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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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(** Equality is decidable on [nat] *)
Local Open Scope nat_scope.
Notation not_eq_sym := not_eq_sym (only parsing).
Implicit Types m n p q : nat.
Definition le_or_le_S := le_le_S_dec.
Definition Pcompare := gt_eq_gt_dec.
Proof le_ge_dec.
Definition lt_or_eq n m := {m > n} + {n = m}.
Proof le_lt_eq_dec.
Proof le_lt_or_eq.

(* By special request of G. Kahn - Used in Group Theory *)
Lemma discrete_nat :
forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + r))).