diff options
Diffstat (limited to 'theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v')
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v | 76 |
1 files changed, 38 insertions, 38 deletions
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v index 269d62bb..a7e55671 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v @@ -8,7 +8,7 @@ (* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) (************************************************************************) -(*i $Id: DoubleSub.v 10964 2008-05-22 11:08:13Z letouzey $ i*) +(*i $Id$ i*) Set Implicit Arguments. @@ -17,7 +17,7 @@ Require Import BigNumPrelude. Require Import DoubleType. Require Import DoubleBase. -Open Local Scope Z_scope. +Local Open Scope Z_scope. Section DoubleSub. Variable w : Type. @@ -39,7 +39,7 @@ Section DoubleSub. Definition ww_opp_c x := match x with | W0 => C0 W0 - | WW xh xl => + | WW xh xl => match w_opp_c xl with | C0 _ => match w_opp_c xh with @@ -53,7 +53,7 @@ Section DoubleSub. Definition ww_opp x := match x with | W0 => W0 - | WW xh xl => + | WW xh xl => match w_opp_c xl with | C0 _ => WW (w_opp xh) w_0 | C1 l => WW (w_opp_carry xh) l @@ -72,14 +72,14 @@ Section DoubleSub. | WW xh xl => match w_pred_c xl with | C0 l => C0 (w_WW xh l) - | C1 _ => - match w_pred_c xh with + | C1 _ => + match w_pred_c xh with | C0 h => C0 (WW h w_Bm1) | C1 _ => C1 ww_Bm1 end end end. - + Definition ww_pred x := match x with | W0 => ww_Bm1 @@ -89,19 +89,19 @@ Section DoubleSub. | C1 l => WW (w_pred xh) w_Bm1 end end. - + Definition ww_sub_c x y := match y, x with | W0, _ => C0 x | WW yh yl, W0 => ww_opp_c (WW yh yl) | WW yh yl, WW xh xl => match w_sub_c xl yl with - | C0 l => + | C0 l => match w_sub_c xh yh with | C0 h => C0 (w_WW h l) | C1 h => C1 (WW h l) end - | C1 l => + | C1 l => match w_sub_carry_c xh yh with | C0 h => C0 (WW h l) | C1 h => C1 (WW h l) @@ -109,7 +109,7 @@ Section DoubleSub. end end. - Definition ww_sub x y := + Definition ww_sub x y := match y, x with | W0, _ => x | WW yh yl, W0 => ww_opp (WW yh yl) @@ -127,7 +127,7 @@ Section DoubleSub. | WW yh yl, W0 => C1 (ww_opp_carry (WW yh yl)) | WW yh yl, WW xh xl => match w_sub_carry_c xl yl with - | C0 l => + | C0 l => match w_sub_c xh yh with | C0 h => C0 (w_WW h l) | C1 h => C1 (WW h l) @@ -155,7 +155,7 @@ Section DoubleSub. (*Section DoubleProof.*) Variable w_digits : positive. Variable w_to_Z : w -> Z. - + Notation wB := (base w_digits). Notation wwB := (base (ww_digits w_digits)). @@ -166,13 +166,13 @@ Section DoubleSub. (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99). Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[+[ c ]]" := + (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) + Notation "[-[ c ]]" := + (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) (at level 0, x at level 99). - + Variable spec_w_0 : [|w_0|] = 0. Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. Variable spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1. @@ -187,7 +187,7 @@ Section DoubleSub. Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|]. Variable spec_sub_carry_c : forall x y, [-|w_sub_carry_c x y|] = [|x|] - [|y|] - 1. - + Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB. Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. Variable spec_sub_carry : @@ -197,12 +197,12 @@ Section DoubleSub. Lemma spec_ww_opp_c : forall x, [-[ww_opp_c x]] = -[[x]]. Proof. destruct x as [ |xh xl];simpl. reflexivity. - rewrite Zopp_plus_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) + rewrite Zopp_plus_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H; - rewrite Zopp_mult_distr_l. + rewrite Zopp_mult_distr_l. assert ([|l|] = 0). assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh) + rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh) as [h|h];intros H1;unfold interp_carry in *;rewrite <- H1. assert ([|h|] = 0). assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. @@ -216,7 +216,7 @@ Section DoubleSub. Proof. destruct x as [ |xh xl];simpl. reflexivity. rewrite Zopp_plus_distr;rewrite Zopp_mult_distr_l. - generalize (spec_opp_c xl);destruct (w_opp_c xl) + generalize (spec_opp_c xl);destruct (w_opp_c xl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. rewrite spec_w_0;rewrite Zplus_0_r;rewrite wwB_wBwB. assert ([|l|] = 0). @@ -247,7 +247,7 @@ Section DoubleSub. assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. rewrite H0;change ([|xh|] + -1) with ([|xh|] - 1). generalize (spec_pred_c xh);destruct (w_pred_c xh) as [h|h]; - intros H1;unfold interp_carry in H1;rewrite <- H1. + intros H1;unfold interp_carry in H1;rewrite <- H1. simpl;rewrite spec_w_Bm1;ring. assert ([|h|] = wB - 1). assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. @@ -258,14 +258,14 @@ Section DoubleSub. Proof. destruct y as [ |yh yl];simpl. ring. destruct x as [ |xh xl];simpl. exact (spec_ww_opp_c (WW yh yl)). - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|])). 2:ring. generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl) as [l|l];intros H; unfold interp_carry in H;rewrite <- H. generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1;simpl ww_to_Z; @@ -275,37 +275,37 @@ Section DoubleSub. Lemma spec_ww_sub_carry_c : forall x y, [-[ww_sub_carry_c x y]] = [[x]] - [[y]] - 1. Proof. - destruct y as [ |yh yl];simpl. + destruct y as [ |yh yl];simpl. unfold Zminus;simpl;rewrite Zplus_0_r;exact (spec_ww_pred_c x). destruct x as [ |xh xl]. unfold interp_carry;rewrite spec_w_WW;simpl ww_to_Z;rewrite wwB_wBwB; repeat rewrite spec_opp_carry;ring. simpl ww_to_Z. - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|]-1)). 2:ring. - generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl) + generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H. generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1;try rewrite spec_w_WW; simpl ww_to_Z; try rewrite wwB_wBwB;ring. - Qed. - + Qed. + Lemma spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. Proof. - destruct x as [ |xh xl];simpl. + destruct x as [ |xh xl];simpl. apply Zmod_unique with (-1). apply spec_ww_to_Z;trivial. rewrite spec_ww_Bm1;ring. replace ([|xh|]*wB + [|xl|] - 1) with ([|xh|]*wB + ([|xl|] - 1)). 2:ring. generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l];intro H; unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. - rewrite Zmod_small. apply spec_w_WW. + rewrite Zmod_small. apply spec_w_WW. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh l)). - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. change ([|xh|] + -1) with ([|xh|] - 1). assert ([|l|] = wB - 1). assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega. @@ -318,7 +318,7 @@ Section DoubleSub. destruct y as [ |yh yl];simpl. ring_simplify ([[x]] - 0);rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. destruct x as [ |xh xl];simpl. exact (spec_ww_opp (WW yh yl)). - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|])) with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|])). 2:ring. generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl)as[l|l];intros H; unfold interp_carry in H;rewrite <- H. @@ -338,7 +338,7 @@ Section DoubleSub. apply spec_ww_to_Z;trivial. fold (ww_opp_carry (WW yh yl)). rewrite (spec_ww_opp_carry (WW yh yl));simpl ww_to_Z;ring. - replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) + replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1) with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|] - 1)). 2:ring. generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl)as[l|l]; intros H;unfold interp_carry in H;rewrite <- H;rewrite spec_w_WW. @@ -354,4 +354,4 @@ End DoubleSub. - + |