-
Notifications
You must be signed in to change notification settings - Fork 0
/
euler_sieve.mlw
971 lines (868 loc) · 42.1 KB
/
euler_sieve.mlw
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
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
(** Euler's Sieve
This is a variant of Eratosthenes's sieve with complexity O(N),
where N is the upper limit of the sieve.
Cf https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes#Euler's_sieve
Euler's Sieve makes use of a linked list of all integers,
and marks composite numbers as in Eratosthenes's sieve.
In the program below, the linked list is implemented with an array
(where each integer is mapped to the next one) and negative values
are used to represent marked integers.
In addition, the list only contains the odd integers (to save half
of space).
Author: Josué Moreau (Université Paris-Saclay)
*)
module ArithmeticResults
use int.Int
use number.Divisibility
use number.Prime
use int.EuclideanDivision
let lemma mult_croissance_locale (n a: int)
requires { n > 0 /\ 0 <= a }
ensures { n * a < n * (a + 1) } = ()
let rec lemma mult_croissance (n a b: int)
requires { n > 0 /\ 0 <= a < b }
ensures { n * a < n * b }
variant { b - a }
= if a + 1 = b then ()
else mult_croissance n (a + 1) b
let lemma comp_mult_2 (n k: int)
requires { n > 0 /\ k >= 2 }
ensures { n * k >= n * 2 } = ()
let lemma div_croissance_locale1 (i n: int)
requires { 0 <= i /\ n > 0 }
ensures { div i n <= div (i + 1) n } = ()
let rec lemma div_croissance1 (i j n: int)
requires { 0 <= i <= j /\ n > 0 }
ensures { div i n <= div j n }
variant { j - i }
= if i < j then
div_croissance1 i (j - 1) n
let rec lemma div_croissance_locale2 (m i: int)
requires { i > 0 /\ m >= 0 }
ensures { div m (i + 1) <= div m i }
variant { m }
= if m > 0 then div_croissance_locale2 (m - 1) i
let rec lemma div_croissance2 (m i j: int)
requires { 0 < i <= j /\ m >= 0 }
ensures { div m i >= div m j }
variant { j - i }
= if i < j then div_croissance2 m i (j - 1)
let lemma div_mult_1 (n k max: int)
requires { n > 0 /\ max >= n /\ n * k <= max }
ensures { k = div (n * k) n <= div max n } = ()
let lemma mult_borne_sous_exp (n a b: int)
requires { a >= 1 /\ b >= 1 /\ n >= 1 /\ a * b < n }
ensures { a < n /\ b < n } = ()
let rec lemma sq_ineq (a b: int)
requires { a >= 0 /\ b >= 0 }
requires { a * a < b * b }
ensures { a < b }
variant { b * b - a * a }
= if (b - 1) * (b - 1) > a * a then
sq_ineq a (b - 1)
end
module DivisibilityResults
use int.Int
use int.EuclideanDivision
use number.Divisibility
use number.Prime
let lemma divides_div (n m k: int)
requires { 2 <= n /\ 2 <= m /\ 2 <= k < n /\ divides n m /\ not divides k m }
ensures { not divides k (div m n) } = ()
let lemma divides_inf (n m: int)
requires { 2 <= n /\ 2 <= m /\ divides n m /\
forall k. 2 <= k < n -> not divides k m }
ensures { forall k. 2 <= k < n -> not divides k (div m n) } = ()
let lemma not_prime_divider_limits (n: int)
requires { not prime n /\ 2 <= n }
ensures { exists i. 2 <= i /\ i * i <= n /\ divides i n } = ()
let lemma no_prod_impl_no_divider (n: int)
requires { n >= 0 }
ensures { forall i.
2 <= i < n * n ->
not (exists k l. 2 <= k < n /\ 2 <= l < i /\ k * l = i ) ->
not (exists k. 2 <= k < n /\ k <> i /\ divides k i) } = ()
use ArithmeticResults
let lemma not_prime_impl_divisor_under_sqrt (n: int)
requires { n >= 2 }
ensures { forall i.
2 <= i < n * n ->
not prime i ->
exists j. 2 <= j < i /\ ((j < n)
by j * j < n * n /\ j >= 0) /\ divides j i } = ()
end
module EulerSieveSpec
use int.Int
use number.Divisibility
use number.Prime
use seq.Seq
use int.EuclideanDivision
use ArithmeticResults
use DivisibilityResults
(*******************************************************************************)
(* *)
(* INVARIANTS SUR LES RELATIONS ENTRE STRUCTURES DE DONNEES *)
(* *)
(*******************************************************************************)
(* Borne sur le suivant de chaque élément de la liste chaînée *)
predicate inv_nexts (nexts: seq int) (n: int) =
forall i. 0 <= i < n ->
i < nexts[i] <= n
(* Tous les éléments éliminés de la liste chaînée sont marqués *)
predicate all_eliminated_marked (marked: seq bool)
(nexts: seq int) =
marked.length = nexts.length /\
forall i. 0 <= i < marked.length ->
forall j. i < j < nexts[i] ->
marked[j]
(* Tous les éléments éliminés, à partir d'un certain indice, sont marqués *)
predicate all_eliminated_marked_partial (marked: seq bool)
(nexts: seq int)
(min: int) =
marked.length = nexts.length /\
forall i. min <= i < marked.length ->
forall j. i < j < nexts[i] ->
marked[j]
(* L'élément suivant d'un élément non marqué inférieur à max / n est
non marqué lorsqu'il est lui-même inférieur à max / n *)
predicate not_marked_impl_next_not_marked (marked_old: seq bool)
(nexts: seq int)
(n: int) =
marked_old.length = nexts.length /\
marked_old.length >= 2 /\
n >= 2 /\
forall i. 2 <= i <= div (marked_old.length - 1) n ->
nexts[i] <= div (marked_old.length - 1) n ->
not marked_old[i] ->
not marked_old[nexts[i]]
(* Les deux tableaux marked sont identiques *)
predicate is_copy (marked: seq bool)
(marked_old: seq bool) =
marked.length = marked_old.length /\
forall i. 0 <= i < marked.length ->
not marked[i] ->
not marked_old[i]
(* L'élément suivant d'un élément non marqué inférieur à p <= max / n est
non marqué lorsqu'il est inférieur à max / n *)
predicate not_marked_impl_next_not_marked_partial (marked: seq bool)
(nexts: seq int)
(n: int)
(p: int) =
marked.length = nexts.length /\
marked.length >= 2 /\
n >= 2 /\
p <= div (marked.length - 1) n /\
forall i. 2 <= i < p ->
nexts[i] <= div (marked.length - 1) n ->
not marked[i] ->
not marked[nexts[i]]
(*******************************************************************************)
(* *)
(* PROPRIETES LIEES AUX NOMBRES PREMIERS *)
(* *)
(*******************************************************************************)
(* Tous les nombres jusqu'à n sont non marqués si et seulement si ils sont
premiers *)
predicate all_primes (marked: seq bool) (n: int) =
forall i. 0 <= i < n -> not marked[i] <-> prime i
(* Tous les multiples de i sont marqués *)
predicate all_multiples_marked (marked: seq bool) (i max: int) =
2 <= i < marked.length /\
forall k. 2 <= k < max ->
i * k < marked.length ->
marked[i * k]
(* Tous les multiples des 2 <= i < n sont marqués *)
predicate previously_marked_multiples (marked: seq bool) (n: int) =
forall i. 2 <= i < n ->
all_multiples_marked marked i marked.length
(* Les multiples des 2 <= i < n sont les uniques nombres marqués *)
predicate only_multiples_marked (marked: seq bool) (n: int) =
forall k. 2 <= k < marked.length ->
marked[k] ->
exists i j. 2 <= i < n /\ 2 <= j < marked.length /\ i * j = k
(* Les multiples de n avec les éléments non marqués précédement sont
marqués *)
predicate prime_multiples_marked (marked_old: seq bool)
(marked: seq bool)
(n max: int) =
marked_old.length = marked.length /\
n < max <= marked.length /\
forall i. n <= i < max ->
not marked_old[i] ->
n * i < marked_old.length ->
marked[n * i]
(* Invariants de base de remove_products *)
predicate inv_remove_products (nexts: seq int)
(marked: seq bool)
(n: int) =
nexts.length = marked.length /\
not marked[2] /\
all_primes marked n /\
prime n /\
not marked[n] /\
inv_nexts nexts nexts.length
(*******************************************************************************)
(* *)
(* QUELQUES LEMMES DE CONSERVATION DES STRUCTURES *)
(* *)
(*******************************************************************************)
let lemma conservation_all_eliminated_marked_on_marked_change
(marked: seq bool)
(nexts: seq int)
(i: int)
requires { marked.length = nexts.length }
requires { inv_nexts nexts nexts.length }
requires { all_eliminated_marked marked nexts }
requires { 0 <= i < marked.length }
ensures { all_eliminated_marked marked[i <- true] nexts } = ()
let lemma conservation_all_eliminated_marked_on_nexts_change
(marked: seq bool)
(nexts: seq int)
(i v: int)
requires { marked.length = nexts.length }
requires { all_eliminated_marked marked nexts }
requires { inv_nexts nexts marked.length }
requires { 0 <= i < marked.length }
requires { i < v <= marked.length }
requires { forall j. i < j < v -> marked[j] }
ensures { all_eliminated_marked marked nexts[i <- v] } = ()
(*******************************************************************************)
(* *)
(* QUELQUES PREDICATS UTILES *)
(* *)
(*******************************************************************************)
predicate ordered (a: seq int) (n: int) =
forall i j. 0 <= i < j < n -> a[i] < a[j]
predicate all_inf_or_eq (a: seq int) (n k: int) =
forall i. 0 <= i < n -> a[i] <= k
end
module EulerSieve
use int.Int
use number.Divisibility
use number.Prime
use seq.Seq
use int.EuclideanDivision
use ArithmeticResults
use DivisibilityResults
use EulerSieveSpec
(*******************************************************************************)
(* *)
(* PROPRIETES LIEES AUX NOMBRES PREMIERS *)
(* *)
(*******************************************************************************)
(* Si tous les multiples des nombres i < n sont marqués,
alors les mutliples de ces multiples sont marqués *)
let lemma multiples_of_marked_are_marked (marked: seq bool) (n: int)
requires { 2 <= n <= marked.length }
requires { previously_marked_multiples marked n }
ensures { forall k i.
2 <= k < n ->
2 <= i < marked.length ->
k * i < marked.length ->
forall j.
1 <= j < marked.length ->
k * i * j < marked.length ->
(marked[k * i * j]
by 1 <= i * j < marked.length
by k >= 2 /\ i >= 2 /\ j >= 1 /\ k * (i * j) < marked.length) }
= ()
(* Lemme essentiel : si tous les multiples des nombres i < n sont marqués
et qu'on a marqué tous les produits de n avec les nombres
non marqués
alors tous les multiples de n sont marqués *)
let lemma prev_and_new_impl_all_multiples_marked
(marked_old marked: seq bool)
(n max: int)
requires { 2 <= n < marked.length /\ 2 <= max < marked.length /\
marked_old.length = marked.length }
requires { is_copy marked marked_old }
requires { previously_marked_multiples marked_old n }
requires { previously_marked_multiples marked n }
requires { only_multiples_marked marked_old n }
requires { prime_multiples_marked marked_old marked n marked.length }
ensures { all_multiples_marked marked n marked.length } =
assert { forall i. 2 <= i < n /\ 2 <= i < marked.length ->
not marked_old[i] ->
i * n < marked.length ->
marked[i * n] };
assert { forall k. 2 <= k < marked_old.length ->
marked_old[k] ->
n * k < marked_old.length ->
exists i j. 2 <= i < n /\ 2 <= j < marked_old.length /\
i * j = k /\ marked_old[i * j] };
multiples_of_marked_are_marked marked_old n;
multiples_of_marked_are_marked marked n;
assert { marked.length = marked_old.length };
assert { forall k. 2 <= k < marked_old.length ->
marked_old[k] ->
n * k < marked_old.length ->
marked[k * n]
by forall k. 2 <= k < marked_old.length ->
marked_old[k] ->
n * k < marked_old.length ->
marked_old[k * n] };
assert { prime_multiples_marked marked_old marked n marked.length };
assert { (forall k. 2 <= k < marked.length ->
not marked_old[k] -> n * k < marked.length ->
marked[n * k]) }
(*******************************************************************************)
(* *)
(* LEMMES DE CONSERVATION LIES AUX NOMBRES PREMIERS *)
(* *)
(*******************************************************************************)
let lemma conservation_only_multiples_marked (marked: seq bool) (n i j: int)
requires { 2 <= i < n /\ 2 <= j < marked.length /\ i * j < marked.length }
requires { only_multiples_marked marked n }
ensures { only_multiples_marked marked[(i * j) <- true] n } =
assert { forall k. 0 <= k < marked.length ->
k <> i * j ->
marked[(i * j) <- true][k] = marked[k] };
assert { forall k.
2 <= k < marked.length ->
k <> i * j ->
marked[(i * j) <- true][k] ->
exists x y. 2 <= x < n /\ 2 <= y < marked.length /\ k = x * y };
assert { marked[i * j] -> 2 <= i < n -> 2 <= j < marked.length ->
exists x y. 2 <= x < n /\ 2 <= y < marked.length /\ x * y = i * j }
let lemma conservation_previously_marked_multiples (marked: seq bool)
(nexts: seq int)
(n: int)
requires { 2 <= n < marked.length /\ marked.length = nexts.length /\
nexts[n] <= marked.length }
requires { previously_marked_multiples marked n }
requires { only_multiples_marked marked (n + 1) }
requires { all_eliminated_marked marked nexts }
requires { not_marked_impl_next_not_marked marked nexts nexts[n] }
requires { all_multiples_marked marked n marked.length }
ensures { previously_marked_multiples marked nexts[n] } =
assert { previously_marked_multiples marked (n + 1) };
assert { forall i. n < i < nexts[n] -> marked[i] };
assert { forall i. n < i < nexts[n] ->
((exists k l. 2 <= k <= n /\ 2 <= l < marked.length /\ k * l = i)
by marked[i]) };
multiples_of_marked_are_marked marked (n + 1);
assert { forall i j.
n < i < nexts[n] ->
2 <= j < marked.length ->
i * j < marked.length ->
(marked[i * j]
by (exists k l. 2 <= k <= n /\ 2 <= l < marked.length /\ k * l = i)) };
assert { forall i.
n < i < nexts[n] ->
all_multiples_marked marked i marked.length }
lemma conservation_previously_marked_multiples_on_marked_change:
forall marked n.
previously_marked_multiples marked n ->
forall i. 0 <= i < marked.length ->
previously_marked_multiples marked[i <- true] n
(*******************************************************************************)
(* *)
(* INVARIANTS SIMPLES DE FONCTIONS *)
(* *)
(*******************************************************************************)
let lemma conservation_not_marked_impl_next_not_marked
(marked: seq bool)
(nexts: seq int)
(max n p: int)
requires { not_marked_impl_next_not_marked marked nexts n /\
nexts[n] > n > 0 /\
div max nexts[n] <= div max n
by forall i. p < i < nexts[p] -> marked[i] /\
not_marked_impl_next_not_marked_partial marked nexts n p }
ensures { not_marked_impl_next_not_marked marked nexts nexts[n] } = ()
let lemma unchanged_other_elements (s1: seq 'a) (s2: seq 'a) (i: int) (v: 'a)
requires { 0 <= i < length s1 /\ length s1 = length s2 }
requires { s1 = s2[i <- v] }
ensures { forall j. 0 <= j < length s1 -> i <> j -> s1[j] = s2[j] } = ()
(*******************************************************************************)
(* *)
(* DEFINITION DE LA STRUCTURE DE DONNEE ABSTRAITE *)
(* *)
(*******************************************************************************)
use mach.int.Int63
type t = private {
mutable ghost nexts: seq int;
mutable ghost marked: seq bool;
max: int63;
}
invariant { max_int > max >= 3 }
invariant { nexts.length = marked.length = max + 1 }
invariant { inv_nexts nexts nexts.length }
invariant { all_eliminated_marked marked nexts }
invariant { forall i. 3 <= i <= max -> mod i 2 = 0 -> marked[i] }
invariant { forall i. 3 <= i < max - 1 ->
mod i 2 = 1 ->
mod (Seq.get nexts i) 2 = 1 \/ Seq.get nexts i = max + 1 }
invariant { Seq.get nexts max = max + 1 /\
(mod (max - 1) 2 = 0 -> Seq.get nexts (max - 1) = max) /\
(mod (max - 1) 2 = 1 -> Seq.get nexts (max - 1) = max + 1) }
by { nexts = Seq.create 4 (fun i -> i + 1);
marked = Seq.create 4 (fun i -> i < 2);
max = 3 }
val create (max: int63) : t
requires { max_int > max >= 3 }
ensures { result.max = max }
ensures { Seq.get result.marked 0 = Seq.get result.marked 1 = true /\
not Seq.get result.marked 2 }
ensures { forall i. 3 <= i <= max ->
mod i 2 = 0 <-> Seq.get result.marked i }
ensures { forall i. 3 <= i < max - 1 ->
mod i 2 = 0 -> Seq.get result.nexts i = i + 1 }
ensures { forall i. 3 <= i < max - 1 ->
mod i 2 = 1 -> Seq.get result.nexts i = i + 2 }
ensures { forall i. 0 <= i <= max ->
Seq.get result.marked i -> i < 2 \/ divides 2 i }
val set_next (t: t) (i v: int63) : unit
requires { 0 <= i <= t.max /\ i < v <= t.max + 1 }
requires { mod i 2 = 1 }
requires { forall j. i < j < v -> Seq.get t.marked j }
requires { not Seq.get t.marked i }
requires { mod v 2 = 1 \/ v = t.max + 1 }
writes { t.nexts }
ensures { t.nexts = (old t.nexts)[i <- v] }
val get_next (t: t) (i: int63) : int63
requires { 3 <= i <= t.max }
requires { mod i 2 = 1 }
ensures { 3 <= result <= t.max + 1 }
ensures { result = t.nexts[i] }
ensures { mod result 2 = 1 \/ result = t.max + 1 }
val set_mark (t: t) (i: int63) : unit
requires { 0 <= i <= t.max }
requires { mod i 2 = 1 }
writes { t.marked }
ensures { t.marked = (old t.marked)[i <- true] }
val get_mark (t: t) (i: int63) : bool
requires { 0 <= i <= t.max }
requires { mod i 2 = 1 }
ensures { result = t.marked[i] }
val get_max (t: t) : int63
ensures { result = t.max /\ result >= 2 }
(*******************************************************************************)
(* *)
(* PREUVE DE REMOVE_PRODUCTS DANS LA STRUCTURE DE DONNEE ABSTRAITE *)
(* *)
(*******************************************************************************)
let remove_products (t: t) (n: int63) : unit
requires { 3 <= n <= t.max /\ n * n <= t.max }
requires { inv_remove_products t.nexts t.marked n }
requires { previously_marked_multiples t.marked n }
requires { only_multiples_marked t.marked n }
requires { not_marked_impl_next_not_marked t.marked t.nexts n }
ensures { inv_remove_products t.nexts t.marked n }
ensures { not_marked_impl_next_not_marked t.marked t.nexts t.nexts[n] }
ensures { previously_marked_multiples t.marked t.nexts[n] }
ensures { only_multiples_marked t.marked t.nexts[n] }
= let d = get_max t / n in
let ghost max = t.max in
let ghost marked_old = t.marked in
let rec loop (p: int63) (ghost x: int) =
requires { n <= p <= max /\ 3 <= n <= max /\ mod p 2 = 1 /\
p <= x < t.nexts[p] /\ t.nexts[x] = t.nexts[p] /\
t.marked[n * n] }
requires { inv_remove_products t.nexts t.marked n }
requires { previously_marked_multiples t.marked n }
requires { not t.marked[p] }
requires { is_copy t.marked marked_old }
requires { all_eliminated_marked_partial marked_old t.nexts x }
requires { not_marked_impl_next_not_marked marked_old t.nexts n }
requires { prime_multiples_marked marked_old t.marked n t.nexts[x] }
requires { not_marked_impl_next_not_marked_partial t.marked t.nexts n p }
requires { only_multiples_marked t.marked (n + 1) }
ensures { inv_remove_products t.nexts t.marked n }
ensures { not_marked_impl_next_not_marked t.marked t.nexts t.nexts[n] }
ensures { is_copy t.marked marked_old }
ensures { previously_marked_multiples t.marked n }
ensures { prime_multiples_marked marked_old t.marked n t.marked.length }
ensures { only_multiples_marked t.marked (n + 1) }
variant { max - t.nexts[p] }
let next = get_next t p in
if 0 <= next <= get_max t then
if next <= d then begin
assert { n * next <= max by n * next <= n * d by next <= d };
ghost (conservation_only_multiples_marked t.marked (to_int n + 1)
(to_int n) t.nexts[to_int p]);
let ghost marked_copy = t.marked in
set_mark t (n * next);
unchanged_other_elements t.marked marked_copy
(to_int n * to_int next) true;
assert { p < 2 * p < 2 * t.nexts[p] <= n * t.nexts[p] = n * next };
assert { not t.marked[p] by n * next > p /\ p <= length t.marked };
assert { not marked_old[t.nexts[p]] by 2 <= p < next <= div max n };
assert { prime_multiples_marked marked_old t.marked n t.nexts[next]
by prime_multiples_marked marked_old t.marked n t.nexts[x] };
assert { forall i. 0 <= i < p -> t.nexts[i] <= p
by forall i. 0 <= i < p ->
forall j. i < j < t.nexts[i] -> t.marked[j] };
if get_mark t next then begin
assert { t.nexts[p] <> t.marked.length };
let ghost nexts_copy = t.nexts in
set_next t p (get_next t next);
unchanged_other_elements t.nexts nexts_copy (to_int p)
nexts_copy[to_int next];
assert { all_eliminated_marked_partial marked_old t.nexts next
by p <= x < next /\
forall j. next <= j < length t.nexts ->
t.nexts[j] = nexts_copy[j] };
loop p (to_int next)
end else begin
assert { forall i. p < i < t.nexts[p] -> t.marked[i] };
loop next (to_int next)
end
end else begin
assert { n * next > max
by n * (div max n + 1) > max >= n * div max n /\
n * next >= n * (d + 1) by next >= d + 1 };
ghost (conservation_not_marked_impl_next_not_marked t.marked t.nexts
(to_int max) (to_int n) (to_int p))
end else
ghost (conservation_not_marked_impl_next_not_marked t.marked t.nexts
(to_int max) (to_int n) (to_int p)) in
ghost (conservation_only_multiples_marked t.marked (to_int n + 1)
(to_int n) (to_int n));
let ghost marked_copy = t.marked in
set_mark t (n * n);
unchanged_other_elements t.marked marked_copy (to_int n * to_int n) true;
assert { forall i. 0 <= i < n -> t.nexts[i] <> n * n
by forall i. 0 <= i < n -> t.nexts[i] <= n < n * n
by not t.marked[n] /\
forall i. 0 <= i < n ->
forall j. i < j < t.nexts[i] -> t.marked[j] };
loop n (to_int n);
ghost (prev_and_new_impl_all_multiples_marked marked_old t.marked
(to_int n) (to_int max));
ghost (conservation_previously_marked_multiples t.marked t.nexts
(to_int n))
(*******************************************************************************)
(* *)
(* QUELQUES LEMMES NECESSAIRES A LA SUITE DE LA PREUVE *)
(* *)
(*******************************************************************************)
let lemma previously_marked_multiples_impl_prime (marked: seq bool) (n: int)
requires { 2 <= n < marked.length /\ not marked[n] }
requires { previously_marked_multiples marked n }
ensures { prime n } =
assert { forall k.
2 <= k < n -> not divides k n
by forall k.
2 <= k < n ->
not (exists i. 2 <= i < marked.length /\ n = k * i) }
let lemma only_multiples_marked_impl_not_marked (marked: seq bool)
(nexts: seq int)
(n: int)
requires { 2 <= n < marked.length }
requires { only_multiples_marked marked nexts[n] }
requires { prime n }
ensures { not marked[n] } =
assert { forall i j. 2 <= i < n -> 2 <= j < marked.length -> i * j <> n }
end
module EulerSieveImpl
use int.Int
use seq.Seq
use mach.int.Int63
use mach.array.ArrayInt63
use int.Abs
use int.EuclideanDivision
use number.Divisibility
use number.Prime
use DivisibilityResults
use EulerSieveSpec
(*******************************************************************************)
(* *)
(* LEMMES DE CONSERVATION LIES A L'IMPLEMENTATION *)
(* *)
(*******************************************************************************)
let lemma conservation_inv_arr_on_mark (arr: seq int) (i: int)
requires { forall j k. 0 <= j < length arr ->
j < k < div (abs arr[j]) 2 -> arr[k] < 0 }
requires { forall i. 0 <= i < length arr ->
i < div (abs arr[i]) 2 <= length arr }
requires { 0 <= i < length arr }
requires { arr[i] >= 0 }
ensures { forall j k. 0 <= j < length arr ->
j < k < div (abs arr[i <- - arr[i]][j]) 2 ->
arr[i <- - arr[i]][k] < 0 } =
assert { forall j. 0 <= j < length arr ->
arr[j] < i -> arr[j] = arr[i <- - arr[i]][j] };
assert { forall j. 0 <= j < length arr ->
arr[i] < j -> arr[j] = arr[i <- - arr[i]][j] }
let lemma conservation_inv_arr_on_jump (arr: seq int) (min i: int)
requires { min >= 0 }
requires { forall j k. min <= j < length arr ->
j < k < div (abs arr[j]) 2 -> arr[k] < 0 }
requires { forall i. min <= i < length arr ->
i < div (abs arr[i]) 2 <= length arr }
requires { min <= i < length arr }
requires { 0 <= div arr[i] 2 < length arr }
requires { arr[div arr[i] 2] < 0 }
ensures { forall j k. min <= j < length arr ->
j < k < div (abs arr[i <- - arr[div arr[i] 2]][j]) 2 ->
arr[i <- - arr[div arr[i] 2]][k] < 0 } =
let ghost next = div (Seq.get arr i) 2 in
let ghost arr1 = arr[i <- - Seq.get arr next] in
assert { forall j. min <= j < i -> arr[j] = arr1[j] };
assert { forall j. i < j < length arr -> arr[j] = arr1[j] }
(*******************************************************************************)
(* *)
(* DEFINITION DE L'IMPLEMENTATION DE LA STRUCTURE DE DONNEE *)
(* *)
(*******************************************************************************)
type t = {
mutable ghost nexts: seq int;
mutable ghost marked: seq bool;
arr: array63;
max: int63;
max_arr: int63
}
invariant { max_int > max >= 3 }
invariant { Seq.length nexts = Seq.length marked = max + 1 }
invariant { div (max - 1) 2 = max_arr }
invariant { length arr = max_arr + 1 }
invariant { inv_nexts nexts (Seq.length nexts) }
invariant { all_eliminated_marked marked nexts }
invariant { forall i. 3 <= i <= max -> mod i 2 = 0 -> Seq.get marked i }
invariant { forall i. 3 <= i < max - 1 ->
mod i 2 = 1 ->
mod (Seq.get nexts i) 2 = 1 \/ Seq.get nexts i = max + 1 }
invariant { Seq.get nexts max = max + 1 /\
(mod (max - 1) 2 = 0 -> Seq.get nexts (max - 1) = max) /\
(mod (max - 1) 2 = 1 -> Seq.get nexts (max - 1) = max + 1) }
(* glueing invariant *)
invariant { forall i. 0 <= i <= max_arr -> -(max + 1) <= arr[i] <= max + 1 }
invariant { forall i. 0 <= i <= max_arr ->
Seq.get marked (2 * i + 1) <-> arr[i] < 0 }
invariant { forall i. 0 <= i <= max_arr ->
not Seq.get marked (2 * i + 1) ->
arr[i] = Seq.get nexts (2 * i + 1) }
invariant { forall i. 0 <= i <= max_arr ->
Seq.get marked (2 * i + 1) ->
arr[i] = - Seq.get nexts (2 * i + 1) }
invariant { forall i. 0 <= i <= max_arr ->
i < div (abs arr[i]) 2 <= max_arr + 1 /\
abs arr[i] <= max + 1 }
invariant { forall i j. 0 <= i <= max_arr ->
i < j < div (abs arr[i]) 2 -> arr[j] < 0 }
by { nexts = Seq.create 4 (fun i -> i + 1);
marked = Seq.create 4 (fun i -> i < 2);
arr = ArrayInt63.set (ArrayInt63.make 2 (-2)) 1 4;
max = 3; max_arr = 1 }
let create (max: int63) : t
requires { max_int > max >= 3 }
ensures { result.max = max }
ensures { Seq.get result.marked 0 = Seq.get result.marked 1 = true /\
not Seq.get result.marked 2 }
ensures { forall i. 1 <= i <= div (max - 1) 2 ->
not Seq.get result.marked (2 * i + 1) }
ensures { forall i. 2 <= i <= div (max + 1) 2 ->
2 * i <= max ->
Seq.get result.marked (2 * i) }
ensures { forall i. 1 <= i <= div (max - 1) 2 ->
2 * i + 1 < max - 1 ->
Seq.get result.nexts (2 * i + 1) = 2 * i + 3 }
ensures { forall i. 2 <= i <= div (max - 1) 2 ->
2 * i < max - 1 ->
Seq.get result.nexts (2 * i) = 2 * i + 1 }
ensures { forall i. 0 <= i <= max ->
Seq.get result.marked i -> i < 2 \/ divides 2 i }
= let ghost len = pure { max + 1 } in
let len_arr = (max - 1) / 2 + 1 in
let ghost nexts = Seq.create len (pure { fun (i: int) ->
if i = max then max + 1
else if i = max - 1 then if mod i 2 = 0 then max else max + 1
else if i < 3 || mod i 2 = 0 then i + 1
else i + 2}) in
let ghost marked = Seq.create len (fun i ->
i = 0 || i = 1 || (i > 2 && mod i 2 = 0)) in
let ghost f = pure { fun i -> if i = len_arr - 1 then max + 1
else if i = 0 then -2
else 2 * i + 3 } in
let arr = ArrayInt63.make len_arr (-2) in
for i = 1 to len_arr - 1 do
invariant { forall j. 0 <= j < i -> arr[j] = f j }
arr[i] <- if i = len_arr - 1 then max + 1 else 2 * i + 3
done;
{ nexts = nexts;
marked = marked;
arr = arr;
max = max;
max_arr = (max - 1) / 2 }
let set_next (t: t) (i v: int63) : unit
requires { 0 <= i <= t.max /\ i < v <= t.max + 1 }
requires { mod i 2 = 1 }
requires { forall j. i < j < v -> Seq.get t.marked j }
requires { not Seq.get t.marked i }
requires { mod v 2 = 1 \/ v = t.max + 1 }
writes { t.nexts, t.arr }
ensures { t.nexts = Seq.set (old t.nexts) i v }
= ghost ( conservation_all_eliminated_marked_on_nexts_change
t.marked t.nexts (to_int i) (to_int v) );
t.arr[i / 2] <- v;
ghost (t.nexts <- Seq.set t.nexts (to_int i) (to_int v))
let get_next (t: t) (i: int63) : int63
requires { 3 <= i <= t.max }
requires { mod i 2 = 1 }
ensures { 3 <= result <= t.max + 1 }
ensures { result = Seq.get t.nexts i }
ensures { mod result 2 = 1 \/ result = t.max + 1 }
= let x = i / 2 in
if t.arr[x] < 0 then - t.arr[x] else t.arr[x]
let set_mark (t: t) (i: int63) : unit
requires { 0 <= i <= t.max }
requires { mod i 2 = 1 }
writes { t.marked, t.arr }
ensures { t.marked = Seq.set (old t.marked) i true }
= let x = i / 2 in
ghost ( conservation_all_eliminated_marked_on_marked_change
t.marked t.nexts (to_int i));
if t.arr[x] >= 0 then begin
ghost ( conservation_inv_arr_on_mark t.arr.elts (to_int x) );
t.arr[x] <- - t.arr[x]
end;
ghost (t.marked <- Seq.set t.marked (to_int i) true)
let get_mark (t: t) (i: int63) : bool
requires { 0 <= i <= t.max }
requires { mod i 2 = 1 }
ensures { result = Seq.get t.marked i }
= t.arr[i / 2] < 0
let get_max (t: t) : int63
ensures { result = t.max /\ result >= 2 }
= t.max
clone EulerSieve with type t = t,
val create = create,
val set_next = set_next, val get_next = get_next,
val set_mark = set_mark, val get_mark = get_mark,
val get_max = get_max
predicate inv_count (arr: seq int) (min: int) =
forall i. min <= i < arr.length ->
(i < div (abs arr[i]) 2 <= arr.length /\
- max_int <= arr[i] <= max_int /\
(forall j. i < j < div (abs arr[i]) 2 -> arr[j] < 0) /\
(forall j. 2 * i + 1 < j < abs arr[i] -> not prime j))
(*******************************************************************************)
(* *)
(* PREUVE DU CRIBLE D'EULER *)
(* *)
(*******************************************************************************)
let euler_sieve (max: int63) : array63
requires { max_int > max >= 3 }
ensures { forall i j. 0 <= i < j < result.length -> result[i] < result[j] }
ensures { forall i. 0 <= i < result.length -> 2 <= result[i] <= max }
ensures { forall i. 0 <= i < result.length -> prime result[i] }
ensures { forall i. 2 <= i <= max -> prime i ->
exists j. 0 <= j < result.length /\ result[j] = i }
= let t = create max in
let rec loop (n: int63) =
requires { 3 <= n <= max }
requires { n * n <= max }
requires { previously_marked_multiples t.marked n }
requires { only_multiples_marked t.marked n }
requires { not_marked_impl_next_not_marked t.marked t.nexts n }
requires { inv_remove_products t.nexts t.marked n }
ensures { forall i. 0 <= i < Seq.length t.marked ->
not Seq.get t.marked i <-> prime i }
variant { max + 1 - n }
remove_products t n;
let nn = get_next t n in
if nn <= max / nn then begin
assert { nn * nn <= max };
ghost (previously_marked_multiples_impl_prime t.marked (to_int nn));
assert { (forall i. n < i < nn -> not prime i)
by (forall i.
n < i < nn ->
(exists j k.
2 <= j < nn /\
2 <= k < Seq.length t.marked /\
j * k = i)
by Seq.get t.marked i) };
loop (nn)
end else begin
ghost (not_prime_impl_divisor_under_sqrt (to_int nn));
assert { forall i.
2 <= i < Seq.length t.marked ->
not Seq.get t.marked i ->
not (exists k l. 2 <= k < nn /\ 2 <= l < i /\
k * l = i) };
ghost (no_prod_impl_no_divider (to_int nn));
assert { Seq.length t.marked <= nn * nn
by nn * nn > max
by nn * (div max nn + 1) > max >= nn * div max nn
/\ nn * nn >= nn * (div max nn + 1)
by nn >= div max nn + 1 };
assert { forall i. 2 <= i < Seq.length t.marked ->
not Seq.get t.marked i ->
prime i
by forall i.
2 <= i < Seq.length t.marked ->
not Seq.get t.marked i ->
(not (exists k. 2 <= k < nn /\ k <> i /\
divides k i)
by forall k.
2 <= k ->
k <> i ->
(divides k i <-> exists l. 2 <= l < i /\ k * l = i)
by divides k i -> 2 <= div i k < i) }
end in
if max >= 9 then loop 3;
assert { forall i. 1 <= i <= t.max_arr -> t.arr[i] >= 0 <-> prime (2 * i + 1) };
assert { forall p i. 1 <= p <= t.max_arr ->
2 * p + 1 < i < abs t.arr[p] -> not prime i
by mod i 2 = 0 -> (Seq.get t.marked i
by Seq.get t.nexts (2 * i + 1) = t.arr[i]) };
assert { forall j. 1 <= j <= t.max_arr -> t.arr[0 <- 2][j] = t.arr[j] };
let ref cnt = 1:int63 in
let ref p = 1 in
t.arr[0] <- 2;
while 2 * p + 1 <= max do
invariant { 1 <= p <= t.max_arr + 1 }
invariant { 1 <= cnt <= p }
invariant { 2 * p + 1 <= max -> t.arr[p] >= 0 }
invariant { 2 * p + 1 <= max -> prime (2 * p + 1) }
invariant { inv_count t.arr cnt }
invariant { ordered t.arr cnt }
invariant { all_inf_or_eq t.arr cnt (2 * p) }
invariant { forall i. cnt <= i <= t.max_arr ->
t.arr[i] >= 0 <-> prime (2 * i + 1) }
invariant { forall i. 0 <= i < cnt -> prime t.arr[i] }
invariant { forall i. 2 <= i < 2 * p + 1 -> prime i ->
exists j. 0 <= j < cnt /\ t.arr[j] = i }
variant { t.max_arr + 1 - p , max + 1 - t.arr[p] }
let next = t.arr[p] / 2 in
if next <= t.max_arr then begin
assert { 1 <= p <= t.max_arr /\ t.arr[p] >= 0 /\ prime (2 * p + 1) };
if t.arr[next] < 0 then begin
label BeforeAssign in
t.arr[p] <- - t.arr[next];
unchanged_other_elements t.arr.elts (pure { t.arr.elts at BeforeAssign })
(to_int p) (- to_int t.arr[next]);
assert { forall i. p < i < div t.arr[p] 2 -> t.arr[i] < 0 };
assert { t.arr[p] = abs (t.arr at BeforeAssign)[next]
> (t.arr at BeforeAssign)[p] > p };
conservation_inv_arr_on_jump (pure { t.arr.elts at BeforeAssign })
(to_int cnt) (to_int p);
assert { cnt <= next < length t.arr };
assert { (forall j. 2 * p + 1 < j < (t.arr at BeforeAssign)[p] ->
not prime j) /\
(forall j. 2 * next + 1 < j < abs (t.arr at BeforeAssign)[next] ->
not prime j) }
end else begin
label BeforeAssign in
t.arr[cnt] <- 2 * p + 1;
unchanged_other_elements t.arr.elts (pure { t.arr.elts at BeforeAssign })
(to_int cnt) (2 * to_int p + 1);
cnt <- cnt + 1;
p <- next;
assert { forall k. cnt <= k < length (t.arr at BeforeAssign) ->
t.arr[k] = (t.arr at BeforeAssign)[k] }
end
end else begin
label BeforeAssign in
t.arr[cnt] <- 2 * p + 1;
unchanged_other_elements t.arr.elts (pure { t.arr.elts at BeforeAssign })
(to_int cnt) (2 * to_int p + 1);
cnt <- cnt + 1;
p <- t.max_arr + 1;
assert { forall k. cnt <= k < length (t.arr at BeforeAssign) ->
t.arr[k] = (t.arr at BeforeAssign)[k] }
end
done;
sub t.arr 0 cnt
end