 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.
Theorem eq_nat_refl : forall n, eq_nat n n.
Hint Resolve eq_nat_refl: arith v62.
eq restricted to nat and eq_nat are equivalent
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}.

# Boolean equality on nat

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.