diff options
Diffstat (limited to 'theories/Numbers/Cyclic/DoubleCyclic')
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v | 317 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v | 437 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v | 966 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v | 1494 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v | 519 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v | 475 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v | 621 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v | 1369 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v | 356 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v | 70 |
10 files changed, 0 insertions, 6624 deletions
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v deleted file mode 100644 index 407bcca4..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v +++ /dev/null @@ -1,317 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleAdd. - Variable w : Type. - Variable w_0 : w. - Variable w_1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable ww_1 : zn2z w. - Variable w_succ_c : w -> carry w. - Variable w_add_c : w -> w -> carry w. - Variable w_add_carry_c : w -> w -> carry w. - Variable w_succ : w -> w. - Variable w_add : w -> w -> w. - Variable w_add_carry : w -> w -> w. - - Definition ww_succ_c x := - match x with - | W0 => C0 ww_1 - | WW xh xl => - match w_succ_c xl with - | C0 l => C0 (WW xh l) - | C1 l => - match w_succ_c xh with - | C0 h => C0 (WW h w_0) - | C1 h => C1 W0 - end - end - end. - - Definition ww_succ x := - match x with - | W0 => ww_1 - | WW xh xl => - match w_succ_c xl with - | C0 l => WW xh l - | C1 l => w_W0 (w_succ xh) - end - end. - - Definition ww_add_c x y := - match x, y with - | W0, _ => C0 y - | _, W0 => C0 x - | WW xh xl, WW yh yl => - match w_add_c xl yl with - | C0 l => - match w_add_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (w_WW h l) - end - | C1 l => - match w_add_carry_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (w_WW h l) - end - end - end. - - Variable R : Type. - Variable f0 f1 : zn2z w -> R. - - Definition ww_add_c_cont x y := - match x, y with - | W0, _ => f0 y - | _, W0 => f0 x - | WW xh xl, WW yh yl => - match w_add_c xl yl with - | C0 l => - match w_add_c xh yh with - | C0 h => f0 (WW h l) - | C1 h => f1 (w_WW h l) - end - | C1 l => - match w_add_carry_c xh yh with - | C0 h => f0 (WW h l) - | C1 h => f1 (w_WW h l) - end - end - end. - - (* ww_add et ww_add_carry conserve la forme normale s'il n'y a pas - de debordement *) - Definition ww_add x y := - match x, y with - | W0, _ => y - | _, W0 => x - | WW xh xl, WW yh yl => - match w_add_c xl yl with - | C0 l => WW (w_add xh yh) l - | C1 l => WW (w_add_carry xh yh) l - end - end. - - Definition ww_add_carry_c x y := - match x, y with - | W0, W0 => C0 ww_1 - | W0, WW yh yl => ww_succ_c (WW yh yl) - | WW xh xl, W0 => ww_succ_c (WW xh xl) - | WW xh xl, WW yh yl => - match w_add_carry_c xl yl with - | C0 l => - match w_add_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (WW h l) - end - | C1 l => - match w_add_carry_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (w_WW h l) - end - end - end. - - Definition ww_add_carry x y := - match x, y with - | W0, W0 => ww_1 - | W0, WW yh yl => ww_succ (WW yh yl) - | WW xh xl, W0 => ww_succ (WW xh xl) - | WW xh xl, WW yh yl => - match w_add_carry_c xl yl with - | C0 l => WW (w_add xh yh) l - | C1 l => WW (w_add_carry xh yh) l - end - end. - - (*Section DoubleProof.*) - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c 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) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_succ_c : forall x, [+|w_succ_c x|] = [|x|] + 1. - 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. - Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB. - Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. - Variable spec_w_add_carry : - forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB. - - Lemma spec_ww_succ_c : forall x, [+[ww_succ_c x]] = [[x]] + 1. - Proof. - destruct x as [ |xh xl];simpl. apply spec_ww_1. - generalize (spec_w_succ_c xl);destruct (w_succ_c xl) as [l|l]; - intro H;unfold interp_carry in H. simpl;rewrite H;ring. - rewrite <- Z.add_assoc;rewrite <- H;rewrite Z.mul_1_l. - assert ([|l|] = 0). generalize (spec_to_Z xl)(spec_to_Z l);omega. - rewrite H0;generalize (spec_w_succ_c xh);destruct (w_succ_c xh) as [h|h]; - intro H1;unfold interp_carry in H1. - simpl;rewrite H1;rewrite spec_w_0;ring. - unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB. - assert ([|xh|] = wB - 1). generalize (spec_to_Z xh)(spec_to_Z h);omega. - rewrite H2;ring. - Qed. - - Lemma spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. - Proof. - destruct x as [ |xh xl];trivial. - destruct y as [ |yh yl]. rewrite Z.add_0_r;trivial. - simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring. - generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; - intros H;unfold interp_carry in H;rewrite <- H. - generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *;rewrite <- H1. trivial. - repeat rewrite Z.mul_1_l;rewrite spec_w_WW;rewrite wwB_wBwB; ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1. - simpl;ring. - repeat rewrite Z.mul_1_l;rewrite wwB_wBwB;rewrite spec_w_WW;ring. - Qed. - - Section Cont. - Variable P : zn2z w -> zn2z w -> R -> Prop. - Variable x y : zn2z w. - Variable spec_f0 : forall r, [[r]] = [[x]] + [[y]] -> P x y (f0 r). - Variable spec_f1 : forall r, wwB + [[r]] = [[x]] + [[y]] -> P x y (f1 r). - - Lemma spec_ww_add_c_cont : P x y (ww_add_c_cont x y). - Proof. - destruct x as [ |xh xl];trivial. - apply spec_f0;trivial. - destruct y as [ |yh yl]. - apply spec_f0;rewrite Z.add_0_r;trivial. - simpl. - generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; - intros H;unfold interp_carry in H. - generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in *. - apply spec_f0. simpl;rewrite H;rewrite H1;ring. - apply spec_f1. simpl;rewrite spec_w_WW;rewrite H. - rewrite Z.add_assoc;rewrite wwB_wBwB. rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r. - rewrite Z.mul_1_l in H1;rewrite H1;ring. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h]; intros H1;unfold interp_carry in *. - apply spec_f0;simpl;rewrite H1. rewrite Z.mul_add_distr_r. - rewrite <- Z.add_assoc;rewrite H;ring. - apply spec_f1. rewrite spec_w_WW;rewrite wwB_wBwB. - rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r. - rewrite Z.mul_1_l in H1;rewrite H1. rewrite Z.mul_add_distr_r. - rewrite <- Z.add_assoc;rewrite H; simpl; ring. - Qed. - - End Cont. - - Lemma spec_ww_add_carry_c : - forall x y, [+[ww_add_carry_c x y]] = [[x]] + [[y]] + 1. - Proof. - destruct x as [ |xh xl];intro y. - exact (spec_ww_succ_c y). - destruct y as [ |yh yl]. - rewrite Z.add_0_r;exact (spec_ww_succ_c (WW xh xl)). - simpl; replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. - generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) - as [l|l];intros H;unfold interp_carry in H;rewrite <- H. - generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; - intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. - unfold interp_carry;repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) - as [h|h];intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. - unfold interp_carry;rewrite spec_w_WW; - repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_succ : forall x, [[ww_succ x]] = ([[x]] + 1) mod wwB. - Proof. - destruct x as [ |xh xl];simpl. - rewrite spec_ww_1;rewrite Zmod_small;trivial. - split;[intro;discriminate|apply wwB_pos]. - rewrite <- Z.add_assoc;generalize (spec_w_succ_c xl); - destruct (w_succ_c xl) as[l|l];intro H;unfold interp_carry in H;rewrite <-H. - rewrite Zmod_small;trivial. - rewrite wwB_wBwB;apply beta_mult;apply spec_to_Z. - assert ([|l|] = 0). clear spec_ww_1 spec_w_1 spec_w_0. - assert (H1:= spec_to_Z l); assert (H2:= spec_to_Z xl); omega. - rewrite H0;rewrite Z.add_0_r;rewrite <- Z.mul_add_distr_r;rewrite wwB_wBwB. - rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;try apply lt_0_wB. - rewrite spec_w_W0;rewrite spec_w_succ;trivial. - Qed. - - Lemma spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB. - Proof. - destruct x as [ |xh xl];intros y. - rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. - destruct y as [ |yh yl]. - change [[W0]] with 0;rewrite Z.add_0_r. - rewrite Zmod_small;trivial. - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)). - simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring. - generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; - unfold interp_carry;intros H;simpl;rewrite <- H. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. - Qed. - - Lemma spec_ww_add_carry : - forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB. - Proof. - destruct x as [ |xh xl];intros y. - exact (spec_ww_succ y). - destruct y as [ |yh yl]. - change [[W0]] with 0;rewrite Z.add_0_r. exact (spec_ww_succ (WW xh xl)). - simpl;replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) - with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. - generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) - as [l|l];unfold interp_carry;intros H;rewrite <- H;simpl ww_to_Z. - rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. - Qed. - -(* End DoubleProof. *) -End DoubleAdd. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v deleted file mode 100644 index e94a891d..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ /dev/null @@ -1,437 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith Ndigits. -Require Import BigNumPrelude. -Require Import DoubleType. - -Local Open Scope Z_scope. - -Local Infix "<<" := Pos.shiftl_nat (at level 30). - -Section DoubleBase. - Variable w : Type. - Variable w_0 : w. - Variable w_1 : w. - Variable w_Bm1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_digits : positive. - Variable w_zdigits: w. - Variable w_add: w -> w -> zn2z w. - Variable w_to_Z : w -> Z. - Variable w_compare : w -> w -> comparison. - - Definition ww_digits := xO w_digits. - - Definition ww_zdigits := w_add w_zdigits w_zdigits. - - Definition ww_to_Z := zn2z_to_Z (base w_digits) w_to_Z. - - Definition ww_1 := WW w_0 w_1. - - Definition ww_Bm1 := WW w_Bm1 w_Bm1. - - Definition ww_WW xh xl : zn2z (zn2z w) := - match xh, xl with - | W0, W0 => W0 - | _, _ => WW xh xl - end. - - Definition ww_W0 h : zn2z (zn2z w) := - match h with - | W0 => W0 - | _ => WW h W0 - end. - - Definition ww_0W l : zn2z (zn2z w) := - match l with - | W0 => W0 - | _ => WW W0 l - end. - - Definition double_WW (n:nat) := - match n return word w n -> word w n -> word w (S n) with - | O => w_WW - | S n => - fun (h l : zn2z (word w n)) => - match h, l with - | W0, W0 => W0 - | _, _ => WW h l - end - end. - - Definition double_wB n := base (w_digits << n). - - Fixpoint double_to_Z (n:nat) : word w n -> Z := - match n return word w n -> Z with - | O => w_to_Z - | S n => zn2z_to_Z (double_wB n) (double_to_Z n) - end. - - Fixpoint extend_aux (n:nat) (x:zn2z w) {struct n}: word w (S n) := - match n return word w (S n) with - | O => x - | S n1 => WW W0 (extend_aux n1 x) - end. - - Definition extend (n:nat) (x:w) : word w (S n) := - let r := w_0W x in - match r with - | W0 => W0 - | _ => extend_aux n r - end. - - Definition double_0 n : word w n := - match n return word w n with - | O => w_0 - | S _ => W0 - end. - - Definition double_split (n:nat) (x:zn2z (word w n)) := - match x with - | W0 => - match n return word w n * word w n with - | O => (w_0,w_0) - | S _ => (W0, W0) - end - | WW h l => (h,l) - end. - - Definition ww_compare x y := - match x, y with - | W0, W0 => Eq - | W0, WW yh yl => - match w_compare w_0 yh with - | Eq => w_compare w_0 yl - | _ => Lt - end - | WW xh xl, W0 => - match w_compare xh w_0 with - | Eq => w_compare xl w_0 - | _ => Gt - end - | WW xh xl, WW yh yl => - match w_compare xh yh with - | Eq => w_compare xl yl - | Lt => Lt - | Gt => Gt - end - end. - - - (* Return the low part of the composed word*) - Fixpoint get_low (n : nat) {struct n}: - word w n -> w := - match n return (word w n -> w) with - | 0%nat => fun x => x - | S n1 => - fun x => - match x with - | W0 => w_0 - | WW _ x1 => get_low n1 x1 - end - end. - - - Section DoubleProof. - Notation wB := (base w_digits). - Notation wwB := (base ww_digits). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[[ x ]]" := (ww_to_Z x) (at level 0, x at level 99). - Notation "[+[ c ]]" := - (interp_carry 1 wwB ww_to_Z c) (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB ww_to_Z c) (at level 0, c at level 99). - Notation "[! n | x !]" := (double_to_Z n x) (at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - 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_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_compare : forall x y, - w_compare x y = Z.compare [|x|] [|y|]. - - Lemma wwB_wBwB : wwB = wB^2. - Proof. - unfold base, ww_digits;rewrite Z.pow_2_r; rewrite (Pos2Z.inj_xO w_digits). - replace (2 * Zpos w_digits) with (Zpos w_digits + Zpos w_digits). - apply Zpower_exp; unfold Z.ge;simpl;intros;discriminate. - ring. - Qed. - - Lemma spec_ww_1 : [[ww_1]] = 1. - Proof. simpl;rewrite spec_w_0;rewrite spec_w_1;ring. Qed. - - Lemma spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1. - Proof. simpl;rewrite spec_w_Bm1;rewrite wwB_wBwB;ring. Qed. - - Lemma lt_0_wB : 0 < wB. - Proof. - unfold base;apply Z.pow_pos_nonneg. unfold Z.lt;reflexivity. - unfold Z.le;intros H;discriminate H. - Qed. - - Lemma lt_0_wwB : 0 < wwB. - Proof. rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_pos_pos;apply lt_0_wB. Qed. - - Lemma wB_pos: 1 < wB. - Proof. - unfold base;apply Z.lt_le_trans with (2^1). unfold Z.lt;reflexivity. - apply Zpower_le_monotone. unfold Z.lt;reflexivity. - split;unfold Z.le;intros H. discriminate H. - clear spec_w_0W w_0W spec_w_Bm1 spec_to_Z spec_w_WW w_WW. - destruct w_digits; discriminate H. - Qed. - - Lemma wwB_pos: 1 < wwB. - Proof. - assert (H:= wB_pos);rewrite wwB_wBwB;rewrite <-(Z.mul_1_r 1). - rewrite Z.pow_2_r. - apply Zmult_lt_compat2;(split;[unfold Z.lt;reflexivity|trivial]). - apply Z.lt_le_incl;trivial. - Qed. - - Theorem wB_div_2: 2 * (wB / 2) = wB. - Proof. - clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W - spec_to_Z;unfold base. - assert (2 ^ Zpos w_digits = 2 * (2 ^ (Zpos w_digits - 1))). - pattern 2 at 2; rewrite <- Z.pow_1_r. - rewrite <- Zpower_exp; auto with zarith. - f_equal; auto with zarith. - case w_digits; compute; intros; discriminate. - rewrite H; f_equal; auto with zarith. - rewrite Z.mul_comm; apply Z_div_mult; auto with zarith. - Qed. - - Theorem wwB_div_2 : wwB / 2 = wB / 2 * wB. - Proof. - clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W - spec_to_Z. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - pattern wB at 1; rewrite <- wB_div_2; auto. - rewrite <- Z.mul_assoc. - repeat (rewrite (Z.mul_comm 2); rewrite Z_div_mult); auto with zarith. - Qed. - - Lemma mod_wwB : forall z x, - (z*wB + [|x|]) mod wwB = (z mod wB)*wB + [|x|]. - Proof. - intros z x. - rewrite Zplus_mod. - pattern wwB at 1;rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite Zmult_mod_distr_r;try apply lt_0_wB. - rewrite (Zmod_small [|x|]). - apply Zmod_small;rewrite wwB_wBwB;apply beta_mult;try apply spec_to_Z. - apply Z_mod_lt;apply Z.lt_gt;apply lt_0_wB. - destruct (spec_to_Z x);split;trivial. - change [|x|] with (0*wB+[|x|]). rewrite wwB_wBwB. - rewrite Z.pow_2_r;rewrite <- (Z.add_0_r (wB*wB));apply beta_lex_inv. - apply lt_0_wB. apply spec_to_Z. split;[apply Z.le_refl | apply lt_0_wB]. - Qed. - - Lemma wB_div : forall x y, ([|x|] * wB + [|y|]) / wB = [|x|]. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - intros x y;unfold base;rewrite Zdiv_shift_r;auto with zarith. - rewrite Z_div_mult;auto with zarith. - destruct (spec_to_Z x);trivial. - Qed. - - Lemma wB_div_plus : forall x y p, - 0 <= p -> - ([|x|]*wB + [|y|]) / 2^(Zpos w_digits + p) = [|x|] / 2^p. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - intros x y p Hp;rewrite Zpower_exp;auto with zarith. - rewrite <- Zdiv_Zdiv;auto with zarith. - rewrite wB_div;trivial. - Qed. - - Lemma lt_wB_wwB : wB < wwB. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - unfold base;apply Zpower_lt_monotone;auto with zarith. - assert (0 < Zpos w_digits). compute;reflexivity. - unfold ww_digits;rewrite Pos2Z.inj_xO;auto with zarith. - Qed. - - Lemma w_to_Z_wwB : forall x, x < wB -> x < wwB. - Proof. - intros x H;apply Z.lt_trans with wB;trivial;apply lt_wB_wwB. - Qed. - - Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - destruct x as [ |h l];simpl. - split;[apply Z.le_refl|apply lt_0_wwB]. - assert (H:=spec_to_Z h);assert (L:=spec_to_Z l);split. - apply Z.add_nonneg_nonneg;auto with zarith. - rewrite <- (Z.add_0_r wwB);rewrite wwB_wBwB; rewrite Z.pow_2_r; - apply beta_lex_inv;auto with zarith. - Qed. - - Lemma double_wB_wwB : forall n, double_wB n * double_wB n = double_wB (S n). - Proof. - intros n;unfold double_wB;simpl. - unfold base. rewrite (Pos2Z.inj_xO (_ << _)). - replace (2 * Zpos (w_digits << n)) with - (Zpos (w_digits << n) + Zpos (w_digits << n)) by ring. - symmetry; apply Zpower_exp;intro;discriminate. - Qed. - - Lemma double_wB_pos: - forall n, 0 <= double_wB n. - Proof. - intros n; unfold double_wB, base; auto with zarith. - Qed. - - Lemma double_wB_more_digits: - forall n, wB <= double_wB n. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - intros n; elim n; clear n; auto. - unfold double_wB, "<<"; auto with zarith. - intros n H1; rewrite <- double_wB_wwB. - apply Z.le_trans with (wB * 1). - rewrite Z.mul_1_r; apply Z.le_refl. - unfold base; auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - apply Z.le_trans with wB; auto with zarith. - unfold base. - rewrite <- (Z.pow_0_r 2). - apply Z.pow_le_mono_r; auto with zarith. - Qed. - - Lemma spec_double_to_Z : - forall n (x:word w n), 0 <= [!n | x!] < double_wB n. - Proof. - clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. - induction n;intros. exact (spec_to_Z x). - unfold double_to_Z;fold double_to_Z. - destruct x;unfold zn2z_to_Z. - unfold double_wB,base;split;auto with zarith. - assert (U0:= IHn w0);assert (U1:= IHn w1). - split;auto with zarith. - apply Z.lt_le_trans with ((double_wB n - 1) * double_wB n + double_wB n). - assert (double_to_Z n w0*double_wB n <= (double_wB n - 1)*double_wB n). - apply Z.mul_le_mono_nonneg_r;auto with zarith. - auto with zarith. - rewrite <- double_wB_wwB. - replace ((double_wB n - 1) * double_wB n + double_wB n) with (double_wB n * double_wB n); - [auto with zarith | ring]. - Qed. - - Lemma spec_get_low: - forall n x, - [!n | x!] < wB -> [|get_low n x|] = [!n | x!]. - Proof. - clear spec_w_1 spec_w_Bm1. - intros n; elim n; auto; clear n. - intros n Hrec x; case x; clear x; auto. - intros xx yy; simpl. - destruct (spec_double_to_Z n xx) as [F1 _]. Z.le_elim F1. - - (* 0 < [!n | xx!] *) - intros; exfalso. - assert (F3 := double_wB_more_digits n). - destruct (spec_double_to_Z n yy) as [F4 _]. - assert (F5: 1 * wB <= [!n | xx!] * double_wB n); - auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - unfold base; auto with zarith. - - (* 0 = [!n | xx!] *) - rewrite <- F1; rewrite Z.mul_0_l, Z.add_0_l. - intros; apply Hrec; auto. - Qed. - - Lemma spec_double_WW : forall n (h l : word w n), - [!S n|double_WW n h l!] = [!n|h!] * double_wB n + [!n|l!]. - Proof. - induction n;simpl;intros;trivial. - destruct h;auto. - destruct l;auto. - Qed. - - Lemma spec_extend_aux : forall n x, [!S n|extend_aux n x!] = [[x]]. - Proof. induction n;simpl;trivial. Qed. - - Lemma spec_extend : forall n x, [!S n|extend n x!] = [|x|]. - Proof. - intros n x;assert (H:= spec_w_0W x);unfold extend. - destruct (w_0W x);simpl;trivial. - rewrite <- H;exact (spec_extend_aux n (WW w0 w1)). - Qed. - - Lemma spec_double_0 : forall n, [!n|double_0 n!] = 0. - Proof. destruct n;trivial. Qed. - - Lemma spec_double_split : forall n x, - let (h,l) := double_split n x in - [!S n|x!] = [!n|h!] * double_wB n + [!n|l!]. - Proof. - destruct x;simpl;auto. - destruct n;simpl;trivial. - rewrite spec_w_0;trivial. - Qed. - - Lemma wB_lex_inv: forall a b c d, - a < c -> - a * wB + [|b|] < c * wB + [|d|]. - Proof. - intros a b c d H1; apply beta_lex_inv with (1 := H1); auto. - Qed. - - Ltac comp2ord := match goal with - | |- Lt = (?x ?= ?y) => symmetry; change (x < y) - | |- Gt = (?x ?= ?y) => symmetry; change (x > y); apply Z.lt_gt - end. - - Lemma spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Proof. - destruct x as [ |xh xl];destruct y as [ |yh yl];simpl;trivial. - (* 1st case *) - rewrite 2 spec_w_compare, spec_w_0. - destruct (Z.compare_spec 0 [|yh|]) as [H|H|H]. - rewrite <- H;simpl. reflexivity. - symmetry. change (0 < [|yh|]*wB+[|yl|]). - change 0 with (0*wB+0). rewrite <- spec_w_0 at 2. - apply wB_lex_inv;trivial. - absurd (0 <= [|yh|]). apply Z.lt_nge; trivial. - destruct (spec_to_Z yh);trivial. - (* 2nd case *) - rewrite 2 spec_w_compare, spec_w_0. - destruct (Z.compare_spec [|xh|] 0) as [H|H|H]. - rewrite H;simpl;reflexivity. - absurd (0 <= [|xh|]). apply Z.lt_nge; trivial. - destruct (spec_to_Z xh);trivial. - comp2ord. - change 0 with (0*wB+0). rewrite <- spec_w_0 at 2. - apply wB_lex_inv;trivial. - (* 3rd case *) - rewrite 2 spec_w_compare. - destruct (Z.compare_spec [|xh|] [|yh|]) as [H|H|H]. - rewrite H. - symmetry. apply Z.add_compare_mono_l. - comp2ord. apply wB_lex_inv;trivial. - comp2ord. apply wB_lex_inv;trivial. - Qed. - - - End DoubleProof. - -End DoubleBase. - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v deleted file mode 100644 index 4ebe8fac..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v +++ /dev/null @@ -1,966 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. -Require Import DoubleAdd. -Require Import DoubleSub. -Require Import DoubleMul. -Require Import DoubleSqrt. -Require Import DoubleLift. -Require Import DoubleDivn1. -Require Import DoubleDiv. -Require Import CyclicAxioms. - -Local Open Scope Z_scope. - - -Section Z_2nZ. - - Context {t : Type}{ops : ZnZ.Ops t}. - - Let w_digits := ZnZ.digits. - Let w_zdigits := ZnZ.zdigits. - - Let w_to_Z := ZnZ.to_Z. - Let w_of_pos := ZnZ.of_pos. - Let w_head0 := ZnZ.head0. - Let w_tail0 := ZnZ.tail0. - - Let w_0 := ZnZ.zero. - Let w_1 := ZnZ.one. - Let w_Bm1 := ZnZ.minus_one. - - Let w_compare := ZnZ.compare. - Let w_eq0 := ZnZ.eq0. - - Let w_opp_c := ZnZ.opp_c. - Let w_opp := ZnZ.opp. - Let w_opp_carry := ZnZ.opp_carry. - - Let w_succ_c := ZnZ.succ_c. - Let w_add_c := ZnZ.add_c. - Let w_add_carry_c := ZnZ.add_carry_c. - Let w_succ := ZnZ.succ. - Let w_add := ZnZ.add. - Let w_add_carry := ZnZ.add_carry. - - Let w_pred_c := ZnZ.pred_c. - Let w_sub_c := ZnZ.sub_c. - Let w_sub_carry_c := ZnZ.sub_carry_c. - Let w_pred := ZnZ.pred. - Let w_sub := ZnZ.sub. - Let w_sub_carry := ZnZ.sub_carry. - - - Let w_mul_c := ZnZ.mul_c. - Let w_mul := ZnZ.mul. - Let w_square_c := ZnZ.square_c. - - Let w_div21 := ZnZ.div21. - Let w_div_gt := ZnZ.div_gt. - Let w_div := ZnZ.div. - - Let w_mod_gt := ZnZ.modulo_gt. - Let w_mod := ZnZ.modulo. - - Let w_gcd_gt := ZnZ.gcd_gt. - Let w_gcd := ZnZ.gcd. - - Let w_add_mul_div := ZnZ.add_mul_div. - - Let w_pos_mod := ZnZ.pos_mod. - - Let w_is_even := ZnZ.is_even. - Let w_sqrt2 := ZnZ.sqrt2. - Let w_sqrt := ZnZ.sqrt. - - Let _zn2z := zn2z t. - - Let wB := base w_digits. - - Let w_Bm2 := w_pred w_Bm1. - - Let ww_1 := ww_1 w_0 w_1. - Let ww_Bm1 := ww_Bm1 w_Bm1. - - Let w_add2 a b := match w_add_c a b with C0 p => WW w_0 p | C1 p => WW w_1 p end. - - Let _ww_digits := xO w_digits. - - Let _ww_zdigits := w_add2 w_zdigits w_zdigits. - - Let to_Z := zn2z_to_Z wB w_to_Z. - - Let w_W0 := ZnZ.WO. - Let w_0W := ZnZ.OW. - Let w_WW := ZnZ.WW. - - Let ww_of_pos p := - match w_of_pos p with - | (N0, l) => (N0, WW w_0 l) - | (Npos ph,l) => - let (n,h) := w_of_pos ph in (n, w_WW h l) - end. - - Let head0 := - Eval lazy beta delta [ww_head0] in - ww_head0 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits. - - Let tail0 := - Eval lazy beta delta [ww_tail0] in - ww_tail0 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits. - - Let ww_WW := Eval lazy beta delta [ww_WW] in (@ww_WW t). - Let ww_0W := Eval lazy beta delta [ww_0W] in (@ww_0W t). - Let ww_W0 := Eval lazy beta delta [ww_W0] in (@ww_W0 t). - - (* ** Comparison ** *) - Let compare := - Eval lazy beta delta[ww_compare] in ww_compare w_0 w_compare. - - Let eq0 (x:zn2z t) := - match x with - | W0 => true - | _ => false - end. - - (* ** Opposites ** *) - Let opp_c := - Eval lazy beta delta [ww_opp_c] in ww_opp_c w_0 w_opp_c w_opp_carry. - - Let opp := - Eval lazy beta delta [ww_opp] in ww_opp w_0 w_opp_c w_opp_carry w_opp. - - Let opp_carry := - Eval lazy beta delta [ww_opp_carry] in ww_opp_carry w_WW ww_Bm1 w_opp_carry. - - (* ** Additions ** *) - - Let succ_c := - Eval lazy beta delta [ww_succ_c] in ww_succ_c w_0 ww_1 w_succ_c. - - Let add_c := - Eval lazy beta delta [ww_add_c] in ww_add_c w_WW w_add_c w_add_carry_c. - - Let add_carry_c := - Eval lazy beta iota delta [ww_add_carry_c ww_succ_c] in - ww_add_carry_c w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c. - - Let succ := - Eval lazy beta delta [ww_succ] in ww_succ w_W0 ww_1 w_succ_c w_succ. - - Let add := - Eval lazy beta delta [ww_add] in ww_add w_add_c w_add w_add_carry. - - Let add_carry := - Eval lazy beta iota delta [ww_add_carry ww_succ] in - ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ w_add w_add_carry. - - (* ** Subtractions ** *) - - Let pred_c := - Eval lazy beta delta [ww_pred_c] in ww_pred_c w_Bm1 w_WW ww_Bm1 w_pred_c. - - Let sub_c := - Eval lazy beta iota delta [ww_sub_c ww_opp_c] in - ww_sub_c w_0 w_WW w_opp_c w_opp_carry w_sub_c w_sub_carry_c. - - Let sub_carry_c := - Eval lazy beta iota delta [ww_sub_carry_c ww_pred_c ww_opp_carry] in - ww_sub_carry_c w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_c w_sub_carry_c. - - Let pred := - Eval lazy beta delta [ww_pred] in ww_pred w_Bm1 w_WW ww_Bm1 w_pred_c w_pred. - - Let sub := - Eval lazy beta iota delta [ww_sub ww_opp] in - ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry. - - Let sub_carry := - Eval lazy beta iota delta [ww_sub_carry ww_pred ww_opp_carry] in - ww_sub_carry w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_carry_c w_pred - w_sub w_sub_carry. - - - (* ** Multiplication ** *) - - Let mul_c := - Eval lazy beta iota delta [ww_mul_c double_mul_c] in - ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry. - - Let karatsuba_c := - Eval lazy beta iota delta [ww_karatsuba_c double_mul_c kara_prod] in - ww_karatsuba_c w_0 w_1 w_WW w_W0 w_compare w_add w_sub w_mul_c - add_c add add_carry sub_c sub. - - Let mul := - Eval lazy beta delta [ww_mul] in - ww_mul w_W0 w_add w_mul_c w_mul add. - - Let square_c := - Eval lazy beta delta [ww_square_c] in - ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add add_carry. - - (* Division operation *) - - Let div32 := - Eval lazy beta iota delta [w_div32] in - w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c - w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c. - - Let div21 := - Eval lazy beta iota delta [ww_div21] in - ww_div21 w_0 w_0W div32 ww_1 compare sub. - - Let low (p: zn2z t) := match p with WW _ p1 => p1 | _ => w_0 end. - - Let add_mul_div := - Eval lazy beta delta [ww_add_mul_div] in - ww_add_mul_div w_0 w_WW w_W0 w_0W compare w_add_mul_div sub w_zdigits low. - - Let div_gt := - Eval lazy beta delta [ww_div_gt] in - ww_div_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp - w_opp_carry w_sub_c w_sub w_sub_carry - w_div_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits. - - Let div := - Eval lazy beta delta [ww_div] in ww_div ww_1 compare div_gt. - - Let mod_gt := - Eval lazy beta delta [ww_mod_gt] in - ww_mod_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry - w_mod_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div w_zdigits. - - Let mod_ := - Eval lazy beta delta [ww_mod] in ww_mod compare mod_gt. - - Let pos_mod := - Eval lazy beta delta [ww_pos_mod] in - ww_pos_mod w_0 w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits. - - Let is_even := - Eval lazy beta delta [ww_is_even] in ww_is_even w_is_even. - - Let sqrt2 := - Eval lazy beta delta [ww_sqrt2] in - ww_sqrt2 w_is_even w_compare w_0 w_1 w_Bm1 w_0W w_sub w_square_c - w_div21 w_add_mul_div w_zdigits w_add_c w_sqrt2 w_pred pred_c - pred add_c add sub_c add_mul_div. - - Let sqrt := - Eval lazy beta delta [ww_sqrt] in - ww_sqrt w_is_even w_0 w_sub w_add_mul_div w_zdigits - _ww_zdigits w_sqrt2 pred add_mul_div head0 compare low. - - Let gcd_gt_fix := - Eval cbv beta delta [ww_gcd_gt_aux ww_gcd_gt_body] in - ww_gcd_gt_aux w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry - w_sub_c w_sub w_sub_carry w_gcd_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div - w_zdigits. - - Let gcd_cont := - Eval lazy beta delta [gcd_cont] in gcd_cont ww_1 w_1 w_compare. - - Let gcd_gt := - Eval lazy beta delta [ww_gcd_gt] in - ww_gcd_gt w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont. - - Let gcd := - Eval lazy beta delta [ww_gcd] in - ww_gcd compare w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont. - - Definition lor (x y : zn2z t) := - match x, y with - | W0, _ => y - | _, W0 => x - | WW hx lx, WW hy ly => WW (ZnZ.lor hx hy) (ZnZ.lor lx ly) - end. - - Definition land (x y : zn2z t) := - match x, y with - | W0, _ => W0 - | _, W0 => W0 - | WW hx lx, WW hy ly => WW (ZnZ.land hx hy) (ZnZ.land lx ly) - end. - - Definition lxor (x y : zn2z t) := - match x, y with - | W0, _ => y - | _, W0 => x - | WW hx lx, WW hy ly => WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly) - end. - - (* ** Record of operators on 2 words *) - - Global Instance mk_zn2z_ops : ZnZ.Ops (zn2z t) | 1 := - ZnZ.MkOps _ww_digits _ww_zdigits - to_Z ww_of_pos head0 tail0 - W0 ww_1 ww_Bm1 - compare eq0 - opp_c opp opp_carry - succ_c add_c add_carry_c - succ add add_carry - pred_c sub_c sub_carry_c - pred sub sub_carry - mul_c mul square_c - div21 div_gt div - mod_gt mod_ - gcd_gt gcd - add_mul_div - pos_mod - is_even - sqrt2 - sqrt - lor - land - lxor. - - Global Instance mk_zn2z_ops_karatsuba : ZnZ.Ops (zn2z t) | 2 := - ZnZ.MkOps _ww_digits _ww_zdigits - to_Z ww_of_pos head0 tail0 - W0 ww_1 ww_Bm1 - compare eq0 - opp_c opp opp_carry - succ_c add_c add_carry_c - succ add add_carry - pred_c sub_c sub_carry_c - pred sub sub_carry - karatsuba_c mul square_c - div21 div_gt div - mod_gt mod_ - gcd_gt gcd - add_mul_div - pos_mod - is_even - sqrt2 - sqrt - lor - land - lxor. - - (* Proof *) - Context {specs : ZnZ.Specs ops}. - - Create HintDb ZnZ. - - Hint Resolve - ZnZ.spec_to_Z - ZnZ.spec_of_pos - ZnZ.spec_0 - ZnZ.spec_1 - ZnZ.spec_m1 - ZnZ.spec_compare - ZnZ.spec_eq0 - ZnZ.spec_opp_c - ZnZ.spec_opp - ZnZ.spec_opp_carry - ZnZ.spec_succ_c - ZnZ.spec_add_c - ZnZ.spec_add_carry_c - ZnZ.spec_succ - ZnZ.spec_add - ZnZ.spec_add_carry - ZnZ.spec_pred_c - ZnZ.spec_sub_c - ZnZ.spec_sub_carry_c - ZnZ.spec_pred - ZnZ.spec_sub - ZnZ.spec_sub_carry - ZnZ.spec_mul_c - ZnZ.spec_mul - ZnZ.spec_square_c - ZnZ.spec_div21 - ZnZ.spec_div_gt - ZnZ.spec_div - ZnZ.spec_modulo_gt - ZnZ.spec_modulo - ZnZ.spec_gcd_gt - ZnZ.spec_gcd - ZnZ.spec_head0 - ZnZ.spec_tail0 - ZnZ.spec_add_mul_div - ZnZ.spec_pos_mod - ZnZ.spec_is_even - ZnZ.spec_sqrt2 - ZnZ.spec_sqrt - ZnZ.spec_WO - ZnZ.spec_OW - ZnZ.spec_WW : ZnZ. - - Ltac wwauto := unfold ww_to_Z; eauto with ZnZ. - - Let wwB := base _ww_digits. - - Notation "[| x |]" := (to_Z x) (at level 0, x at level 99). - - Notation "[+| c |]" := - (interp_carry 1 wwB to_Z c) (at level 0, c at level 99). - - Notation "[-| c |]" := - (interp_carry (-1) wwB to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (zn2z_to_Z wwB to_Z x) (at level 0, x at level 99). - - Let spec_ww_to_Z : forall x, 0 <= [| x |] < wwB. - Proof. refine (spec_ww_to_Z w_digits w_to_Z _); wwauto. Qed. - - Let spec_ww_of_pos : forall p, - Zpos p = (Z.of_N (fst (ww_of_pos p)))*wwB + [|(snd (ww_of_pos p))|]. - Proof. - unfold ww_of_pos;intros. - rewrite (ZnZ.spec_of_pos p). unfold w_of_pos. - case (ZnZ.of_pos p); intros. simpl. - destruct n; simpl ZnZ.to_Z. - simpl;unfold w_to_Z,w_0; rewrite ZnZ.spec_0;trivial. - unfold Z.of_N. - rewrite (ZnZ.spec_of_pos p0). - case (ZnZ.of_pos p0); intros. simpl. - unfold fst, snd,Z.of_N, to_Z, wB, w_digits, w_to_Z, w_WW. - rewrite ZnZ.spec_WW. - replace wwB with (wB*wB). - unfold wB,w_to_Z,w_digits;destruct n;ring. - symmetry. rewrite <- Z.pow_2_r; exact (wwB_wBwB w_digits). - Qed. - - Let spec_ww_0 : [|W0|] = 0. - Proof. reflexivity. Qed. - - Let spec_ww_1 : [|ww_1|] = 1. - Proof. refine (spec_ww_1 w_0 w_1 w_digits w_to_Z _ _);wwauto. Qed. - - Let spec_ww_Bm1 : [|ww_Bm1|] = wwB - 1. - Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);wwauto. Qed. - - Let spec_ww_compare : - forall x y, compare x y = Z.compare [|x|] [|y|]. - Proof. - refine (spec_ww_compare w_0 w_digits w_to_Z w_compare _ _ _);wwauto. - Qed. - - Let spec_ww_eq0 : forall x, eq0 x = true -> [|x|] = 0. - Proof. destruct x;simpl;intros;trivial;discriminate. Qed. - - Let spec_ww_opp_c : forall x, [-|opp_c x|] = -[|x|]. - Proof. - refine(spec_ww_opp_c w_0 w_0 W0 w_opp_c w_opp_carry w_digits w_to_Z _ _ _ _); - wwauto. - Qed. - - Let spec_ww_opp : forall x, [|opp x|] = (-[|x|]) mod wwB. - Proof. - refine(spec_ww_opp w_0 w_0 W0 w_opp_c w_opp_carry w_opp - w_digits w_to_Z _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_opp_carry : forall x, [|opp_carry x|] = wwB - [|x|] - 1. - Proof. - refine (spec_ww_opp_carry w_WW ww_Bm1 w_opp_carry w_digits w_to_Z _ _ _); - wwauto. - Qed. - - Let spec_ww_succ_c : forall x, [+|succ_c x|] = [|x|] + 1. - Proof. - refine (spec_ww_succ_c w_0 w_0 ww_1 w_succ_c w_digits w_to_Z _ _ _ _);wwauto. - Qed. - - Let spec_ww_add_c : forall x y, [+|add_c x y|] = [|x|] + [|y|]. - Proof. - refine (spec_ww_add_c w_WW w_add_c w_add_carry_c w_digits w_to_Z _ _ _);wwauto. - Qed. - - Let spec_ww_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|]+[|y|]+1. - Proof. - refine (spec_ww_add_carry_c w_0 w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c - w_digits w_to_Z _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_succ : forall x, [|succ x|] = ([|x|] + 1) mod wwB. - Proof. - refine (spec_ww_succ w_W0 ww_1 w_succ_c w_succ w_digits w_to_Z _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wwB. - Proof. - refine (spec_ww_add w_add_c w_add w_add_carry w_digits w_to_Z _ _ _ _);wwauto. - Qed. - - Let spec_ww_add_carry : forall x y, [|add_carry x y|]=([|x|]+[|y|]+1)mod wwB. - Proof. - refine (spec_ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ - w_add w_add_carry w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_pred_c : forall x, [-|pred_c x|] = [|x|] - 1. - Proof. - refine (spec_ww_pred_c w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_digits w_to_Z - _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|]. - Proof. - refine (spec_ww_sub_c w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c - w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|]-[|y|]-1. - Proof. - refine (spec_ww_sub_carry_c w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c - w_sub_c w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_pred : forall x, [|pred x|] = ([|x|] - 1) mod wwB. - Proof. - refine (spec_ww_pred w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_pred w_digits w_to_Z - _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wwB. - Proof. - refine (spec_ww_sub w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c w_opp - w_sub w_sub_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_sub_carry : forall x y, [|sub_carry x y|]=([|x|]-[|y|]-1) mod wwB. - Proof. - refine (spec_ww_sub_carry w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c - w_sub_carry_c w_pred w_sub w_sub_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_mul_c : forall x y, [[mul_c x y ]] = [|x|] * [|y|]. - Proof. - refine (spec_ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry w_digits - w_to_Z _ _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_ww_karatsuba_c : forall x y, [[karatsuba_c x y ]] = [|x|] * [|y|]. - Proof. - refine (spec_ww_karatsuba_c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _); wwauto. - unfold w_digits; apply ZnZ.spec_more_than_1_digit; auto. - Qed. - - Let spec_ww_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wwB. - Proof. - refine (spec_ww_mul w_W0 w_add w_mul_c w_mul add w_digits w_to_Z _ _ _ _ _); - wwauto. - Qed. - - Let spec_ww_square_c : forall x, [[square_c x]] = [|x|] * [|x|]. - Proof. - refine (spec_ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add - add_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_w_div32 : forall a1 a2 a3 b1 b2, - wB / 2 <= (w_to_Z b1) -> - [|WW a1 a2|] < [|WW b1 b2|] -> - let (q, r) := div32 a1 a2 a3 b1 b2 in - (w_to_Z a1) * wwB + (w_to_Z a2) * wB + (w_to_Z a3) = - (w_to_Z q) * ((w_to_Z b1)*wB + (w_to_Z b2)) + [|r|] /\ - 0 <= [|r|] < (w_to_Z b1)*wB + w_to_Z b2. - Proof. - refine (spec_w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c - w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c w_digits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - unfold w_Bm2, w_to_Z, w_pred, w_Bm1. - rewrite ZnZ.spec_pred, ZnZ.spec_m1. - unfold w_digits;rewrite Zmod_small. ring. - assert (H:= wB_pos(ZnZ.digits)). omega. - exact ZnZ.spec_div21. - Qed. - - Let spec_ww_div21 : forall a1 a2 b, - wwB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := div21 a1 a2 b in - [|a1|] *wwB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Proof. - refine (spec_ww_div21 w_0 w_0W div32 ww_1 compare sub w_digits w_to_Z - _ _ _ _ _ _ _);wwauto. - Qed. - - Let spec_add2: forall x y, - [|w_add2 x y|] = w_to_Z x + w_to_Z y. - unfold w_add2. - intros xh xl; generalize (ZnZ.spec_add_c xh xl). - unfold w_add_c; case ZnZ.add_c; unfold interp_carry; simpl ww_to_Z. - intros w0 Hw0; simpl; unfold w_to_Z; rewrite Hw0. - unfold w_0; rewrite ZnZ.spec_0; simpl; auto with zarith. - intros w0; rewrite Z.mul_1_l; simpl. - unfold w_to_Z, w_1; rewrite ZnZ.spec_1; auto with zarith. - rewrite Z.mul_1_l; auto. - Qed. - - Let spec_low: forall x, - w_to_Z (low x) = [|x|] mod wB. - intros x; case x; simpl low. - unfold ww_to_Z, w_to_Z, w_0; rewrite ZnZ.spec_0; simpl; wwauto. - intros xh xl; simpl. - rewrite Z.add_comm; rewrite Z_mod_plus; auto with zarith. - rewrite Zmod_small; auto with zarith. - unfold wB, base; eauto with ZnZ zarith. - unfold wB, base; eauto with ZnZ zarith. - Qed. - - Let spec_ww_digits: - [|_ww_zdigits|] = Zpos (xO w_digits). - Proof. - unfold w_to_Z, _ww_zdigits. - rewrite spec_add2. - unfold w_to_Z, w_zdigits, w_digits. - rewrite ZnZ.spec_zdigits; auto. - rewrite Pos2Z.inj_xO; auto with zarith. - Qed. - - - Let spec_ww_head00 : forall x, [|x|] = 0 -> [|head0 x|] = Zpos _ww_digits. - Proof. - refine (spec_ww_head00 w_0 w_0W - w_compare w_head0 w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto. - exact ZnZ.spec_head00. - exact ZnZ.spec_zdigits. - Qed. - - Let spec_ww_head0 : forall x, 0 < [|x|] -> - wwB/ 2 <= 2 ^ [|head0 x|] * [|x|] < wwB. - Proof. - refine (spec_ww_head0 w_0 w_0W w_compare w_head0 - w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - Qed. - - Let spec_ww_tail00 : forall x, [|x|] = 0 -> [|tail0 x|] = Zpos _ww_digits. - Proof. - refine (spec_ww_tail00 w_0 w_0W - w_compare w_tail0 w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto. - exact ZnZ.spec_tail00. - exact ZnZ.spec_zdigits. - Qed. - - - Let spec_ww_tail0 : forall x, 0 < [|x|] -> - exists y, 0 <= y /\ [|x|] = (2 * y + 1) * 2 ^ [|tail0 x|]. - Proof. - refine (spec_ww_tail0 (w_digits := w_digits) w_0 w_0W w_compare w_tail0 - w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - Qed. - - Lemma spec_ww_add_mul_div : forall x y p, - [|p|] <= Zpos _ww_digits -> - [| add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos _ww_digits) - [|p|]))) mod wwB. - Proof. - refine (@spec_ww_add_mul_div t w_0 w_WW w_W0 w_0W compare w_add_mul_div - sub w_digits w_zdigits low w_to_Z - _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - Qed. - - Let spec_ww_div_gt : forall a b, - [|a|] > [|b|] -> 0 < [|b|] -> - let (q,r) := div_gt a b in - [|a|] = [|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|]. - Proof. -refine -(@spec_ww_div_gt t w_digits w_0 w_WW w_0W w_compare w_eq0 - w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -). - exact ZnZ.spec_0. - exact ZnZ.spec_to_Z. - wwauto. - wwauto. - exact ZnZ.spec_compare. - exact ZnZ.spec_eq0. - exact ZnZ.spec_opp_c. - exact ZnZ.spec_opp. - exact ZnZ.spec_opp_carry. - exact ZnZ.spec_sub_c. - exact ZnZ.spec_sub. - exact ZnZ.spec_sub_carry. - exact ZnZ.spec_div_gt. - exact ZnZ.spec_add_mul_div. - exact ZnZ.spec_head0. - exact ZnZ.spec_div21. - exact spec_w_div32. - exact ZnZ.spec_zdigits. - exact spec_ww_digits. - exact spec_ww_1. - exact spec_ww_add_mul_div. - Qed. - - Let spec_ww_div : forall a b, 0 < [|b|] -> - let (q,r) := div a b in - [|a|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Proof. - refine (spec_ww_div w_digits ww_1 compare div_gt w_to_Z _ _ _ _);wwauto. - Qed. - - Let spec_ww_mod_gt : forall a b, - [|a|] > [|b|] -> 0 < [|b|] -> - [|mod_gt a b|] = [|a|] mod [|b|]. - Proof. - refine (@spec_ww_mod_gt t w_digits w_0 w_WW w_0W w_compare w_eq0 - w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt w_mod_gt - w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div - w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_div_gt. - exact ZnZ.spec_div21. - exact ZnZ.spec_zdigits. - exact spec_ww_add_mul_div. - Qed. - - Let spec_ww_mod : forall a b, 0 < [|b|] -> [|mod_ a b|] = [|a|] mod [|b|]. - Proof. - refine (spec_ww_mod w_digits W0 compare mod_gt w_to_Z _ _ _);wwauto. - Qed. - - Let spec_ww_gcd_gt : forall a b, [|a|] > [|b|] -> - Zis_gcd [|a|] [|b|] [|gcd_gt a b|]. - Proof. - refine (@spec_ww_gcd_gt t w_digits W0 w_to_Z _ - w_0 w_0 w_eq0 w_gcd_gt _ww_digits - _ gcd_gt_fix _ _ _ _ gcd_cont _);wwauto. - refine (@spec_ww_gcd_gt_aux t w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp - w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0 - w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_div21. - exact ZnZ.spec_zdigits. - exact spec_ww_add_mul_div. - refine (@spec_gcd_cont t w_digits ww_1 w_to_Z _ _ w_0 w_1 w_compare - _ _);wwauto. - Qed. - - Let spec_ww_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|]. - Proof. - refine (@spec_ww_gcd t w_digits W0 compare w_to_Z _ _ w_0 w_0 w_eq0 w_gcd_gt - _ww_digits _ gcd_gt_fix _ _ _ _ gcd_cont _);wwauto. - refine (@spec_ww_gcd_gt_aux t w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp - w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0 - w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_div21. - exact ZnZ.spec_zdigits. - exact spec_ww_add_mul_div. - refine (@spec_gcd_cont t w_digits ww_1 w_to_Z _ _ w_0 w_1 w_compare - _ _);wwauto. - Qed. - - Let spec_ww_is_even : forall x, - match is_even x with - true => [|x|] mod 2 = 0 - | false => [|x|] mod 2 = 1 - end. - Proof. - refine (@spec_ww_is_even t w_is_even w_digits _ _ ). - exact ZnZ.spec_is_even. - Qed. - - Let spec_ww_sqrt2 : forall x y, - wwB/ 4 <= [|x|] -> - let (s,r) := sqrt2 x y in - [[WW x y]] = [|s|] ^ 2 + [+|r|] /\ - [+|r|] <= 2 * [|s|]. - Proof. - intros x y H. - refine (@spec_ww_sqrt2 t w_is_even w_compare w_0 w_1 w_Bm1 - w_0W w_sub w_square_c w_div21 w_add_mul_div w_digits w_zdigits - _ww_zdigits - w_add_c w_sqrt2 w_pred pred_c pred add_c add sub_c add_mul_div - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto. - exact ZnZ.spec_zdigits. - exact ZnZ.spec_more_than_1_digit. - exact ZnZ.spec_is_even. - exact ZnZ.spec_div21. - exact spec_ww_add_mul_div. - exact ZnZ.spec_sqrt2. - Qed. - - Let spec_ww_sqrt : forall x, - [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2. - Proof. - refine (@spec_ww_sqrt t w_is_even w_0 w_1 w_Bm1 - w_sub w_add_mul_div w_digits w_zdigits _ww_zdigits - w_sqrt2 pred add_mul_div head0 compare - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto. - exact ZnZ.spec_zdigits. - exact ZnZ.spec_more_than_1_digit. - exact ZnZ.spec_is_even. - exact spec_ww_add_mul_div. - exact ZnZ.spec_sqrt2. - Qed. - - Let wB_pos : 0 < wB. - Proof. - unfold wB, base; apply Z.pow_pos_nonneg; auto with zarith. - Qed. - - Hint Transparent ww_to_Z. - - Let ww_testbit_high n x y : Z.pos w_digits <= n -> - Z.testbit [|WW x y|] n = - Z.testbit (ZnZ.to_Z x) (n - Z.pos w_digits). - Proof. - intros Hn. - assert (E : ZnZ.to_Z x = [|WW x y|] / wB). - { simpl. - rewrite Z.div_add_l; eauto with ZnZ zarith. - now rewrite Z.div_small, Z.add_0_r; wwauto. } - rewrite E. - unfold wB, base. rewrite Z.div_pow2_bits. - - f_equal; auto with zarith. - - easy. - - auto with zarith. - Qed. - - Let ww_testbit_low n x y : 0 <= n < Z.pos w_digits -> - Z.testbit [|WW x y|] n = Z.testbit (ZnZ.to_Z y) n. - Proof. - intros (Hn,Hn'). - assert (E : ZnZ.to_Z y = [|WW x y|] mod wB). - { simpl; symmetry. - rewrite Z.add_comm, Z.mod_add; auto with zarith nocore. - apply Z.mod_small; eauto with ZnZ zarith. } - rewrite E. - unfold wB, base. symmetry. apply Z.mod_pow2_bits_low; auto. - Qed. - - Let spec_lor x y : [|lor x y|] = Z.lor [|x|] [|y|]. - Proof. - destruct x as [ |hx lx]. trivial. - destruct y as [ |hy ly]. now rewrite Z.lor_comm. - change ([|WW (ZnZ.lor hx hy) (ZnZ.lor lx ly)|] = - Z.lor [|WW hx lx|] [|WW hy ly|]). - apply Z.bits_inj'; intros n Hn. - rewrite Z.lor_spec. - destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT]. - - now rewrite !ww_testbit_high, ZnZ.spec_lor, Z.lor_spec. - - rewrite !ww_testbit_low; auto. - now rewrite ZnZ.spec_lor, Z.lor_spec. - Qed. - - Let spec_land x y : [|land x y|] = Z.land [|x|] [|y|]. - Proof. - destruct x as [ |hx lx]. trivial. - destruct y as [ |hy ly]. now rewrite Z.land_comm. - change ([|WW (ZnZ.land hx hy) (ZnZ.land lx ly)|] = - Z.land [|WW hx lx|] [|WW hy ly|]). - apply Z.bits_inj'; intros n Hn. - rewrite Z.land_spec. - destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT]. - - now rewrite !ww_testbit_high, ZnZ.spec_land, Z.land_spec. - - rewrite !ww_testbit_low; auto. - now rewrite ZnZ.spec_land, Z.land_spec. - Qed. - - Let spec_lxor x y : [|lxor x y|] = Z.lxor [|x|] [|y|]. - Proof. - destruct x as [ |hx lx]. trivial. - destruct y as [ |hy ly]. now rewrite Z.lxor_comm. - change ([|WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly)|] = - Z.lxor [|WW hx lx|] [|WW hy ly|]). - apply Z.bits_inj'; intros n Hn. - rewrite Z.lxor_spec. - destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT]. - - now rewrite !ww_testbit_high, ZnZ.spec_lxor, Z.lxor_spec. - - rewrite !ww_testbit_low; auto. - now rewrite ZnZ.spec_lxor, Z.lxor_spec. - Qed. - - Global Instance mk_zn2z_specs : ZnZ.Specs mk_zn2z_ops. - Proof. - apply ZnZ.MkSpecs; auto. - exact spec_ww_add_mul_div. - - refine (@spec_ww_pos_mod t w_0 w_digits w_zdigits w_WW - w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - unfold w_to_Z, w_zdigits. - rewrite ZnZ.spec_zdigits. - rewrite <- Pos2Z.inj_xO; exact spec_ww_digits. - Qed. - - Global Instance mk_zn2z_specs_karatsuba : ZnZ.Specs mk_zn2z_ops_karatsuba. - Proof. - apply ZnZ.MkSpecs; auto. - exact spec_ww_add_mul_div. - refine (@spec_ww_pos_mod t w_0 w_digits w_zdigits w_WW - w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z - _ _ _ _ _ _ _ _ _ _ _ _);wwauto. - exact ZnZ.spec_zdigits. - unfold w_to_Z, w_zdigits. - rewrite ZnZ.spec_zdigits. - rewrite <- Pos2Z.inj_xO; exact spec_ww_digits. - Qed. - -End Z_2nZ. - - -Section MulAdd. - - Context {t : Type}{ops : ZnZ.Ops t}{specs : ZnZ.Specs ops}. - - Definition mul_add:= w_mul_add ZnZ.zero ZnZ.succ ZnZ.add_c ZnZ.mul_c. - - Notation "[| x |]" := (ZnZ.to_Z x) (at level 0, x at level 99). - - Notation "[|| x ||]" := - (zn2z_to_Z (base ZnZ.digits) ZnZ.to_Z x) (at level 0, x at level 99). - - Lemma spec_mul_add: forall x y z, - let (zh, zl) := mul_add x y z in - [||WW zh zl||] = [|x|] * [|y|] + [|z|]. - Proof. - intros x y z. - refine (spec_w_mul_add _ _ _ _ _ _ _ _ _ _ _ _ x y z); auto. - exact ZnZ.spec_0. - exact ZnZ.spec_to_Z. - exact ZnZ.spec_succ. - exact ZnZ.spec_add_c. - exact ZnZ.spec_mul_c. - Qed. - -End MulAdd. - - -(** Modular versions of DoubleCyclic *) - -Module DoubleCyclic (C:CyclicType) <: CyclicType. - Definition t := zn2z C.t. - Instance ops : ZnZ.Ops t := mk_zn2z_ops. - Instance specs : ZnZ.Specs ops := mk_zn2z_specs. -End DoubleCyclic. - -Module DoubleCyclicKaratsuba (C:CyclicType) <: CyclicType. - Definition t := zn2z C.t. - Definition ops : ZnZ.Ops t := mk_zn2z_ops_karatsuba. - Definition specs : ZnZ.Specs ops := mk_zn2z_specs_karatsuba. -End DoubleCyclicKaratsuba. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v deleted file mode 100644 index 09d7329b..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ /dev/null @@ -1,1494 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. -Require Import DoubleDivn1. -Require Import DoubleAdd. -Require Import DoubleSub. - -Local Open Scope Z_scope. - -Ltac zarith := auto with zarith. - - -Section POS_MOD. - - Variable w:Type. - Variable w_0 : w. - Variable w_digits : positive. - Variable w_zdigits : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_pos_mod : w -> w -> w. - Variable w_compare : w -> w -> comparison. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable w_0W : w -> zn2z w. - Variable low: zn2z w -> w. - Variable ww_sub: zn2z w -> zn2z w -> zn2z w. - Variable ww_zdigits : zn2z w. - - - Definition ww_pos_mod p x := - let zdigits := w_0W w_zdigits in - match x with - | W0 => W0 - | WW xh xl => - match ww_compare p zdigits with - | Eq => w_WW w_0 xl - | Lt => w_WW w_0 (w_pos_mod (low p) xl) - | Gt => - match ww_compare p ww_zdigits with - | Lt => - let n := low (ww_sub p zdigits) in - w_WW (w_pos_mod n xh) xl - | _ => x - end - end - end. - - - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (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). - - - Variable spec_w_0 : [|w_0|] = 0. - - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - - Variable spec_pos_mod : forall w p, - [|w_pos_mod p w|] = [|w|] mod (2 ^ [|p|]). - - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_sub: forall x y, - [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - - Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. - Variable spec_low: forall x, [| low x|] = [[x]] mod wB. - Variable spec_ww_zdigits : [[ww_zdigits]] = 2 * [|w_zdigits|]. - Variable spec_ww_digits : ww_digits w_digits = xO w_digits. - - - Hint Rewrite spec_w_0 spec_w_WW : w_rewrite. - - Lemma spec_ww_pos_mod : forall w p, - [[ww_pos_mod p w]] = [[w]] mod (2 ^ [[p]]). - assert (HHHHH:= lt_0_wB w_digits). - assert (F0: forall x y, x - y + y = x); auto with zarith. - intros w1 p; case (spec_to_w_Z p); intros HH1 HH2. - unfold ww_pos_mod; case w1. reflexivity. - intros xh xl; rewrite spec_ww_compare. - case Z.compare_spec; - rewrite spec_w_0W; rewrite spec_zdigits; fold wB; - intros H1. - rewrite H1; simpl ww_to_Z. - autorewrite with w_rewrite rm10. - rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_mult; auto with zarith. - autorewrite with rm10. - rewrite Zmod_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - autorewrite with w_rewrite rm10. - simpl ww_to_Z. - rewrite spec_pos_mod. - assert (HH0: [|low p|] = [[p]]). - rewrite spec_low. - apply Zmod_small; auto with zarith. - case (spec_to_w_Z p); intros HHH1 HHH2; split; auto with zarith. - apply Z.lt_le_trans with (1 := H1). - unfold base; apply Zpower2_le_lin; auto with zarith. - rewrite HH0. - rewrite Zplus_mod; auto with zarith. - unfold base. - rewrite <- (F0 (Zpos w_digits) [[p]]). - rewrite Zpower_exp; auto with zarith. - rewrite Z.mul_assoc. - rewrite Z_mod_mult; auto with zarith. - autorewrite with w_rewrite rm10. - rewrite Zmod_mod; auto with zarith. - rewrite spec_ww_compare. - case Z.compare_spec; rewrite spec_ww_zdigits; - rewrite spec_zdigits; intros H2. - replace (2^[[p]]) with wwB. - rewrite Zmod_small; auto with zarith. - unfold base; rewrite H2. - rewrite spec_ww_digits; auto. - assert (HH0: [|low (ww_sub p (w_0W w_zdigits))|] = - [[p]] - Zpos w_digits). - rewrite spec_low. - rewrite spec_ww_sub. - rewrite spec_w_0W; rewrite spec_zdigits. - rewrite <- Zmod_div_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - exists wB; unfold base; rewrite <- Zpower_exp; auto with zarith. - rewrite spec_ww_digits; - apply f_equal with (f := Z.pow 2); rewrite Pos2Z.inj_xO; auto with zarith. - simpl ww_to_Z; autorewrite with w_rewrite. - rewrite spec_pos_mod; rewrite HH0. - pattern [|xh|] at 2; - rewrite Z_div_mod_eq with (b := 2 ^ ([[p]] - Zpos w_digits)); - auto with zarith. - rewrite (fun x => (Z.mul_comm (2 ^ x))); rewrite Z.mul_add_distr_r. - unfold base; rewrite <- Z.mul_assoc; rewrite <- Zpower_exp; - auto with zarith. - rewrite F0; auto with zarith. - rewrite <- Z.add_assoc; rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_mult; auto with zarith. - autorewrite with rm10. - rewrite Zmod_mod; auto with zarith. - symmetry; apply Zmod_small; auto with zarith. - case (spec_to_Z xh); intros U1 U2. - case (spec_to_Z xl); intros U3 U4. - split; auto with zarith. - apply Z.add_nonneg_nonneg; auto with zarith. - apply Z.mul_nonneg_nonneg; auto with zarith. - match goal with |- 0 <= ?X mod ?Y => - case (Z_mod_lt X Y); auto with zarith - end. - match goal with |- ?X mod ?Y * ?U + ?Z < ?T => - apply Z.le_lt_trans with ((Y - 1) * U + Z ); - [case (Z_mod_lt X Y); auto with zarith | idtac] - end. - match goal with |- ?X * ?U + ?Y < ?Z => - apply Z.le_lt_trans with (X * U + (U - 1)) - end. - apply Z.add_le_mono_l; auto with zarith. - case (spec_to_Z xl); unfold base; auto with zarith. - rewrite Z.mul_sub_distr_r; rewrite <- Zpower_exp; auto with zarith. - rewrite F0; auto with zarith. - rewrite Zmod_small; auto with zarith. - case (spec_to_w_Z (WW xh xl)); intros U1 U2. - split; auto with zarith. - apply Z.lt_le_trans with (1:= U2). - unfold base; rewrite spec_ww_digits. - apply Zpower_le_monotone; auto with zarith. - split; auto with zarith. - rewrite Pos2Z.inj_xO; auto with zarith. - Qed. - -End POS_MOD. - -Section DoubleDiv32. - - Variable w : Type. - Variable w_0 : w. - Variable w_Bm1 : w. - Variable w_Bm2 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable w_add_c : w -> w -> carry w. - Variable w_add_carry_c : w -> w -> carry w. - Variable w_add : w -> w -> w. - Variable w_add_carry : w -> w -> w. - Variable w_pred : w -> w. - Variable w_sub : w -> w -> w. - Variable w_mul_c : w -> w -> zn2z w. - Variable w_div21 : w -> w -> w -> w*w. - Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w). - - Definition w_div32_body a1 a2 a3 b1 b2 := - match w_compare a1 b1 with - | Lt => - let (q,r) := w_div21 a1 a2 b1 in - match ww_sub_c (w_WW r a3) (w_mul_c q b2) with - | C0 r1 => (q,r1) - | C1 r1 => - let q := w_pred q in - ww_add_c_cont w_WW w_add_c w_add_carry_c - (fun r2=>(w_pred q, ww_add w_add_c w_add w_add_carry r2 (WW b1 b2))) - (fun r2 => (q,r2)) - r1 (WW b1 b2) - end - | Eq => - ww_add_c_cont w_WW w_add_c w_add_carry_c - (fun r => (w_Bm2, ww_add w_add_c w_add w_add_carry r (WW b1 b2))) - (fun r => (w_Bm1,r)) - (WW (w_sub a2 b2) a3) (WW b1 b2) - | Gt => (w_0, W0) (* cas absurde *) - end. - - Definition w_div32 a1 a2 a3 b1 b2 := - Eval lazy beta iota delta [ww_add_c_cont ww_add w_div32_body] in - w_div32_body a1 a2 a3 b1 b2. - - (* Proof *) - - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c 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) - (at level 0, c at level 99). - - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - Variable spec_w_Bm2 : [|w_Bm2|] = wB - 2. - - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|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. - - Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. - Variable spec_w_add_carry : - forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB. - - 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_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|]. - Variable spec_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - - Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Theorem wB_div2: forall x, wB/2 <= x -> wB <= 2 * x. - intros x H; rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. - Qed. - - Lemma Zmult_lt_0_reg_r_2 : forall n m : Z, 0 <= n -> 0 < m * n -> 0 < m. - Proof. - intros n m H1 H2;apply Z.mul_pos_cancel_r with n;trivial. - Z.le_elim H1; trivial. - subst;rewrite Z.mul_0_r in H2;discriminate H2. - Qed. - - Theorem spec_w_div32 : forall a1 a2 a3 b1 b2, - wB/2 <= [|b1|] -> - [[WW a1 a2]] < [[WW b1 b2]] -> - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]. - Proof. - intros a1 a2 a3 b1 b2 Hle Hlt. - assert (U:= lt_0_wB w_digits); assert (U1:= lt_0_wwB w_digits). - Spec_w_to_Z a1;Spec_w_to_Z a2;Spec_w_to_Z a3;Spec_w_to_Z b1;Spec_w_to_Z b2. - rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Z.mul_assoc;rewrite <- Z.mul_add_distr_r. - change (w_div32 a1 a2 a3 b1 b2) with (w_div32_body a1 a2 a3 b1 b2). - unfold w_div32_body. - rewrite spec_compare. case Z.compare_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). - simpl;rewrite spec_sub. - assert ([|a2|] - [|b2|] = wB*(-1) + ([|a2|] - [|b2|] + wB)). ring. - assert (0 <= [|a2|] - [|b2|] + wB < wB). omega. - rewrite <-(Zmod_unique ([|a2|]-[|b2|]) wB (-1) ([|a2|]-[|b2|]+wB) H1 H0). - rewrite wwB_wBwB;ring. - assert (U2 := wB_pos w_digits). - eapply spec_ww_add_c_cont with (P := - fun (x y:zn2z w) (res:w*zn2z w) => - let (q, r) := res in - ([|a1|] * wB + [|a2|]) * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]);eauto. - rewrite H0;intros r. - repeat - (rewrite spec_ww_add;eauto || rewrite spec_w_Bm1 || rewrite spec_w_Bm2); - simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1. - assert (0<= ([[r]] + ([|b1|] * wB + [|b2|])) - wwB < [|b1|] * wB + [|b2|]). - Spec_ww_to_Z r;split;zarith. - rewrite H1. - assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. - assert (-wwB < ([|a2|] - [|b2|]) * wB + [|a3|] < 0). - split. apply Z.lt_le_trans with (([|a2|] - [|b2|]) * wB);zarith. - rewrite wwB_wBwB;replace (-(wB^2)) with (-wB*wB);[zarith | ring]. - apply Z.mul_lt_mono_pos_r;zarith. - apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. - replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with - (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith | ring]. - assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith. - replace 0 with (0*wB);zarith. - replace (([|a2|] - [|b2|]) * wB + [|a3|] + wwB + ([|b1|] * wB + [|b2|]) + - ([|b1|] * wB + [|b2|]) - wwB) with - (([|a2|] - [|b2|]) * wB + [|a3|] + 2*[|b1|] * wB + 2*[|b2|]); - [zarith | ring]. - rewrite <- (Zmod_unique ([[r]] + ([|b1|] * wB + [|b2|])) wwB - 1 ([[r]] + ([|b1|] * wB + [|b2|]) - wwB));zarith;try (ring;fail). - split. rewrite H1;rewrite Hcmp;ring. trivial. - Spec_ww_to_Z (WW b1 b2). simpl in HH4;zarith. - rewrite H0;intros r;repeat - (rewrite spec_w_Bm1 || rewrite spec_w_Bm2); - simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1. - assert ([[r]]=([|a2|]-[|b2|])*wB+[|a3|]+([|b1|]*wB+[|b2|])). zarith. - split. rewrite H2;rewrite Hcmp;ring. - split. Spec_ww_to_Z r;zarith. - rewrite H2. - assert (([|a2|] - [|b2|]) * wB + [|a3|] < 0);zarith. - apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. - replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with - (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith|ring]. - assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith. - replace 0 with (0*wB);zarith. - (* Cas Lt *) - assert (Hdiv21 := spec_div21 a2 Hle Hcmp); - destruct (w_div21 a1 a2 b1) as (q, r);destruct Hdiv21. - rewrite H. - assert (Hq := spec_to_Z q). - generalize - (spec_ww_sub_c (w_WW r a3) (w_mul_c q b2)); - destruct (ww_sub_c (w_WW r a3) (w_mul_c q b2)) - as [r1|r1];repeat (rewrite spec_w_WW || rewrite spec_mul_c); - unfold interp_carry;intros H1. - rewrite H1. - split. ring. split. - rewrite <- H1;destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z r1);trivial. - apply Z.le_lt_trans with ([|r|] * wB + [|a3|]). - assert ( 0 <= [|q|] * [|b2|]);zarith. - apply beta_lex_inv;zarith. - assert ([[r1]] = [|r|] * wB + [|a3|] - [|q|] * [|b2|] + wwB). - rewrite <- H1;ring. - Spec_ww_to_Z r1; assert (0 <= [|r|]*wB). zarith. - assert (0 < [|q|] * [|b2|]). zarith. - assert (0 < [|q|]). - apply Zmult_lt_0_reg_r_2 with [|b2|];zarith. - eapply spec_ww_add_c_cont with (P := - fun (x y:zn2z w) (res:w*zn2z w) => - let (q0, r0) := res in - ([|q|] * [|b1|] + [|r|]) * wB + [|a3|] = - [|q0|] * ([|b1|] * wB + [|b2|]) + [[r0]] /\ - 0 <= [[r0]] < [|b1|] * wB + [|b2|]);eauto. - intros r2;repeat (rewrite spec_pred || rewrite spec_ww_add;eauto); - simpl ww_to_Z;intros H7. - assert (0 < [|q|] - 1). - assert (H6 : 1 <= [|q|]) by zarith. - Z.le_elim H6;zarith. - rewrite <- H6 in H2;rewrite H2 in H7. - assert (0 < [|b1|]*wB). apply Z.mul_pos_pos;zarith. - Spec_ww_to_Z r2. zarith. - rewrite (Zmod_small ([|q|] -1));zarith. - rewrite (Zmod_small ([|q|] -1 -1));zarith. - assert ([[r2]] + ([|b1|] * wB + [|b2|]) = - wwB * 1 + - ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2 * ([|b1|] * wB + [|b2|]))). - rewrite H7;rewrite H2;ring. - assert - ([|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) - < [|b1|]*wB + [|b2|]). - Spec_ww_to_Z r2;omega. - Spec_ww_to_Z (WW b1 b2). simpl in HH5. - assert - (0 <= [|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|]) - < wwB). split;try omega. - replace (2*([|b1|]*wB+[|b2|])) with ((2*[|b1|])*wB+2*[|b2|]). 2:ring. - assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. omega. - rewrite <- (Zmod_unique - ([[r2]] + ([|b1|] * wB + [|b2|])) - wwB - 1 - ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2*([|b1|] * wB + [|b2|])) - H10 H8). - split. ring. zarith. - intros r2;repeat (rewrite spec_pred);simpl ww_to_Z;intros H7. - rewrite (Zmod_small ([|q|] -1));zarith. - split. - replace [[r2]] with ([[r1]] + ([|b1|] * wB + [|b2|]) -wwB). - rewrite H2; ring. rewrite <- H7; ring. - Spec_ww_to_Z r2;Spec_ww_to_Z r1. omega. - simpl in Hlt. - assert ([|a1|] * wB + [|a2|] <= [|b1|] * wB + [|b2|]). zarith. - assert (H1 := beta_lex _ _ _ _ _ H HH0 HH3). rewrite spec_w_0;simpl;zarith. - Qed. - - -End DoubleDiv32. - -Section DoubleDiv21. - Variable w : Type. - Variable w_0 : w. - - Variable w_0W : w -> zn2z w. - Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w. - - Variable ww_1 : zn2z w. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable ww_sub : zn2z w -> zn2z w -> zn2z w. - - - Definition ww_div21 a1 a2 b := - match a1 with - | W0 => - match ww_compare a2 b with - | Gt => (ww_1, ww_sub a2 b) - | Eq => (ww_1, W0) - | Lt => (W0, a2) - end - | WW a1h a1l => - match a2 with - | W0 => - match b with - | W0 => (W0,W0) (* cas absurde *) - | WW b1 b2 => - let (q1, r) := w_div32 a1h a1l w_0 b1 b2 in - match r with - | W0 => (WW q1 w_0, W0) - | WW r1 r2 => - let (q2, s) := w_div32 r1 r2 w_0 b1 b2 in - (WW q1 q2, s) - end - end - | WW a2h a2l => - match b with - | W0 => (W0,W0) (* cas absurde *) - | WW b1 b2 => - let (q1, r) := w_div32 a1h a1l a2h b1 b2 in - match r with - | W0 => (WW q1 w_0, w_0W a2l) - | WW r1 r2 => - let (q2, s) := w_div32 r1 r2 a2l b1 b2 in - (WW q1 q2, s) - end - end - end - end. - - (* Proof *) - - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (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) - (at level 0, c at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_w_div32 : forall a1 a2 a3 b1 b2, - wB/2 <= [|b1|] -> - [[WW a1 a2]] < [[WW b1 b2]] -> - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]. - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - - Theorem wwB_div: wwB = 2 * (wwB / 2). - Proof. - rewrite wwB_div_2; rewrite Z.mul_assoc; rewrite wB_div_2; auto. - rewrite <- Z.pow_2_r; apply wwB_wBwB. - Qed. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Theorem spec_ww_div21 : forall a1 a2 b, - wwB/2 <= [[b]] -> - [[a1]] < [[b]] -> - let (q,r) := ww_div21 a1 a2 b in - [[a1]] *wwB+[[a2]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]. - Proof. - assert (U:= lt_0_wB w_digits). - assert (U1:= lt_0_wwB w_digits). - 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. - rewrite spec_ww_compare. case Z.compare_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. - rewrite wwB_div;zarith. - intros a1h a1l. Spec_w_to_Z a1h;Spec_w_to_Z a1l. Spec_ww_to_Z a2. - destruct a2 as [ |a3 a4]; - (destruct b as [ |b1 b2];[unfold Z.le in Eq;discriminate Eq|idtac]); - try (Spec_w_to_Z a3; Spec_w_to_Z a4); Spec_w_to_Z b1; Spec_w_to_Z b2; - intros Hlt H; match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => - generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); - intros q1 r H0 - end; (assert (Eq1: wB / 2 <= [|b1|]);[ - apply (@beta_lex (wB / 2) 0 [|b1|] [|b2|] wB); auto with zarith; - autorewrite with rm10;repeat rewrite (Z.mul_comm wB); - rewrite <- wwB_div_2; trivial - | generalize (H0 Eq1 Hlt);clear H0;destruct r as [ |r1 r2];simpl; - try rewrite spec_w_0; try rewrite spec_w_0W;repeat rewrite Z.add_0_r; - intros (H1,H2) ]). - split;[rewrite wwB_wBwB; rewrite Z.pow_2_r | trivial]. - rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc; - rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H1;ring. - destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => - generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); - intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end. - split;[rewrite wwB_wBwB | trivial]. - rewrite Z.pow_2_r. - rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc; - rewrite <- Z.pow_2_r. - rewrite <- wwB_wBwB;rewrite H1. - rewrite spec_w_0 in H4;rewrite Z.add_0_r in H4. - repeat rewrite Z.mul_add_distr_r. rewrite <- (Z.mul_assoc [|r1|]). - rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H4;simpl;ring. - split;[rewrite wwB_wBwB | split;zarith]. - replace (([|a1h|] * wB + [|a1l|]) * wB^2 + ([|a3|] * wB + [|a4|])) - with (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB+ [|a4|]). - rewrite H1;ring. rewrite wwB_wBwB;ring. - change [|a4|] with (0*wB+[|a4|]);apply beta_lex_inv;zarith. - assert (1 <= wB/2);zarith. - assert (H_:= wB_pos w_digits);apply Zdiv_le_lower_bound;zarith. - destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => - generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); - intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end. - split;trivial. - replace (([|a1h|] * wB + [|a1l|]) * wwB + ([|a3|] * wB + [|a4|])) with - (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB + [|a4|]); - [rewrite H1 | rewrite wwB_wBwB;ring]. - replace (([|q1|]*([|b1|]*wB+[|b2|])+([|r1|]*wB+[|r2|]))*wB+[|a4|]) with - (([|q1|]*([|b1|]*wB+[|b2|]))*wB+([|r1|]*wwB+[|r2|]*wB+[|a4|])); - [rewrite H4;simpl|rewrite wwB_wBwB];ring. - Qed. - -End DoubleDiv21. - -Section DoubleDivGt. - Variable w : Type. - Variable w_digits : positive. - Variable w_0 : w. - - Variable w_WW : w -> w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable w_eq0 : w -> bool. - Variable w_opp_c : w -> carry w. - Variable w_opp w_opp_carry : w -> w. - Variable w_sub_c : w -> w -> carry w. - Variable w_sub w_sub_carry : w -> w -> w. - - Variable w_div_gt : w -> w -> w*w. - Variable w_mod_gt : w -> w -> w. - Variable w_gcd_gt : w -> w -> w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable w_head0 : w -> w. - Variable w_div21 : w -> w -> w -> w * w. - Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w. - - - Variable _ww_zdigits : zn2z w. - Variable ww_1 : zn2z w. - Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w. - - Variable w_zdigits : w. - - Definition ww_div_gt_aux ah al bh bl := - Eval lazy beta iota delta [ww_sub ww_opp] in - let p := w_head0 bh in - match w_compare p w_0 with - | Gt => - let b1 := w_add_mul_div p bh bl in - let b2 := w_add_mul_div p bl w_0 in - let a1 := w_add_mul_div p w_0 ah in - let a2 := w_add_mul_div p ah al in - let a3 := w_add_mul_div p al w_0 in - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - (WW w_0 q, ww_add_mul_div - (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r) - | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)) - end. - - Definition ww_div_gt a b := - Eval lazy beta iota delta [ww_div_gt_aux double_divn1 - double_divn1_p double_divn1_p_aux double_divn1_0 double_divn1_0_aux - double_split double_0 double_WW] in - match a, b with - | W0, _ => (W0,W0) - | _, W0 => (W0,W0) - | WW ah al, WW bh bl => - if w_eq0 ah then - let (q,r) := w_div_gt al bl in - (WW w_0 q, w_0W r) - else - match w_compare w_0 bh with - | Eq => - let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl in - (q, w_0W r) - | Lt => ww_div_gt_aux ah al bh bl - | Gt => (W0,W0) (* cas absurde *) - end - end. - - Definition ww_mod_gt_aux ah al bh bl := - Eval lazy beta iota delta [ww_sub ww_opp] in - let p := w_head0 bh in - match w_compare p w_0 with - | Gt => - let b1 := w_add_mul_div p bh bl in - let b2 := w_add_mul_div p bl w_0 in - let a1 := w_add_mul_div p w_0 ah in - let a2 := w_add_mul_div p ah al in - let a3 := w_add_mul_div p al w_0 in - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r - | _ => - ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry (WW ah al) (WW bh bl) - end. - - Definition ww_mod_gt a b := - Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 - double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux - double_split double_0 double_WW snd] in - match a, b with - | W0, _ => W0 - | _, W0 => W0 - | WW ah al, WW bh bl => - if w_eq0 ah then w_0W (w_mod_gt al bl) - else - match w_compare w_0 bh with - | Eq => - w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl) - | Lt => ww_mod_gt_aux ah al bh bl - | Gt => W0 (* cas absurde *) - end - end. - - Definition ww_gcd_gt_body (cont: w->w->w->w->zn2z w) (ah al bh bl: w) := - Eval lazy beta iota delta [ww_mod_gt_aux double_modn1 - double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux - double_split double_0 double_WW snd] in - match w_compare w_0 bh with - | Eq => - match w_compare w_0 bl with - | Eq => WW ah al (* normalement n'arrive pas si forme normale *) - | Lt => - let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW ah al) bl in - WW w_0 (w_gcd_gt bl m) - | Gt => W0 (* absurde *) - end - | Lt => - let m := ww_mod_gt_aux ah al bh bl in - match m with - | W0 => WW bh bl - | WW mh ml => - match w_compare w_0 mh with - | Eq => - match w_compare w_0 ml with - | Eq => WW bh bl - | _ => - let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW bh bl) ml in - WW w_0 (w_gcd_gt ml r) - end - | Lt => - let r := ww_mod_gt_aux bh bl mh ml in - match r with - | W0 => m - | WW rh rl => cont mh ml rh rl - end - | Gt => W0 (* absurde *) - end - end - | Gt => W0 (* absurde *) - end. - - Fixpoint ww_gcd_gt_aux - (p:positive) (cont: w -> w -> w -> w -> zn2z w) (ah al bh bl : w) - {struct p} : zn2z w := - ww_gcd_gt_body - (fun mh ml rh rl => match p with - | xH => cont mh ml rh rl - | xO p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl - | xI p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl - end) ah al bh bl. - - - (* Proof *) - - Variable w_to_Z : w -> Z. - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99). - - Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - - 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, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. - - Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. - Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB. - Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1. - - Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|]. - Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - Variable spec_sub_carry : - forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB. - - Variable spec_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> - let (q,r) := w_div_gt a b in - [|a|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Variable spec_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] -> - [|w_mod_gt a b|] = [|a|] mod [|b|]. - Variable spec_gcd_gt : forall a b, [|a|] > [|b|] -> - Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]. - - Variable spec_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ ([|p|])) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB. - - Variable spec_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - - Variable spec_w_div32 : forall a1 a2 a3 b1 b2, - wB/2 <= [|b1|] -> - [[WW a1 a2]] < [[WW b1 b2]] -> - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - [|a1|] * wwB + [|a2|] * wB + [|a3|] = - [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\ - 0 <= [[r]] < [|b1|] * wB + [|b2|]. - - Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits. - - Variable spec_ww_digits_ : [[_ww_zdigits]] = Zpos (xO w_digits). - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_ww_add_mul_div : forall x y p, - [[p]] <= Zpos (xO w_digits) -> - [[ ww_add_mul_div p x y ]] = - ([[x]] * (2^[[p]]) + - [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Lemma to_Z_div_minus_p : forall x p, - 0 < [|p|] < Zpos w_digits -> - 0 <= [|x|] / 2 ^ (Zpos w_digits - [|p|]) < 2 ^ [|p|]. - Proof. - intros x p H;Spec_w_to_Z x. - split. apply Zdiv_le_lower_bound;zarith. - apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - ring_simplify ([|p|] + (Zpos w_digits - [|p|])); unfold base in HH;zarith. - Qed. - Hint Resolve to_Z_div_minus_p : zarith. - - Lemma spec_ww_div_gt_aux : forall ah al bh bl, - [[WW ah al]] > [[WW bh bl]] -> - 0 < [|bh|] -> - let (q,r) := ww_div_gt_aux ah al bh bl in - [[WW ah al]] = [[q]] * [[WW bh bl]] + [[r]] /\ - 0 <= [[r]] < [[WW bh bl]]. - Proof. - intros ah al bh bl Hgt Hpos;unfold ww_div_gt_aux. - change - (let (q, r) := let p := w_head0 bh in - match w_compare p w_0 with - | Gt => - let b1 := w_add_mul_div p bh bl in - let b2 := w_add_mul_div p bl w_0 in - let a1 := w_add_mul_div p w_0 ah in - let a2 := w_add_mul_div p ah al in - let a3 := w_add_mul_div p al w_0 in - let (q,r) := w_div32 a1 a2 a3 b1 b2 in - (WW w_0 q, ww_add_mul_div - (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r) - | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c - w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)) - end in [[WW ah al]]=[[q]]*[[WW bh bl]]+[[r]] /\ 0 <=[[r]]< [[WW bh bl]]). - assert (Hh := spec_head0 Hpos). - lazy zeta. - rewrite spec_compare; case Z.compare_spec; - rewrite spec_w_0; intros HH. - generalize Hh; rewrite HH; simpl Z.pow; - rewrite Z.mul_1_l; intros (HH1, HH2); clear HH. - assert (wwB <= 2*[[WW bh bl]]). - apply Z.le_trans with (2*[|bh|]*wB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg_r; zarith. - rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; zarith. - simpl ww_to_Z;rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc. - Spec_w_to_Z bl;zarith. - Spec_ww_to_Z (WW ah al). - rewrite spec_ww_sub;eauto. - simpl;rewrite spec_ww_1;rewrite Z.mul_1_l;simpl. - simpl ww_to_Z in Hgt, H, HH;rewrite Zmod_small;split;zarith. - case (spec_to_Z (w_head0 bh)); auto with zarith. - assert ([|w_head0 bh|] < Zpos w_digits). - destruct (Z_lt_ge_dec [|w_head0 bh|] (Zpos w_digits));trivial. - exfalso. - assert (2 ^ [|w_head0 bh|] * [|bh|] >= wB);auto with zarith. - apply Z.le_ge; replace wB with (wB * 1);try ring. - Spec_w_to_Z bh;apply Z.mul_le_mono_nonneg;zarith. - unfold base;apply Zpower_le_monotone;zarith. - assert (HHHH : 0 < [|w_head0 bh|] < Zpos w_digits); auto with zarith. - assert (Hb:= Z.lt_le_incl _ _ H). - generalize (spec_add_mul_div w_0 ah Hb) - (spec_add_mul_div ah al Hb) - (spec_add_mul_div al w_0 Hb) - (spec_add_mul_div bh bl Hb) - (spec_add_mul_div bl w_0 Hb); - rewrite spec_w_0; repeat rewrite Z.mul_0_l;repeat rewrite Z.add_0_l; - rewrite Zdiv_0_l;repeat rewrite Z.add_0_r. - Spec_w_to_Z ah;Spec_w_to_Z bh. - unfold base;repeat rewrite Zmod_shift_r;zarith. - assert (H3:=to_Z_div_minus_p ah HHHH);assert(H4:=to_Z_div_minus_p al HHHH); - assert (H5:=to_Z_div_minus_p bl HHHH). - rewrite Z.mul_comm in Hh. - assert (2^[|w_head0 bh|] < wB). unfold base;apply Zpower_lt_monotone;zarith. - unfold base in H0;rewrite Zmod_small;zarith. - fold wB; rewrite (Zmod_small ([|bh|] * 2 ^ [|w_head0 bh|]));zarith. - intros U1 U2 U3 V1 V2. - generalize (@spec_w_div32 (w_add_mul_div (w_head0 bh) w_0 ah) - (w_add_mul_div (w_head0 bh) ah al) - (w_add_mul_div (w_head0 bh) al w_0) - (w_add_mul_div (w_head0 bh) bh bl) - (w_add_mul_div (w_head0 bh) bl w_0)). - destruct (w_div32 (w_add_mul_div (w_head0 bh) w_0 ah) - (w_add_mul_div (w_head0 bh) ah al) - (w_add_mul_div (w_head0 bh) al w_0) - (w_add_mul_div (w_head0 bh) bh bl) - (w_add_mul_div (w_head0 bh) bl w_0)) as (q,r). - rewrite V1;rewrite V2. rewrite Z.mul_add_distr_r. - rewrite <- (Z.add_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)). - unfold base;rewrite <- shift_unshift_mod;zarith. fold wB. - replace ([|bh|] * 2 ^ [|w_head0 bh|] * wB + [|bl|] * 2 ^ [|w_head0 bh|]) with - ([[WW bh bl]] * 2^[|w_head0 bh|]). 2:simpl;ring. - fold wwB. rewrite wwB_wBwB. rewrite Z.pow_2_r. rewrite U1;rewrite U2;rewrite U3. - rewrite Z.mul_assoc. rewrite Z.mul_add_distr_r. - rewrite (Z.add_assoc ([|ah|] / 2^(Zpos(w_digits) - [|w_head0 bh|])*wB * wB)). - rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc. - unfold base;repeat rewrite <- shift_unshift_mod;zarith. fold wB. - replace ([|ah|] * 2 ^ [|w_head0 bh|] * wB + [|al|] * 2 ^ [|w_head0 bh|]) with - ([[WW ah al]] * 2^[|w_head0 bh|]). 2:simpl;ring. - intros Hd;destruct Hd;zarith. - simpl. apply beta_lex_inv;zarith. rewrite U1;rewrite V1. - assert ([|ah|] / 2 ^ (Zpos (w_digits) - [|w_head0 bh|]) < wB/2);zarith. - apply Zdiv_lt_upper_bound;zarith. - unfold base. - replace (2^Zpos (w_digits)) with (2^(Zpos (w_digits) - 1)*2). - rewrite Z_div_mult;zarith. rewrite <- Zpower_exp;zarith. - apply Z.lt_le_trans with wB;zarith. - unfold base;apply Zpower_le_monotone;zarith. - pattern 2 at 2;replace 2 with (2^1);trivial. - rewrite <- Zpower_exp;zarith. ring_simplify (Zpos (w_digits) - 1 + 1);trivial. - change [[WW w_0 q]] with ([|w_0|]*wB+[|q|]);rewrite spec_w_0;rewrite - Z.mul_0_l;rewrite Z.add_0_l. - replace [[ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry - _ww_zdigits (w_0W (w_head0 bh))) W0 r]] with ([[r]]/2^[|w_head0 bh|]). - assert (0 < 2^[|w_head0 bh|]). apply Z.pow_pos_nonneg;zarith. - split. - rewrite <- (Z_div_mult [[WW ah al]] (2^[|w_head0 bh|]));zarith. - rewrite H1;rewrite Z.mul_assoc;apply Z_div_plus_l;trivial. - split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith. - rewrite spec_ww_add_mul_div. - rewrite spec_ww_sub; auto with zarith. - rewrite spec_ww_digits_. - change (Zpos (xO (w_digits))) with (2*Zpos (w_digits));zarith. - simpl ww_to_Z;rewrite Z.mul_0_l;rewrite Z.add_0_l. - rewrite spec_w_0W. - rewrite (fun x y => Zmod_small (x-y)); auto with zarith. - ring_simplify (2 * Zpos w_digits - (2 * Zpos w_digits - [|w_head0 bh|])). - rewrite Zmod_small;zarith. - split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith. - Spec_ww_to_Z r. - apply Z.lt_le_trans with wwB;zarith. - rewrite <- (Z.mul_1_r wwB);apply Z.mul_le_mono_nonneg;zarith. - split; auto with zarith. - apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith. - unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits). - apply Zpower2_lt_lin; auto with zarith. - rewrite spec_ww_sub; auto with zarith. - rewrite spec_ww_digits_; rewrite spec_w_0W. - rewrite Zmod_small;zarith. - rewrite Pos2Z.inj_xO; split; auto with zarith. - apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith. - unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits). - apply Zpower2_lt_lin; auto with zarith. - Qed. - - Lemma spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - let (q,r) := ww_div_gt a b in - [[a]] = [[q]] * [[b]] + [[r]] /\ - 0 <= [[r]] < [[b]]. - Proof. - intros a b Hgt Hpos;unfold ww_div_gt. - change (let (q,r) := match a, b with - | W0, _ => (W0,W0) - | _, W0 => (W0,W0) - | WW ah al, WW bh bl => - if w_eq0 ah then - let (q,r) := w_div_gt al bl in - (WW w_0 q, w_0W r) - else - match w_compare w_0 bh with - | Eq => - let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl in - (q, w_0W r) - | Lt => ww_div_gt_aux ah al bh bl - | Gt => (W0,W0) (* cas absurde *) - end - end in [[a]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]). - destruct a as [ |ah al]. simpl in Hgt;omega. - destruct b as [ |bh bl]. simpl in Hpos;omega. - Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl. - assert (H:=@spec_eq0 ah);destruct (w_eq0 ah). - simpl ww_to_Z;rewrite H;trivial. simpl in Hgt;rewrite H in Hgt;trivial. - assert ([|bh|] <= 0). - apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. - assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;rewrite H1;simpl in Hgt. - simpl. simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos. - 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. - rewrite spec_compare; case Z.compare_spec; intros Hcmp. - rewrite spec_w_0 in Hcmp. change [[WW bh bl]] with ([|bh|]*wB+[|bl|]). - rewrite <- Hcmp;rewrite Z.mul_0_l;rewrite Z.add_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). - 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). - rewrite spec_w_0W;unfold ww_to_Z;trivial. - apply spec_ww_div_gt_aux;trivial. rewrite spec_w_0 in Hcmp;trivial. - rewrite spec_w_0 in Hcmp;exfalso;omega. - Qed. - - Lemma spec_ww_mod_gt_aux_eq : forall ah al bh bl, - ww_mod_gt_aux ah al bh bl = snd (ww_div_gt_aux ah al bh bl). - Proof. - intros ah al bh bl. unfold ww_mod_gt_aux, ww_div_gt_aux. - case w_compare; auto. - case w_div32; auto. - Qed. - - Lemma spec_ww_mod_gt_aux : forall ah al bh bl, - [[WW ah al]] > [[WW bh bl]] -> - 0 < [|bh|] -> - [[ww_mod_gt_aux ah al bh bl]] = [[WW ah al]] mod [[WW bh bl]]. - Proof. - intros. rewrite spec_ww_mod_gt_aux_eq;trivial. - assert (H3 := spec_ww_div_gt_aux ah al bl H H0). - destruct (ww_div_gt_aux ah al bh bl) as (q,r);simpl. simpl in H,H3. - destruct H3;apply Zmod_unique with [[q]];zarith. - rewrite H1;ring. - Qed. - - Lemma spec_w_mod_gt_eq : forall a b, [|a|] > [|b|] -> 0 <[|b|] -> - [|w_mod_gt a b|] = [|snd (w_div_gt a b)|]. - Proof. - intros a b Hgt Hpos. - rewrite spec_mod_gt;trivial. - assert (H:=spec_div_gt Hgt Hpos). - destruct (w_div_gt a b) as (q,r);simpl. - rewrite Z.mul_comm in H;destruct H. - symmetry;apply Zmod_unique with [|q|];trivial. - Qed. - - Lemma spec_ww_mod_gt_eq : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - [[ww_mod_gt a b]] = [[snd (ww_div_gt a b)]]. - Proof. - intros a b Hgt Hpos. - change (ww_mod_gt a b) with - (match a, b with - | W0, _ => W0 - | _, W0 => W0 - | WW ah al, WW bh bl => - if w_eq0 ah then w_0W (w_mod_gt al bl) - else - match w_compare w_0 bh with - | Eq => - w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl) - | Lt => ww_mod_gt_aux ah al bh bl - | Gt => W0 (* cas absurde *) - end end). - change (ww_div_gt a b) with - (match a, b with - | W0, _ => (W0,W0) - | _, W0 => (W0,W0) - | WW ah al, WW bh bl => - if w_eq0 ah then - let (q,r) := w_div_gt al bl in - (WW w_0 q, w_0W r) - else - match w_compare w_0 bh with - | Eq => - let(q,r):= - double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 a bl in - (q, w_0W r) - | Lt => ww_div_gt_aux ah al bh bl - | Gt => (W0,W0) (* cas absurde *) - end - end). - destruct a as [ |ah al];trivial. - destruct b as [ |bh bl];trivial. - Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl. - assert (H:=@spec_eq0 ah);destruct (w_eq0 ah). - simpl in Hgt;rewrite H in Hgt;trivial. - assert ([|bh|] <= 0). - apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. - assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. - simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos. - 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. - rewrite spec_compare; case Z.compare_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 - (WW ah al) bl);simpl;trivial. - rewrite spec_ww_mod_gt_aux_eq;trivial;symmetry;trivial. - trivial. - Qed. - - Lemma spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - [[ww_mod_gt a b]] = [[a]] mod [[b]]. - Proof. - intros a b Hgt Hpos. - assert (H:= spec_ww_div_gt a b Hgt Hpos). - rewrite (spec_ww_mod_gt_eq a b Hgt Hpos). - destruct (ww_div_gt a b)as(q,r);destruct H. - apply Zmod_unique with[[q]];simpl;trivial. - rewrite Z.mul_comm;trivial. - Qed. - - Lemma Zis_gcd_mod : forall a b d, - 0 < b -> Zis_gcd b (a mod b) d -> Zis_gcd a b d. - Proof. - intros a b d H H1; apply Zis_gcd_for_euclid with (a/b). - pattern a at 1;rewrite (Z_div_mod_eq a b). - ring_simplify (b * (a / b) + a mod b - a / b * b);trivial. zarith. - Qed. - - Lemma spec_ww_gcd_gt_aux_body : - forall ah al bh bl n cont, - [[WW bh bl]] <= 2^n -> - [[WW ah al]] > [[WW bh bl]] -> - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^(n-1) -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> - Zis_gcd [[WW ah al]] [[WW bh bl]] [[ww_gcd_gt_body cont ah al bh bl]]. - Proof. - intros ah al bh bl n cont Hlog Hgt Hcont. - change (ww_gcd_gt_body cont ah al bh bl) with (match w_compare w_0 bh with - | Eq => - match w_compare w_0 bl with - | Eq => WW ah al (* normalement n'arrive pas si forme normale *) - | Lt => - let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW ah al) bl in - WW w_0 (w_gcd_gt bl m) - | Gt => W0 (* absurde *) - end - | Lt => - let m := ww_mod_gt_aux ah al bh bl in - match m with - | W0 => WW bh bl - | WW mh ml => - match w_compare w_0 mh with - | Eq => - match w_compare w_0 ml with - | Eq => WW bh bl - | _ => - let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21 - w_compare w_sub 1 (WW bh bl) ml in - WW w_0 (w_gcd_gt ml r) - end - | Lt => - let r := ww_mod_gt_aux bh bl mh ml in - match r with - | W0 => m - | WW rh rl => cont mh ml rh rl - end - | Gt => W0 (* absurde *) - end - end - | Gt => W0 (* absurde *) - end). - rewrite spec_compare, spec_w_0. - case Z.compare_spec; intros Hbh. - simpl ww_to_Z in *. rewrite <- Hbh. - rewrite Z.mul_0_l;rewrite Z.add_0_l. - rewrite spec_compare, spec_w_0. - case Z.compare_spec; intros Hbl. - rewrite <- Hbl;apply Zis_gcd_0. - simpl;rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. - 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 - 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 Hbl). - apply spec_gcd_gt. - rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - 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 Z.lt_le_trans with ([|bh|]*wB);zarith. - apply Z.mul_pos_pos;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. - rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hmh. - simpl;rewrite <- Hmh;simpl. - rewrite spec_compare, spec_w_0; case Z.compare_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 - spec_div21 spec_compare spec_sub 1 (WW bh bl) ml Hml). - apply spec_gcd_gt. - rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - Spec_w_to_Z ml;exfalso;omega. - assert ([[WW bh bl]] > [[WW mh ml]]). - rewrite H;simpl; apply Z.lt_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). - assert (H3 : 0 < [[WW mh ml]]). - simpl;Spec_w_to_Z ml. apply Z.lt_le_trans with ([|mh|]*wB);zarith. - apply Z.mul_pos_pos;zarith. - apply Zis_gcd_mod;zarith. simpl in *;rewrite <- H1. - destruct (ww_mod_gt_aux bh bl mh ml) as [ |rh rl]. simpl; apply Zis_gcd_0. - simpl;apply Hcont. simpl in H1;rewrite H1. - apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => - destruct (Z_mod_lt x y);zarith end. - apply Z.le_trans with (2^n/2). - apply Zdiv_le_lower_bound;zarith. - apply Z.le_trans with ([|bh|] * wB + [|bl|]);zarith. - assert (H3' := Z_div_mod_eq [[WW bh bl]] [[WW mh ml]] (Z.lt_gt _ _ H3)). - assert (H4 : 0 <= [[WW bh bl]]/[[WW mh ml]]). - apply Z.ge_le;apply Z_div_ge0;zarith. simpl in *;rewrite H1. - pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3'. - Z.le_elim H4. - assert (H6' : [[WW bh bl]] mod [[WW mh ml]] = - [[WW bh bl]] - [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). - simpl;pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3';ring. simpl in H6'. - assert ([[WW mh ml]] <= [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). - simpl;pattern ([|mh|]*wB+[|ml|]) at 1;rewrite <- Z.mul_1_r;zarith. - simpl in *;assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in H8; - zarith. - assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in *;zarith. - rewrite <- H4 in H3';rewrite Z.mul_0_r in H3';simpl in H3';zarith. - pattern n at 1;replace n with (n-1+1);try ring. - rewrite Zpower_exp;zarith. change (2^1) with 2. - 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. - Spec_w_to_Z mh;exfalso;zarith. - Spec_w_to_Z bh;exfalso;zarith. - Qed. - - Lemma spec_ww_gcd_gt_aux : - forall p cont n, - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 2^n -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> - forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] -> - [[WW bh bl]] <= 2^(Zpos p + n) -> - Zis_gcd [[WW ah al]] [[WW bh bl]] - [[ww_gcd_gt_aux p cont ah al bh bl]]. - Proof. - induction p;intros cont n Hcont ah al bh bl Hgt Hs;simpl ww_gcd_gt_aux. - assert (0 < Zpos p). unfold Z.lt;reflexivity. - apply spec_ww_gcd_gt_aux_body with (n := Zpos (xI p) + n); - trivial;rewrite Pos2Z.inj_xI. - intros. apply IHp with (n := Zpos p + n);zarith. - intros. apply IHp with (n := n );zarith. - apply Z.le_trans with (2 ^ (2* Zpos p + 1+ n -1));zarith. - apply Z.pow_le_mono_r;zarith. - assert (0 < Zpos p). unfold Z.lt;reflexivity. - apply spec_ww_gcd_gt_aux_body with (n := Zpos (xO p) + n );trivial. - rewrite (Pos2Z.inj_xO p). - intros. apply IHp with (n := Zpos p + n - 1);zarith. - intros. apply IHp with (n := n -1 );zarith. - intros;apply Hcont;zarith. - apply Z.le_trans with (2^(n-1));zarith. - apply Z.pow_le_mono_r;zarith. - apply Z.le_trans with (2 ^ (Zpos p + n -1));zarith. - apply Z.pow_le_mono_r;zarith. - apply Z.le_trans with (2 ^ (2*Zpos p + n -1));zarith. - apply Z.pow_le_mono_r;zarith. - apply spec_ww_gcd_gt_aux_body with (n := n+1);trivial. - rewrite Z.add_comm;trivial. - ring_simplify (n + 1 - 1);trivial. - Qed. - -End DoubleDivGt. - -Section DoubleDiv. - - Variable w : Type. - Variable w_digits : positive. - Variable ww_1 : zn2z w. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - - Variable ww_div_gt : zn2z w -> zn2z w -> zn2z w * zn2z w. - Variable ww_mod_gt : zn2z w -> zn2z w -> zn2z w. - - Definition ww_div a b := - match ww_compare a b with - | Gt => ww_div_gt a b - | Eq => (ww_1, W0) - | Lt => (W0, a) - end. - - Definition ww_mod a b := - match ww_compare a b with - | Gt => ww_mod_gt a b - | Eq => W0 - | Lt => a - end. - - Variable w_to_Z : w -> Z. - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (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). - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_ww_1 : [[ww_1]] = 1. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[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]] /\ - 0 <= [[r]] < [[b]]. - Variable spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> - [[ww_mod_gt a b]] = [[a]] mod [[b]]. - - Ltac Spec_w_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_to_Z x). - - Ltac Spec_ww_to_Z x := - let H:= fresh "HH" in - assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). - - Lemma spec_ww_div : forall a b, 0 < [[b]] -> - let (q,r) := ww_div a b in - [[a]] = [[q]] * [[b]] + [[r]] /\ - 0 <= [[r]] < [[b]]. - Proof. - intros a b Hpos;unfold ww_div. - rewrite spec_ww_compare; case Z.compare_spec; intros. - simpl;rewrite spec_ww_1;split;zarith. - simpl;split;[ring|Spec_ww_to_Z a;zarith]. - 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. - rewrite spec_ww_compare; case Z.compare_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;auto with zarith. - Qed. - - - Variable w_0 : w. - Variable w_1 : w. - Variable w_compare : w -> w -> comparison. - Variable w_eq0 : w -> bool. - Variable w_gcd_gt : w -> w -> w. - Variable _ww_digits : positive. - Variable spec_ww_digits_ : _ww_digits = xO w_digits. - Variable ww_gcd_gt_fix : - positive -> (w -> w -> w -> w -> zn2z w) -> - w -> w -> w -> w -> zn2z w. - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|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|]. - Variable spec_gcd_gt_fix : - forall p cont n, - (forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 2^n -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) -> - forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] -> - [[WW bh bl]] <= 2^(Zpos p + n) -> - Zis_gcd [[WW ah al]] [[WW bh bl]] - [[ww_gcd_gt_fix p cont ah al bh bl]]. - - Definition gcd_cont (xh xl yh yl:w) := - match w_compare w_1 yl with - | Eq => ww_1 - | _ => WW xh xl - end. - - Lemma spec_gcd_cont : forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 1 -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[gcd_cont xh xl yh yl]]. - Proof. - intros xh xl yh yl Hgt' Hle. simpl in Hle. - assert ([|yh|] = 0). - change 1 with (0*wB+1) in Hle. - 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; rewrite spec_compare, spec_w_1. - case Z.compare_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 (H0 : [|yl|] = 0) by (Spec_w_to_Z yl;zarith). - simpl. rewrite H0, H;simpl;apply Zis_gcd_0;trivial. - Qed. - - - Variable cont : w -> w -> w -> w -> zn2z w. - Variable spec_cont : forall xh xl yh yl, - [[WW xh xl]] > [[WW yh yl]] -> - [[WW yh yl]] <= 1 -> - Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]. - - Definition ww_gcd_gt a b := - match a, b with - | W0, _ => b - | _, W0 => a - | WW ah al, WW bh bl => - if w_eq0 ah then (WW w_0 (w_gcd_gt al bl)) - else ww_gcd_gt_fix _ww_digits cont ah al bh bl - end. - - Definition ww_gcd a b := - Eval lazy beta delta [ww_gcd_gt] in - match ww_compare a b with - | Gt => ww_gcd_gt a b - | Eq => a - | Lt => ww_gcd_gt b a - end. - - Lemma spec_ww_gcd_gt : forall a b, [[a]] > [[b]] -> - Zis_gcd [[a]] [[b]] [[ww_gcd_gt a b]]. - Proof. - intros a b Hgt;unfold ww_gcd_gt. - destruct a as [ |ah al]. simpl;apply Zis_gcd_sym;apply Zis_gcd_0. - destruct b as [ |bh bl]. simpl;apply Zis_gcd_0. - simpl in Hgt. generalize (@spec_eq0 ah);destruct (w_eq0 ah);intros. - simpl;rewrite H in Hgt;trivial;rewrite H;trivial;rewrite spec_w_0;simpl. - assert ([|bh|] <= 0). - apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith. - Spec_w_to_Z bh;assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. - rewrite H1;simpl;auto. clear H. - apply spec_gcd_gt_fix with (n:= 0);trivial. - rewrite Z.add_0_r;rewrite spec_ww_digits_. - change (2 ^ Zpos (xO w_digits)) with wwB. Spec_ww_to_Z (WW bh bl);zarith. - Qed. - - Lemma spec_ww_gcd : forall a b, Zis_gcd [[a]] [[b]] [[ww_gcd a b]]. - Proof. - intros a b. - change (ww_gcd a b) with - (match ww_compare a b with - | Gt => ww_gcd_gt a b - | Eq => a - | Lt => ww_gcd_gt b a - end). - rewrite spec_ww_compare; case Z.compare_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. - apply Zis_gcd_sym;apply spec_ww_gcd_gt;zarith. - apply spec_ww_gcd_gt;zarith. - Qed. - -End DoubleDiv. - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v deleted file mode 100644 index 195527dd..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ /dev/null @@ -1,519 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith Ndigits. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Local Infix "<<" := Pos.shiftl_nat (at level 30). - -Section GENDIVN1. - - Variable w : Type. - Variable w_digits : positive. - Variable w_zdigits : w. - Variable w_0 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_head0 : w -> w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable w_div21 : w -> w -> w -> w * w. - Variable w_compare : w -> w -> comparison. - Variable w_sub : w -> w -> w. - - - - (* ** For proofs ** *) - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) - (at level 0, x at level 99). - Notation "[[ x ]]" := (zn2z_to_Z wB w_to_Z x) (at level 0, x at level 99). - - Variable spec_to_Z : forall x, 0 <= [| x |] < wB. - Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits. - Variable spec_0 : [|w_0|] = 0. - Variable spec_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB. - Variable spec_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Variable spec_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_sub: forall x y, - [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - - - - Section DIVAUX. - Variable b2p : w. - Variable b2p_le : wB/2 <= [|b2p|]. - - Definition double_divn1_0_aux n (divn1: w -> word w n -> word w n * w) r h := - let (hh,hl) := double_split w_0 n h in - let (qh,rh) := divn1 r hh in - let (ql,rl) := divn1 rh hl in - (double_WW w_WW n qh ql, rl). - - Fixpoint double_divn1_0 (n:nat) : w -> word w n -> word w n * w := - match n return w -> word w n -> word w n * w with - | O => fun r x => w_div21 r x b2p - | S n => double_divn1_0_aux n (double_divn1_0 n) - end. - - Lemma spec_split : forall (n : nat) (x : zn2z (word w n)), - let (h, l) := double_split w_0 n x in - [!S n | x!] = [!n | h!] * double_wB w_digits n + [!n | l!]. - Proof (spec_double_split w_0 w_digits w_to_Z spec_0). - - Lemma spec_double_divn1_0 : forall n r a, - [|r|] < [|b2p|] -> - let (q,r') := double_divn1_0 n r a in - [|r|] * double_wB w_digits n + [!n|a!] = [!n|q!] * [|b2p|] + [|r'|] /\ - 0 <= [|r'|] < [|b2p|]. - Proof. - induction n;intros. - exact (spec_div21 a b2p_le H). - simpl (double_divn1_0 (S n) r a); unfold double_divn1_0_aux. - assert (H1 := spec_split n a);destruct (double_split w_0 n a) as (hh,hl). - rewrite H1. - assert (H2 := IHn r hh H);destruct (double_divn1_0 n r hh) as (qh,rh). - destruct H2. - assert ([|rh|] < [|b2p|]). omega. - assert (H4 := IHn rh hl H3);destruct (double_divn1_0 n rh hl) as (ql,rl). - destruct H4;split;trivial. - rewrite spec_double_WW;trivial. - rewrite <- double_wB_wwB. - rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite H0;rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc. - rewrite H4;ring. - Qed. - - Definition double_modn1_0_aux n (modn1:w -> word w n -> w) r h := - let (hh,hl) := double_split w_0 n h in modn1 (modn1 r hh) hl. - - Fixpoint double_modn1_0 (n:nat) : w -> word w n -> w := - match n return w -> word w n -> w with - | O => fun r x => snd (w_div21 r x b2p) - | S n => double_modn1_0_aux n (double_modn1_0 n) - end. - - Lemma spec_double_modn1_0 : forall n r x, - double_modn1_0 n r x = snd (double_divn1_0 n r x). - Proof. - induction n;simpl;intros;trivial. - unfold double_modn1_0_aux, double_divn1_0_aux. - destruct (double_split w_0 n x) as (hh,hl). - rewrite (IHn r hh). - destruct (double_divn1_0 n r hh) as (qh,rh);simpl. - rewrite IHn. destruct (double_divn1_0 n rh hl);trivial. - Qed. - - Variable p : w. - Variable p_bounded : [|p|] <= Zpos w_digits. - - Lemma spec_add_mul_divp : forall x y, - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Proof. - intros;apply spec_add_mul_div;auto. - Qed. - - Definition double_divn1_p_aux n - (divn1 : w -> word w n -> word w n -> word w n * w) r h l := - let (hh,hl) := double_split w_0 n h in - let (lh,ll) := double_split w_0 n l in - let (qh,rh) := divn1 r hh hl in - let (ql,rl) := divn1 rh hl lh in - (double_WW w_WW n qh ql, rl). - - Fixpoint double_divn1_p (n:nat) : w -> word w n -> word w n -> word w n * w := - match n return w -> word w n -> word w n -> word w n * w with - | O => fun r h l => w_div21 r (w_add_mul_div p h l) b2p - | S n => double_divn1_p_aux n (double_divn1_p n) - end. - - Lemma p_lt_double_digits : forall n, [|p|] <= Zpos (w_digits << n). - Proof. - induction n;simpl. trivial. - case (spec_to_Z p); rewrite Pos2Z.inj_xO;auto with zarith. - Qed. - - Lemma spec_double_divn1_p : forall n r h l, - [|r|] < [|b2p|] -> - let (q,r') := double_divn1_p n r h l in - [|r|] * double_wB w_digits n + - ([!n|h!]*2^[|p|] + - [!n|l!] / (2^(Zpos(w_digits << n) - [|p|]))) - mod double_wB w_digits n = [!n|q!] * [|b2p|] + [|r'|] /\ - 0 <= [|r'|] < [|b2p|]. - Proof. - case (spec_to_Z p); intros HH0 HH1. - induction n;intros. - simpl (double_divn1_p 0 r h l). - unfold double_to_Z, double_wB, "<<". - rewrite <- spec_add_mul_divp. - exact (spec_div21 (w_add_mul_div p h l) b2p_le H). - simpl (double_divn1_p (S n) r h l). - unfold double_divn1_p_aux. - assert (H1 := spec_split n h);destruct (double_split w_0 n h) as (hh,hl). - rewrite H1. rewrite <- double_wB_wwB. - assert (H2 := spec_split n l);destruct (double_split w_0 n l) as (lh,ll). - rewrite H2. - replace ([|r|] * (double_wB w_digits n * double_wB w_digits n) + - (([!n|hh!] * double_wB w_digits n + [!n|hl!]) * 2 ^ [|p|] + - ([!n|lh!] * double_wB w_digits n + [!n|ll!]) / - 2^(Zpos (w_digits << (S n)) - [|p|])) mod - (double_wB w_digits n * double_wB w_digits n)) with - (([|r|] * double_wB w_digits n + ([!n|hh!] * 2^[|p|] + - [!n|hl!] / 2^(Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n) * double_wB w_digits n + - ([!n|hl!] * 2^[|p|] + - [!n|lh!] / 2^(Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n). - generalize (IHn r hh hl H);destruct (double_divn1_p n r hh hl) as (qh,rh); - intros (H3,H4);rewrite H3. - assert ([|rh|] < [|b2p|]). omega. - replace (([!n|qh!] * [|b2p|] + [|rh|]) * double_wB w_digits n + - ([!n|hl!] * 2 ^ [|p|] + - [!n|lh!] / 2 ^ (Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n) with - ([!n|qh!] * [|b2p|] *double_wB w_digits n + ([|rh|]*double_wB w_digits n + - ([!n|hl!] * 2 ^ [|p|] + - [!n|lh!] / 2 ^ (Zpos (w_digits << n) - [|p|])) mod - double_wB w_digits n)). 2:ring. - generalize (IHn rh hl lh H0);destruct (double_divn1_p n rh hl lh) as (ql,rl); - intros (H5,H6);rewrite H5. - split;[rewrite spec_double_WW;trivial;ring|trivial]. - assert (Uhh := spec_double_to_Z w_digits w_to_Z spec_to_Z n hh); - unfold double_wB,base in Uhh. - assert (Uhl := spec_double_to_Z w_digits w_to_Z spec_to_Z n hl); - unfold double_wB,base in Uhl. - assert (Ulh := spec_double_to_Z w_digits w_to_Z spec_to_Z n lh); - unfold double_wB,base in Ulh. - assert (Ull := spec_double_to_Z w_digits w_to_Z spec_to_Z n ll); - unfold double_wB,base in Ull. - unfold double_wB,base. - assert (UU:=p_lt_double_digits n). - rewrite Zdiv_shift_r;auto with zarith. - 2:change (Zpos (w_digits << (S n))) - with (2*Zpos (w_digits << n));auto with zarith. - replace (2 ^ (Zpos (w_digits << (S n)) - [|p|])) with - (2^(Zpos (w_digits << n) - [|p|])*2^Zpos (w_digits << n)). - rewrite Zdiv_mult_cancel_r;auto with zarith. - rewrite Z.mul_add_distr_r with (p:= 2^[|p|]). - pattern ([!n|hl!] * 2^[|p|]) at 2; - rewrite (shift_unshift_mod (Zpos(w_digits << n))([|p|])([!n|hl!])); - auto with zarith. - rewrite Z.add_assoc. - replace - ([!n|hh!] * 2^Zpos (w_digits << n)* 2^[|p|] + - ([!n|hl!] / 2^(Zpos (w_digits << n)-[|p|])* - 2^Zpos(w_digits << n))) - with - (([!n|hh!] *2^[|p|] + double_to_Z w_digits w_to_Z n hl / - 2^(Zpos (w_digits << n)-[|p|])) - * 2^Zpos(w_digits << n));try (ring;fail). - rewrite <- Z.add_assoc. - rewrite <- (Zmod_shift_r ([|p|]));auto with zarith. - replace - (2 ^ Zpos (w_digits << n) * 2 ^ Zpos (w_digits << n)) with - (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n))). - rewrite (Zmod_shift_r (Zpos (w_digits << n)));auto with zarith. - replace (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n))) - with (2^Zpos(w_digits << n) *2^Zpos(w_digits << n)). - rewrite (Z.mul_comm (([!n|hh!] * 2 ^ [|p|] + - [!n|hl!] / 2 ^ (Zpos (w_digits << n) - [|p|])))). - rewrite Zmult_mod_distr_l;auto with zarith. - ring. - rewrite Zpower_exp;auto with zarith. - assert (0 < Zpos (w_digits << n)). unfold Z.lt;reflexivity. - auto with zarith. - apply Z_mod_lt;auto with zarith. - rewrite Zpower_exp;auto with zarith. - split;auto with zarith. - apply Zdiv_lt_upper_bound;auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace ([|p|] + (Zpos (w_digits << n) - [|p|])) with - (Zpos(w_digits << n));auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (w_digits << (S n)) - [|p|]) with - (Zpos (w_digits << n) - [|p|] + - Zpos (w_digits << n));trivial. - change (Zpos (w_digits << (S n))) with - (2*Zpos (w_digits << n)). ring. - Qed. - - Definition double_modn1_p_aux n (modn1 : w -> word w n -> word w n -> w) r h l:= - let (hh,hl) := double_split w_0 n h in - let (lh,ll) := double_split w_0 n l in - modn1 (modn1 r hh hl) hl lh. - - Fixpoint double_modn1_p (n:nat) : w -> word w n -> word w n -> w := - match n return w -> word w n -> word w n -> w with - | O => fun r h l => snd (w_div21 r (w_add_mul_div p h l) b2p) - | S n => double_modn1_p_aux n (double_modn1_p n) - end. - - Lemma spec_double_modn1_p : forall n r h l , - double_modn1_p n r h l = snd (double_divn1_p n r h l). - Proof. - induction n;simpl;intros;trivial. - unfold double_modn1_p_aux, double_divn1_p_aux. - destruct(double_split w_0 n h)as(hh,hl);destruct(double_split w_0 n l) as (lh,ll). - rewrite (IHn r hh hl);destruct (double_divn1_p n r hh hl) as (qh,rh). - rewrite IHn;simpl;destruct (double_divn1_p n rh hl lh);trivial. - Qed. - - End DIVAUX. - - Fixpoint high (n:nat) : word w n -> w := - match n return word w n -> w with - | O => fun a => a - | S n => - fun (a:zn2z (word w n)) => - match a with - | W0 => w_0 - | WW h l => high n h - end - end. - - Lemma spec_double_digits:forall n, Zpos w_digits <= Zpos (w_digits << n). - Proof. - induction n;simpl;auto with zarith. - change (Zpos (xO (w_digits << n))) with - (2*Zpos (w_digits << n)). - assert (0 < Zpos w_digits) by reflexivity. - auto with zarith. - Qed. - - Lemma spec_high : forall n (x:word w n), - [|high n x|] = [!n|x!] / 2^(Zpos (w_digits << n) - Zpos w_digits). - Proof. - induction n;intros. - unfold high,double_to_Z. rewrite Pshiftl_nat_0. - replace (Zpos w_digits - Zpos w_digits) with 0;try ring. - simpl. rewrite <- (Zdiv_unique [|x|] 1 [|x|] 0);auto with zarith. - assert (U2 := spec_double_digits n). - assert (U3 : 0 < Zpos w_digits). exact (eq_refl Lt). - destruct x;unfold high;fold high. - unfold double_to_Z,zn2z_to_Z;rewrite spec_0. - rewrite Zdiv_0_l;trivial. - assert (U0 := spec_double_to_Z w_digits w_to_Z spec_to_Z n w0); - assert (U1 := spec_double_to_Z w_digits w_to_Z spec_to_Z n w1). - simpl [!S n|WW w0 w1!]. - unfold double_wB,base;rewrite Zdiv_shift_r;auto with zarith. - replace (2 ^ (Zpos (w_digits << (S n)) - Zpos w_digits)) with - (2^(Zpos (w_digits << n) - Zpos w_digits) * - 2^Zpos (w_digits << n)). - rewrite Zdiv_mult_cancel_r;auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (w_digits << n) - Zpos w_digits + - Zpos (w_digits << n)) with - (Zpos (w_digits << (S n)) - Zpos w_digits);trivial. - change (Zpos (w_digits << (S n))) with - (2*Zpos (w_digits << n));ring. - change (Zpos (w_digits << (S n))) with - (2*Zpos (w_digits << n)); auto with zarith. - Qed. - - Definition double_divn1 (n:nat) (a:word w n) (b:w) := - let p := w_head0 b in - match w_compare p w_0 with - | Gt => - let b2p := w_add_mul_div p b w_0 in - let ha := high n a in - let k := w_sub w_zdigits p in - let lsr_n := w_add_mul_div k w_0 in - let r0 := w_add_mul_div p w_0 ha in - let (q,r) := double_divn1_p b2p p n r0 a (double_0 w_0 n) in - (q, lsr_n r) - | _ => double_divn1_0 b n w_0 a - end. - - Lemma spec_double_divn1 : forall n a b, - 0 < [|b|] -> - let (q,r) := double_divn1 n a b in - [!n|a!] = [!n|q!] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Proof. - intros n a b H. unfold double_divn1. - case (spec_head0 H); intros H0 H1. - case (spec_to_Z (w_head0 b)); intros HH1 HH2. - rewrite spec_compare; case Z.compare_spec; - rewrite spec_0; intros H2; auto with zarith. - assert (Hv1: wB/2 <= [|b|]). - generalize H0; rewrite H2; rewrite Z.pow_0_r; - rewrite Z.mul_1_l; auto. - assert (Hv2: [|w_0|] < [|b|]). - rewrite spec_0; auto. - generalize (spec_double_divn1_0 Hv1 n a Hv2). - rewrite spec_0;rewrite Z.mul_0_l; rewrite Z.add_0_l; auto. - contradict H2; auto with zarith. - assert (HHHH : 0 < [|w_head0 b|]); auto with zarith. - assert ([|w_head0 b|] < Zpos w_digits). - case (Z.le_gt_cases (Zpos w_digits) [|w_head0 b|]); auto; intros HH. - assert (2 ^ [|w_head0 b|] < wB). - apply Z.le_lt_trans with (2 ^ [|w_head0 b|] * [|b|]);auto with zarith. - replace (2 ^ [|w_head0 b|]) with (2^[|w_head0 b|] * 1);try (ring;fail). - apply Z.mul_le_mono_nonneg;auto with zarith. - assert (wB <= 2^[|w_head0 b|]). - unfold base;apply Zpower_le_monotone;auto with zarith. omega. - assert ([|w_add_mul_div (w_head0 b) b w_0|] = - 2 ^ [|w_head0 b|] * [|b|]). - rewrite (spec_add_mul_div b w_0); auto with zarith. - rewrite spec_0;rewrite Zdiv_0_l; try omega. - rewrite Z.add_0_r; rewrite Z.mul_comm. - rewrite Zmod_small; auto with zarith. - assert (H5 := spec_to_Z (high n a)). - assert - ([|w_add_mul_div (w_head0 b) w_0 (high n a)|] - <[|w_add_mul_div (w_head0 b) b w_0|]). - rewrite H4. - rewrite spec_add_mul_div;auto with zarith. - rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. - assert (([|high n a|]/2^(Zpos w_digits - [|w_head0 b|])) < wB). - apply Zdiv_lt_upper_bound;auto with zarith. - apply Z.lt_le_trans with wB;auto with zarith. - pattern wB at 1;replace wB with (wB*1);try ring. - apply Z.mul_le_mono_nonneg;auto with zarith. - assert (H6 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|])); - auto with zarith. - rewrite Zmod_small;auto with zarith. - apply Zdiv_lt_upper_bound;auto with zarith. - apply Z.lt_le_trans with wB;auto with zarith. - apply Z.le_trans with (2 ^ [|w_head0 b|] * [|b|] * 2). - rewrite <- wB_div_2; try omega. - apply Z.mul_le_mono_nonneg;auto with zarith. - pattern 2 at 1;rewrite <- Z.pow_1_r. - apply Zpower_le_monotone;split;auto with zarith. - rewrite <- H4 in H0. - assert (Hb3: [|w_head0 b|] <= Zpos w_digits); auto with zarith. - assert (H7:= spec_double_divn1_p H0 Hb3 n a (double_0 w_0 n) H6). - destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n - (w_add_mul_div (w_head0 b) w_0 (high n a)) a - (double_0 w_0 n)) as (q,r). - assert (U:= spec_double_digits n). - rewrite spec_double_0 in H7;trivial;rewrite Zdiv_0_l in H7. - rewrite Z.add_0_r in H7. - rewrite spec_add_mul_div in H7;auto with zarith. - rewrite spec_0 in H7;rewrite Z.mul_0_l in H7;rewrite Z.add_0_l in H7. - assert (([|high n a|] / 2 ^ (Zpos w_digits - [|w_head0 b|])) mod wB - = [!n|a!] / 2^(Zpos (w_digits << n) - [|w_head0 b|])). - rewrite Zmod_small;auto with zarith. - rewrite spec_high. rewrite Zdiv_Zdiv;auto with zarith. - rewrite <- Zpower_exp;auto with zarith. - replace (Zpos (w_digits << n) - Zpos w_digits + - (Zpos w_digits - [|w_head0 b|])) - with (Zpos (w_digits << n) - [|w_head0 b|]);trivial;ring. - assert (H8 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|]));auto with zarith. - split;auto with zarith. - apply Z.le_lt_trans with ([|high n a|]);auto with zarith. - apply Zdiv_le_upper_bound;auto with zarith. - pattern ([|high n a|]) at 1;rewrite <- Z.mul_1_r. - apply Z.mul_le_mono_nonneg;auto with zarith. - rewrite H8 in H7;unfold double_wB,base in H7. - rewrite <- shift_unshift_mod in H7;auto with zarith. - rewrite H4 in H7. - assert ([|w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r|] - = [|r|]/2^[|w_head0 b|]). - rewrite spec_add_mul_div. - rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. - replace (Zpos w_digits - [|w_sub w_zdigits (w_head0 b)|]) - with ([|w_head0 b|]). - rewrite Zmod_small;auto with zarith. - assert (H9 := spec_to_Z r). - split;auto with zarith. - apply Z.le_lt_trans with ([|r|]);auto with zarith. - apply Zdiv_le_upper_bound;auto with zarith. - pattern ([|r|]) at 1;rewrite <- Z.mul_1_r. - apply Z.mul_le_mono_nonneg;auto with zarith. - assert (H10 := Z.pow_pos_nonneg 2 ([|w_head0 b|]));auto with zarith. - rewrite spec_sub. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - case (spec_to_Z w_zdigits); auto with zarith. - rewrite spec_sub. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - case (spec_to_Z w_zdigits); auto with zarith. - case H7; intros H71 H72. - split. - rewrite <- (Z_div_mult [!n|a!] (2^[|w_head0 b|]));auto with zarith. - rewrite H71;rewrite H9. - replace ([!n|q!] * (2 ^ [|w_head0 b|] * [|b|])) - with ([!n|q!] *[|b|] * 2^[|w_head0 b|]); - try (ring;fail). - rewrite Z_div_plus_l;auto with zarith. - assert (H10 := spec_to_Z - (w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r));split; - auto with zarith. - rewrite H9. - apply Zdiv_lt_upper_bound;auto with zarith. - rewrite Z.mul_comm;auto with zarith. - exact (spec_double_to_Z w_digits w_to_Z spec_to_Z n a). - Qed. - - - Definition double_modn1 (n:nat) (a:word w n) (b:w) := - let p := w_head0 b in - match w_compare p w_0 with - | Gt => - let b2p := w_add_mul_div p b w_0 in - let ha := high n a in - let k := w_sub w_zdigits p in - let lsr_n := w_add_mul_div k w_0 in - let r0 := w_add_mul_div p w_0 ha in - let r := double_modn1_p b2p p n r0 a (double_0 w_0 n) in - lsr_n r - | _ => double_modn1_0 b n w_0 a - end. - - Lemma spec_double_modn1_aux : forall n a b, - double_modn1 n a b = snd (double_divn1 n a b). - Proof. - intros n a b;unfold double_divn1,double_modn1. - rewrite spec_compare; case Z.compare_spec; - rewrite spec_0; intros H2; auto with zarith. - apply spec_double_modn1_0. - apply spec_double_modn1_0. - rewrite spec_double_modn1_p. - destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n - (w_add_mul_div (w_head0 b) w_0 (high n a)) a (double_0 w_0 n));simpl;trivial. - Qed. - - Lemma spec_double_modn1 : forall n a b, 0 < [|b|] -> - [|double_modn1 n a b|] = [!n|a!] mod [|b|]. - Proof. - intros n a b H;assert (H1 := spec_double_divn1 n a H). - assert (H2 := spec_double_modn1_aux n a b). - rewrite H2;destruct (double_divn1 n a b) as (q,r). - simpl;apply Zmod_unique with (double_to_Z w_digits w_to_Z n q);auto with zarith. - destruct H1 as (h1,h2);rewrite h1;ring. - Qed. - -End GENDIVN1. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v deleted file mode 100644 index f65b47c8..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v +++ /dev/null @@ -1,475 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleLift. - Variable w : Type. - Variable w_0 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable w_head0 : w -> w. - Variable w_tail0 : w -> w. - Variable w_add: w -> w -> zn2z w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable ww_sub: zn2z w -> zn2z w -> zn2z w. - Variable w_digits : positive. - Variable ww_Digits : positive. - Variable w_zdigits : w. - Variable ww_zdigits : zn2z w. - Variable low: zn2z w -> w. - - Definition ww_head0 x := - match x with - | W0 => ww_zdigits - | WW xh xl => - match w_compare w_0 xh with - | Eq => w_add w_zdigits (w_head0 xl) - | _ => w_0W (w_head0 xh) - end - end. - - - Definition ww_tail0 x := - match x with - | W0 => ww_zdigits - | WW xh xl => - match w_compare w_0 xl with - | Eq => w_add w_zdigits (w_tail0 xh) - | _ => w_0W (w_tail0 xl) - end - end. - - - (* 0 < p < ww_digits *) - Definition ww_add_mul_div p x y := - let zdigits := w_0W w_zdigits in - match x, y with - | W0, W0 => W0 - | W0, WW yh yl => - match ww_compare p zdigits with - | Eq => w_0W yh - | Lt => w_0W (w_add_mul_div (low p) w_0 yh) - | Gt => - let n := low (ww_sub p zdigits) in - w_WW (w_add_mul_div n w_0 yh) (w_add_mul_div n yh yl) - end - | WW xh xl, W0 => - match ww_compare p zdigits with - | Eq => w_W0 xl - | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl w_0) - | Gt => - let n := low (ww_sub p zdigits) in - w_W0 (w_add_mul_div n xl w_0) - end - | WW xh xl, WW yh yl => - match ww_compare p zdigits with - | Eq => w_WW xl yh - | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl yh) - | Gt => - let n := low (ww_sub p zdigits) in - w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl) - end - end. - - Section DoubleProof. - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (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). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_compare : forall x y, - w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_digits : ww_Digits = xO w_digits. - Variable spec_w_head00 : forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits. - Variable spec_w_head0 : forall x, 0 < [|x|] -> - wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB. - Variable spec_w_tail00 : forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits. - Variable spec_w_tail0 : forall x, 0 < [|x|] -> - exists y, 0 <= y /\ [|x|] = (2* y + 1) * (2 ^ [|w_tail0 x|]). - Variable spec_w_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_w_add: forall x y, - [[w_add x y]] = [|x|] + [|y|]. - Variable spec_ww_sub: forall x y, - [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - - Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. - Variable spec_low: forall x, [| low x|] = [[x]] mod wB. - - Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos ww_Digits. - - Hint Resolve div_le_0 div_lt w_to_Z_wwB: lift. - Ltac zarith := auto with zarith lift. - - Lemma spec_ww_head00 : forall x, [[x]] = 0 -> [[ww_head0 x]] = Zpos ww_Digits. - Proof. - intros x; case x; unfold ww_head0. - intros HH; rewrite spec_ww_zdigits; auto. - intros xh xl; simpl; intros Hx. - case (spec_to_Z xh); intros Hx1 Hx2. - case (spec_to_Z xl); intros Hy1 Hy2. - assert (F1: [|xh|] = 0). - { Z.le_elim Hy1; auto. - - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - apply Z.lt_le_trans with (1 := Hy1); auto with zarith. - pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]). - apply Z.add_le_mono_r; auto with zarith. - - Z.le_elim Hx1; auto. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. } - rewrite spec_compare. case Z.compare_spec. - intros H; simpl. - rewrite spec_w_add; rewrite spec_w_head00. - rewrite spec_zdigits; rewrite spec_ww_digits. - rewrite Pos2Z.inj_xO; auto with zarith. - rewrite F1 in Hx; auto with zarith. - rewrite spec_w_0; auto with zarith. - rewrite spec_w_0; auto with zarith. - Qed. - - Lemma spec_ww_head0 : forall x, 0 < [[x]] -> - wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. - Proof. - clear spec_ww_zdigits. - rewrite wwB_div_2;rewrite Z.mul_comm;rewrite wwB_wBwB. - assert (U:= lt_0_wB w_digits); destruct x as [ |xh xl];simpl ww_to_Z;intros H. - unfold Z.lt in H;discriminate H. - rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0. - rewrite <- H0 in *. simpl Z.add. simpl in H. - case (spec_to_Z w_zdigits); - case (spec_to_Z (w_head0 xl)); intros HH1 HH2 HH3 HH4. - rewrite spec_w_add. - rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. - case (spec_w_head0 H); intros H1 H2. - rewrite Z.pow_2_r; fold wB; rewrite <- Z.mul_assoc; split. - apply Z.mul_le_mono_nonneg_l; auto with zarith. - apply Z.mul_lt_mono_pos_l; auto with zarith. - assert (H1 := spec_w_head0 H0). - rewrite spec_w_0W. - split. - rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc. - apply Z.le_trans with (2 ^ [|w_head0 xh|] * [|xh|] * wB). - rewrite Z.mul_comm; zarith. - assert (0 <= 2 ^ [|w_head0 xh|] * [|xl|]);zarith. - assert (H2:=spec_to_Z xl);apply Z.mul_nonneg_nonneg;zarith. - case (spec_to_Z (w_head0 xh)); intros H2 _. - generalize ([|w_head0 xh|]) H1 H2;clear H1 H2; - intros p H1 H2. - assert (Eq1 : 2^p < wB). - rewrite <- (Z.mul_1_r (2^p));apply Z.le_lt_trans with (2^p*[|xh|]);zarith. - assert (Eq2: p < Zpos w_digits). - destruct (Z.le_gt_cases (Zpos w_digits) p);trivial;contradict Eq1. - apply Z.le_ngt;unfold base;apply Zpower_le_monotone;zarith. - assert (Zpos w_digits = p + (Zpos w_digits - p)). ring. - rewrite Z.pow_2_r. - unfold base at 2;rewrite H3;rewrite Zpower_exp;zarith. - rewrite <- Z.mul_assoc; apply Z.mul_lt_mono_pos_l; zarith. - rewrite <- (Z.add_0_r (2^(Zpos w_digits - p)*wB));apply beta_lex_inv;zarith. - apply Z.mul_lt_mono_pos_r with (2 ^ p); zarith. - rewrite <- Zpower_exp;zarith. - rewrite Z.mul_comm;ring_simplify (Zpos w_digits - p + p);fold wB;zarith. - assert (H1 := spec_to_Z xh);zarith. - Qed. - - Lemma spec_ww_tail00 : forall x, [[x]] = 0 -> [[ww_tail0 x]] = Zpos ww_Digits. - Proof. - intros x; case x; unfold ww_tail0. - intros HH; rewrite spec_ww_zdigits; auto. - intros xh xl; simpl; intros Hx. - case (spec_to_Z xh); intros Hx1 Hx2. - case (spec_to_Z xl); intros Hy1 Hy2. - assert (F1: [|xh|] = 0). - { Z.le_elim Hy1; auto. - - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - apply Z.lt_le_trans with (1 := Hy1); auto with zarith. - pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]). - apply Z.add_le_mono_r; auto with zarith. - - Z.le_elim Hx1; auto. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith. - apply Z.mul_pos_pos; auto with zarith. } - assert (F2: [|xl|] = 0). - rewrite F1 in Hx; auto with zarith. - rewrite spec_compare; case Z.compare_spec. - intros H; simpl. - rewrite spec_w_add; rewrite spec_w_tail00; auto. - rewrite spec_zdigits; rewrite spec_ww_digits. - rewrite Pos2Z.inj_xO; auto with zarith. - rewrite spec_w_0; auto with zarith. - rewrite spec_w_0; auto with zarith. - Qed. - - Lemma spec_ww_tail0 : forall x, 0 < [[x]] -> - exists y, 0 <= y /\ [[x]] = (2 * y + 1) * 2 ^ [[ww_tail0 x]]. - Proof. - clear spec_ww_zdigits. - destruct x as [ |xh xl];simpl ww_to_Z;intros H. - unfold Z.lt in H;discriminate H. - rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0. - rewrite <- H0; rewrite Z.add_0_r. - case (spec_to_Z (w_tail0 xh)); intros HH1 HH2. - generalize H; rewrite <- H0; rewrite Z.add_0_r; clear H; intros H. - case (@spec_w_tail0 xh). - apply Z.mul_lt_mono_pos_r with wB; auto with zarith. - unfold base; auto with zarith. - intros z (Hz1, Hz2); exists z; split; auto. - rewrite spec_w_add; rewrite (fun x => Z.add_comm [|x|]). - rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. - rewrite Z.mul_assoc; rewrite <- Hz2; auto. - - case (spec_to_Z (w_tail0 xh)); intros HH1 HH2. - case (spec_w_tail0 H0); intros z (Hz1, Hz2). - assert (Hp: [|w_tail0 xl|] < Zpos w_digits). - case (Z.le_gt_cases (Zpos w_digits) [|w_tail0 xl|]); auto; intros H1. - absurd (2 ^ (Zpos w_digits) <= 2 ^ [|w_tail0 xl|]). - apply Z.lt_nge. - case (spec_to_Z xl); intros HH3 HH4. - apply Z.le_lt_trans with (2 := HH4). - apply Z.le_trans with (1 * 2 ^ [|w_tail0 xl|]); auto with zarith. - rewrite Hz2. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - apply Zpower_le_monotone; auto with zarith. - exists ([|xh|] * (2 ^ ((Zpos w_digits - [|w_tail0 xl|]) - 1)) + z); split. - apply Z.add_nonneg_nonneg; auto. - apply Z.mul_nonneg_nonneg; auto with zarith. - case (spec_to_Z xh); auto. - rewrite spec_w_0W. - rewrite (Z.mul_add_distr_l 2); rewrite <- Z.add_assoc. - rewrite Z.mul_add_distr_r; rewrite <- Hz2. - apply f_equal2 with (f := Z.add); auto. - rewrite (Z.mul_comm 2). - repeat rewrite <- Z.mul_assoc. - apply f_equal2 with (f := Z.mul); auto. - case (spec_to_Z (w_tail0 xl)); intros HH3 HH4. - pattern 2 at 2; rewrite <- Z.pow_1_r. - lazy beta; repeat rewrite <- Zpower_exp; auto with zarith. - unfold base; apply f_equal with (f := Z.pow 2); auto with zarith. - - contradict H0; case (spec_to_Z xl); auto with zarith. - Qed. - - Hint Rewrite Zdiv_0_l Z.mul_0_l Z.add_0_l Z.mul_0_r Z.add_0_r - spec_w_W0 spec_w_0W spec_w_WW spec_w_0 - (wB_div w_digits w_to_Z spec_to_Z) - (wB_div_plus w_digits w_to_Z spec_to_Z) : w_rewrite. - Ltac w_rewrite := autorewrite with w_rewrite;trivial. - - Lemma spec_ww_add_mul_div_aux : forall xh xl yh yl p, - let zdigits := w_0W w_zdigits in - [[p]] <= Zpos (xO w_digits) -> - [[match ww_compare p zdigits with - | Eq => w_WW xl yh - | Lt => w_WW (w_add_mul_div (low p) xh xl) - (w_add_mul_div (low p) xl yh) - | Gt => - let n := low (ww_sub p zdigits) in - w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl) - end]] = - ([[WW xh xl]] * (2^[[p]]) + - [[WW yh yl]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. - Proof. - clear spec_ww_zdigits. - intros xh xl yh yl p zdigits;assert (HwwB := wwB_pos w_digits). - case (spec_to_w_Z p); intros Hv1 Hv2. - replace (Zpos (xO w_digits)) with (Zpos w_digits + Zpos w_digits). - 2 : rewrite Pos2Z.inj_xO;ring. - replace (Zpos w_digits + Zpos w_digits - [[p]]) with - (Zpos w_digits + (Zpos w_digits - [[p]])). 2:ring. - intros Hp; assert (Hxh := spec_to_Z xh);assert (Hxl:=spec_to_Z xl); - assert (Hx := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)); - simpl in Hx;assert (Hyh := spec_to_Z yh);assert (Hyl:=spec_to_Z yl); - assert (Hy:=spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW yh yl));simpl in Hy. - rewrite spec_ww_compare; case Z.compare_spec; intros H1. - rewrite H1; unfold zdigits; rewrite spec_w_0W. - rewrite spec_zdigits; rewrite Z.sub_diag; rewrite Z.add_0_r. - simpl ww_to_Z; w_rewrite;zarith. - fold wB. - rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc;rewrite <- Z.add_assoc. - rewrite <- Z.pow_2_r. - rewrite <- wwB_wBwB;apply Zmod_unique with [|xh|]. - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xl yh)). ring. - simpl ww_to_Z; w_rewrite;zarith. - assert (HH0: [|low p|] = [[p]]). - rewrite spec_low. - apply Zmod_small. - case (spec_to_w_Z p); intros HH1 HH2; split; auto. - generalize H1; unfold zdigits; rewrite spec_w_0W; - rewrite spec_zdigits; intros tmp. - apply Z.lt_le_trans with (1 := tmp). - unfold base. - apply Zpower2_le_lin; auto with zarith. - 2: generalize H1; unfold zdigits; rewrite spec_w_0W; - rewrite spec_zdigits; auto with zarith. - generalize H1; unfold zdigits; rewrite spec_w_0W; - rewrite spec_zdigits; auto; clear H1; intros H1. - assert (HH: [|low p|] <= Zpos w_digits). - rewrite HH0; auto with zarith. - repeat rewrite spec_w_add_mul_div with (1 := HH). - rewrite HH0. - rewrite Z.mul_add_distr_r. - pattern ([|xl|] * 2 ^ [[p]]) at 2; - rewrite shift_unshift_mod with (n:= Zpos w_digits);fold wB;zarith. - replace ([|xh|] * wB * 2^[[p]]) with ([|xh|] * 2^[[p]] * wB). 2:ring. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc. - unfold base at 5;rewrite <- Zmod_shift_r;zarith. - unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); - fold wB;fold wwB;zarith. - rewrite wwB_wBwB;rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;zarith. - unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. apply Z_mod_lt;zarith. - split;zarith. apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - ring_simplify ([[p]] + (Zpos w_digits - [[p]]));fold wB;zarith. - assert (Hv: [[p]] > Zpos w_digits). - generalize H1; clear H1. - unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits; auto with zarith. - clear H1. - assert (HH0: [|low (ww_sub p zdigits)|] = [[p]] - Zpos w_digits). - rewrite spec_low. - rewrite spec_ww_sub. - unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits. - rewrite <- Zmod_div_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - exists wB; unfold base. - unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits). - rewrite <- Zpower_exp; auto with zarith. - apply f_equal with (f := fun x => 2 ^ x); auto with zarith. - assert (HH: [|low (ww_sub p zdigits)|] <= Zpos w_digits). - rewrite HH0; auto with zarith. - replace (Zpos w_digits + (Zpos w_digits - [[p]])) with - (Zpos w_digits - ([[p]] - Zpos w_digits)); zarith. - lazy zeta; simpl ww_to_Z; w_rewrite;zarith. - repeat rewrite spec_w_add_mul_div;zarith. - rewrite HH0. - pattern wB at 5;replace wB with - (2^(([[p]] - Zpos w_digits) - + (Zpos w_digits - ([[p]] - Zpos w_digits)))). - rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. - rewrite Z_div_plus_l;zarith. - rewrite shift_unshift_mod with (a:= [|yh|]) (p:= [[p]] - Zpos w_digits) - (n := Zpos w_digits);zarith. fold wB. - set (u := [[p]] - Zpos w_digits). - replace [[p]] with (u + Zpos w_digits);zarith. - rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. fold wB. - repeat rewrite Z.add_assoc. rewrite <- Z.mul_add_distr_r. - repeat rewrite <- Z.add_assoc. - unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); - fold wB;fold wwB;zarith. - unfold base;rewrite Zmod_shift_r with (a:= Zpos w_digits) - (b:= Zpos w_digits);fold wB;fold wwB;zarith. - rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;zarith. - rewrite Z.mul_add_distr_r. - replace ([|xh|] * wB * 2 ^ u) with - ([|xh|]*2^u*wB). 2:ring. - repeat rewrite <- Z.add_assoc. - rewrite (Z.add_comm ([|xh|] * 2 ^ u * wB)). - rewrite Z_mod_plus;zarith. rewrite Z_mod_mult;zarith. - unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. - unfold u; split;zarith. - split;zarith. unfold u; apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - fold u. - ring_simplify (u + (Zpos w_digits - u)); fold - wB;zarith. unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. - unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. - unfold u; split;zarith. - unfold u; split;zarith. - apply Zdiv_lt_upper_bound;zarith. - rewrite <- Zpower_exp;zarith. - fold u. - ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith. - unfold u;zarith. - unfold u;zarith. - set (u := [[p]] - Zpos w_digits). - ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith. - Qed. - - Lemma spec_ww_add_mul_div : forall x y p, - [[p]] <= Zpos (xO w_digits) -> - [[ ww_add_mul_div p x y ]] = - ([[x]] * (2^[[p]]) + - [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. - Proof. - clear spec_ww_zdigits. - intros x y p H. - destruct x as [ |xh xl]; - [assert (H1 := @spec_ww_add_mul_div_aux w_0 w_0) - |assert (H1 := @spec_ww_add_mul_div_aux xh xl)]; - (destruct y as [ |yh yl]; - [generalize (H1 w_0 w_0 p H) | generalize (H1 yh yl p H)]; - clear H1;w_rewrite);simpl ww_add_mul_div. - replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. - intros Heq;rewrite <- Heq;clear Heq; auto. - rewrite spec_ww_compare. case Z.compare_spec; intros H1; w_rewrite. - rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. - generalize H1; w_rewrite; rewrite spec_zdigits; clear H1; intros H1. - assert (HH0: [|low p|] = [[p]]). - rewrite spec_low. - apply Zmod_small. - case (spec_to_w_Z p); intros HH1 HH2; split; auto. - apply Z.lt_le_trans with (1 := H1). - unfold base; apply Zpower2_le_lin; auto with zarith. - rewrite HH0; auto with zarith. - replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. - intros Heq;rewrite <- Heq;clear Heq. - generalize (spec_ww_compare p (w_0W w_zdigits)); - case ww_compare; intros H1; w_rewrite. - rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. - rewrite Pos2Z.inj_xO in H;zarith. - assert (HH: [|low (ww_sub p (w_0W w_zdigits)) |] = [[p]] - Zpos w_digits). - symmetry in H1; change ([[p]] > [[w_0W w_zdigits]]) in H1. - revert H1. - rewrite spec_low. - rewrite spec_ww_sub; w_rewrite; intros H1. - rewrite <- Zmod_div_mod; auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - unfold base; auto with zarith. - unfold base; auto with zarith. - exists wB; unfold base. - unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits). - rewrite <- Zpower_exp; auto with zarith. - apply f_equal with (f := fun x => 2 ^ x); auto with zarith. - case (spec_to_Z xh); auto with zarith. - Qed. - - End DoubleProof. - -End DoubleLift. - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v deleted file mode 100644 index b9901390..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v +++ /dev/null @@ -1,621 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleMul. - Variable w : Type. - Variable w_0 : w. - Variable w_1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_compare : w -> w -> comparison. - Variable w_succ : w -> w. - Variable w_add_c : w -> w -> carry w. - Variable w_add : w -> w -> w. - Variable w_sub: w -> w -> w. - Variable w_mul_c : w -> w -> zn2z w. - Variable w_mul : w -> w -> w. - Variable w_square_c : w -> zn2z w. - Variable ww_add_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_add : zn2z w -> zn2z w -> zn2z w. - Variable ww_add_carry : zn2z w -> zn2z w -> zn2z w. - Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_sub : zn2z w -> zn2z w -> zn2z w. - - (* ** Multiplication ** *) - - (* (xh*B+xl) (yh*B + yl) - xh*yh = hh = |hhh|hhl|B2 - xh*yl +xl*yh = cc = |cch|ccl|B - xl*yl = ll = |llh|lll - *) - - Definition double_mul_c (cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w) x y := - match x, y with - | W0, _ => W0 - | _, W0 => W0 - | WW xh xl, WW yh yl => - let hh := w_mul_c xh yh in - let ll := w_mul_c xl yl in - let (wc,cc) := cross xh xl yh yl hh ll in - match cc with - | W0 => WW (ww_add hh (w_W0 wc)) ll - | WW cch ccl => - match ww_add_c (w_W0 ccl) ll with - | C0 l => WW (ww_add hh (w_WW wc cch)) l - | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l - end - end - end. - - Definition ww_mul_c := - double_mul_c - (fun xh xl yh yl hh ll=> - match ww_add_c (w_mul_c xh yl) (w_mul_c xl yh) with - | C0 cc => (w_0, cc) - | C1 cc => (w_1, cc) - end). - - Definition w_2 := w_add w_1 w_1. - - Definition kara_prod xh xl yh yl hh ll := - match ww_add_c hh ll with - C0 m => - match w_compare xl xh with - Eq => (w_0, m) - | Lt => - match w_compare yl yh with - Eq => (w_0, m) - | Lt => (w_0, ww_sub m (w_mul_c (w_sub xh xl) (w_sub yh yl))) - | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with - C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1) - end - end - | Gt => - match w_compare yl yh with - Eq => (w_0, m) - | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with - C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1) - end - | Gt => (w_0, ww_sub m (w_mul_c (w_sub xl xh) (w_sub yl yh))) - end - end - | C1 m => - match w_compare xl xh with - Eq => (w_1, m) - | Lt => - match w_compare yl yh with - Eq => (w_1, m) - | Lt => match ww_sub_c m (w_mul_c (w_sub xh xl) (w_sub yh yl)) with - C0 m1 => (w_1, m1) | C1 m1 => (w_0, m1) - end - | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with - C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1) - end - end - | Gt => - match w_compare yl yh with - Eq => (w_1, m) - | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with - C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1) - end - | Gt => match ww_sub_c m (w_mul_c (w_sub xl xh) (w_sub yl yh)) with - C1 m1 => (w_0, m1) | C0 m1 => (w_1, m1) - end - end - end - end. - - Definition ww_karatsuba_c := double_mul_c kara_prod. - - Definition ww_mul x y := - match x, y with - | W0, _ => W0 - | _, W0 => W0 - | WW xh xl, WW yh yl => - let ccl := w_add (w_mul xh yl) (w_mul xl yh) in - ww_add (w_W0 ccl) (w_mul_c xl yl) - end. - - Definition ww_square_c x := - match x with - | W0 => W0 - | WW xh xl => - let hh := w_square_c xh in - let ll := w_square_c xl in - let xhxl := w_mul_c xh xl in - let (wc,cc) := - match ww_add_c xhxl xhxl with - | C0 cc => (w_0, cc) - | C1 cc => (w_1, cc) - end in - match cc with - | W0 => WW (ww_add hh (w_W0 wc)) ll - | WW cch ccl => - match ww_add_c (w_W0 ccl) ll with - | C0 l => WW (ww_add hh (w_WW wc cch)) l - | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l - end - end - end. - - Section DoubleMulAddn1. - Variable w_mul_add : w -> w -> w -> w * w. - - Fixpoint double_mul_add_n1 (n:nat) : word w n -> w -> w -> w * word w n := - match n return word w n -> w -> w -> w * word w n with - | O => w_mul_add - | S n1 => - let mul_add := double_mul_add_n1 n1 in - fun x y r => - match x with - | W0 => (w_0,extend w_0W n1 r) - | WW xh xl => - let (rl,l) := mul_add xl y r in - let (rh,h) := mul_add xh y rl in - (rh, double_WW w_WW n1 h l) - end - end. - - End DoubleMulAddn1. - - Section DoubleMulAddmn1. - Variable wn: Type. - Variable extend_n : w -> wn. - Variable wn_0W : wn -> zn2z wn. - Variable wn_WW : wn -> wn -> zn2z wn. - Variable w_mul_add_n1 : wn -> w -> w -> w*wn. - Fixpoint double_mul_add_mn1 (m:nat) : - word wn m -> w -> w -> w*word wn m := - match m return word wn m -> w -> w -> w*word wn m with - | O => w_mul_add_n1 - | S m1 => - let mul_add := double_mul_add_mn1 m1 in - fun x y r => - match x with - | W0 => (w_0,extend wn_0W m1 (extend_n r)) - | WW xh xl => - let (rl,l) := mul_add xl y r in - let (rh,h) := mul_add xh y rl in - (rh, double_WW wn_WW m1 h l) - end - end. - - End DoubleMulAddmn1. - - Definition w_mul_add x y r := - match w_mul_c x y with - | W0 => (w_0, r) - | WW h l => - match w_add_c l r with - | C0 lr => (h,lr) - | C1 lr => (w_succ h, lr) - end - end. - - - (*Section DoubleProof. *) - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c 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) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Notation "[|| x ||]" := - (zn2z_to_Z wwB (ww_to_Z w_digits w_to_Z) x) (at level 0, x at level 99). - - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) - (at level 0, x at level 99). - - Variable spec_more_than_1_digit: 1 < Zpos w_digits. - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_w_compare : - forall x y, w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB. - Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. - Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - - Variable spec_w_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|]. - Variable spec_w_mul : forall x y, [|w_mul x y|] = ([|x|] * [|y|]) mod wB. - Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|]. - - Variable spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. - Variable spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB. - Variable spec_ww_add_carry : - forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB. - Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - - - Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB. - Proof. intros x;apply spec_ww_to_Z;auto. Qed. - - Lemma spec_ww_to_Z_wBwB : forall x, 0 <= [[x]] < wB^2. - Proof. rewrite <- wwB_wBwB;apply spec_ww_to_Z. Qed. - - Hint Resolve spec_ww_to_Z spec_ww_to_Z_wBwB : mult. - Ltac zarith := auto with zarith mult. - - Lemma wBwB_lex: forall a b c d, - a * wB^2 + [[b]] <= c * wB^2 + [[d]] -> - a <= c. - Proof. - intros a b c d H; apply beta_lex with [[b]] [[d]] (wB^2);zarith. - Qed. - - Lemma wBwB_lex_inv: forall a b c d, - a < c -> - a * wB^2 + [[b]] < c * wB^2 + [[d]]. - Proof. - intros a b c d H; apply beta_lex_inv; zarith. - Qed. - - Lemma sum_mul_carry : forall xh xl yh yl wc cc, - [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] -> - 0 <= [|wc|] <= 1. - Proof. - intros. - apply (sum_mul_carry [|xh|] [|xl|] [|yh|] [|yl|] [|wc|][[cc]] wB);zarith. - apply wB_pos. - Qed. - - Theorem mult_add_ineq: forall xH yH crossH, - 0 <= [|xH|] * [|yH|] + [|crossH|] < wwB. - Proof. - intros;rewrite wwB_wBwB;apply mult_add_ineq;zarith. - Qed. - - Hint Resolve mult_add_ineq : mult. - - Lemma spec_mul_aux : forall xh xl yh yl wc (cc:zn2z w) hh ll, - [[hh]] = [|xh|] * [|yh|] -> - [[ll]] = [|xl|] * [|yl|] -> - [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] -> - [||match cc with - | W0 => WW (ww_add hh (w_W0 wc)) ll - | WW cch ccl => - match ww_add_c (w_W0 ccl) ll with - | C0 l => WW (ww_add hh (w_WW wc cch)) l - | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l - end - end||] = ([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|]). - Proof. - intros;assert (U1 := wB_pos w_digits). - replace (([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|])) with - ([|xh|]*[|yh|]*wB^2 + ([|xh|]*[|yl|] + [|xl|]*[|yh|])*wB + [|xl|]*[|yl|]). - 2:ring. rewrite <- H1;rewrite <- H;rewrite <- H0. - assert (H2 := sum_mul_carry _ _ _ _ _ _ H1). - destruct cc as [ | cch ccl]; simpl zn2z_to_Z; simpl ww_to_Z. - rewrite spec_ww_add;rewrite spec_w_W0;rewrite Zmod_small; - rewrite wwB_wBwB. ring. - rewrite <- (Z.add_0_r ([|wc|]*wB));rewrite H;apply mult_add_ineq3;zarith. - simpl ww_to_Z in H1. assert (U:=spec_to_Z cch). - assert ([|wc|]*wB + [|cch|] <= 2*wB - 3). - destruct (Z_le_gt_dec ([|wc|]*wB + [|cch|]) (2*wB - 3)) as [Hle|Hgt];trivial. - assert ([|xh|] * [|yl|] + [|xl|] * [|yh|] <= (2*wB - 4)*wB + 2). - ring_simplify ((2*wB - 4)*wB + 2). - assert (H4 := Zmult_lt_b _ _ _ (spec_to_Z xh) (spec_to_Z yl)). - assert (H5 := Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)). - omega. - generalize H3;clear H3;rewrite <- H1. - rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite Z.mul_assoc; - rewrite <- Z.mul_add_distr_r. - assert (((2 * wB - 4) + 2)*wB <= ([|wc|] * wB + [|cch|])*wB). - apply Z.mul_le_mono_nonneg;zarith. - rewrite Z.mul_add_distr_r in H3. - intros. assert (U2 := spec_to_Z ccl);omega. - generalize (spec_ww_add_c (w_W0 ccl) ll);destruct (ww_add_c (w_W0 ccl) ll) - as [l|l];unfold interp_carry;rewrite spec_w_W0;try rewrite Z.mul_1_l; - simpl zn2z_to_Z; - try rewrite spec_ww_add;try rewrite spec_ww_add_carry;rewrite spec_w_WW; - rewrite Zmod_small;rewrite wwB_wBwB;intros. - rewrite H4;ring. rewrite H;apply mult_add_ineq2;zarith. - rewrite Z.add_assoc;rewrite Z.mul_add_distr_r. - rewrite Z.mul_1_l;rewrite <- Z.add_assoc;rewrite H4;ring. - repeat rewrite <- Z.add_assoc;rewrite H;apply mult_add_ineq2;zarith. - Qed. - - Lemma spec_double_mul_c : forall cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w, - (forall xh xl yh yl hh ll, - [[hh]] = [|xh|]*[|yh|] -> - [[ll]] = [|xl|]*[|yl|] -> - let (wc,cc) := cross xh xl yh yl hh ll in - [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]) -> - forall x y, [||double_mul_c cross x y||] = [[x]] * [[y]]. - Proof. - intros cross Hcross x y;destruct x as [ |xh xl];simpl;trivial. - destruct y as [ |yh yl];simpl. rewrite Z.mul_0_r;trivial. - assert (H1:= spec_w_mul_c xh yh);assert (H2:= spec_w_mul_c xl yl). - generalize (Hcross _ _ _ _ _ _ H1 H2). - destruct (cross xh xl yh yl (w_mul_c xh yh) (w_mul_c xl yl)) as (wc,cc). - intros;apply spec_mul_aux;trivial. - rewrite <- wwB_wBwB;trivial. - Qed. - - Lemma spec_ww_mul_c : forall x y, [||ww_mul_c x y||] = [[x]] * [[y]]. - Proof. - intros x y;unfold ww_mul_c;apply spec_double_mul_c. - intros xh xl yh yl hh ll H1 H2. - generalize (spec_ww_add_c (w_mul_c xh yl) (w_mul_c xl yh)); - destruct (ww_add_c (w_mul_c xh yl) (w_mul_c xl yh)) as [c|c]; - unfold interp_carry;repeat rewrite spec_w_mul_c;intros H; - (rewrite spec_w_0 || rewrite spec_w_1);rewrite H;ring. - Qed. - - Lemma spec_w_2: [|w_2|] = 2. - unfold w_2; rewrite spec_w_add; rewrite spec_w_1; simpl. - apply Zmod_small; split; auto with zarith. - rewrite <- (Z.pow_1_r 2); unfold base; apply Zpower_lt_monotone; auto with zarith. - Qed. - - Lemma kara_prod_aux : forall xh xl yh yl, - xh*yh + xl*yl - (xh-xl)*(yh-yl) = xh*yl + xl*yh. - Proof. intros;ring. Qed. - - Lemma spec_kara_prod : forall xh xl yh yl hh ll, - [[hh]] = [|xh|]*[|yh|] -> - [[ll]] = [|xl|]*[|yl|] -> - let (wc,cc) := kara_prod xh xl yh yl hh ll in - [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]. - Proof. - intros xh xl yh yl hh ll H H0; rewrite <- kara_prod_aux; - rewrite <- H; rewrite <- H0; unfold kara_prod. - assert (Hxh := (spec_to_Z xh)); assert (Hxl := (spec_to_Z xl)); - assert (Hyh := (spec_to_Z yh)); assert (Hyl := (spec_to_Z yl)). - generalize (spec_ww_add_c hh ll); case (ww_add_c hh ll); - intros z Hz; rewrite <- Hz; unfold interp_carry; assert (Hz1 := (spec_ww_to_Z z)). - rewrite spec_w_compare; case Z.compare_spec; intros Hxlh; - try rewrite Hxlh; try rewrite spec_w_0; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh. - rewrite Hylh; rewrite spec_w_0; try (ring; fail). - rewrite spec_w_0; try (ring; fail). - repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - split; auto with zarith. - simpl in Hz; rewrite Hz; rewrite H; rewrite H0. - rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith. - apply Z.le_lt_trans with ([[z]]-0); auto with zarith. - unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive. - apply Z.mul_nonneg_nonneg; auto with zarith. - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2; - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh. - rewrite Hylh; rewrite spec_w_0; try (ring; fail). - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2; - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; try (ring; fail). - repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - split. - match goal with |- context[(?x - ?y) * (?z - ?t)] => - replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring] - end. - simpl in Hz; rewrite Hz; rewrite H; rewrite H0. - rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith. - apply Z.le_lt_trans with ([[z]]-0); auto with zarith. - unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive. - apply Z.mul_nonneg_nonneg; auto with zarith. - (** there is a carry in hh + ll **) - rewrite Z.mul_1_l. - rewrite spec_w_compare; case Z.compare_spec; intros Hxlh; - try rewrite Hxlh; try rewrite spec_w_1; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh; - try rewrite Hylh; try rewrite spec_w_1; try (ring; fail). - match goal with |- context[ww_sub_c ?x ?y] => - generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l. - generalize Hz2; clear Hz2; unfold interp_carry. - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_2; unfold interp_carry in Hz2. - transitivity (wwB + (1 * wwB + [[z1]])). - ring. - rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_compare; case Z.compare_spec; intros Hylh; - try rewrite Hylh; try rewrite spec_w_1; try (ring; fail). - match goal with |- context[ww_add_c ?x ?y] => - generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_2; unfold interp_carry in Hz2. - transitivity (wwB + (1 * wwB + [[z1]])). - ring. - rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - match goal with |- context[ww_sub_c ?x ?y] => - generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1; - intros z1 Hz2 - end. - simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l. - match goal with |- context[(?x - ?y) * (?z - ?t)] => - replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring] - end. - generalize Hz2; clear Hz2; unfold interp_carry. - repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). - repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - Qed. - - Lemma sub_carry : forall xh xl yh yl z, - 0 <= z -> - [|xh|]*[|yl|] + [|xl|]*[|yh|] = wwB + z -> - z < wwB. - Proof. - intros xh xl yh yl z Hle Heq. - destruct (Z_le_gt_dec wwB z);auto with zarith. - generalize (Zmult_lt_b _ _ _ (spec_to_Z xh) (spec_to_Z yl)). - generalize (Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)). - rewrite <- wwB_wBwB;intros H1 H2. - assert (H3 := wB_pos w_digits). - assert (2*wB <= wwB). - rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg;zarith. - omega. - Qed. - - Ltac Spec_ww_to_Z x := - let H:= fresh "H" in - assert (H:= spec_ww_to_Z x). - - Ltac Zmult_lt_b x y := - let H := fresh "H" in - assert (H := Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)). - - Lemma spec_ww_karatsuba_c : forall x y, [||ww_karatsuba_c x y||]=[[x]]*[[y]]. - Proof. - intros x y; unfold ww_karatsuba_c;apply spec_double_mul_c. - intros; apply spec_kara_prod; auto. - Qed. - - Lemma spec_ww_mul : forall x y, [[ww_mul x y]] = [[x]]*[[y]] mod wwB. - Proof. - assert (U:= lt_0_wB w_digits). - assert (U1:= lt_0_wwB w_digits). - intros x y; case x; auto; intros xh xl. - case y; auto. - simpl; rewrite Z.mul_0_r; rewrite Zmod_small; auto with zarith. - intros yh yl;simpl. - repeat (rewrite spec_ww_add || rewrite spec_w_W0 || rewrite spec_w_mul_c - || rewrite spec_w_add || rewrite spec_w_mul). - rewrite <- Zplus_mod; auto with zarith. - repeat (rewrite Z.mul_add_distr_r || rewrite Z.mul_add_distr_l). - rewrite <- Zmult_mod_distr_r; auto with zarith. - rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB; auto with zarith. - rewrite Zplus_mod; auto with zarith. - rewrite Zmod_mod; auto with zarith. - rewrite <- Zplus_mod; auto with zarith. - match goal with |- ?X mod _ = _ => - rewrite <- Z_mod_plus with (a := X) (b := [|xh|] * [|yh|]) - end; auto with zarith. - f_equal; auto; rewrite wwB_wBwB; ring. - Qed. - - Lemma spec_ww_square_c : forall x, [||ww_square_c x||] = [[x]]*[[x]]. - Proof. - destruct x as [ |xh xl];simpl;trivial. - case_eq match ww_add_c (w_mul_c xh xl) (w_mul_c xh xl) with - | C0 cc => (w_0, cc) - | C1 cc => (w_1, cc) - end;intros wc cc Heq. - apply (spec_mul_aux xh xl xh xl wc cc);trivial. - generalize Heq (spec_ww_add_c (w_mul_c xh xl) (w_mul_c xh xl));clear Heq. - rewrite spec_w_mul_c;destruct (ww_add_c (w_mul_c xh xl) (w_mul_c xh xl)); - unfold interp_carry;try rewrite Z.mul_1_l;intros Heq Heq';inversion Heq; - rewrite (Z.mul_comm [|xl|]);subst. - rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l;trivial. - rewrite spec_w_1;rewrite Z.mul_1_l;rewrite <- wwB_wBwB;trivial. - Qed. - - Section DoubleMulAddn1Proof. - - Variable w_mul_add : w -> w -> w -> w * w. - Variable spec_w_mul_add : forall x y r, - let (h,l):= w_mul_add x y r in - [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. - - Lemma spec_double_mul_add_n1 : forall n x y r, - let (h,l) := double_mul_add_n1 w_mul_add n x y r in - [|h|]*double_wB w_digits n + [!n|l!] = [!n|x!]*[|y|]+[|r|]. - Proof. - induction n;intros x y r;trivial. - exact (spec_w_mul_add x y r). - unfold double_mul_add_n1;destruct x as[ |xh xl]; - fold(double_mul_add_n1 w_mul_add). - rewrite spec_w_0;rewrite spec_extend;simpl;trivial. - assert(H:=IHn xl y r);destruct (double_mul_add_n1 w_mul_add n xl y r)as(rl,l). - assert(U:=IHn xh y rl);destruct(double_mul_add_n1 w_mul_add n xh y rl)as(rh,h). - rewrite <- double_wB_wwB. rewrite spec_double_WW;simpl;trivial. - rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc;rewrite <- H. - rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite U;ring. - Qed. - - End DoubleMulAddn1Proof. - - Lemma spec_w_mul_add : forall x y r, - let (h,l):= w_mul_add x y r in - [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|]. - Proof. - intros x y r;unfold w_mul_add;assert (H:=spec_w_mul_c x y); - destruct (w_mul_c x y) as [ |h l];simpl;rewrite <- H. - rewrite spec_w_0;trivial. - assert (U:=spec_w_add_c l r);destruct (w_add_c l r) as [lr|lr];unfold - interp_carry in U;try rewrite Z.mul_1_l in H;simpl. - rewrite U;ring. rewrite spec_w_succ. rewrite Zmod_small. - rewrite <- Z.add_assoc;rewrite <- U;ring. - simpl in H;assert (H1:= Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)). - rewrite <- H in H1. - assert (H2:=spec_to_Z h);split;zarith. - case H1;clear H1;intro H1;clear H1. - replace (wB ^ 2 - 2 * wB) with ((wB - 2)*wB). 2:ring. - intros H0;assert (U1:= wB_pos w_digits). - assert (H1 := beta_lex _ _ _ _ _ H0 (spec_to_Z l));zarith. - Qed. - -(* End DoubleProof. *) - -End DoubleMul. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v deleted file mode 100644 index d07ce301..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v +++ /dev/null @@ -1,1369 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleSqrt. - Variable w : Type. - Variable w_is_even : w -> bool. - Variable w_compare : w -> w -> comparison. - Variable w_0 : w. - Variable w_1 : w. - Variable w_Bm1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable w_W0 : w -> zn2z w. - Variable w_0W : w -> zn2z w. - Variable w_sub : w -> w -> w. - Variable w_sub_c : w -> w -> carry w. - Variable w_square_c : w -> zn2z w. - Variable w_div21 : w -> w -> w -> w * w. - Variable w_add_mul_div : w -> w -> w -> w. - Variable w_digits : positive. - Variable w_zdigits : w. - Variable ww_zdigits : zn2z w. - Variable w_add_c : w -> w -> carry w. - Variable w_sqrt2 : w -> w -> w * carry w. - Variable w_pred : w -> w. - Variable ww_pred_c : zn2z w -> carry (zn2z w). - Variable ww_pred : zn2z w -> zn2z w. - Variable ww_add_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_add : zn2z w -> zn2z w -> zn2z w. - Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w). - Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w. - Variable ww_head0 : zn2z w -> zn2z w. - Variable ww_compare : zn2z w -> zn2z w -> comparison. - Variable low : zn2z w -> w. - - Let wwBm1 := ww_Bm1 w_Bm1. - - Definition ww_is_even x := - match x with - | W0 => true - | WW xh xl => w_is_even xl - end. - - Let w_div21c x y z := - match w_compare x z with - | Eq => - match w_compare y z with - Eq => (C1 w_1, w_0) - | Gt => (C1 w_1, w_sub y z) - | Lt => (C1 w_0, y) - end - | Gt => - let x1 := w_sub x z in - let (q, r) := w_div21 x1 y z in - (C1 q, r) - | Lt => - let (q, r) := w_div21 x y z in - (C0 q, r) - end. - - Let w_div2s x y s := - match x with - C1 x1 => - let x2 := w_sub x1 s in - let (q, r) := w_div21c x2 y s in - match q with - C0 q1 => - if w_is_even q1 then - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), C0 r) - else - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), w_add_c r s) - | C1 q1 => - if w_is_even q1 then - (C1 (w_add_mul_div (w_pred w_zdigits) w_0 q1), C0 r) - else - (C1 (w_add_mul_div (w_pred w_zdigits) w_0 q1), w_add_c r s) - end - | C0 x1 => - let (q, r) := w_div21c x1 y s in - match q with - C0 q1 => - if w_is_even q1 then - (C0 (w_add_mul_div (w_pred w_zdigits) w_0 q1), C0 r) - else - (C0 (w_add_mul_div (w_pred w_zdigits) w_0 q1), w_add_c r s) - | C1 q1 => - if w_is_even q1 then - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), C0 r) - else - (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), w_add_c r s) - end - end. - - Definition split x := - match x with - | W0 => (w_0,w_0) - | WW h l => (h,l) - end. - - Definition ww_sqrt2 x y := - let (x1, x2) := split x in - let (y1, y2) := split y in - let ( q, r) := w_sqrt2 x1 x2 in - let (q1, r1) := w_div2s r y1 q in - match q1 with - C0 q1 => - let q2 := w_square_c q1 in - let a := WW q q1 in - match r1 with - C1 r2 => - match ww_sub_c (WW r2 y2) q2 with - C0 r3 => (a, C1 r3) - | C1 r3 => (a, C0 r3) - end - | C0 r2 => - match ww_sub_c (WW r2 y2) q2 with - C0 r3 => (a, C0 r3) - | C1 r3 => - let a2 := ww_add_mul_div (w_0W w_1) a W0 in - match ww_pred_c a2 with - C0 a3 => - (ww_pred a, ww_add_c a3 r3) - | C1 a3 => - (ww_pred a, C0 (ww_add a3 r3)) - end - end - end - | C1 q1 => - let a1 := WW q w_Bm1 in - let a2 := ww_add_mul_div (w_0W w_1) a1 wwBm1 in - (a1, ww_add_c a2 y) - end. - - Definition ww_is_zero x := - match ww_compare W0 x with - Eq => true - | _ => false - end. - - Definition ww_head1 x := - let p := ww_head0 x in - if (ww_is_even p) then p else ww_pred p. - - Definition ww_sqrt x := - if (ww_is_zero x) then W0 - else - let p := ww_head1 x in - match ww_compare p W0 with - | Gt => - match ww_add_mul_div p x W0 with - W0 => W0 - | WW x1 x2 => - let (r, _) := w_sqrt2 x1 x2 in - WW w_0 (w_add_mul_div - (w_sub w_zdigits - (low (ww_add_mul_div (ww_pred ww_zdigits) - W0 p))) w_0 r) - end - | _ => - match x with - W0 => W0 - | WW x1 x2 => WW w_0 (fst (w_sqrt2 x1 x2)) - end - end. - - - Variable w_to_Z : w -> Z. - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c 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) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c at level 99). - - Notation "[|| x ||]" := - (zn2z_to_Z wwB (ww_to_Z w_digits w_to_Z) x) (at level 0, x at level 99). - - Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x) - (at level 0, x at level 99). - - Variable spec_w_0 : [|w_0|] = 0. - Variable spec_w_1 : [|w_1|] = 1. - Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1. - Variable spec_w_zdigits : [|w_zdigits|] = Zpos w_digits. - Variable spec_more_than_1_digit: 1 < Zpos w_digits. - - Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos (xO w_digits). - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB. - - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. - Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. - Variable spec_w_is_even : forall x, - if w_is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1. - Variable spec_w_compare : forall x y, - w_compare x y = Z.compare [|x|] [|y|]. - Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. - Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|]. - Variable spec_w_div21 : forall a1 a2 b, - wB/2 <= [|b|] -> - [|a1|] < [|b|] -> - let (q,r) := w_div21 a1 a2 b in - [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ - 0 <= [|r|] < [|b|]. - Variable spec_w_add_mul_div : forall x y p, - [|p|] <= Zpos w_digits -> - [| w_add_mul_div p x y |] = - ([|x|] * (2 ^ [|p|]) + - [|y|] / (Z.pow 2 ((Zpos w_digits) - [|p|]))) mod wB. - Variable spec_ww_add_mul_div : forall x y p, - [[p]] <= Zpos (xO w_digits) -> - [[ ww_add_mul_div p x y ]] = - ([[x]] * (2^ [[p]]) + - [[y]] / (2^ (Zpos (xO w_digits) - [[p]]))) mod wwB. - Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. - Variable spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB. - Variable spec_w_sqrt2 : forall x y, - wB/ 4 <= [|x|] -> - let (s,r) := w_sqrt2 x y in - [[WW x y]] = [|s|] ^ 2 + [+|r|] /\ - [+|r|] <= 2 * [|s|]. - Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - Variable spec_ww_pred_c : forall x, [-[ww_pred_c x]] = [[x]] - 1. - Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB. - Variable spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. - Variable spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. - Variable spec_ww_compare : forall x y, - ww_compare x y = Z.compare [[x]] [[y]]. - Variable spec_ww_head0 : forall x, 0 < [[x]] -> - wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. - Variable spec_low: forall x, [|low x|] = [[x]] mod wB. - - Let spec_ww_Bm1 : [[wwBm1]] = wwB - 1. - Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);auto. Qed. - - Hint Rewrite spec_w_0 spec_w_1 spec_w_WW spec_w_sub - spec_w_add_mul_div spec_ww_Bm1 spec_w_add_c : w_rewrite. - - Lemma spec_ww_is_even : forall x, - if ww_is_even x then [[x]] mod 2 = 0 else [[x]] mod 2 = 1. -clear spec_more_than_1_digit. -intros x; case x; simpl ww_is_even. - reflexivity. - simpl. - intros w1 w2; simpl. - unfold base. - rewrite Zplus_mod; auto with zarith. - rewrite (fun x y => (Zdivide_mod (x * y))); auto with zarith. - rewrite Z.add_0_l; rewrite Zmod_mod; auto with zarith. - apply spec_w_is_even; auto with zarith. - apply Z.divide_mul_r; apply Zpower_divide; auto with zarith. - Qed. - - - Theorem spec_w_div21c : forall a1 a2 b, - wB/2 <= [|b|] -> - let (q,r) := w_div21c a1 a2 b in - [|a1|] * wB + [|a2|] = [+|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|]. - intros a1 a2 b Hb; unfold w_div21c. - assert (H: 0 < [|b|]); auto with zarith. - assert (U := wB_pos w_digits). - apply Z.lt_le_trans with (2 := Hb); auto with zarith. - apply Z.lt_le_trans with 1; auto with zarith. - apply Zdiv_le_lower_bound; auto with zarith. - rewrite !spec_w_compare. repeat case Z.compare_spec. - intros H1 H2; split. - unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. - rewrite H1; rewrite H2; ring. - autorewrite with w_rewrite; auto with zarith. - intros H1 H2; split. - unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. - rewrite H2; ring. - destruct (spec_to_Z a2);auto with zarith. - intros H1 H2; split. - unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. - rewrite H2; rewrite Zmod_small; auto with zarith. - ring. - destruct (spec_to_Z a2);auto with zarith. - rewrite spec_w_sub; auto with zarith. - destruct (spec_to_Z a2) as [H3 H4];auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - assert ([|a2|] < 2 * [|b|]); auto with zarith. - apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. - rewrite wB_div_2; auto. - intros H1. - match goal with |- context[w_div21 ?y ?z ?t] => - generalize (@spec_w_div21 y z t Hb H1); - case (w_div21 y z t); simpl; autorewrite with w_rewrite; - auto - end. - intros H1. - assert (H2: [|w_sub a1 b|] < [|b|]). - rewrite spec_w_sub; auto with zarith. - rewrite Zmod_small; auto with zarith. - assert ([|a1|] < 2 * [|b|]); auto with zarith. - apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. - rewrite wB_div_2; auto. - destruct (spec_to_Z a1);auto with zarith. - destruct (spec_to_Z a1);auto with zarith. - match goal with |- context[w_div21 ?y ?z ?t] => - generalize (@spec_w_div21 y z t Hb H2); - case (w_div21 y z t); autorewrite with w_rewrite; - auto - end. - intros w0 w1; replace [+|C1 w0|] with (wB + [|w0|]). - rewrite Zmod_small; auto with zarith. - intros (H3, H4); split; auto. - rewrite Z.mul_add_distr_r. - rewrite <- Z.add_assoc; rewrite <- H3; ring. - split; auto with zarith. - assert ([|a1|] < 2 * [|b|]); auto with zarith. - apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. - rewrite wB_div_2; auto. - destruct (spec_to_Z a1);auto with zarith. - destruct (spec_to_Z a1);auto with zarith. - simpl; case wB; auto. - Qed. - - Theorem C0_id: forall p, [+|C0 p|] = [|p|]. - intros p; simpl; auto. - Qed. - - Theorem add_mult_div_2: forall w, - [|w_add_mul_div (w_pred w_zdigits) w_0 w|] = [|w|] / 2. - intros w1. - assert (Hp: [|w_pred w_zdigits|] = Zpos w_digits - 1). - rewrite spec_pred; rewrite spec_w_zdigits. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - rewrite spec_w_add_mul_div; auto with zarith. - autorewrite with w_rewrite rm10. - match goal with |- context[?X - ?Y] => - replace (X - Y) with 1 - end. - rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith. - destruct (spec_to_Z w1) as [H1 H2];auto with zarith. - split; auto with zarith. - apply Zdiv_lt_upper_bound; auto with zarith. - rewrite Hp; ring. - Qed. - - Theorem add_mult_div_2_plus_1: forall w, - [|w_add_mul_div (w_pred w_zdigits) w_1 w|] = - [|w|] / 2 + 2 ^ Zpos (w_digits - 1). - intros w1. - assert (Hp: [|w_pred w_zdigits|] = Zpos w_digits - 1). - rewrite spec_pred; rewrite spec_w_zdigits. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - autorewrite with w_rewrite rm10; auto with zarith. - match goal with |- context[?X - ?Y] => - replace (X - Y) with 1 - end; rewrite Hp; try ring. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith. - destruct (spec_to_Z w1) as [H1 H2];auto with zarith. - split; auto with zarith. - unfold base. - match goal with |- _ < _ ^ ?X => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp - end. - rewrite Zpower_exp; try rewrite Z.pow_1_r; auto with zarith. - assert (tmp: forall p, 1 + (p -1) - 1 = p - 1); auto with zarith; - rewrite tmp; clear tmp; auto with zarith. - match goal with |- ?X + ?Y < _ => - assert (Y < X); auto with zarith - end. - apply Zdiv_lt_upper_bound; auto with zarith. - pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp; - auto with zarith. - assert (tmp: forall p, (p - 1) + 1 = p); auto with zarith; - rewrite tmp; clear tmp; auto with zarith. - Qed. - - Theorem add_mult_mult_2: forall w, - [|w_add_mul_div w_1 w w_0|] = 2 * [|w|] mod wB. - intros w1. - autorewrite with w_rewrite rm10; auto with zarith. - rewrite Z.pow_1_r; auto with zarith. - rewrite Z.mul_comm; auto. - Qed. - - Theorem ww_add_mult_mult_2: forall w, - [[ww_add_mul_div (w_0W w_1) w W0]] = 2 * [[w]] mod wwB. - intros w1. - rewrite spec_ww_add_mul_div; auto with zarith. - autorewrite with w_rewrite rm10. - rewrite spec_w_0W; rewrite spec_w_1. - rewrite Z.pow_1_r; auto with zarith. - rewrite Z.mul_comm; auto. - rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - red; simpl; intros; discriminate. - Qed. - - Theorem ww_add_mult_mult_2_plus_1: forall w, - [[ww_add_mul_div (w_0W w_1) w wwBm1]] = - (2 * [[w]] + 1) mod wwB. - intros w1. - rewrite spec_ww_add_mul_div; auto with zarith. - rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - rewrite Z.pow_1_r; auto with zarith. - f_equal; auto. - rewrite Z.mul_comm; f_equal; auto. - autorewrite with w_rewrite rm10. - unfold ww_digits, base. - symmetry; apply Zdiv_unique with (r := 2 ^ (Zpos (ww_digits w_digits) - 1) -1); - auto with zarith. - unfold ww_digits; split; auto with zarith. - match goal with |- 0 <= ?X - 1 => - assert (0 < X); auto with zarith - end. - apply Z.pow_pos_nonneg; auto with zarith. - match goal with |- 0 <= ?X - 1 => - assert (0 < X); auto with zarith; red; reflexivity - end. - unfold ww_digits; autorewrite with rm10. - assert (tmp: forall p q r, p + (q - r) = p + q - r); auto with zarith; - rewrite tmp; clear tmp. - assert (tmp: forall p, p + p = 2 * p); auto with zarith; - rewrite tmp; clear tmp. - f_equal; auto. - pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp; - auto with zarith. - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite tmp; clear tmp; auto. - match goal with |- ?X - 1 >= 0 => - assert (0 < X); auto with zarith; red; reflexivity - end. - rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - red; simpl; intros; discriminate. - Qed. - - Theorem Zplus_mod_one: forall a1 b1, 0 < b1 -> (a1 + b1) mod b1 = a1 mod b1. - intros a1 b1 H; rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_same; try rewrite Z.add_0_r; auto with zarith. - apply Zmod_mod; auto. - Qed. - - Lemma C1_plus_wB: forall x, [+|C1 x|] = wB + [|x|]. - unfold interp_carry; auto with zarith. - Qed. - - Theorem spec_w_div2s : forall a1 a2 b, - wB/2 <= [|b|] -> [+|a1|] <= 2 * [|b|] -> - let (q,r) := w_div2s a1 a2 b in - [+|a1|] * wB + [|a2|] = [+|q|] * (2 * [|b|]) + [+|r|] /\ 0 <= [+|r|] < 2 * [|b|]. - intros a1 a2 b H. - assert (HH: 0 < [|b|]); auto with zarith. - assert (U := wB_pos w_digits). - apply Z.lt_le_trans with (2 := H); auto with zarith. - apply Z.lt_le_trans with 1; auto with zarith. - apply Zdiv_le_lower_bound; auto with zarith. - unfold w_div2s; case a1; intros w0 H0. - match goal with |- context[w_div21c ?y ?z ?t] => - generalize (@spec_w_div21c y z t H); - case (w_div21c y z t); autorewrite with w_rewrite; - auto - end. - intros c w1; case c. - simpl interp_carry; intros w2 (Hw1, Hw2). - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat rewrite C0_id. - rewrite add_mult_div_2. - intros H1; split; auto with zarith. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - repeat rewrite C0_id. - rewrite add_mult_div_2. - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - intros w2; rewrite C1_plus_wB. - intros (Hw1, Hw2). - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat rewrite C0_id. - intros H1; split; auto with zarith. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1. - repeat rewrite C0_id. - rewrite add_mult_div_2_plus_1; unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - repeat rewrite C0_id. - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - rewrite add_mult_div_2_plus_1. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1. - unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - repeat rewrite C1_plus_wB in H0. - rewrite C1_plus_wB. - match goal with |- context[w_div21c ?y ?z ?t] => - generalize (@spec_w_div21c y z t H); - case (w_div21c y z t); autorewrite with w_rewrite; - auto - end. - intros c w1; case c. - intros w2 (Hw1, Hw2); rewrite C0_id in Hw1. - rewrite <- Zplus_mod_one in Hw1; auto with zarith. - rewrite Zmod_small in Hw1; auto with zarith. - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat rewrite C0_id. - intros H1; split; auto with zarith. - rewrite add_mult_div_2_plus_1. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - repeat rewrite C0_id. - rewrite add_mult_div_2_plus_1. - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; unfold base. - match goal with |- context[_ ^ ?X] => - assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; - rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith - end. - rewrite Pos2Z.inj_sub_max; auto with zarith. - rewrite Z.max_r; auto with zarith. - ring. - split; auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z w0);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - intros w2; rewrite C1_plus_wB. - rewrite <- Zplus_mod_one; auto with zarith. - rewrite Zmod_small; auto with zarith. - intros (Hw1, Hw2). - match goal with |- context[w_is_even ?y] => - generalize (spec_w_is_even y); - case (w_is_even y) - end. - repeat (rewrite C0_id || rewrite C1_plus_wB). - intros H1; split; auto with zarith. - rewrite add_mult_div_2. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - repeat (rewrite C0_id || rewrite C1_plus_wB). - rewrite spec_w_add_c; auto with zarith. - intros H1; split; auto with zarith. - rewrite add_mult_div_2. - replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); - auto with zarith. - rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. - rewrite Hw1. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); - auto with zarith. - rewrite H1; ring. - split; auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z w0);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - destruct (spec_to_Z b);auto with zarith. - Qed. - - Theorem wB_div_4: 4 * (wB / 4) = wB. - Proof. - unfold base. - assert (2 ^ Zpos w_digits = - 4 * (2 ^ (Zpos w_digits - 2))). - change 4 with (2 ^ 2). - rewrite <- Zpower_exp; auto with zarith. - f_equal; auto with zarith. - rewrite H. - rewrite (fun x => (Z.mul_comm 4 (2 ^x))). - rewrite Z_div_mult; auto with zarith. - Qed. - - Theorem Zsquare_mult: forall p, p ^ 2 = p * p. - intros p; change 2 with (1 + 1); rewrite Zpower_exp; - try rewrite Z.pow_1_r; auto with zarith. - Qed. - - Theorem Zsquare_pos: forall p, 0 <= p ^ 2. - intros p; case (Z.le_gt_cases 0 p); intros H1. - rewrite Zsquare_mult; apply Z.mul_nonneg_nonneg; auto with zarith. - rewrite Zsquare_mult; replace (p * p) with ((- p) * (- p)); try ring. - apply Z.mul_nonneg_nonneg; auto with zarith. - Qed. - - Lemma spec_split: forall x, - [|fst (split x)|] * wB + [|snd (split x)|] = [[x]]. - intros x; case x; simpl; autorewrite with w_rewrite; - auto with zarith. - Qed. - - Theorem mult_wwB: forall x y, [|x|] * [|y|] < wwB. - Proof. - intros x y; rewrite wwB_wBwB; rewrite Z.pow_2_r. - generalize (spec_to_Z x); intros U. - generalize (spec_to_Z y); intros U1. - apply Z.le_lt_trans with ((wB -1 ) * (wB - 1)); auto with zarith. - apply Z.mul_le_mono_nonneg; auto with zarith. - rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r; auto with zarith. - Qed. - Hint Resolve mult_wwB. - - Lemma spec_ww_sqrt2 : forall x y, - wwB/ 4 <= [[x]] -> - let (s,r) := ww_sqrt2 x y in - [||WW x y||] = [[s]] ^ 2 + [+[r]] /\ - [+[r]] <= 2 * [[s]]. - intros x y H; unfold ww_sqrt2. - repeat match goal with |- context[split ?x] => - generalize (spec_split x); case (split x) - end; simpl @fst; simpl @snd. - intros w0 w1 Hw0 w2 w3 Hw1. - assert (U: wB/4 <= [|w2|]). - case (Z.le_gt_cases (wB / 4) [|w2|]); auto; intros H1. - contradict H; apply Z.lt_nge. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - pattern wB at 1; rewrite <- wB_div_4; rewrite <- Z.mul_assoc; - rewrite Z.mul_comm. - rewrite Z_div_mult; auto with zarith. - rewrite <- Hw1. - match goal with |- _ < ?X => - pattern X; rewrite <- Z.add_0_r; apply beta_lex_inv; - auto with zarith - end. - destruct (spec_to_Z w3);auto with zarith. - generalize (@spec_w_sqrt2 w2 w3 U); case (w_sqrt2 w2 w3). - intros w4 c (H1, H2). - assert (U1: wB/2 <= [|w4|]). - case (Z.le_gt_cases (wB/2) [|w4|]); auto with zarith. - intros U1. - assert (U2 : [|w4|] <= wB/2 -1); auto with zarith. - assert (U3 : [|w4|] ^ 2 <= wB/4 * wB - wB + 1); auto with zarith. - match goal with |- ?X ^ 2 <= ?Y => - rewrite Zsquare_mult; - replace Y with ((wB/2 - 1) * (wB/2 -1)) - end. - apply Z.mul_le_mono_nonneg; auto with zarith. - destruct (spec_to_Z w4);auto with zarith. - destruct (spec_to_Z w4);auto with zarith. - pattern wB at 4 5; rewrite <- wB_div_2. - rewrite Z.mul_assoc. - replace ((wB / 4) * 2) with (wB / 2). - ring. - pattern wB at 1; rewrite <- wB_div_4. - change 4 with (2 * 2). - rewrite <- Z.mul_assoc; rewrite (Z.mul_comm 2). - rewrite Z_div_mult; try ring; auto with zarith. - assert (U4 : [+|c|] <= wB -2); auto with zarith. - apply Z.le_trans with (1 := H2). - match goal with |- ?X <= ?Y => - replace Y with (2 * (wB/ 2 - 1)); auto with zarith - end. - pattern wB at 2; rewrite <- wB_div_2; auto with zarith. - match type of H1 with ?X = _ => - assert (U5: X < wB / 4 * wB) - end. - rewrite H1; auto with zarith. - contradict U; apply Z.lt_nge. - apply Z.mul_lt_mono_pos_r with wB; auto with zarith. - destruct (spec_to_Z w4);auto with zarith. - apply Z.le_lt_trans with (2 := U5). - unfold ww_to_Z, zn2z_to_Z. - destruct (spec_to_Z w3);auto with zarith. - generalize (@spec_w_div2s c w0 w4 U1 H2). - case (w_div2s c w0 w4). - intros c0; case c0; intros w5; - repeat (rewrite C0_id || rewrite C1_plus_wB). - intros c1; case c1; intros w6; - repeat (rewrite C0_id || rewrite C1_plus_wB). - intros (H3, H4). - match goal with |- context [ww_sub_c ?y ?z] => - generalize (spec_ww_sub_c y z); case (ww_sub_c y z) - end. - intros z; change [-[C0 z]] with ([[z]]). - change [+[C0 z]] with ([[z]]). - intros H5; rewrite spec_w_square_c in H5; - auto. - split. - unfold zn2z_to_Z; rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1. rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite H5. - unfold ww_to_Z, zn2z_to_Z. - repeat rewrite Zsquare_mult; ring. - rewrite H5. - unfold ww_to_Z, zn2z_to_Z. - match goal with |- ?X - ?Y * ?Y <= _ => - assert (V := Zsquare_pos Y); - rewrite Zsquare_mult in V; - apply Z.le_trans with X; auto with zarith; - clear V - end. - match goal with |- ?X * wB + ?Y <= 2 * (?Z * wB + ?T) => - apply Z.le_trans with ((2 * Z - 1) * wB + wB); auto with zarith - end. - destruct (spec_to_Z w1);auto with zarith. - match goal with |- ?X <= _ => - replace X with (2 * [|w4|] * wB); auto with zarith - end. - rewrite Z.mul_add_distr_l; rewrite Z.mul_assoc. - destruct (spec_to_Z w5); auto with zarith. - ring. - intros z; replace [-[C1 z]] with (- wwB + [[z]]). - 2: simpl; case wwB; auto with zarith. - intros H5; rewrite spec_w_square_c in H5; - auto. - match goal with |- context [ww_pred_c ?y] => - generalize (spec_ww_pred_c y); case (ww_pred_c y) - end. - intros z1; change [-[C0 z1]] with ([[z1]]). - rewrite ww_add_mult_mult_2. - rewrite spec_ww_add_c. - rewrite spec_ww_pred. - rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]); - auto with zarith. - intros Hz1; rewrite Zmod_small; auto with zarith. - match type of H5 with -?X + ?Y = ?Z => - assert (V: Y = Z + X); - try (rewrite <- H5; ring) - end. - split. - unfold zn2z_to_Z; rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite V. - rewrite Hz1. - unfold ww_to_Z; simpl zn2z_to_Z. - repeat rewrite Zsquare_mult; ring. - rewrite Hz1. - destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. - assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)). - assert (0 < [[WW w4 w5]]); auto with zarith. - apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith. - autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith. - apply Z.mul_lt_mono_pos_r with 2; auto with zarith. - autorewrite with rm10. - rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith. - case (spec_to_Z w5);auto with zarith. - case (spec_to_Z w5);auto with zarith. - simpl. - assert (V2 := spec_to_Z w5);auto with zarith. - assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith. - split; auto with zarith. - assert (wwB <= 2 * [[WW w4 w5]]); auto with zarith. - apply Z.le_trans with (2 * ([|w4|] * wB)). - rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith. - assert (V2 := spec_to_Z w5);auto with zarith. - rewrite <- wB_div_2; auto with zarith. - simpl ww_to_Z; assert (V2 := spec_to_Z w5);auto with zarith. - assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith. - intros z1; change [-[C1 z1]] with (-wwB + [[z1]]). - match goal with |- context[([+[C0 ?z]])] => - change [+[C0 z]] with ([[z]]) - end. - rewrite spec_ww_add; auto with zarith. - rewrite spec_ww_pred; auto with zarith. - rewrite ww_add_mult_mult_2. - rename V1 into VV1. - assert (VV2: 0 < [[WW w4 w5]]); auto with zarith. - apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith. - autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith. - apply Z.mul_lt_mono_pos_r with 2; auto with zarith. - autorewrite with rm10. - rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith. - assert (VV3 := spec_to_Z w5);auto with zarith. - assert (VV3 := spec_to_Z w5);auto with zarith. - simpl. - assert (VV3 := spec_to_Z w5);auto with zarith. - assert (VV3: wwB <= 2 * [[WW w4 w5]]); auto with zarith. - apply Z.le_trans with (2 * ([|w4|] * wB)). - rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith. - case (spec_to_Z w5);auto with zarith. - rewrite <- wB_div_2; auto with zarith. - simpl ww_to_Z; assert (V4 := spec_to_Z w5);auto with zarith. - rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]); - auto with zarith. - intros Hz1; rewrite Zmod_small; auto with zarith. - match type of H5 with -?X + ?Y = ?Z => - assert (V: Y = Z + X); - try (rewrite <- H5; ring) - end. - match type of Hz1 with -?X + ?Y = -?X + ?Z - 1 => - assert (V1: Y = Z - 1); - [replace (Z - 1) with (X + (-X + Z -1)); - [rewrite <- Hz1 | idtac]; ring - | idtac] - end. - rewrite <- Zmod_unique with (q := 1) (r := -wwB + [[z1]] + [[z]]); - auto with zarith. - unfold zn2z_to_Z; rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. - rewrite <- Hw0. - split. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite V. - rewrite Hz1. - unfold ww_to_Z; simpl zn2z_to_Z. - repeat rewrite Zsquare_mult; ring. - assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. - assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. - assert (V3 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z1);auto with zarith. - split; auto with zarith. - rewrite (Z.add_comm (-wwB)); rewrite <- Z.add_assoc. - rewrite H5. - match goal with |- 0 <= ?X + (?Y - ?Z) => - apply Z.le_trans with (X - Z); auto with zarith - end. - 2: generalize (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w6 w1)); unfold ww_to_Z; auto with zarith. - rewrite V1. - match goal with |- 0 <= ?X - 1 - ?Y => - assert (Y < X); auto with zarith - end. - apply Z.lt_le_trans with wwB; auto with zarith. - intros (H3, H4). - match goal with |- context [ww_sub_c ?y ?z] => - generalize (spec_ww_sub_c y z); case (ww_sub_c y z) - end. - intros z; change [-[C0 z]] with ([[z]]). - match goal with |- context[([+[C1 ?z]])] => - replace [+[C1 z]] with (wwB + [[z]]) - end. - 2: simpl; case wwB; auto. - intros H5; rewrite spec_w_square_c in H5; - auto. - split. - change ([||WW x y||]) with ([[x]] * wwB + [[y]]). - rewrite <- Hw1. - unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite H5. - unfold ww_to_Z; simpl zn2z_to_Z. - rewrite wwB_wBwB. - repeat rewrite Zsquare_mult; ring. - simpl ww_to_Z. - rewrite H5. - simpl ww_to_Z. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - match goal with |- ?X * ?Y + (?Z * ?Y + ?T - ?U) <= _ => - apply Z.le_trans with (X * Y + (Z * Y + T - 0)); - auto with zarith - end. - assert (V := Zsquare_pos [|w5|]); - rewrite Zsquare_mult in V; auto with zarith. - autorewrite with rm10. - match goal with |- _ <= 2 * (?U * ?V + ?W) => - apply Z.le_trans with (2 * U * V + 0); - auto with zarith - end. - match goal with |- ?X * ?Y + (?Z * ?Y + ?T) <= _ => - replace (X * Y + (Z * Y + T)) with ((X + Z) * Y + T); - try ring - end. - apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith. - destruct (spec_to_Z w1);auto with zarith. - destruct (spec_to_Z w5);auto with zarith. - rewrite Z.mul_add_distr_l; auto with zarith. - rewrite Z.mul_assoc; auto with zarith. - intros z; replace [-[C1 z]] with (- wwB + [[z]]). - 2: simpl; case wwB; auto with zarith. - intros H5; rewrite spec_w_square_c in H5; - auto. - match goal with |- context[([+[C0 ?z]])] => - change [+[C0 z]] with ([[z]]) - end. - match type of H5 with -?X + ?Y = ?Z => - assert (V: Y = Z + X); - try (rewrite <- H5; ring) - end. - change ([||WW x y||]) with ([[x]] * wwB + [[y]]). - simpl ww_to_Z. - rewrite <- Hw1. - simpl ww_to_Z in H1; rewrite H1. - rewrite <- Hw0. - split. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H3. - rewrite V. - simpl ww_to_Z. - rewrite wwB_wBwB. - repeat rewrite Zsquare_mult; ring. - rewrite V. - simpl ww_to_Z. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - match goal with |- (?Z * ?Y + ?T - ?U) + ?X * ?Y <= _ => - apply Z.le_trans with ((Z * Y + T - 0) + X * Y); - auto with zarith - end. - assert (V1 := Zsquare_pos [|w5|]); - rewrite Zsquare_mult in V1; auto with zarith. - autorewrite with rm10. - match goal with |- _ <= 2 * (?U * ?V + ?W) => - apply Z.le_trans with (2 * U * V + 0); - auto with zarith - end. - match goal with |- (?Z * ?Y + ?T) + ?X * ?Y <= _ => - replace ((Z * Y + T) + X * Y) with ((X + Z) * Y + T); - try ring - end. - apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith. - destruct (spec_to_Z w1);auto with zarith. - destruct (spec_to_Z w5);auto with zarith. - rewrite Z.mul_add_distr_l; auto with zarith. - rewrite Z.mul_assoc; auto with zarith. - Z.le_elim H2. - intros c1 (H3, H4). - match type of H3 with ?X = ?Y => absurd (X < Y) end. - apply Z.le_ngt; rewrite <- H3; auto with zarith. - rewrite Z.mul_add_distr_r. - apply Z.lt_le_trans with ((2 * [|w4|]) * wB + 0); - auto with zarith. - apply beta_lex_inv; auto with zarith. - destruct (spec_to_Z w0);auto with zarith. - assert (V1 := spec_to_Z w5);auto with zarith. - rewrite (Z.mul_comm wB); auto with zarith. - assert (0 <= [|w5|] * (2 * [|w4|])); auto with zarith. - intros c1 (H3, H4); rewrite H2 in H3. - match type of H3 with ?X + ?Y = (?Z + ?T) * ?U + ?V => - assert (VV: (Y = (T * U) + V)); - [replace Y with ((X + Y) - X); - [rewrite H3; ring | ring] | idtac] - end. - assert (V1 := spec_to_Z w0);auto with zarith. - assert (V2 := spec_to_Z w5);auto with zarith. - case V2; intros V3 _. - Z.le_elim V3; auto with zarith. - match type of VV with ?X = ?Y => absurd (X < Y) end. - apply Z.le_ngt; rewrite <- VV; auto with zarith. - apply Z.lt_le_trans with wB; auto with zarith. - match goal with |- _ <= ?X + _ => - apply Z.le_trans with X; auto with zarith - end. - match goal with |- _ <= _ * ?X => - apply Z.le_trans with (1 * X); auto with zarith - end. - autorewrite with rm10. - rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite <- V3 in VV; generalize VV; autorewrite with rm10; - clear VV; intros VV. - rewrite spec_ww_add_c; auto with zarith. - rewrite ww_add_mult_mult_2_plus_1. - match goal with |- context[?X mod wwB] => - rewrite <- Zmod_unique with (q := 1) (r := -wwB + X) - end; auto with zarith. - simpl ww_to_Z. - rewrite spec_w_Bm1; auto with zarith. - split. - change ([||WW x y||]) with ([[x]] * wwB + [[y]]). - rewrite <- Hw1. - simpl ww_to_Z in H1; rewrite H1. - rewrite <- Hw0. - match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) - end. - repeat rewrite Zsquare_mult. - rewrite wwB_wBwB; ring. - rewrite H2. - rewrite wwB_wBwB. - repeat rewrite Zsquare_mult; ring. - assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z y);auto with zarith. - assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z y);auto with zarith. - simpl ww_to_Z; unfold ww_to_Z. - rewrite spec_w_Bm1; auto with zarith. - split. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - match goal with |- _ <= -?X + (2 * (?Z * ?T + ?U) + ?V) => - assert (X <= 2 * Z * T); auto with zarith - end. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite Z.mul_add_distr_l; auto with zarith. - rewrite Z.mul_assoc; auto with zarith. - match goal with |- _ + ?X < _ => - replace X with ((2 * (([|w4|]) + 1) * wB) - 1); try ring - end. - assert (2 * ([|w4|] + 1) * wB <= 2 * wwB); auto with zarith. - rewrite <- Z.mul_assoc; apply Z.mul_le_mono_nonneg_l; auto with zarith. - rewrite wwB_wBwB; rewrite Z.pow_2_r. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - case (spec_to_Z w4);auto with zarith. -Qed. - - Lemma spec_ww_is_zero: forall x, - if ww_is_zero x then [[x]] = 0 else 0 < [[x]]. - intro x; unfold ww_is_zero. - rewrite spec_ww_compare. case Z.compare_spec; - auto with zarith. - simpl ww_to_Z. - assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z x);auto with zarith. - Qed. - - Lemma wwB_4_2: 2 * (wwB / 4) = wwB/ 2. - pattern wwB at 1; rewrite wwB_wBwB; rewrite Z.pow_2_r. - rewrite <- wB_div_2. - match goal with |- context[(2 * ?X) * (2 * ?Z)] => - replace ((2 * X) * (2 * Z)) with ((X * Z) * 4); try ring - end. - rewrite Z_div_mult; auto with zarith. - rewrite Z.mul_assoc; rewrite wB_div_2. - rewrite wwB_div_2; ring. - Qed. - - - Lemma spec_ww_head1 - : forall x : zn2z w, - (ww_is_even (ww_head1 x) = true) /\ - (0 < [[x]] -> wwB / 4 <= 2 ^ [[ww_head1 x]] * [[x]] < wwB). - assert (U := wB_pos w_digits). - intros x; unfold ww_head1. - generalize (spec_ww_is_even (ww_head0 x)); case_eq (ww_is_even (ww_head0 x)). - intros HH H1; rewrite HH; split; auto. - intros H2. - generalize (spec_ww_head0 x H2); case (ww_head0 x); autorewrite with rm10. - intros (H3, H4); split; auto with zarith. - apply Z.le_trans with (2 := H3). - apply Zdiv_le_compat_l; auto with zarith. - intros xh xl (H3, H4); split; auto with zarith. - apply Z.le_trans with (2 := H3). - apply Zdiv_le_compat_l; auto with zarith. - intros H1. - case (spec_to_w_Z (ww_head0 x)); intros Hv1 Hv2. - assert (Hp0: 0 < [[ww_head0 x]]). - generalize (spec_ww_is_even (ww_head0 x)); rewrite H1. - generalize Hv1; case [[ww_head0 x]]. - rewrite Zmod_small; auto with zarith. - intros; assert (0 < Zpos p); auto with zarith. - red; simpl; auto. - intros p H2; case H2; auto. - assert (Hp: [[ww_pred (ww_head0 x)]] = [[ww_head0 x]] - 1). - rewrite spec_ww_pred. - rewrite Zmod_small; auto with zarith. - intros H2; split. - generalize (spec_ww_is_even (ww_pred (ww_head0 x))); - case ww_is_even; auto. - rewrite Hp. - rewrite Zminus_mod; auto with zarith. - rewrite H2; repeat rewrite Zmod_small; auto with zarith. - intros H3; rewrite Hp. - case (spec_ww_head0 x); auto; intros Hv3 Hv4. - assert (Hu: forall u, 0 < u -> 2 * 2 ^ (u - 1) = 2 ^u). - intros u Hu. - pattern 2 at 1; rewrite <- Z.pow_1_r. - rewrite <- Zpower_exp; auto with zarith. - ring_simplify (1 + (u - 1)); auto with zarith. - split; auto with zarith. - apply Z.mul_le_mono_pos_r with 2; auto with zarith. - repeat rewrite (fun x => Z.mul_comm x 2). - rewrite wwB_4_2. - rewrite Z.mul_assoc; rewrite Hu; auto with zarith. - apply Z.le_lt_trans with (2 * 2 ^ ([[ww_head0 x]] - 1) * [[x]]); auto with zarith; - rewrite Hu; auto with zarith. - apply Z.mul_le_mono_nonneg_r; auto with zarith. - apply Zpower_le_monotone; auto with zarith. - Qed. - - Theorem wwB_4_wB_4: wwB / 4 = wB / 4 * wB. - Proof. - symmetry; apply Zdiv_unique with 0; auto with zarith. - rewrite Z.mul_assoc; rewrite wB_div_4; auto with zarith. - rewrite wwB_wBwB; ring. - Qed. - - Lemma spec_ww_sqrt : forall x, - [[ww_sqrt x]] ^ 2 <= [[x]] < ([[ww_sqrt x]] + 1) ^ 2. - assert (U := wB_pos w_digits). - intro x; unfold ww_sqrt. - generalize (spec_ww_is_zero x); case (ww_is_zero x). - simpl ww_to_Z; simpl Z.pow; unfold Z.pow_pos; simpl; - auto with zarith. - intros H1. - rewrite spec_ww_compare. case Z.compare_spec; - simpl ww_to_Z; autorewrite with rm10. - generalize H1; case x. - intros HH; contradict HH; simpl ww_to_Z; auto with zarith. - intros w0 w1; simpl ww_to_Z; autorewrite with w_rewrite rm10. - intros H2; case (spec_ww_head1 (WW w0 w1)); intros H3 H4 H5. - generalize (H4 H2); clear H4; rewrite H5; clear H5; autorewrite with rm10. - intros (H4, H5). - assert (V: wB/4 <= [|w0|]). - apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10. - rewrite <- wwB_4_wB_4; auto. - generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith. - case (w_sqrt2 w0 w1); intros w2 c. - simpl ww_to_Z; simpl @fst. - case c; unfold interp_carry; autorewrite with rm10. - intros w3 (H6, H7); rewrite H6. - assert (V1 := spec_to_Z w3);auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. - match goal with |- ?X < ?Z => - replace Z with (X + 1); auto with zarith - end. - repeat rewrite Zsquare_mult; ring. - intros w3 (H6, H7); rewrite H6. - assert (V1 := spec_to_Z w3);auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. - match goal with |- ?X < ?Z => - replace Z with (X + 1); auto with zarith - end. - repeat rewrite Zsquare_mult; ring. - intros HH; case (spec_to_w_Z (ww_head1 x)); auto with zarith. - intros Hv1. - case (spec_ww_head1 x); intros Hp1 Hp2. - generalize (Hp2 H1); clear Hp2; intros Hp2. - assert (Hv2: [[ww_head1 x]] <= Zpos (xO w_digits)). - case (Z.le_gt_cases (Zpos (xO w_digits)) [[ww_head1 x]]); auto with zarith; intros HH1. - case Hp2; intros _ HH2; contradict HH2. - apply Z.le_ngt; unfold base. - apply Z.le_trans with (2 ^ [[ww_head1 x]]). - apply Zpower_le_monotone; auto with zarith. - pattern (2 ^ [[ww_head1 x]]) at 1; - rewrite <- (Z.mul_1_r (2 ^ [[ww_head1 x]])). - apply Z.mul_le_mono_nonneg_l; auto with zarith. - generalize (spec_ww_add_mul_div x W0 (ww_head1 x) Hv2); - case ww_add_mul_div. - simpl ww_to_Z; autorewrite with w_rewrite rm10. - rewrite Zmod_small; auto with zarith. - intros H2. symmetry in H2. rewrite Z.mul_eq_0 in H2. destruct H2 as [H2|H2]. - rewrite H2; unfold Z.pow, Z.pow_pos; simpl; auto with zarith. - match type of H2 with ?X = ?Y => - absurd (Y < X); try (rewrite H2; auto with zarith; fail) - end. - apply Z.pow_pos_nonneg; auto with zarith. - split; auto with zarith. - case Hp2; intros _ tmp; apply Z.le_lt_trans with (2 := tmp); - clear tmp. - rewrite Z.mul_comm; apply Z.mul_le_mono_nonneg_r; auto with zarith. - assert (Hv0: [[ww_head1 x]] = 2 * ([[ww_head1 x]]/2)). - pattern [[ww_head1 x]] at 1; rewrite (Z_div_mod_eq [[ww_head1 x]] 2); - auto with zarith. - generalize (spec_ww_is_even (ww_head1 x)); rewrite Hp1; - intros tmp; rewrite tmp; rewrite Z.add_0_r; auto. - intros w0 w1; autorewrite with w_rewrite rm10. - rewrite Zmod_small; auto with zarith. - 2: rewrite Z.mul_comm; auto with zarith. - intros H2. - assert (V: wB/4 <= [|w0|]). - apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10. - simpl ww_to_Z in H2; rewrite H2. - rewrite <- wwB_4_wB_4; auto with zarith. - rewrite Z.mul_comm; auto with zarith. - assert (V1 := spec_to_Z w1);auto with zarith. - generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith. - case (w_sqrt2 w0 w1); intros w2 c. - case (spec_to_Z w2); intros HH1 HH2. - simpl ww_to_Z; simpl @fst. - assert (Hv3: [[ww_pred ww_zdigits]] - = Zpos (xO w_digits) - 1). - rewrite spec_ww_pred; rewrite spec_ww_zdigits. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (Zpos (xO w_digits)); auto with zarith. - unfold base; apply Zpower2_le_lin; auto with zarith. - assert (Hv4: [[ww_head1 x]]/2 < wB). - apply Z.le_lt_trans with (Zpos w_digits). - apply Z.mul_le_mono_pos_r with 2; auto with zarith. - repeat rewrite (fun x => Z.mul_comm x 2). - rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto. - unfold base; apply Zpower2_lt_lin; auto with zarith. - assert (Hv5: [[(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))]] - = [[ww_head1 x]]/2). - rewrite spec_ww_add_mul_div. - simpl ww_to_Z; autorewrite with rm10. - rewrite Hv3. - ring_simplify (Zpos (xO w_digits) - (Zpos (xO w_digits) - 1)). - rewrite Z.pow_1_r. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - apply Z.lt_le_trans with (1 := Hv4); auto with zarith. - unfold base; apply Zpower_le_monotone; auto with zarith. - split; unfold ww_digits; try rewrite Pos2Z.inj_xO; auto with zarith. - rewrite Hv3; auto with zarith. - assert (Hv6: [|low(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))|] - = [[ww_head1 x]]/2). - rewrite spec_low. - rewrite Hv5; rewrite Zmod_small; auto with zarith. - rewrite spec_w_add_mul_div; auto with zarith. - rewrite spec_w_sub; auto with zarith. - rewrite spec_w_0. - simpl ww_to_Z; autorewrite with rm10. - rewrite Hv6; rewrite spec_w_zdigits. - rewrite (fun x y => Zmod_small (x - y)). - ring_simplify (Zpos w_digits - (Zpos w_digits - [[ww_head1 x]] / 2)). - rewrite Zmod_small. - simpl ww_to_Z in H2; rewrite H2; auto with zarith. - intros (H4, H5); split. - apply Z.mul_le_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith. - rewrite H4. - apply Z.le_trans with ([|w2|] ^ 2); auto with zarith. - rewrite Z.mul_comm. - pattern [[ww_head1 x]] at 1; - rewrite Hv0; auto with zarith. - rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; - auto with zarith. - assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2); - try (intros; repeat rewrite Zsquare_mult; ring); - rewrite tmp; clear tmp. - apply Zpower_le_monotone3; auto with zarith. - split; auto with zarith. - pattern [|w2|] at 2; - rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]] / 2))); - auto with zarith. - match goal with |- ?X <= ?X + ?Y => - assert (0 <= Y); auto with zarith - end. - case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]] / 2))); auto with zarith. - case c; unfold interp_carry; autorewrite with rm10; - intros w3; assert (V3 := spec_to_Z w3);auto with zarith. - apply Z.mul_lt_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith. - rewrite H4. - apply Z.le_lt_trans with ([|w2|] ^ 2 + 2 * [|w2|]); auto with zarith. - apply Z.lt_le_trans with (([|w2|] + 1) ^ 2); auto with zarith. - match goal with |- ?X < ?Y => - replace Y with (X + 1); auto with zarith - end. - repeat rewrite (Zsquare_mult); ring. - rewrite Z.mul_comm. - pattern [[ww_head1 x]] at 1; rewrite Hv0. - rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; - auto with zarith. - assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2); - try (intros; repeat rewrite Zsquare_mult; ring); - rewrite tmp; clear tmp. - apply Zpower_le_monotone3; auto with zarith. - split; auto with zarith. - pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]]/2))); - auto with zarith. - rewrite <- Z.add_assoc; rewrite Z.mul_add_distr_l. - autorewrite with rm10; apply Z.add_le_mono_l; auto with zarith. - case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]]/2))); auto with zarith. - split; auto with zarith. - apply Z.le_lt_trans with ([|w2|]); auto with zarith. - apply Zdiv_le_upper_bound; auto with zarith. - pattern [|w2|] at 1; replace [|w2|] with ([|w2|] * 2 ^0); - auto with zarith. - apply Z.mul_le_mono_nonneg_l; auto with zarith. - apply Zpower_le_monotone; auto with zarith. - rewrite Z.pow_0_r; autorewrite with rm10; auto. - split; auto with zarith. - rewrite Hv0 in Hv2; rewrite (Pos2Z.inj_xO w_digits) in Hv2; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - rewrite spec_w_sub; auto with zarith. - rewrite Hv6; rewrite spec_w_zdigits; auto with zarith. - assert (Hv7: 0 < [[ww_head1 x]]/2); auto with zarith. - rewrite Zmod_small; auto with zarith. - split; auto with zarith. - assert ([[ww_head1 x]]/2 <= Zpos w_digits); auto with zarith. - apply Z.mul_le_mono_pos_r with 2; auto with zarith. - repeat rewrite (fun x => Z.mul_comm x 2). - rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto with zarith. - apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. - unfold base; apply Zpower2_lt_lin; auto with zarith. - Qed. - -End DoubleSqrt. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v deleted file mode 100644 index a2df2600..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v +++ /dev/null @@ -1,356 +0,0 @@ - -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Require Import BigNumPrelude. -Require Import DoubleType. -Require Import DoubleBase. - -Local Open Scope Z_scope. - -Section DoubleSub. - Variable w : Type. - Variable w_0 : w. - Variable w_Bm1 : w. - Variable w_WW : w -> w -> zn2z w. - Variable ww_Bm1 : zn2z w. - Variable w_opp_c : w -> carry w. - Variable w_opp_carry : w -> w. - Variable w_pred_c : w -> carry w. - Variable w_sub_c : w -> w -> carry w. - Variable w_sub_carry_c : w -> w -> carry w. - Variable w_opp : w -> w. - Variable w_pred : w -> w. - Variable w_sub : w -> w -> w. - Variable w_sub_carry : w -> w -> w. - - (* ** Opposites ** *) - Definition ww_opp_c x := - match x with - | W0 => C0 W0 - | WW xh xl => - match w_opp_c xl with - | C0 _ => - match w_opp_c xh with - | C0 h => C0 W0 - | C1 h => C1 (WW h w_0) - end - | C1 l => C1 (WW (w_opp_carry xh) l) - end - end. - - Definition ww_opp x := - match x with - | W0 => W0 - | WW xh xl => - match w_opp_c xl with - | C0 _ => WW (w_opp xh) w_0 - | C1 l => WW (w_opp_carry xh) l - end - end. - - Definition ww_opp_carry x := - match x with - | W0 => ww_Bm1 - | WW xh xl => w_WW (w_opp_carry xh) (w_opp_carry xl) - end. - - Definition ww_pred_c x := - match x with - | W0 => C1 ww_Bm1 - | WW xh xl => - match w_pred_c xl with - | C0 l => C0 (w_WW xh l) - | 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 - | WW xh xl => - match w_pred_c xl with - | C0 l => w_WW xh l - | 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 => - match w_sub_c xh yh with - | C0 h => C0 (w_WW h l) - | C1 h => C1 (WW h l) - end - | C1 l => - match w_sub_carry_c xh yh with - | C0 h => C0 (WW h l) - | C1 h => C1 (WW h l) - end - end - end. - - Definition ww_sub x y := - match y, x with - | W0, _ => x - | WW yh yl, W0 => ww_opp (WW yh yl) - | WW yh yl, WW xh xl => - match w_sub_c xl yl with - | C0 l => w_WW (w_sub xh yh) l - | C1 l => WW (w_sub_carry xh yh) l - end - end. - - Definition ww_sub_carry_c x y := - match y, x with - | W0, W0 => C1 ww_Bm1 - | W0, WW xh xl => ww_pred_c (WW xh xl) - | 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 => - match w_sub_c xh yh with - | C0 h => C0 (w_WW h l) - | C1 h => C1 (WW h l) - end - | C1 l => - match w_sub_carry_c xh yh with - | C0 h => C0 (w_WW h l) - | C1 h => C1 (w_WW h l) - end - end - end. - - Definition ww_sub_carry x y := - match y, x with - | W0, W0 => ww_Bm1 - | W0, WW xh xl => ww_pred (WW xh xl) - | WW yh yl, W0 => ww_opp_carry (WW yh yl) - | WW yh yl, WW xh xl => - match w_sub_carry_c xl yl with - | C0 l => w_WW (w_sub xh yh) l - | C1 l => w_WW (w_sub_carry xh yh) l - end - end. - - (*Section DoubleProof.*) - Variable w_digits : positive. - Variable w_to_Z : w -> Z. - - - Notation wB := (base w_digits). - Notation wwB := (base (ww_digits w_digits)). - Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99). - Notation "[+| c |]" := - (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99). - Notation "[-| c |]" := - (interp_carry (-1) wB w_to_Z c) (at level 0, c 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) - (at level 0, c at level 99). - Notation "[-[ c ]]" := - (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c) - (at level 0, c 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. - Variable spec_to_Z : forall x, 0 <= [|x|] < wB. - Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. - - Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. - Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB. - Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1. - - Variable spec_pred_c : forall x, [-|w_pred_c x|] = [|x|] - 1. - 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 : - forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB. - - - Lemma spec_ww_opp_c : forall x, [-[ww_opp_c x]] = -[[x]]. - Proof. - destruct x as [ |xh xl];simpl. reflexivity. - rewrite Z.opp_add_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) - as [l|l];intros H;unfold interp_carry in H;rewrite <- H; - rewrite <- Z.mul_opp_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) - 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. - rewrite H2;reflexivity. - simpl ww_to_Z;rewrite wwB_wBwB;rewrite spec_w_0;ring. - unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB;rewrite spec_opp_carry; - ring. - Qed. - - Lemma spec_ww_opp : forall x, [[ww_opp x]] = (-[[x]]) mod wwB. - Proof. - destruct x as [ |xh xl];simpl. reflexivity. - rewrite Z.opp_add_distr, <- Z.mul_opp_l. - 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 Z.add_0_r;rewrite wwB_wBwB. - assert ([|l|] = 0). - assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;rewrite Z.add_0_r; rewrite Z.pow_2_r; - rewrite Zmult_mod_distr_r;try apply lt_0_wB. - rewrite spec_opp;trivial. - apply Zmod_unique with (q:= -1). - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW (w_opp_carry xh) l)). - rewrite spec_opp_carry;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_opp_carry : forall x, [[ww_opp_carry x]] = wwB - [[x]] - 1. - Proof. - destruct x as [ |xh xl];simpl. rewrite spec_ww_Bm1;ring. - rewrite spec_w_WW;simpl;repeat rewrite spec_opp_carry;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_pred_c : forall x, [-[ww_pred_c x]] = [[x]] - 1. - Proof. - destruct x as [ |xh xl];unfold ww_pred_c. - unfold interp_carry;rewrite spec_ww_Bm1;simpl ww_to_Z;ring. - simpl ww_to_Z;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]; - intros H;unfold interp_carry in H;rewrite <- H. simpl;apply spec_w_WW. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - assert ([|l|] = wB - 1). - 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. - simpl;rewrite spec_w_Bm1;ring. - assert ([|h|] = wB - 1). - assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega. - rewrite H2;unfold interp_carry;rewrite spec_ww_Bm1;rewrite wwB_wBwB;ring. - Qed. - - Lemma spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]]. - 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|])) - 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 Z.add_assoc;rewrite <- Z.mul_add_distr_r. - 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; - try rewrite wwB_wBwB;ring. - Qed. - - 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. - unfold Z.sub;simpl;rewrite Z.add_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) - 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. - 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 Z.add_assoc;rewrite <- Z.mul_add_distr_r. - 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. - - Lemma spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. - Proof. - 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. - exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh l)). - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - change ([|xh|] + -1) with ([|xh|] - 1). - assert ([|l|] = wB - 1). - assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega. - rewrite (mod_wwB w_digits w_to_Z);trivial. - rewrite spec_pred;rewrite spec_w_Bm1;rewrite <- H0;trivial. - Qed. - - Lemma spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. - Proof. - 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|])) - 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. - rewrite spec_w_WW;rewrite (mod_wwB w_digits w_to_Z spec_to_Z). - rewrite spec_sub;trivial. - simpl ww_to_Z;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial. - Qed. - - Lemma spec_ww_sub_carry : - forall x y, [[ww_sub_carry x y]] = ([[x]] - [[y]] - 1) mod wwB. - Proof. - destruct y as [ |yh yl];simpl. - ring_simplify ([[x]] - 0);exact (spec_ww_pred x). - destruct x as [ |xh xl];simpl. - apply Zmod_unique with (-1). - 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) - 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. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub;trivial. - rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. - rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial. - Qed. - -(* End DoubleProof. *) - -End DoubleSub. - - - - - diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v deleted file mode 100644 index abd567a8..00000000 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v +++ /dev/null @@ -1,70 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *) -(************************************************************************) - -Set Implicit Arguments. - -Require Import ZArith. -Local Open Scope Z_scope. - -Definition base digits := Z.pow 2 (Zpos digits). -Arguments base digits: simpl never. - -Section Carry. - - Variable A : Type. - - Inductive carry := - | C0 : A -> carry - | C1 : A -> carry. - - Definition interp_carry (sign:Z)(B:Z)(interp:A -> Z) c := - match c with - | C0 x => interp x - | C1 x => sign*B + interp x - end. - -End Carry. - -Section Zn2Z. - - Variable znz : Type. - - (** From a type [znz] representing a cyclic structure Z/nZ, - we produce a representation of Z/2nZ by pairs of elements of [znz] - (plus a special case for zero). High half of the new number comes - first. - *) - - Inductive zn2z := - | W0 : zn2z - | WW : znz -> znz -> zn2z. - - Definition zn2z_to_Z (wB:Z) (w_to_Z:znz->Z) (x:zn2z) := - match x with - | W0 => 0 - | WW xh xl => w_to_Z xh * wB + w_to_Z xl - end. - -End Zn2Z. - -Arguments W0 {znz}. - -(** From a cyclic representation [w], we iterate the [zn2z] construct - [n] times, gaining the type of binary trees of depth at most [n], - whose leafs are either W0 (if depth < n) or elements of w - (if depth = n). -*) - -Fixpoint word (w:Type) (n:nat) : Type := - match n with - | O => w - | S n => zn2z (word w n) - end. - |