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(************************************************************************)
(* 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.
|
