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(************************************************************************)
(* * The Rocq Prover / The Rocq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
From Stdlib Require Export Decidable.
From Stdlib Require Export ZAxioms.
From Stdlib Require Import NZMulOrder.
Module ZBaseProp (Import Z : ZAxiomsMiniSig').
Include NZMulOrderProp Z.
(* Theorems that are true for integers but not for natural numbers *)
Theorem pred_inj : forall n m, P n == P m -> n == m.
Proof.
intros n m H. apply succ_wd in H. now rewrite 2 succ_pred in H.
Qed.
Theorem pred_inj_wd : forall n1 n2, P n1 == P n2 <-> n1 == n2.
Proof.
intros n1 n2; split; [apply pred_inj | intros; now f_equiv].
Qed.
Lemma succ_m1 : S (-1) == 0.
Proof.
now rewrite one_succ, opp_succ, opp_0, succ_pred.
Qed.
End ZBaseProp.
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