Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Arith.EqNat

Require Import PeanoNat.
Local Open Scope nat_scope.
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.
Theorem eq_nat_refl n : eq_nat n n.
eq restricted to nat and eq_nat are equivalent
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}.