# 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}.

# Boolean equality on nat.

We reuse the one already defined in module Nat. In scope nat_scope, the notation "=?" can be used.

Notation beq_nat := Nat.eqb (only parsing).

Notation beq_nat_true_iff := Nat.eqb_eq (only parsing).
Notation beq_nat_false_iff := Nat.eqb_neq (only parsing).

#[local]
Definition beq_nat_refl_stt := fun n => eq_sym (Nat.eqb_refl n).
Opaque beq_nat_refl_stt.
#[deprecated(since="8.16",note="Use Nat.eqb_refl (with symmetry of equality) instead.")]
Notation beq_nat_refl := beq_nat_refl_stt.

#[local]
Definition beq_nat_true_stt := fun n m => proj1 (Nat.eqb_eq n m).
Opaque beq_nat_true_stt.
#[deprecated(since="8.16",note="Use the bidirectional version Nat.eqb_eq instead.")]
Notation beq_nat_true := beq_nat_true_stt.

#[local]
Definition beq_nat_false_stt := fun n m => proj1 (Nat.eqb_neq n m).
Opaque beq_nat_false_stt.
#[deprecated(since="8.16",note="Use the bidirectional version Nat.eqb_neq instead.")]
Notation beq_nat_false := beq_nat_false_stt.

#[local]
Definition beq_nat_eq_stt := fun n m Heq => proj1 (Nat.eqb_eq n m) (eq_sym Heq).
Opaque beq_nat_eq_stt.
#[deprecated(since="8.16",note="Use the bidirectional version Nat.eqb_eq (with symmetry of equality) instead.")]
Notation beq_nat_eq := beq_nat_eq_stt.