diff options
| author | letouzey | 2010-01-08 14:44:56 +0000 |
|---|---|---|
| committer | letouzey | 2010-01-08 14:44:56 +0000 |
| commit | 5db31bb0333810ccdd0a79e9855ae9d2fcdbf2d3 (patch) | |
| tree | dd8cd4a8b4453d96fdcd8fea56c9a56a4f766087 /theories/Numbers/Integer/Abstract/ZDivTrunc.v | |
| parent | c630fdf04db508d5d877a6b1fd93145893377287 (diff) | |
Numbers: axiomatization + generic properties of abs and sgn.
This allow to really finish files about division.
An abs and sgn is added to BigZ.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12644 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZDivTrunc.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivTrunc.v | 27 |
1 files changed, 6 insertions, 21 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v index 0468b3bf58..6b6b717035 100644 --- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v +++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v @@ -35,8 +35,8 @@ End ZDivSpecific. Module Type ZDiv (Z:ZAxiomsSig) := DivMod Z <+ NZDivCommon Z <+ ZDivSpecific Z. -Module Type ZDivSig := ZAxiomsSig <+ ZDiv. -Module Type ZDivSig' := ZAxiomsSig' <+ ZDiv <+ DivModNotation. +Module Type ZDivSig := ZAxiomsExtSig <+ ZDiv. +Module Type ZDivSig' := ZAxiomsExtSig' <+ ZDiv <+ DivModNotation. Module ZDivPropFunct (Import Z : ZDivSig')(Import ZP : ZPropSig Z). @@ -205,31 +205,16 @@ Proof. exact div_pos. Qed. Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b. Proof. exact div_str_pos. Qed. -(** TODO: TO MIGRATE LATER *) -Definition abs z := max z (-z). -Lemma abs_pos : forall z, 0<=z -> abs z == z. -Proof. -intros; apply max_l. apply le_trans with 0; trivial. -now rewrite opp_nonpos_nonneg. -Qed. -Lemma abs_neg : forall z, 0<=-z -> abs z == -z. -Proof. -intros; apply max_r. apply le_trans with 0; trivial. -now rewrite <- opp_nonneg_nonpos. -Qed. - -(** END TODO *) - Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> abs a < abs b). Proof. intros. pos_or_neg a; pos_or_neg b. -rewrite div_small_iff; try order. rewrite 2 abs_pos; intuition; order. +rewrite div_small_iff; try order. rewrite 2 abs_eq; intuition; order. rewrite <- opp_inj_wd, opp_0, <- div_opp_r, div_small_iff by order. - rewrite (abs_pos a), (abs_neg b); intuition; order. + rewrite (abs_eq a), (abs_neq' b); intuition; order. rewrite <- opp_inj_wd, opp_0, <- div_opp_l, div_small_iff by order. - rewrite (abs_neg a), (abs_pos b); intuition; order. + rewrite (abs_neq' a), (abs_eq b); intuition; order. rewrite <- div_opp_opp, div_small_iff by order. - rewrite (abs_neg a), (abs_neg b); intuition; order. + rewrite (abs_neq' a), (abs_neq' b); intuition; order. Qed. Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> abs a < abs b). |