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+(************************************************************************)
+(* v * The Coq Proof Assistant / The Coq Development Team *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(* \VV/ **************************************************************)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(************************************************************************)
+(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *)
+(************************************************************************)
+
+(*i $Id: DoubleSub.v 10964 2008-05-22 11:08:13Z letouzey $ i*)
+
+Set Implicit Arguments.
+
+Require Import ZArith.
+Require Import BigNumPrelude.
+Require Import DoubleType.
+Require Import DoubleBase.
+
+Open Local 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, x at level 99).
+ Notation "[-| c |]" :=
+ (interp_carry (-1) wB w_to_Z c) (at level 0, x at level 99).
+
+ Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
+ Notation "[+[ c ]]" :=
+ (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c)
+ (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, x at level 99).
+
+ Variable spec_w_0 : [|w_0|] = 0.
+ Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1.
+ Variable spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1.
+ 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 Zopp_plus_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl)
+ as [l|l];intros H;unfold interp_carry in H;rewrite <- H;
+ rewrite Zopp_mult_distr_l.
+ 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 Zopp_plus_distr;rewrite Zopp_mult_distr_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 Zplus_0_r;rewrite wwB_wBwB.
+ assert ([|l|] = 0).
+ assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega.
+ rewrite H0;rewrite Zplus_0_r; rewrite Zpower_2;
+ 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 Zplus_assoc;rewrite <- Zmult_plus_distr_l.
+ 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 Zplus_assoc;rewrite <- Zmult_plus_distr_l.
+ change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1).
+ generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h];
+ intros H1;unfold interp_carry in *;rewrite <- H1;simpl ww_to_Z;
+ 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 Zminus;simpl;rewrite Zplus_0_r;exact (spec_ww_pred_c x).
+ destruct x as [ |xh xl].
+ unfold interp_carry;rewrite spec_w_WW;simpl ww_to_Z;rewrite wwB_wBwB;
+ repeat rewrite spec_opp_carry;ring.
+ simpl ww_to_Z.
+ replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1)
+ 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 Zplus_assoc;rewrite <- Zmult_plus_distr_l.
+ change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1).
+ generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h];
+ intros H1;unfold interp_carry in *;rewrite <- H1;try rewrite spec_w_WW;
+ simpl ww_to_Z; try rewrite wwB_wBwB;ring.
+ Qed.
+
+ 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 Zplus_assoc;rewrite <- Zmult_plus_distr_l.
+ change ([|xh|] + -1) with ([|xh|] - 1).
+ assert ([|l|] = wB - 1).
+ assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega.
+ 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 Zplus_assoc;rewrite <- Zmult_plus_distr_l.
+ 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 Zplus_assoc;rewrite <- Zmult_plus_distr_l.
+ rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial.
+ Qed.
+
+(* End DoubleProof. *)
+
+End DoubleSub.
+
+
+
+
+
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