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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014     *)
(*   \VV/  **************************************************************)
(*    //   *      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.

Require Import Arith_base.
Require Import Peano_dec.
Require Import Compare_dec.

Definition le_or_le_S := le_le_S_dec.

Definition Pcompare := gt_eq_gt_dec.

Lemma le_dec : forall n m, {n <= m} + {m <= n}.
Proof le_ge_dec.

Definition lt_or_eq n m := {m > n} + {n = m}.

Lemma le_decide : forall n m, n <= m -> lt_or_eq n m.
Proof le_lt_eq_dec.

Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m.
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))).
Proof.
  intros m n H.
  lapply (lt_le_S m n); auto with arith.
  intro H'; lapply (le_lt_or_eq (S m) n); auto with arith.
  induction 1; auto with arith.
  right; exists (n - S (S m)); simpl.
  rewrite (plus_comm m (n - S (S m))).
  rewrite (plus_n_Sm (n - S (S m)) m).
  rewrite (plus_n_Sm (n - S (S m)) (S m)).
  rewrite (plus_comm (n - S (S m)) (S (S m))); auto with arith.
Qed.

Require Export Wf_nat.

Require Export Min Max.
