ZBase.v

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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
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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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(*                      Evgeny Makarov, INRIA, 2007                     *)
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Require Export Decidable.
Require Export ZAxioms.
Require Import NZProperties.

Module ZBaseProp (Import Z : ZAxiomsMiniSig').
Include NZProp Z.

(* Theorems that are true for integers but not for natural numbers *)

Theorem pred_inj : forall n m, P n == P m -> n == m.forall n m : t, P n == P m -> n == m
Proof.forall n m : t, P n == P m -> n == m
intros n m H.n, m:t
H:P n == P m
n == m apply succ_wd in H.n, m:t
H:S (P n) == S (P m)
n == m now rewrite 2 succ_pred in H.
Qed.

Theorem pred_inj_wd : forall n1 n2, P n1 == P n2 <-> n1 == n2.forall n1 n2 : t, P n1 == P n2 <-> n1 == n2
Proof.forall n1 n2 : t, P n1 == P n2 <-> n1 == n2
intros n1 n2; split; [apply pred_inj | intros; now f_equiv].
Qed.

Lemma succ_m1 : S (-1) == 0.S (-1) == 0
Proof.S (-1) == 0
 now rewrite one_succ, opp_succ, opp_0, succ_pred.
Qed.

End ZBaseProp.