diff options
Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZBase.v')
-rw-r--r-- | theories/Numbers/Integer/Abstract/ZBase.v | 69 |
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. |