diff options
Diffstat (limited to 'theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v')
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v | 122 |
1 files changed, 46 insertions, 76 deletions
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v index 4e6eccea..0cb6848e 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -8,8 +8,6 @@ (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) -(*i $Id: DoubleDiv.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Set Implicit Arguments. Require Import ZArith. @@ -82,11 +80,7 @@ Section POS_MOD. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_ww_compare : forall x y, - match ww_compare x y with - | Eq => [[x]] = [[y]] - | Lt => [[x]] < [[y]] - | Gt => [[x]] > [[y]] - end. + ww_compare x y = Zcompare [[x]] [[y]]. Variable spec_ww_sub: forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. @@ -105,8 +99,8 @@ Section POS_MOD. intros w1 p; case (spec_to_w_Z p); intros HH1 HH2. unfold ww_pos_mod; case w1. simpl; rewrite Zmod_small; split; auto with zarith. - intros xh xl; generalize (spec_ww_compare p (w_0W w_zdigits)); - case ww_compare; + intros xh xl; rewrite spec_ww_compare. + case Zcompare_spec; rewrite spec_w_0W; rewrite spec_zdigits; fold wB; intros H1. rewrite H1; simpl ww_to_Z. @@ -134,8 +128,8 @@ Section POS_MOD. rewrite Z_mod_mult; auto with zarith. autorewrite with w_rewrite rm10. rewrite Zmod_mod; auto with zarith. -generalize (spec_ww_compare p ww_zdigits); - case ww_compare; rewrite spec_ww_zdigits; + rewrite spec_ww_compare. + case Zcompare_spec; rewrite spec_ww_zdigits; rewrite spec_zdigits; intros H2. replace (2^[[p]]) with wwB. rewrite Zmod_small; auto with zarith. @@ -266,12 +260,7 @@ Section DoubleDiv32. Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. Variable spec_compare : - forall x y, - match w_compare x y with - | Eq => [|x|] = [|y|] - | Lt => [|x|] < [|y|] - | Gt => [|x|] > [|y|] - end. + forall x y, w_compare x y = Zcompare [|x|] [|y|]. Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. Variable spec_w_add_carry_c : forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1. @@ -343,7 +332,7 @@ Section DoubleDiv32. (WW (w_sub a2 b2) a3) (WW b1 b2) | Gt => (w_0, W0) (* cas absurde *) end. - assert (Hcmp:=spec_compare a1 b1);destruct (w_compare a1 b1). + rewrite spec_compare. case Zcompare_spec; intro Hcmp. simpl in Hlt. rewrite Hcmp in Hlt;assert ([|a2|] < [|b2|]). omega. assert ([[WW (w_sub a2 b2) a3]] = ([|a2|]-[|b2|])*wB + [|a3|] + wwB). @@ -545,11 +534,7 @@ Section DoubleDiv21. 0 <= [[r]] < [|b1|] * wB + [|b2|]. Variable spec_ww_1 : [[ww_1]] = 1. Variable spec_ww_compare : forall x y, - match ww_compare x y with - | Eq => [[x]] = [[y]] - | Lt => [[x]] < [[y]] - | Gt => [[x]] > [[y]] - end. + ww_compare x y = Zcompare [[x]] [[y]]. Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. Theorem wwB_div: wwB = 2 * (wwB / 2). @@ -576,10 +561,9 @@ Section DoubleDiv21. intros a1 a2 b H Hlt; unfold ww_div21. Spec_ww_to_Z b; assert (Eq: 0 < [[b]]). Spec_ww_to_Z a1;omega. generalize Hlt H ;clear Hlt H;case a1. - intros H1 H2;simpl in H1;Spec_ww_to_Z a2; - match goal with |-context [ww_compare ?Y ?Z] => - generalize (spec_ww_compare Y Z); case (ww_compare Y Z) - end; simpl;try rewrite spec_ww_1;autorewrite with rm10; intros;zarith. + intros H1 H2;simpl in H1;Spec_ww_to_Z a2. + rewrite spec_ww_compare. case Zcompare_spec; + simpl;try rewrite spec_ww_1;autorewrite with rm10; intros;zarith. rewrite spec_ww_sub;simpl. rewrite Zmod_small;zarith. split. ring. assert (wwB <= 2*[[b]]);zarith. @@ -809,12 +793,7 @@ Section DoubleDivGt. Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_compare : - forall x y, - match w_compare x y with - | Eq => [|x|] = [|y|] - | Lt => [|x|] < [|y|] - | Gt => [|x|] > [|y|] - end. + forall x y, w_compare x y = Zcompare [|x|] [|y|]. Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. @@ -914,7 +893,7 @@ Section DoubleDivGt. end in [[WW ah al]]=[[q]]*[[WW bh bl]]+[[r]] /\ 0 <=[[r]]< [[WW bh bl]]). assert (Hh := spec_head0 Hpos). lazy zeta. - generalize (spec_compare (w_head0 bh) w_0); case w_compare; + rewrite spec_compare; case Zcompare_spec; rewrite spec_w_0; intros HH. generalize Hh; rewrite HH; simpl Zpower; rewrite Zmult_1_l; intros (HH1, HH2); clear HH. @@ -1058,14 +1037,13 @@ Section DoubleDivGt. assert (H2:=spec_div_gt Hgt Hpos);destruct (w_div_gt al bl). repeat rewrite spec_w_0W;simpl;rewrite spec_w_0;simpl;trivial. clear H. - assert (Hcmp := spec_compare w_0 bh); destruct (w_compare w_0 bh). + rewrite spec_compare; case Zcompare_spec; intros Hcmp. rewrite spec_w_0 in Hcmp. change [[WW bh bl]] with ([|bh|]*wB+[|bl|]). rewrite <- Hcmp;rewrite Zmult_0_l;rewrite Zplus_0_l. simpl in Hpos;rewrite <- Hcmp in Hpos;simpl in Hpos. assert (H2:= @spec_double_divn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hpos). - unfold double_to_Z,double_wB,double_digits in H2. destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 (WW ah al) bl). @@ -1154,7 +1132,7 @@ Section DoubleDivGt. rewrite spec_w_0W;rewrite spec_w_mod_gt_eq;trivial. destruct (w_div_gt al bl);simpl;rewrite spec_w_0W;trivial. clear H. - assert (H2 := spec_compare w_0 bh);destruct (w_compare w_0 bh). + rewrite spec_compare; case Zcompare_spec; intros H2. rewrite (@spec_double_modn1_aux w w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_w_0 spec_compare 1 (WW ah al) bl). destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 @@ -1227,13 +1205,14 @@ Section DoubleDivGt. end | Gt => W0 (* absurde *) end). - assert (Hbh := spec_compare w_0 bh);destruct (w_compare w_0 bh). - simpl ww_to_Z in *. rewrite spec_w_0 in Hbh;rewrite <- Hbh; + rewrite spec_compare, spec_w_0. + case Zcompare_spec; intros Hbh. + simpl ww_to_Z in *. rewrite <- Hbh. rewrite Zmult_0_l;rewrite Zplus_0_l. - assert (Hbl := spec_compare w_0 bl); destruct (w_compare w_0 bl). - rewrite spec_w_0 in Hbl;rewrite <- Hbl;apply Zis_gcd_0. + rewrite spec_compare, spec_w_0. + case Zcompare_spec; intros Hbl. + rewrite <- Hbl;apply Zis_gcd_0. simpl;rewrite spec_w_0;rewrite Zmult_0_l;rewrite Zplus_0_l. - rewrite spec_w_0 in Hbl. apply Zis_gcd_mod;zarith. change ([|ah|] * wB + [|al|]) with (double_to_Z w_digits w_to_Z 1 (WW ah al)). rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div @@ -1243,20 +1222,20 @@ Section DoubleDivGt. rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. - rewrite spec_w_0 in Hbl;Spec_w_to_Z bl;exfalso;omega. - rewrite spec_w_0 in Hbh;assert (H:= spec_ww_mod_gt_aux _ _ _ Hgt Hbh). + Spec_w_to_Z bl;exfalso;omega. + assert (H:= spec_ww_mod_gt_aux _ _ _ Hgt Hbh). assert (H2 : 0 < [[WW bh bl]]). simpl;Spec_w_to_Z bl. apply Zlt_le_trans with ([|bh|]*wB);zarith. apply Zmult_lt_0_compat;zarith. apply Zis_gcd_mod;trivial. rewrite <- H. simpl in *;destruct (ww_mod_gt_aux ah al bh bl) as [ |mh ml]. simpl;apply Zis_gcd_0;zarith. - assert (Hmh := spec_compare w_0 mh);destruct (w_compare w_0 mh). - simpl;rewrite spec_w_0 in Hmh; rewrite <- Hmh;simpl. - assert (Hml := spec_compare w_0 ml);destruct (w_compare w_0 ml). - rewrite <- Hml;rewrite spec_w_0;simpl;apply Zis_gcd_0. - simpl;rewrite spec_w_0;simpl. - rewrite spec_w_0 in Hml. apply Zis_gcd_mod;zarith. + rewrite spec_compare, spec_w_0; case Zcompare_spec; intros Hmh. + simpl;rewrite <- Hmh;simpl. + rewrite spec_compare, spec_w_0; case Zcompare_spec; intros Hml. + rewrite <- Hml;simpl;apply Zis_gcd_0. + simpl; rewrite spec_w_0; simpl. + apply Zis_gcd_mod;zarith. change ([|bh|] * wB + [|bl|]) with (double_to_Z w_digits w_to_Z 1 (WW bh bl)). rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div @@ -1265,8 +1244,8 @@ Section DoubleDivGt. rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. - rewrite spec_w_0 in Hml;Spec_w_to_Z ml;exfalso;omega. - rewrite spec_w_0 in Hmh. assert ([[WW bh bl]] > [[WW mh ml]]). + Spec_w_to_Z ml;exfalso;omega. + assert ([[WW bh bl]] > [[WW mh ml]]). rewrite H;simpl; apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. assert (H1:= spec_ww_mod_gt_aux _ _ _ H0 Hmh). @@ -1300,8 +1279,8 @@ Section DoubleDivGt. rewrite Z_div_mult;zarith. assert (2^1 <= 2^n). change (2^1) with 2;zarith. assert (H7 := @Zpower_le_monotone_inv 2 1 n);zarith. - rewrite spec_w_0 in Hmh;Spec_w_to_Z mh;exfalso;zarith. - rewrite spec_w_0 in Hbh;Spec_w_to_Z bh;exfalso;zarith. + Spec_w_to_Z mh;exfalso;zarith. + Spec_w_to_Z bh;exfalso;zarith. Qed. Lemma spec_ww_gcd_gt_aux : @@ -1374,11 +1353,7 @@ Section DoubleDiv. Variable spec_to_Z : forall x, 0 <= [|x|] < wB. Variable spec_ww_1 : [[ww_1]] = 1. Variable spec_ww_compare : forall x y, - match ww_compare x y with - | Eq => [[x]] = [[y]] - | Lt => [[x]] < [[y]] - | Gt => [[x]] > [[y]] - end. + ww_compare x y = Zcompare [[x]] [[y]]. Variable spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> let (q,r) := ww_div_gt a b in [[a]] = [[q]] * [[b]] + [[r]] /\ @@ -1400,20 +1375,20 @@ Section DoubleDiv. 0 <= [[r]] < [[b]]. Proof. intros a b Hpos;unfold ww_div. - assert (H:=spec_ww_compare a b);destruct (ww_compare a b). + rewrite spec_ww_compare; case Zcompare_spec; intros. simpl;rewrite spec_ww_1;split;zarith. simpl;split;[ring|Spec_ww_to_Z a;zarith]. - apply spec_ww_div_gt;trivial. + apply spec_ww_div_gt;auto with zarith. Qed. Lemma spec_ww_mod : forall a b, 0 < [[b]] -> [[ww_mod a b]] = [[a]] mod [[b]]. Proof. intros a b Hpos;unfold ww_mod. - assert (H := spec_ww_compare a b);destruct (ww_compare a b). + rewrite spec_ww_compare; case Zcompare_spec; intros. simpl;apply Zmod_unique with 1;try rewrite H;zarith. Spec_ww_to_Z a;symmetry;apply Zmod_small;zarith. - apply spec_ww_mod_gt;trivial. + apply spec_ww_mod_gt;auto with zarith. Qed. @@ -1431,12 +1406,7 @@ Section DoubleDiv. Variable spec_w_0 : [|w_0|] = 0. Variable spec_w_1 : [|w_1|] = 1. Variable spec_compare : - forall x y, - match w_compare x y with - | Eq => [|x|] = [|y|] - | Lt => [|x|] < [|y|] - | Gt => [|x|] > [|y|] - end. + forall x y, w_compare x y = Zcompare [|x|] [|y|]. Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. Variable spec_gcd_gt : forall a b, [|a|] > [|b|] -> Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]. @@ -1468,14 +1438,14 @@ Section DoubleDiv. assert (0 <= 1 < wB). split;zarith. apply wB_pos. assert (H1:= beta_lex _ _ _ _ _ Hle (spec_to_Z yl) H). Spec_w_to_Z yh;zarith. - unfold gcd_cont;assert (Hcmpy:=spec_compare w_1 yl); - rewrite spec_w_1 in Hcmpy. - simpl;rewrite H;simpl;destruct (w_compare w_1 yl). + unfold gcd_cont; rewrite spec_compare, spec_w_1. + case Zcompare_spec; intros Hcmpy. + simpl;rewrite H;simpl; rewrite spec_ww_1;rewrite <- Hcmpy;apply Zis_gcd_mod;zarith. rewrite <- (Zmod_unique ([|xh|]*wB+[|xl|]) 1 ([|xh|]*wB+[|xl|]) 0);zarith. rewrite H in Hle; exfalso;zarith. - assert ([|yl|] = 0). Spec_w_to_Z yl;zarith. - rewrite H0;simpl;apply Zis_gcd_0;trivial. + assert (H0 : [|yl|] = 0) by (Spec_w_to_Z yl;zarith). + simpl. rewrite H0, H;simpl;apply Zis_gcd_0;trivial. Qed. @@ -1528,7 +1498,7 @@ Section DoubleDiv. | Eq => a | Lt => ww_gcd_gt b a end). - assert (Hcmp := spec_ww_compare a b);destruct (ww_compare a b). + rewrite spec_ww_compare; case Zcompare_spec; intros Hcmp. Spec_ww_to_Z b;rewrite Hcmp. apply Zis_gcd_for_euclid with 1;zarith. ring_simplify ([[b]] - 1 * [[b]]). apply Zis_gcd_0;zarith. |