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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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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.
Hint Resolve eq_nat_refl: arith.
(** [eq] restricted to [nat] and [eq_nat] are equivalent *)
Hint Immediate eq_eq_nat eq_nat_eq: arith.

Lemma beq_nat_false n m : (n =? m) = false -> n<>m.