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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
(*i $Id: ZBase.v 11674 2008-12-12 19:48:40Z letouzey $ i*)
Require Export Decidable.
Require Export ZAxioms.
Require Import NZMulOrder.
Module ZBasePropFunct (Import ZAxiomsMod : ZAxiomsSig).
(* Note: writing "Export" instead of "Import" on the previous line leads to
some warnings about hiding repeated declarations and results in the loss of
notations in Zadd and later *)
Open Local Scope IntScope.
Module Export NZMulOrderMod := NZMulOrderPropFunct NZOrdAxiomsMod.
Theorem Zsucc_wd : forall n1 n2 : Z, n1 == n2 -> S n1 == S n2.
Proof NZsucc_wd.
Theorem Zpred_wd : forall n1 n2 : Z, n1 == n2 -> P n1 == P n2.
Proof NZpred_wd.
Theorem Zpred_succ : forall n : Z, P (S n) == n.
Proof NZpred_succ.
Theorem Zeq_refl : forall n : Z, n == n.
Proof (proj1 NZeq_equiv).
Theorem Zeq_sym : forall n m : Z, n == m -> m == n.
Proof (proj2 (proj2 NZeq_equiv)).
Theorem Zeq_trans : forall n m p : Z, n == m -> m == p -> n == p.
Proof (proj1 (proj2 NZeq_equiv)).
Theorem Zneq_sym : forall n m : Z, n ~= m -> m ~= n.
Proof NZneq_sym.
Theorem Zsucc_inj : forall n1 n2 : Z, S n1 == S n2 -> n1 == n2.
Proof NZsucc_inj.
Theorem Zsucc_inj_wd : forall n1 n2 : Z, S n1 == S n2 <-> n1 == n2.
Proof NZsucc_inj_wd.
Theorem Zsucc_inj_wd_neg : forall n m : Z, S n ~= S m <-> n ~= m.
Proof NZsucc_inj_wd_neg.
(* Decidability and stability of equality was proved only in NZOrder, but
since it does not mention order, we'll put it here *)
Theorem Zeq_dec : forall n m : Z, decidable (n == m).
Proof NZeq_dec.
Theorem Zeq_dne : forall n m : Z, ~ ~ n == m <-> n == m.
Proof NZeq_dne.
Theorem Zcentral_induction :
forall A : Z -> Prop, predicate_wd Zeq A ->
forall z : Z, A z ->
(forall n : Z, A n <-> A (S n)) ->
forall n : Z, A n.
Proof NZcentral_induction.
(* Theorems that are true for integers but not for natural numbers *)
Theorem Zpred_inj : forall n m : Z, P n == P m -> n == m.
Proof.
intros n m H. apply NZsucc_wd in H. now do 2 rewrite Zsucc_pred in H.
Qed.
Theorem Zpred_inj_wd : forall n1 n2 : Z, P n1 == P n2 <-> n1 == n2.
Proof.
intros n1 n2; split; [apply Zpred_inj | apply NZpred_wd].
Qed.
End ZBasePropFunct.
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