aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Numbers/BigNumPrelude.v
blob: bd8930872c7c362e7d9a6982c94862a660b7cd4b (plain)
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
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
(************************************************************************)
(*  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             *)
(************************************************************************)

(** * BigNumPrelude *)

(** Auxiliary functions & theorems used for arbitrary precision efficient
    numbers. *)


Require Import ArithRing.
Require Export ZArith.
Require Export Znumtheory.
Require Export Zpow_facts.

Declare ML Module "numbers_syntax_plugin".

(* *** Nota Bene ***
   All results that were general enough have been moved in ZArith.
   Only remain here specialized lemmas and compatibility elements.
   (P.L. 5/11/2007).
*)


Local Open Scope Z_scope.

(* For compatibility of scripts, weaker version of some lemmas of Z.div *)

Lemma Zlt0_not_eq : forall n, 0<n -> n<>0.
Proof.
 auto with zarith.
Qed.

Definition Zdiv_mult_cancel_r a b c H := Zdiv.Zdiv_mult_cancel_r a b c (Zlt0_not_eq _ H).
Definition Zdiv_mult_cancel_l a b c H := Zdiv.Zdiv_mult_cancel_r a b c (Zlt0_not_eq _ H).
Definition Z_div_plus_l a b c H := Zdiv.Z_div_plus_full_l a b c (Zlt0_not_eq _ H).

(* Automation *)

Hint  Extern 2 (Z.le _ _) =>
 (match goal with
   |- Zpos _ <= Zpos _ => exact (eq_refl _)
|   H: _ <=  ?p |- _ <= ?p => apply Z.le_trans with (2 := H)
|   H: _ <  ?p |- _ <= ?p => apply Z.lt_le_incl; apply Z.le_lt_trans with (2 := H)
  end).

Hint  Extern 2 (Z.lt _ _) =>
 (match goal with
   |- Zpos _ < Zpos _ => exact (eq_refl _)
|      H: _ <=  ?p |- _ <= ?p => apply Z.lt_le_trans with (2 := H)
|   H: _ <  ?p |- _ <= ?p => apply Z.le_lt_trans with (2 := H)
  end).


Hint Resolve Z.lt_gt Z.le_ge Z_div_pos: zarith.

(**************************************
   Properties of order and product
 **************************************)

 Theorem beta_lex: forall a b c d beta,
       a * beta + b <= c * beta + d ->
       0 <= b < beta -> 0 <= d < beta ->
       a <= c.
 Proof.
  intros a b c d beta H1 (H3, H4) (H5, H6).
  assert (a - c < 1); auto with zarith.
  apply Z.mul_lt_mono_pos_r with beta; auto with zarith.
  apply Z.le_lt_trans with (d  - b); auto with zarith.
  rewrite Z.mul_sub_distr_r; auto with zarith.
 Qed.

 Theorem beta_lex_inv: forall a b c d beta,
      a < c -> 0 <= b < beta ->
      0 <= d < beta ->
      a * beta + b < c * beta  + d.
 Proof.
  intros a b c d beta H1 (H3, H4) (H5, H6).
  case (Z.le_gt_cases (c * beta + d) (a * beta + b)); auto with zarith.
  intros H7. contradict H1. apply Z.le_ngt. apply beta_lex with (1 := H7); auto.
 Qed.

 Lemma beta_mult : forall h l beta,
   0 <= h < beta -> 0 <= l < beta -> 0 <= h*beta+l < beta^2.
 Proof.
  intros h l beta H1 H2;split. auto with zarith.
  rewrite <- (Z.add_0_r (beta^2)); rewrite Z.pow_2_r;
   apply beta_lex_inv;auto with zarith.
 Qed.

 Lemma Zmult_lt_b :
   forall b x y, 0 <= x < b -> 0 <= y < b -> 0 <= x * y <= b^2 - 2*b + 1.
 Proof.
  intros b x y (Hx1,Hx2) (Hy1,Hy2);split;auto with zarith.
  apply Z.le_trans with ((b-1)*(b-1)).
  apply Z.mul_le_mono_nonneg;auto with zarith.
  apply Z.eq_le_incl; ring.
 Qed.

 Lemma sum_mul_carry : forall xh xl yh yl wc cc beta,
   1 < beta ->
   0 <= wc < beta ->
   0 <= xh < beta ->
   0 <= xl < beta ->
   0 <= yh < beta ->
   0 <= yl < beta ->
   0 <= cc < beta^2 ->
   wc*beta^2 + cc = xh*yl + xl*yh ->
   0 <= wc <= 1.
 Proof.
  intros xh xl yh yl wc cc beta U H1 H2 H3 H4 H5 H6 H7.
  assert (H8 := Zmult_lt_b beta xh yl H2 H5).
  assert (H9 := Zmult_lt_b beta xl yh H3 H4).
  split;auto with zarith.
  apply beta_lex with (cc) (beta^2 - 2) (beta^2); auto with zarith.
 Qed.

 Theorem mult_add_ineq: forall x y cross beta,
   0 <= x < beta ->
   0 <= y < beta ->
   0 <= cross < beta ->
   0 <= x * y + cross < beta^2.
 Proof.
  intros x y cross beta HH HH1 HH2.
  split; auto with zarith.
  apply Z.le_lt_trans with  ((beta-1)*(beta-1)+(beta-1)); auto with zarith.
  apply Z.add_le_mono; auto with zarith.
  apply Z.mul_le_mono_nonneg; auto with zarith.
  rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r, Z.pow_2_r; auto with zarith.
 Qed.

 Theorem mult_add_ineq2: forall x y c cross beta,
   0 <= x < beta ->
   0 <= y < beta ->
   0 <= c*beta + cross <= 2*beta - 2 ->
   0 <= x * y + (c*beta + cross) < beta^2.
 Proof.
  intros x y c cross beta HH HH1 HH2.
  split; auto with zarith.
  apply Z.le_lt_trans with ((beta-1)*(beta-1)+(2*beta-2));auto with zarith.
  apply Z.add_le_mono; auto with zarith.
  apply Z.mul_le_mono_nonneg; auto with zarith.
  rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r, Z.pow_2_r; auto with zarith.
 Qed.

Theorem mult_add_ineq3: forall x y c cross beta,
   0 <= x < beta ->
   0 <= y < beta ->
   0 <= cross <= beta - 2 ->
   0 <= c <= 1 ->
   0 <= x * y + (c*beta + cross) < beta^2.
 Proof.
  intros x y c cross beta HH HH1 HH2 HH3.
  apply mult_add_ineq2;auto with zarith.
  split;auto with zarith.
  apply Z.le_trans with (1*beta+cross);auto with zarith.
 Qed.

Hint Rewrite Z.mul_1_r Z.mul_0_r Z.mul_1_l Z.mul_0_l Z.add_0_l Z.add_0_r Z.sub_0_r: rm10.


(**************************************
 Properties of Z.div and Z.modulo
**************************************)

Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a.
 Proof.
  intros a b H H1;case (Z_mod_lt a b);auto with zarith;intros H2 H3;split;auto.
  case (Z.le_gt_cases b a); intros H4; auto with zarith.
  rewrite Zmod_small; auto with zarith.
 Qed.


 Theorem Zmod_distr: forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a ->
  (2 ^a * r + t) mod (2 ^ b) = (2 ^a * r) mod (2 ^ b) + t.
 Proof.
  intros a b r t (H1, H2) H3 (H4, H5).
  assert (t < 2 ^ b).
  apply Z.lt_le_trans with (1:= H5); auto with zarith.
  apply Zpower_le_monotone; auto with zarith.
  rewrite Zplus_mod; auto with zarith.
  rewrite Zmod_small with (a := t); auto with zarith.
  apply Zmod_small; auto with zarith.
  split; auto with zarith.
  assert (0 <= 2 ^a * r); auto with zarith.
  apply Z.add_nonneg_nonneg; auto with zarith.
  match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
   auto with zarith.
  pattern (2 ^ b) at 2; replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a);
    try ring.
  apply Z.add_le_lt_mono; auto with zarith.
  replace b with ((b - a) + a); try ring.
  rewrite Zpower_exp; auto with zarith.
  pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a));
   try rewrite <- Z.mul_sub_distr_r.
  rewrite (Z.mul_comm (2 ^(b - a))); rewrite  Zmult_mod_distr_l;
   auto with zarith.
  rewrite (Z.mul_comm (2 ^a)); apply Z.mul_le_mono_nonneg_r; auto with zarith.
  match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
   auto with zarith.
 Qed.

 Theorem Zmod_shift_r:
   forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a ->
     (r * 2 ^a + t) mod (2 ^ b) = (r * 2 ^a) mod (2 ^ b) + t.
 Proof.
  intros a b r t (H1, H2) H3 (H4, H5).
  assert (t < 2 ^ b).
  apply Z.lt_le_trans with (1:= H5); auto with zarith.
  apply Zpower_le_monotone; auto with zarith.
  rewrite Zplus_mod; auto with zarith.
  rewrite Zmod_small with (a := t); auto with zarith.
  apply Zmod_small; auto with zarith.
  split; auto with zarith.
  assert (0 <= 2 ^a * r); auto with zarith.
  apply Z.add_nonneg_nonneg; auto with zarith.
  match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
   auto with zarith.
  pattern (2 ^ b) at 2;replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); try ring.
  apply Z.add_le_lt_mono; auto with zarith.
  replace b with ((b - a) + a); try ring.
  rewrite Zpower_exp; auto with zarith.
  pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a));
   try rewrite <- Z.mul_sub_distr_r.
  repeat rewrite (fun x => Z.mul_comm x (2 ^ a)); rewrite Zmult_mod_distr_l;
   auto with zarith.
  apply Z.mul_le_mono_nonneg_l; auto with zarith.
  match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
   auto with zarith.
 Qed.

  Theorem Zdiv_shift_r:
    forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a ->
      (r * 2 ^a + t) /  (2 ^ b) = (r * 2 ^a) / (2 ^ b).
  Proof.
   intros a b r t (H1, H2) H3 (H4, H5).
   assert (Eq: t < 2 ^ b); auto with zarith.
   apply Z.lt_le_trans with (1 := H5); auto with zarith.
   apply Zpower_le_monotone; auto with zarith.
   pattern (r * 2 ^ a) at 1; rewrite Z_div_mod_eq with (b := 2 ^ b);
    auto with zarith.
   rewrite <- Z.add_assoc.
   rewrite <- Zmod_shift_r; auto with zarith.
   rewrite (Z.mul_comm (2 ^ b)); rewrite Z_div_plus_full_l; auto with zarith.
   rewrite (fun x y => @Zdiv_small (x mod y)); auto with zarith.
   match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end;
    auto with zarith.
  Qed.


 Lemma shift_unshift_mod : forall n p a,
     0 <= a < 2^n ->
     0 <= p <= n ->
     a * 2^p = a / 2^(n - p) * 2^n + (a*2^p) mod 2^n.
 Proof.
  intros n p a H1 H2.
  pattern (a*2^p) at 1;replace (a*2^p) with
     (a*2^p/2^n * 2^n + a*2^p mod 2^n).
  2:symmetry;rewrite (Z.mul_comm (a*2^p/2^n));apply Z_div_mod_eq.
  replace (a * 2 ^ p / 2 ^ n) with (a / 2 ^ (n - p));trivial.
  replace (2^n) with (2^(n-p)*2^p).
  symmetry;apply Zdiv_mult_cancel_r.
  destruct H1;trivial.
  cut (0 < 2^p); auto with zarith.
  rewrite <- Zpower_exp.
  replace (n-p+p) with n;trivial. ring.
  omega. omega.
  apply Z.lt_gt. apply Z.pow_pos_nonneg;auto with zarith.
 Qed.


 Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n ->
   ((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) =
   a mod 2 ^ p.
 Proof.
 intros.
 rewrite Zmod_small.
 rewrite Zmod_eq by (auto with zarith).
 unfold Z.sub at 1.
 rewrite Z_div_plus_l by (auto with zarith).
 assert (2^n = 2^(n-p)*2^p).
  rewrite <- Zpower_exp by (auto with zarith).
  replace (n-p+p) with n; auto with zarith.
 rewrite H0.
 rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith).
 rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc.
 rewrite <- Z.mul_opp_l.
 rewrite Z_div_mult by (auto with zarith).
 symmetry; apply Zmod_eq; auto with zarith.

 remember (a * 2 ^ (n - p)) as b.
 destruct (Z_mod_lt b (2^n)); auto with zarith.
 split.
 apply Z_div_pos; auto with zarith.
 apply Zdiv_lt_upper_bound; auto with zarith.
 apply Z.lt_le_trans with (2^n); auto with zarith.
 rewrite <- (Z.mul_1_r (2^n)) at 1.
 apply Z.mul_le_mono_nonneg; auto with zarith.
 cut (0 < 2 ^ (n-p)); auto with zarith.
 Qed.

 Lemma div_le_0 : forall p x, 0 <= x -> 0 <= x / 2 ^ p.
 Proof.
  intros p x Hle;destruct (Z_le_gt_dec 0 p).
  apply  Zdiv_le_lower_bound;auto with zarith.
  replace (2^p) with 0.
  destruct x;compute;intro;discriminate.
  destruct p;trivial;discriminate.
 Qed.

 Lemma div_lt : forall p x y, 0 <= x < y -> x / 2^p < y.
 Proof.
  intros p x y H;destruct (Z_le_gt_dec 0 p).
  apply Zdiv_lt_upper_bound;auto with zarith.
  apply Z.lt_le_trans with y;auto with zarith.
  rewrite <- (Z.mul_1_r y);apply Z.mul_le_mono_nonneg;auto with zarith.
  assert (0 < 2^p);auto with zarith.
  replace (2^p) with 0.
  destruct x;change (0<y);auto with zarith.
  destruct p;trivial;discriminate.
 Qed.

 Theorem Zgcd_div_pos a b:
   0 < b -> 0 < Z.gcd a b -> 0 < b / Z.gcd a b.
 Proof.
 intros Hb Hg.
 assert (H : 0 <= b / Z.gcd a b) by (apply Z.div_pos; auto with zarith).
 Z.le_elim H; trivial.
 rewrite (Zdivide_Zdiv_eq (Z.gcd a b) b), <- H, Z.mul_0_r in Hb;
  auto using Z.gcd_divide_r with zarith.
 Qed.

 Theorem Zdiv_neg a b:
   a < 0 -> 0 < b -> a / b < 0.
 Proof.
 intros Ha Hb.
 assert (b > 0) by omega.
 generalize (Z_mult_div_ge a _ H); intros.
 assert (b * (a / b) < 0)%Z.
  apply Z.le_lt_trans with a; auto with zarith.
 destruct b; try (compute in Hb; discriminate).
 destruct (a/Zpos p)%Z.
 compute in H1; discriminate.
 compute in H1; discriminate.
 compute; auto.
 Qed.

 Lemma Zdiv_gcd_zero : forall a b, b / Z.gcd a b = 0 -> b <> 0 ->
  Z.gcd a b = 0.
 Proof.
 intros.
 generalize (Zgcd_is_gcd a b); destruct 1.
 destruct H2 as (k,Hk).
 generalize H; rewrite Hk at 1.
 destruct (Z.eq_dec (Z.gcd a b) 0) as [H'|H']; auto.
 rewrite Z_div_mult_full; auto.
 intros; subst k; simpl in *; subst b; elim H0; auto.
 Qed.

 Lemma Zgcd_mult_rel_prime : forall a b c,
  Z.gcd a c = 1 -> Z.gcd b c = 1 -> Z.gcd (a*b) c = 1.
 Proof.
 intros.
 rewrite Zgcd_1_rel_prime in *.
 apply rel_prime_sym; apply rel_prime_mult; apply rel_prime_sym; auto.
 Qed.

 Lemma Zcompare_gt : forall (A:Type)(a a':A)(p q:Z),
  match (p?=q)%Z with Gt => a | _ => a' end =
  if Z_le_gt_dec p q then a' else a.
 Proof.
 intros.
 destruct Z_le_gt_dec as [H|H].
 red in H.
 destruct (p?=q)%Z; auto; elim H; auto.
 rewrite H; auto.
 Qed.

Theorem Zbounded_induction :
  (forall Q : Z -> Prop, forall b : Z,
    Q 0 ->
    (forall n, 0 <= n -> n < b - 1 -> Q n -> Q (n + 1)) ->
      forall n, 0 <= n -> n < b -> Q n)%Z.
Proof.
intros Q b Q0 QS.
set (Q' := fun n => (n < b /\ Q n) \/ (b <= n)).
assert (H : forall n, 0 <= n -> Q' n).
apply natlike_rec2; unfold Q'.
destruct (Z.le_gt_cases b 0) as [H | H]. now right. left; now split.
intros n H IH. destruct IH as [[IH1 IH2] | IH].
destruct (Z.le_gt_cases (b - 1) n) as [H1 | H1].
right; auto with zarith.
left. split; [auto with zarith | now apply (QS n)].
right; auto with zarith.
unfold Q' in *; intros n H1 H2. destruct (H n H1) as [[H3 H4] | H3].
assumption. now apply Z.le_ngt in H3.
Qed.

Lemma Zsquare_le x : x <= x*x.
Proof.
destruct (Z.lt_ge_cases 0 x).
- rewrite <- Z.mul_1_l at 1.
  rewrite <- Z.mul_le_mono_pos_r; auto with zarith.
- pose proof (Z.square_nonneg x); auto with zarith.
Qed.