1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
|
(************************************************************************)
(* 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.
|
