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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) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** 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.
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