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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* 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, 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.
|
