1 files changed, 8 insertions, 61 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v
index 0f71f2cc..44bb02ec 100644
--- a/theories/Numbers/Integer/Abstract/ZBase.v
+++ b/theories/Numbers/Integer/Abstract/ZBase.v
@@ -8,78 +8,25 @@
 (*                      Evgeny Makarov, INRIA, 2007                     *)
 (************************************************************************)
 
-(*i $Id: ZBase.v 11674 2008-12-12 19:48:40Z letouzey $ i*)
+(*i $Id$ i*)
 
 Require Export Decidable.
 Require Export ZAxioms.
-Require Import NZMulOrder.
+Require Import NZProperties.
 
-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.
+Module ZBasePropFunct (Import Z : ZAxiomsSig').
+Include NZPropFunct Z.
 
 (* Theorems that are true for integers but not for natural numbers *)
 
-Theorem Zpred_inj : forall n m : Z, P n == P m -> n == m.
+Theorem pred_inj : forall n m, P n == P m -> n == m.
 Proof.
-intros n m H. apply NZsucc_wd in H. now do 2 rewrite Zsucc_pred in H.
+intros n m H. apply succ_wd in H. now do 2 rewrite succ_pred in H.
 Qed.
 
-Theorem Zpred_inj_wd : forall n1 n2 : Z, P n1 == P n2 <-> n1 == n2.
+Theorem pred_inj_wd : forall n1 n2, P n1 == P n2 <-> n1 == n2.
 Proof.
-intros n1 n2; split; [apply Zpred_inj | apply NZpred_wd].
+intros n1 n2; split; [apply pred_inj | apply pred_wd].
 Qed.
 
 End ZBasePropFunct.