diff options
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 0468b3bf5..6b6b71703 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). |