-
Notifications
You must be signed in to change notification settings - Fork 0
/
integers_mod_n.v
1886 lines (1680 loc) · 73.6 KB
/
integers_mod_n.v
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
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
Set Warnings "-ambiguous-paths,-notation-overridden".
Require Export ssreflect ssrbool ssrfun polynomials Setoid.
Open Scope Z_scope.
Definition eqm n a b := n｜b - a.
Notation "a ≡ b (mod n )" := (eqm n a b) (at level 70) : Z_scope.
Theorem eqm_refl : ∀ n a : Z, a ≡ a (mod n).
Proof.
rewrite /eqm /integers.sub /divide => n a.
rewrite integers.A4 -[integers.zero]/(rings.zero ℤ).
auto using div_0_r.
Qed.
Theorem eq_eqm : ∀ a b n, a = b → a ≡ b (mod n).
Proof.
move=> a b n ->.
auto using eqm_refl.
Qed.
Theorem eqm_sym : ∀ n a b : Z, a ≡ b (mod n) → b ≡ a (mod n).
Proof.
rewrite /eqm => n a b /div_mul_l => /(_ (-1)).
by have -> : a-b = -1 * (b-a) by ring.
Qed.
Theorem eqm_gcd : ∀ n a b, a ≡ b (mod n) → gcd(a, n) = 1 → gcd(b, n) = 1.
Proof.
move=> n a b [k /= H] H0.
repeat split; auto using div_1_l with Z => x H1 H2.
apply H0; auto.
have -> : a = (1*b + -(b-a)) by ring.
rewrite H /divide -[integers.add]/(rings.add ℤ) -[integers.mul]/(rings.mul ℤ)
-[integers.neg]/(rings.neg ℤ).
auto using div_add, div_mul_l, div_sign_r_neg, div_mul_l.
Qed.
Theorem n_mod_n_is_0 : ∀ n, n ≡ 0 (mod n).
Proof.
rewrite /eqm /divide /integers.sub -[integers.add]/(rings.add ℤ)
-[integers.neg]/(rings.neg ℤ) -[integers.zero]/(rings.zero ℤ) => n.
auto using div_add, div_0_r, div_sign_r_neg, div_refl.
Qed.
Theorem eqm_trans : ∀ n a b c : Z,
a ≡ b (mod n) → b ≡ c (mod n) → a ≡ c (mod n).
Proof.
rewrite /eqm => n a b c H H0.
have -> : c-a = (c - b) + (b - a) by ring.
by apply div_add.
Qed.
Theorem eqn_zero : ∀ n, n ≡ 0 (mod n).
Proof.
move=> n.
apply eqm_sym.
exists 1 => /=.
now ring_simplify.
Qed.
Theorem eqm_div_n : ∀ n a, n｜a ↔ a ≡ 0 (mod n).
Proof.
split => [H | /eqm_sym]; [apply eqm_sym | ];
rewrite /eqm /integers.sub -[integers.neg]/(rings.neg ℤ)
-[integers.add]/(rings.add ℤ) -neg_0 rings.A3_r //.
Qed.
Section Modular_arithmetic.
Variable n : Z.
Global Add Parametric Relation : Z (eqm n)
reflexivity proved by (eqm_refl n)
symmetry proved by (eqm_sym n)
transitivity proved by (eqm_trans n) as Z_mod_n.
Theorem eqm_sym_iff : ∀ a b : Z, a ≡ b (mod n) ↔ b ≡ a (mod n).
Proof.
now split => [-> | ->].
Qed.
Global Add Morphism integers.add
with signature (eqm n) ==> (eqm n) ==> (eqm n) as Z_add_mod.
Proof.
rewrite /eqm => x y [z /= /(@eq_sym Z) H] x0 y0 [z0 /= /(@eq_sym Z) H0].
exists (z+z0) => /=.
by ring_simplify [H H0].
Qed.
Global Add Morphism integers.mul
with signature (eqm n) ==> (eqm n) ==> (eqm n) as Z_mul_mod.
Proof.
rewrite /eqm => x y [z /= /(@eq_sym Z) H] x0 y0 [z0 /= /(@eq_sym Z) H0].
apply (eqm_trans n _ (y*x0)); [exists (z*x0) | exists (z0*y)] => /=;
by ring_simplify [H H0].
Qed.
Global Add Morphism integers.neg
with signature (eqm n) ==> (eqm n) as Z_neg_mod.
Proof.
rewrite /eqm => x y [z /= /(@eq_sym Z) H].
exists (-z) => /=.
by ring_simplify [H].
Qed.
Global Add Morphism integers.sub
with signature (eqm n) ==> (eqm n) ==> (eqm n) as Z_sub_mod.
Proof.
now rewrite /integers.sub => x y /[swap] x0 /[swap] y0 -> ->.
Qed.
Global Add Morphism (rings.pow ℤ)
with signature (eqm n) ==> (eq) ==> (eqm n) as Z_pow_mod.
Proof.
move=> x y H k.
induction k using Induction;
now rewrite ? rings.pow_0_r ? rings.pow_succ_r ? IHk ? H.
Qed.
Definition modulo : Z → Z.
Proof.
case (excluded_middle_informative (0 < n)) =>
[/[swap] x /(division_algorithm x) /constructive_indefinite_description
[q /constructive_indefinite_description [r H]] | H x].
- exact r.
- exact 0.
Defined.
Theorem modulo_bound : 0 < n → ∀ a, 0 ≤ modulo a < n.
Proof.
rewrite /modulo => H a.
case excluded_middle_informative => [{}H | ] //.
elim constructive_indefinite_description => [q ?].
elim constructive_indefinite_description => [r [? ?]] //.
Qed.
Theorem reduce_mod_eqm : 0 < n → ∀ a, a ≡ modulo a (mod n).
Proof.
rewrite /modulo => H a.
case excluded_middle_informative => [{}H | ] //.
elim constructive_indefinite_description => [q H0].
elim constructive_indefinite_description => [r [<- {}H0]].
now rewrite {2}(n_mod_n_is_0 n) -[integers.add]/(rings.add ℤ)
-[integers.mul]/(rings.mul ℤ) mul_0_l rings.A3.
Qed.
Definition relation_mod := {z of type ℤ × ℤ | π1 z ≡ π2 z (mod n)}.
Theorem equivalence_mod : is_equivalence ℤ relation_mod.
Proof.
(repeat split) => [a H | a b H H0 | a b c H H0 H1].
- have H0: (a, a) ∈ ℤ × ℤ by apply Product_classification; eauto.
rewrite (reify H) (reify H0) Specify_classification despecify
π1_action // π2_action //.
eauto 6 using eqm_refl.
- have H1: (b, a) ∈ ℤ × ℤ by apply Product_classification; eauto.
rewrite (reify H1) ? (Specify_classification (ℤ × ℤ)) => [[]] H2.
rewrite (reify H2) ? despecify ? π1_action // ? π2_action //.
eauto using eqm_sym.
- have H2: (a, c) ∈ ℤ × ℤ by apply Product_classification; eauto.
rewrite (reify H2) ? (Specify_classification (ℤ × ℤ)) =>
[[]] H3 /[swap] [[]] H4.
rewrite (reify H3) (reify H4) ? despecify ? π1_action // ? π2_action //.
eauto using eqm_trans.
Qed.
Declare Scope Zn_scope.
Delimit Scope Zn_scope with Zn.
Open Scope Zn_scope.
Definition Z_ := elts (ℤ / relation_mod)%set.
Bind Scope Zn_scope with Z_.
Definition IZnS := elt_to_set : Z_ → set.
Global Coercion IZnS : Z_ >-> set.
Definition Z_to_Z_n (x : Z) :=
quotient_map relation_mod (mkSet (elts_in_set x)) : Z_.
Global Coercion Z_to_Z_n : Z >-> Z_.
Definition Z_n_to_Z (x : Z_) : Z.
Proof.
elim (constructive_indefinite_description (quotient_lift x)) => [z H].
exact z.
Defined.
Global Coercion Z_n_to_Z : Z_ >-> Z.
Definition add (a b : Z_) := a + b : Z_.
Definition mul (a b : Z_) := a * b : Z_.
Definition neg (a : Z_) := ((-a) : Z_).
Definition sub (a b : Z_) := a - b : Z_.
Infix "+" := add : Zn_scope.
Infix "*" := mul : Zn_scope.
Infix "-" := sub : Zn_scope.
Notation "- a" := (neg a) : Zn_scope.
Theorem IZn_eq : ∀ a b : Z, (a : Z_) = (b : Z_) ↔ a ≡ b (mod n).
Proof.
move=> [a A] [b B].
have H: (a, b) ∈ ℤ × ℤ by apply Product_classification; eauto.
split => [/quotient_equiv /= /(_ equivalence_mod)
/Specify_classification [] H0 | H0];
[ | apply quotient_equiv, Specify_classification, conj;
auto using equivalence_mod ];
rewrite (reify H) despecify π1_action // π2_action //.
Qed.
Theorem Zproj_eq : ∀ a : Z_, a = ((a : Z) : Z_).
Proof.
rewrite /Z_n_to_Z /Z_to_Z_n => a.
elim constructive_indefinite_description => x /[dup] {2}<- H.
by apply /f_equal /set_proj_injective.
Qed.
Theorem Zlift_equiv : ∀ a : Z, a ≡ (a : Z_) : Z (mod n).
Proof.
move=> a.
by rewrite -IZn_eq -Zproj_eq.
Qed.
Theorem modulus_zero : (n : Z_) = 0.
Proof.
apply IZn_eq, eqn_zero.
Qed.
Theorem A1 : ∀ a b : Z_, a + b = b + a.
Proof.
rewrite /add => a b.
by rewrite integers.A1.
Qed.
Theorem A2 : ∀ a b c : Z_, a + (b + c) = (a + b) + c.
Proof.
rewrite /add => a b c.
now rewrite IZn_eq -? Zlift_equiv integers.A2.
Qed.
Theorem A3 : ∀ a : Z_, 0 + a = a.
Proof.
rewrite /add => a.
now rewrite (Zproj_eq a) IZn_eq -? Zlift_equiv integers.A3.
Qed.
Theorem A4 : ∀ a : Z_, a + -a = 0.
Proof.
rewrite /add /neg => a.
now rewrite IZn_eq -Zlift_equiv integers.A4.
Qed.
Theorem sub_is_neg : ∀ a b : Z_, a - b = a + -b.
Proof.
move=> a b.
now rewrite IZn_eq /neg -Zlift_equiv.
Qed.
Theorem M1 : ∀ a b : Z_, a * b = b * a.
Proof.
rewrite /mul => a b.
rewrite integers.M1 //.
Qed.
Theorem M2 : ∀ a b c : Z_, a * (b * c) = (a * b) * c.
Proof.
rewrite /mul => a b c.
now rewrite IZn_eq -? Zlift_equiv integers.M2.
Qed.
Theorem M3 : ∀ a : Z_, 1 * a = a.
Proof.
rewrite /mul => a.
now rewrite (Zproj_eq a) IZn_eq -? Zlift_equiv integers.M3.
Qed.
Theorem D1 : ∀ a b c, (a + b) * c = a * c + b * c.
Proof.
rewrite /add /mul => a b c.
now rewrite IZn_eq -? Zlift_equiv integers.D1.
Qed.
Definition ℤ_ :=
mkRing _ (0 : Z_) (1 : Z_) add mul neg A3 A1 A2 M3 M1 M2 D1 A4.
Add Ring Z_ring_raw : (ringify ℤ_).
Add Ring Z_ring : (ringify ℤ_ : ring_theory (0 : Z_) _ _ _ _ _ eq).
Infix "^" := (rings.pow ℤ_ : Z_ → N → Z_) : Zn_scope.
Notation "a ^ n" := (rings.pow ℤ_ (a : Z_) (n : N) : Z_) : Zn_scope.
Theorem IZn_neg : ∀ a : Z, (-a : Z_) = (-a)%Z.
Proof.
move=> a.
now rewrite IZn_eq -Zlift_equiv.
Qed.
Theorem IZn_mul : ∀ a b : Z, (a * b : Z_) = (a * b)%Z.
Proof.
move=> a b.
now rewrite IZn_eq -? Zlift_equiv.
Qed.
Theorem IZn_add : ∀ a b : Z, (a + b : Z_) = (a + b)%Z.
Proof.
move=> a b.
now rewrite IZn_eq -? Zlift_equiv.
Qed.
Theorem IZn_pow : ∀ (a : Z) k, (a^k) = (a^k : Z)%Z.
Proof.
move=> a.
induction k using Induction.
- rewrite ? rings.pow_0_r //.
- rewrite ? rings.pow_succ_r IHk -IZn_mul //.
Qed.
Theorem injective_mod_n_on_interval :
∀ a b, 0 ≤ a < n → 0 ≤ b < n → a ≡ b (mod n) → a = b.
Proof.
move=> a b.
wlog: a b / a ≤ b.
- (case (classic (a ≤ b)); auto) => /lt_not_ge /or_introl =>
/(_ (b = a)) /[swap] /[apply] /[apply] /[apply] /[swap] /eqm_sym
/[swap] /[apply] //.
- (rewrite /eqm /integers.sub => [[/lt_shift /= H | H]] //) =>
[[H0 H1]] [H2 H3] => /div_le => /(_ H) /le_not_gt /= [].
eapply (lt_le_trans ℤ_order); try apply O1_r; eauto.
rewrite -{2}(rings.A3_r ℤ n).
by apply add_le_l, neg_le_0.
Qed.
Theorem units_in_ℤ_ : ∀ a : Z_, @rings.unit ℤ_ a ↔ gcd(a, n) = 1.
Proof.
split => [[x /IZn_eq [y /= H]] | /Euclidean_algorithm [x [y H]]].
- (repeat split; try apply div_1_l) => z H1 H2.
have -> : (1 = a * (Z_n_to_Z x) + n * (-y))%Z; try by apply div_mul_add.
have -> : (n*(-y) = -(y*n))%Z by ring.
rewrite -H.
ring.
- exists (x : Z_) => /=.
rewrite H -IZn_add -? IZn_mul modulus_zero
-[mul]/(rings.mul ℤ_) mul_0_l /= A1 A3 M1 -Zproj_eq //.
Qed.
Section Positive_modulus.
Hypothesis modulus_pos : 0 < n.
Theorem surjective_mod_n_on_interval :
∀ a : Z_, exists ! x : Z, 0 ≤ x < n ∧ a = x.
Proof.
move=> a.
exists ( modulo a).
(split; try split; auto using modulo_bound) => [ | x' [H H0]].
- now rewrite {1}(Zproj_eq a) IZn_eq -reduce_mod_eqm.
- apply injective_mod_n_on_interval; auto using modulo_bound.
rewrite -reduce_mod_eqm // -IZn_eq -H0 -(Zproj_eq a) //.
Qed.
Definition modulus_in_N : N.
Proof.
apply lt_def in modulus_pos.
elim (constructive_indefinite_description modulus_pos) => [k [H H0]].
exact k.
Defined.
Theorem modulus_in_Z : n = modulus_in_N.
Proof.
rewrite /modulus_in_N /sig_rect.
elim constructive_indefinite_description => x [H ->].
ring.
Qed.
Definition map_to_N (x : elts modulus_in_N) : N.
Proof.
move: (elts_in_set x) => /elements_of_naturals_are_naturals =>
/(_ (elts_in_set modulus_in_N)) H.
exact (mkSet H).
Defined.
Theorem map_to_lt_n : ∀ x, map_to_N x < n.
Proof.
move=> x.
rewrite modulus_in_Z.
apply INZ_lt, lt_is_in, (elts_in_set x).
Qed.
Theorem map_to_ge_0 : ∀ x, 0 ≤ map_to_N x.
Proof.
move=> x.
apply INZ_le, zero_le.
Qed.
Definition map_to_mod_n (x : elts modulus_in_N) := map_to_N x : Z_.
Theorem bijective_map_to_mod_n : bijective (sets.functionify map_to_mod_n).
Proof.
rewrite /bijective Injective_classification Surjective_classification
sets.functionify_domain sets.functionify_range.
split => [x y H H0 | y H].
- rewrite (reify H) (reify H0) ? functionify_action =>
/set_proj_injective /IZn_eq H1.
apply injective_mod_n_on_interval, INZ_eq in H1;
auto using map_to_ge_0, map_to_lt_n.
inversion H1 as [H2].
destruct H2.
by have -> : H = H0 by now apply proof_irrelevance.
- rewrite (reify H).
elim (surjective_mod_n_on_interval (mkSet H)) =>
[x [[[/le_def [ξ ->]]]]].
rewrite integers.A3 => H1 H2 H3.
exists ξ.
have H4: ξ ∈ modulus_in_N by rewrite -lt_is_in -INZ_lt -modulus_in_Z.
split; auto.
rewrite (reify H4) functionify_action H2.
by apply f_equal, IZn_eq, eq_eqm, INZ_eq, set_proj_injective.
Qed.
Theorem bijection_of_Z_mod : (ℤ_ ~ modulus_in_N)%set.
Proof.
symmetry.
exists (sets.functionify map_to_mod_n).
rewrite sets.functionify_domain sets.functionify_range.
auto using bijective_map_to_mod_n.
Qed.
Theorem finite_Z_mod : finite ℤ_.
Proof.
exists modulus_in_N.
auto using bijection_of_Z_mod.
Qed.
Theorem Z_mod_card : # ℤ_ = modulus_in_N.
Proof.
auto using equivalence_to_card, bijection_of_Z_mod.
Qed.
Definition Euler_Phi_set := {x of type ℤ_ | gcd(x : Z_, n) = 1}.
Definition Euler_Phi := # Euler_Phi_set.
Definition 𝐔_ := {x of type ℤ_ | rings.unit x}.
Theorem Euler_Phi_unit : Euler_Phi_set = 𝐔_.
Proof.
apply Extensionality.
split => /Specify_classification [] H;
rewrite Specify_classification (reify H) ? despecify;
split; try apply units_in_ℤ_; eauto using eqm_gcd.
Qed.
Theorem unit_classification : ∀ a : Z_, a ∈ 𝐔_ ↔ @rings.unit ℤ_ a.
Proof.
(split; rewrite ? Specify_classification despecify //) =>
[[] | ]; eauto using elts_in_set.
Qed.
Theorem Euler_Phi_finite : finite Euler_Phi_set.
Proof.
eapply subsets_of_finites_are_finite; eauto using finite_Z_mod =>
x /Specify_classification [] //.
Qed.
Theorem Euler_Phi_nonzero : Euler_Phi ≠ 0%N.
Proof.
rewrite /Euler_Phi => /finite_empty => /(_ Euler_Phi_finite).
rewrite -/(Euler_Phi_set ≠ ∅) Nonempty_classification Euler_Phi_unit.
exists (1 : Z_).
rewrite Specify_classification despecify.
eauto using elts_in_set, (one_unit ℤ_).
Qed.
Corollary Euler_Phi_ge_1 : (1 ≤ Euler_Phi)%N.
Proof.
apply naturals.le_not_gt => /le_lt_succ.
auto using Euler_Phi_nonzero, naturals.le_antisymm, zero_le.
Qed.
Theorem Euler_Phi_helper : ∀ f,
range f = Euler_Phi_set → ∀ x, x ∈ domain f → f x ∈ ℤ_.
Proof.
move: function_maps_domain_to_range =>
/[swap] f /(_ f) /[swap] -> /[swap] x /[apply] /Specify_subset //.
Qed.
Definition Euler_Phi_list : N → Z_.
Proof.
move: Euler_Phi_finite =>
/constructive_indefinite_description
[m /cardinality_sym /constructive_indefinite_description
[f [H [H0 ?]]]] x.
case (excluded_middle_informative (x < m)%N) => [/lt_is_in | H2].
- rewrite -H => /(Euler_Phi_helper _ H0) H2.
exact (mkSet H2).
- exact 0.
Defined.
Lemma Euler_Phi_set_classification :
∀ a : Z_, a ∈ Euler_Phi_set ↔ gcd(a, n) = 1.
Proof.
split => [/Specify_classification [H] | H].
- rewrite (reify H) despecify.
eapply eqm_gcd; eauto.
rewrite -IZn_eq -Zproj_eq (set_proj_injective _ (mkSet H) a)
// -Zproj_eq //.
- rewrite Specify_classification despecify.
eauto using elts_in_set.
Qed.
Lemma Euler_Phi_list_unit :
∀ i, (0 ≤ i ≤ Euler_Phi - 1)%N → @rings.unit ℤ_ (Euler_Phi_list i).
Proof.
rewrite /Euler_Phi_list /eq_rect => i [H].
elim constructive_indefinite_description =>
x /[dup] /equivalence_to_card <- H0.
elim constructive_indefinite_description => f [H1 [H2 H3]].
case excluded_middle_informative => [H4 | ].
- destruct lt_is_in, H1.
rewrite units_in_ℤ_ -Euler_Phi_set_classification /= -H2.
auto using function_maps_domain_to_range.
- by rewrite le_lt_succ -add_1_r add_comm sub_abab;
auto using Euler_Phi_ge_1.
Qed.
Lemma Euler_Phi_list_surj :
∀ a : Z_, @rings.unit ℤ_ a → ∃ i,
(0 ≤ i ≤ Euler_Phi - 1)%N ∧ a = Euler_Phi_list i.
Proof.
rewrite /Euler_Phi_list /eq_rect => a /units_in_ℤ_ H.
elim constructive_indefinite_description =>
[x /[dup] /equivalence_to_card <- H0].
have H1: a ∈ Euler_Phi_set by
rewrite Specify_classification despecify; eauto using elts_in_set.
destruct constructive_indefinite_description as [f [H2 [H3 H4]]].
have H5: (inverse f) a ∈ # Euler_Phi_set.
{ move: H2 H3 (function_maps_domain_to_range (inverse f)).
rewrite inverse_range // inverse_domain // => -> -> /(_ a H1) //. }
have H6: (inverse f) a ∈ ω by
eapply elements_of_naturals_are_naturals; eauto using elts_in_set.
exists (mkSet H6).
have H7: (mkSet H6 < # Euler_Phi_set)%N by now apply lt_is_in.
case excluded_middle_informative => [{}H8 | H8] //.
repeat split; auto using zero_le.
- rewrite le_lt_succ -add_1_r add_comm sub_abab //; apply Euler_Phi_ge_1.
- destruct lt_is_in, H2.
apply set_proj_injective => /=.
rewrite right_inverse ? inverse_domain ? H3 //.
Qed.
Lemma Euler_Phi_list_inj :
∀ i j : N, (0 ≤ i ≤ Euler_Phi - 1)%N → (0 ≤ j ≤ Euler_Phi - 1)%N →
Euler_Phi_list i = Euler_Phi_list j → i = j.
Proof.
rewrite /Euler_Phi_list /eq_rect => i j [H H0] [H1 H2].
elim constructive_indefinite_description =>
[m /[dup] /equivalence_to_card <- H3].
elim constructive_indefinite_description =>
[f [H4 [H5 [/Injective_classification H6 _]]]].
(repeat case excluded_middle_informative) =>
[H7 H8 | [] | _ [] | []]; try move: H0 H2 =>
/le_lt_succ /[swap] /le_lt_succ;
rewrite -add_1_r add_comm sub_abab //; auto using Euler_Phi_ge_1.
(repeat destruct lt_is_in) => ? ?.
destruct H4 => H4.
inversion H4.
apply set_proj_injective, H6; auto.
Qed.
Definition Euler_Phi_product := prod ℤ_ Euler_Phi_list 0 (Euler_Phi - 1).
Lemma Euler_Phi_product_unit : @rings.unit ℤ_ Euler_Phi_product.
Proof.
eauto using unit_prod_closure, Euler_Phi_list_unit.
Qed.
Section Euler's_Theorem.
Variable a : Z_.
Hypothesis unit_a : @rings.unit ℤ_ a.
Definition Euler_Phi_product_shifted :=
prod ℤ_ (λ x, a * (Euler_Phi_list x)) 0 (Euler_Phi - 1).
Lemma Euler_Phi_equal : Euler_Phi_product = Euler_Phi_product_shifted.
Proof.
rewrite /Euler_Phi_product /Euler_Phi_product_shifted.
apply iterate_bijection; auto using M1, M2 => z H.
- elim (Euler_Phi_list_surj (a * Euler_Phi_list z)) => [i [H0 H1] | ].
+ exists i.
split; auto => y [H2 H3].
apply Euler_Phi_list_inj; auto; congruence.
+ apply unit_closure; auto using Euler_Phi_list_unit.
- move: unit_a => [x /= H0].
elim (Euler_Phi_list_surj (x * Euler_Phi_list z)) => [j [H1 H2] | ].
+ exists j.
(split; try by split) => [ | y [H3 H4]].
* rewrite -H2 M2 (M1 a) -H0 M3 //.
* apply Euler_Phi_list_inj; auto.
rewrite -H2 H4 M2 -H0 M3 //.
+ apply unit_closure; auto using Euler_Phi_list_unit.
exists a => /=.
rewrite H0 M1 //.
Qed.
Theorem Euler : a^Euler_Phi = (1 : Z_).
Proof.
move: Euler_Phi_equal.
rewrite /Euler_Phi_product_shifted -prod_mul; auto using zero_le.
(rewrite -/Euler_Phi_product -add_1_r naturals.add_comm sub_0_r sub_abab
-1 ? {1}(M3 Euler_Phi_product) /= -? (M1 Euler_Phi_product);
auto using Euler_Phi_ge_1) => /(unit_cancel ℤ_) -> //.
auto using Euler_Phi_product_unit.
Qed.
End Euler's_Theorem.
Theorem order_construction : ∀ a : Z_, a ∈ 𝐔_ → ∃ m : N,
(a^m = 1 ∧ m ≠ 0%N) ∧ (∀ k : N, (a^k = 1 ∧ k ≠ 0%N) → (m ≤ k)%N).
Proof.
move=> a /Specify_classification [H].
rewrite despecify => /Euler.
eauto using naturals.WOP, Euler_Phi_nonzero.
Qed.
Definition order : Z_ → N.
Proof.
move: excluded_middle_informative => /[swap] a /(_ (a ∈ 𝐔_)) =>
[[/order_construction /constructive_indefinite_description [m H] | H]].
- exact m.
- exact 0%N.
Defined.
Theorem order_pos : ∀ a : Z_, a ∈ 𝐔_ → 0 < order a.
Proof.
rewrite /order => a H.
apply INZ_lt, nonzero_lt.
case excluded_middle_informative => [{}H | ] //.
elim constructive_indefinite_description; intuition.
Qed.
Theorem order_pow : ∀ a : Z_, a ∈ 𝐔_ → a^(order a) = 1.
Proof.
rewrite /order => a H.
case excluded_middle_informative => [{}H | ] //.
elim constructive_indefinite_description; intuition.
Qed.
Theorem div_order : ∀ (a : Z_) (k : N), a ∈ 𝐔_ → order a｜k ↔ a^k = 1.
Proof.
split => [[x] | H0].
- (case (classic (k = 0%N)) => [-> | /succ_0 [m ->]]);
rewrite ? rings.pow_0_r 1 ? rings.M1 // => /[dup] H0.
have: 0 ≤ x => [ | /le_def [c ->]].
{ eapply pos_mul_nonneg; try apply (order_pos a) => //; rewrite -H0.
apply INZ_le, zero_le. }
rewrite integers.A3 /= INZ_mul INZ_eq => ->.
rewrite rings.pow_mul_r order_pow // rings.pow_1_l //.
- move: (division_algorithm k (order a)) H0.
rewrite -integer_powers.pow_nonneg.
(elim; auto using order_pos) => [q [r [<- [/le_def [c ->]]]]].
rewrite integers.A3.
case (classic (c = 0)%N) => [-> | H0 /(lt_not_ge ℤ_order) /[swap] H1].
+ exists q.
rewrite integers.A1 integers.A3 integers.M1 //.
+ move: (H) H1 => /unit_classification H1.
rewrite /order.
case excluded_middle_informative => [{}H | ] //.
elim constructive_indefinite_description => [m [[H2 H3] H4]] H5 [].
apply INZ_le, H4, conj => //.
rewrite -H5 integer_powers.pow_add_r // integer_powers.pow_mul_r
// ? integer_powers.pow_nonneg H2 integer_powers.pow_1_l
rings.M3 //.
Qed.
Theorem order_one : order 1 = 1%N.
Proof.
apply naturals.le_antisymm; apply INZ_le.
- apply div_le, div_order; try apply zero_lt_1;
try apply unit_classification, one_unit.
rewrite rings.pow_1_r //.
- apply lt_0_le_1, order_pos, unit_classification, one_unit.
Qed.
Theorem order_upper_bound : ∀ a : Z_, a ∈ 𝐔_ → (order a ≤ Euler_Phi)%N.
Proof.
move=> a H.
apply INZ_le, div_le.
- apply INZ_lt, nonzero_lt, Euler_Phi_nonzero.
- by apply div_order, Euler, unit_classification.
Qed.
Theorem mul_order : ∀ a b : Z_,
a ∈ 𝐔_ → b ∈ 𝐔_ →
gcd(order a, order b) = 1 → order (a * b) = (order a * order b)%N.
Proof.
have L: ∀ a b : Z_,
a ∈ 𝐔_ → b ∈ 𝐔_ → gcd(order a, order b) = 1 → order a｜order (a * b)
=> a b /[dup] ? /unit_classification ? /[dup] ? /unit_classification *.
- eapply FTA; eauto.
rewrite INZ_mul.
apply div_order; auto.
rewrite -(M3 (a^(order b * order (a * b))%N)) M1
-[1 : Z_]/(rings.one ℤ_) -(rings.pow_1_l ℤ_ (order (a*b))) /=
-(order_pow b) // -rings.pow_mul_r -[mul]/(rings.mul ℤ_)
-rings.pow_mul_l {1}mul_comm ? rings.pow_mul_r ? order_pow
? rings.pow_1_l //.
by apply unit_classification, unit_closure.
- apply assoc_N, conj.
+ apply div_order.
* by apply unit_classification, unit_closure.
* rewrite rings.pow_mul_l rings.pow_mul_r mul_comm rings.pow_mul_r
? order_pow ? rings.pow_1_l ? rings.M3 //.
+ rewrite -INZ_mul.
apply rel_prime_mul; auto.
rewrite M1.
auto using is_gcd_sym.
Qed.
Theorem pow_order :
∀ (k : N) (a : Z_), a ∈ 𝐔_ → order a / gcd k (order a) = order (a^k).
Proof.
move: gcd_sym => /[swap] k /[swap] a -> /[dup] H /[dup] =>
/unit_classification H0 /order_pos /[dup] H1 /INZ_lt /nonzero_lt /[dup]
H2 /INZ_eq H3.
have NZ: gcd (order a) k ≠ 0 by move: H1 => /[swap] /gcd_pos //.
have: 0 ≤ order a / gcd (order a) k => [ | /[dup] H4 /le_def [c H5]].
{ apply div_nonneg; first by left.
case (gcd_nonneg (order a) k) => [? | /(@eq_sym Z)] //. }
have H6: (a^k)%Zn ∈ 𝐔_ by
apply unit_classification, (unit_prod_closure ℤ_).
apply pm_pos; auto; last by apply or_introl, order_pos.
apply assoc_pm, conj.
- apply inv_div_l; auto using gcd_l_div.
have /le_def [d]: 0 ≤ gcd (order a) k by now apply gcd_nonneg.
elim (Euclidean_gcd (order a) k) => [x [y]] <-.
move: integers.A3 INZ_mul => -> /[swap] /[dup] H7 -> ->.
apply div_order; auto.
rewrite -pow_nonneg -INZ_mul -H7 integers.D1 integer_powers.pow_add_r//.
rewrite -? integers.M2 integer_powers.pow_mul_r // (integers.M1 y)
integers.M2 INZ_mul ? integer_powers.pow_mul_r
? pow_nonneg ? order_pow // ? rings.pow_mul_r ? order_pow //
? integer_powers.pow_1_l ? rings.pow_1_l ? rings.M3 //.
auto using (one_unit ℤ_).
- move: integers.A3 H5 -> => /[dup] H5 ->.
apply div_order; auto.
rewrite -? pow_nonneg -H5 -integer_powers.pow_mul_r // mul_div;
rewrite 1 ? integers.M1 -? mul_div ? integer_powers.pow_mul_r
? pow_nonneg ? order_pow ? integer_powers.pow_1_l;
auto using gcd_r_div, gcd_l_div.
Qed.
Theorem order_lcm_closed : ∀ a b : Z_,
a ∈ 𝐔_ → b ∈ 𝐔_ → ∃ c : Z_, c ∈ 𝐔_ ∧ lcm (order a) (order b) = order c.
Proof.
move=> a b.
remember ((order a) * (order b))%N as m.
elim/Strong_Induction: m a b Heqm =>
m H a b Heqm /[dup] H0 /[dup] /unit_classification H1 /order_pos /[dup] H2
/lt_0_le_1 [/exists_prime_divisor [p [H3 [H4 H5]]] | <-] /[dup] H6
/[dup] /unit_classification H7 /order_pos H8; subst;
last by rewrite lcm_l_1; try left; eauto.
set (k := v p (order a)).
set (l := v p (order b)).
(have /lt_def [x]: 0 < order a / p^k; have /lt_def [y]: 0 < order b / p^l)
=> [ | _ | | ]; eauto using val_quot_positive, order_pos.
(have /lt_def [z]: 0 < p^k; have /lt_def [w]: 0 < p^l) =>
[ | _ | | ]; try by apply (ordered_rings.pow_pos ℤ_order).
have: (order a / p^k) * (order b / p^l) < order a * order b.
{ apply (lt_le_cross_mul ℤ_order) =>
/=; try apply val_quot_bound; try apply quot_le_bound;
eauto using order_pos, val_div, val_quot_positive, order_pos. }
rewrite ? integers.A3 /k /l =>
/[swap] [[H9 H10]] /[swap] [[H11 H12]] /[swap]
[[H13 /[dup] H14 ->]] /[swap] [[H15 /[dup] H16 ->]] H17.
move: H14 H16 => /[dup] H14 /[swap] /[dup] H16.
(rewrite -(gcd_val p H4 (order a)) // -1 ? (gcd_val p H4 (order b)) //);
try apply (pos_ne_0 ℤ_order), order_pos; auto => H18 H19.
move: (H10) (H12) (H18) (H19) (H14) (H16) H17 => -> ->.
rewrite ? pow_order ? INZ_mul // => /[dup] H20 =>
/INZ_eq <- /[dup] H21 /INZ_eq <- H17 H22 /INZ_lt /H /(_(rings.pow ℤ_ a z))
/(_ (rings.pow ℤ_ b w)) => [[ | | | c [H23 H24]]] =>
//; try by apply unit_classification, unit_prod_closure.
move: (H10) (H12) H17 H22 H24 <- => <- <- <- H17 {H k l}.
wlog: a b x y z w H0 H1 H2 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16 H17
H18 H19 H20 H21 / (v p (order b) ≤ v p (order a))%N => [H | H].
- case (le_trichotomy (v p (order a)) (v p (order b))); eauto => H22.
eapply (H b a y x w z) in H22 as [d H22]; eauto =>
{H}; try (exists d); rewrite 1 ? lcm_sym //.
move: H5; rewrite -(rings.pow_1_r ℤ p) ? val_lower_bound;
eauto using naturals.le_trans; by apply (pos_ne_0 ℤ_order), order_pos.
- erewrite <-val_lcm_r; eauto using order_pos.
have H22: order (a^x) = z.
{ rewrite -INZ_eq -pow_order -? H13 // -H16 div_l_gcd.
- rewrite H16 INZ_le; auto using zero_le.
- exists (p^v p (order a))%Z => /=.
apply eq_sym, div_inv_l; auto using val_div.
- rewrite div_div; auto using val_div, prime_power_nonzero.
now apply (pos_ne_0 ℤ_order), order_pos. }
have H24: (a^x)%Zn ∈ 𝐔_ by apply unit_classification, unit_prod_closure.
exists (a^x * c).
rewrite H17 H12 -H22 mul_order // ? INZ_mul; auto.
+ rewrite -H17 H22 -H12.
apply val_lcm_r_rel_prime; auto using order_pos.
+ split; auto; apply unit_classification, unit_closure;
by apply unit_classification.
Qed.
Definition max_order : N.
Proof.
move: (lub (λ x, ∃ a : Z_, a ∈ 𝐔_ ∧ order a = x)) => H.
apply constructive_indefinite_description in H as [x [H H0]].
- exact x.
- exists 1%N, 1.
apply conj, order_one.
apply unit_classification, one_unit.
- exists Euler_Phi => n0 [a [H0 <-]].
by apply order_upper_bound.
Defined.
Theorem max_order_ex : ∃ a : Z_, a ∈ 𝐔_ ∧ order a = max_order.
Proof.
rewrite /max_order.
elim constructive_indefinite_description => [x [[a H] H0]].
eauto.
Qed.
Theorem max_order_bound : ∀ a : Z_, a ∈ 𝐔_ → (order a ≤ max_order)%N.
Proof.
rewrite /max_order => a H.
elim constructive_indefinite_description => [x [[b H0] H1]].
eauto.
Qed.
Theorem max_order_div : ∀ a : Z_, a ∈ 𝐔_ → order a｜max_order.
Proof.
move: max_order_ex => [b [H H0]] a H1.
move: (order_lcm_closed a b) => [ | | c [H2]] //.
move: H0 -> => /[dup] H3.
have -> : order c = max_order => [ | <-]; auto using lcm_div_l.
apply naturals.le_antisymm; auto using max_order_bound.
rewrite -INZ_le -H3.
now apply lcm_bound, order_pos.
Qed.
Theorem max_order_pow : ∀ a : Z_, a ∈ 𝐔_ → a^max_order = 1.
Proof.
move=> a H.
apply div_order; auto using max_order_div.
Qed.
End Positive_modulus.
Definition square (a : Z_) := a * a.
Definition square_function := sets.functionify square.
Definition QR := {x of type ℤ_ | rings.unit x ∧ ∃ a, square a = x}.
Definition QNR := {x of type ℤ_ | rings.unit x ∧ (x : Z_) ∉ QR}.
Definition legendre_symbol (a : Z_) : Z.
Proof.
case (excluded_middle_informative (a ∈ QR)) => [H | H].
- exact 1.
- case (excluded_middle_informative (a ∈ QNR)) => [H0 | H0].
+ exact (-(1%Z))%Z.
+ exact 0.
Defined.
Theorem legendre_square : ∀ a, @rings.unit ℤ_ a → legendre_symbol (a * a) = 1.
Proof.
rewrite /legendre_symbol => a H.
case excluded_middle_informative; auto => [[]].
rewrite Specify_classification despecify /square.
eauto using elts_in_set, (unit_closure ℤ_).
Qed.
Section Prime_modulus.
Notation p := n.
Hypothesis prime_modulus : prime p.
Hypothesis positive_prime : 0 < p.
Notation p_in_N := ( modulus_in_N positive_prime).
Theorem Z_mod_prime_is_ID : is_integral_domain ℤ_.
Proof.
(split => [? ? |] /IZn_eq /(iffRL (eqm_div_n _ _)) => [/Euclid's_lemma |]
=> [/(_ prime_modulus) [/eqm_div_n /IZn_eq | /eqm_div_n /IZn_eq] |];
rewrite -? Zproj_eq) => [-> | -> |]; elim prime_modulus; tauto.
Qed.
Definition ℤ_ID := integral_domain_from_ring ℤ_ Z_mod_prime_is_ID.
Lemma nonzero_unit : ∀ a : Z_, a ≠ 0 → @rings.unit ℤ_ a.
Proof.
move=> a H.
apply /units_in_ℤ_ /is_gcd_sym /prime_rel_prime; auto.
move: H => /[swap] /eqm_div_n.
rewrite -IZn_eq -Zproj_eq => -> //.
Qed.
Definition inv : Z_ → Z_.
Proof.
move: excluded_middle_informative => /[swap] a /(_ (a = 0)) =>
[[H | /nonzero_unit /constructive_indefinite_description [x H]]].
- exact 0.
- exact x.
Defined.
Theorem inv_l : ∀ a : Z_, a ≠ 0 → inv a * a = 1.
Proof.
rewrite /inv => a H.
(case excluded_middle_informative; try tauto) => {}H.
by elim constructive_indefinite_description.
Qed.
Definition 𝔽 := mkField ℤ_ inv inv_l (Logic.proj2 Z_mod_prime_is_ID).
Theorem QR_QNR_0 : ∀ a : Z_, a ∉ QR → a ∉ QNR → a = 0.
Proof.
move=> a H H0.
apply NNPP.
move: H0 => /[swap] /nonzero_unit H0 [].
rewrite Specify_classification despecify.
eauto using elts_in_set.
Qed.
Theorem Euler_Criterion_zero : ∀ a, legendre_symbol a = 0 ↔ a = 0.
Proof.
(((split; rewrite /legendre_symbol) =>
[ | ->]; case excluded_middle_informative) =>
[_ /integers.zero_ne_1 | H | | ]
//; try (case excluded_middle_informative; auto using QR_QNR_0)) =>
[_ /(integral_domains.minus_one_nonzero integers.ℤ_ID) | | ] // =>
/Specify_classification; rewrite despecify =>
[[]] _ [] [x]; move: mul_0_r Z_mod_prime_is_ID => -> [] _ /[apply] //.
Qed.
Theorem Prime_Euler_Phi : (Euler_Phi = p_in_N - 1)%N.
Proof.
(rewrite -(singleton_card (0 : Z_)) -Z_mod_card -complement_card;
auto using singletons_are_finite) =>
[z /Singleton_classification -> | ]; eauto using elts_in_set.
apply f_equal, Extensionality => z.
split => [/Specify_classification [] | /Complement_classification []] =>
H; elim prime_modulus => [H0 H1].
- rewrite (reify H) despecify Complement_classification
Singleton_classification => [[_ [_ H2]]].
(split; auto) => /set_proj_injective H3.
contradict H0.
apply H2; eauto using (divide_refl ℤ) with Z.
now rewrite eqm_div_n H3 -Zlift_equiv.
- rewrite (reify H) Singleton_classification
Specify_classification despecify => H2.
(repeat split; auto; try apply div_1_l) => d ? /H1 [? | [? ?]]; auto.
contradict H2.
apply f_equal.
rewrite (Zproj_eq (mkSet H)) IZn_eq -eqm_div_n.
eapply div_trans; eauto.
Qed.
Theorem Prime_Euler_Phi_Z : (p - 1 = Euler_Phi)%Z.
Proof.
rewrite /integers.one ( modulus_in_Z positive_prime) INZ_sub.
- rewrite -lt_0_le_1 -( modulus_in_Z positive_prime); auto.
- apply /INZ_eq /eq_sym /Prime_Euler_Phi.
Qed.
Theorem QR_Euler_Phi : QR ⊂ Euler_Phi_set.
Proof.
move=> x /Specify_classification [H].
rewrite ? Specify_classification (reify H) ? despecify =>
[[/units_in_ℤ_ H0 H1]] //.
Qed.
Theorem QNR_QR_c : QNR = Euler_Phi_set \ QR.
Proof.
apply Extensionality => z.
(split => [/Specify_classification | /Complement_classification] [] =>
[H | /Specify_classification [H] /[swap] ?];
rewrite (reify H) ? Complement_classification Specify_classification
? despecify) => [[/units_in_ℤ_] | /units_in_ℤ_] //.
Qed.
Definition unit_square_function := restriction square_function 𝐔_.
Lemma domain_usf : domain unit_square_function = 𝐔_.
Proof.
rewrite /unit_square_function /square_function
restriction_domain sets.functionify_domain.
apply Intersection_subset => x /Specify_classification [] //.
Qed.
Lemma image_usf : image unit_square_function = QR.