-
Notifications
You must be signed in to change notification settings - Fork 4
/
Copy pathWith_N.v
1305 lines (1004 loc) · 60 KB
/
With_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
(*****************************************************************************)
(* Proof that Coq R is a fourcolor.model of real numbers. *)
(*****************************************************************************)
Require Import HOLLight_Real_With_N.mappings Coq.Reals.Rbase Coq.Reals.Rdefinitions Coq.Reals.Rbasic_fun.
Open Scope R_scope.
Definition R' := {| type := R; el := 0%R |}.
Canonical R'.
Require Import Coq.Logic.ClassicalEpsilon.
Definition Rsup : (R -> Prop) -> R.
Proof.
intro E. case (excluded_middle_informative (bound E)); intro h.
case (excluded_middle_informative (exists x, E x)); intro i.
destruct (completeness E h i) as [b j]. exact b.
exact 0. exact 0.
Defined.
Lemma is_lub_Rsup E : bound E -> (exists x, E x) -> is_lub E (Rsup E).
Proof.
intros h i. unfold Rsup. case (excluded_middle_informative (bound E)); intro h'.
case (excluded_middle_informative (exists x, E x)); intro i'.
destruct (completeness E h' i') as [b j]. exact j. contradiction. contradiction.
Qed.
Require Import fourcolor.reals.real.
Import Real.
Require Import Coq.Init.Datatypes.
Definition R_struct : structure := {|
val := R;
le := Rle;
sup := Rsup;
add := Rplus;
zero := R0;
opp := Ropp;
mul := Rmult;
one := R1;
inv := Rinv
|}.
Canonical R_struct.
Lemma Rsup_upper_bound E : has_sup E -> ub E (Rsup E).
Proof.
intros [i j]. unfold Rsup. case (excluded_middle_informative (bound E)); intro c.
case (excluded_middle_informative (exists x : R, E x)); intro d.
destruct (completeness E c d) as [b [k l]]. intros x h. apply k. exact h.
intros x h. assert (exists x : R, E x). exists x. exact h. contradiction.
intros x h. assert (exists x : R, E x). exists x. exact h. contradiction.
Qed.
Lemma Rsup_total E x : has_sup E -> down E x \/ Rle (sup E) x.
Proof.
intros [i [b j]]. case (classic (down E x)); intro k. auto. right.
assert (l : bound E). exists b. exact j.
generalize (is_lub_Rsup E l i); intros [m n]. apply n.
intros y hy.
unfold down in k. rewrite ex2_eq, not_exists_eq in k.
generalize (k y); intro k'. rewrite not_conj_eq, <- imp_eq_disj in k'.
unfold Rle. left. apply Rnot_le_lt. apply k'. exact hy.
Qed.
(* Remark: in fourcolor, le is primitive and eq is defined as the
intersection of le and the inverse of le, but in coq, lt is primitive
and le is defined from lt and Logic.eq. *)
Lemma eq_R_struct : @eq R_struct = @Logic.eq R.
Proof.
apply fun_ext; intro x. apply fun_ext; intro y.
apply prop_ext; intro h. destruct h as [h i]. apply Rle_antisym; auto.
subst y. split; apply Rle_refl.
Qed.
Lemma R_axioms : axioms R_struct.
Proof.
apply Axioms.
apply Rle_refl.
apply Rle_trans.
apply Rsup_upper_bound.
apply Rsup_total.
apply Rplus_le_compat_l.
intros x y. rewrite eq_R_struct. apply Rplus_comm.
intros x y z. rewrite eq_R_struct. rewrite Rplus_assoc. reflexivity.
intro x. rewrite eq_R_struct. apply Rplus_0_l.
intro x. rewrite eq_R_struct. apply Rplus_opp_r.
apply Rmult_le_compat_l.
intros x y. rewrite eq_R_struct. apply Rmult_comm.
intros x y z. rewrite eq_R_struct. rewrite Rmult_assoc. reflexivity.
intros x y z. rewrite eq_R_struct. apply Rmult_plus_distr_l.
intro x. rewrite eq_R_struct. apply Rmult_1_l.
intro x. rewrite eq_R_struct. apply Rinv_r.
rewrite eq_R_struct. apply R1_neq_R0.
Qed.
Definition R_model : model := {|
model_structure := R_struct;
model_axioms := R_axioms;
|}.
Lemma eq_R_model :
@eq (model_structure R_model) = @Logic.eq (val (model_structure R_model)).
Proof. exact eq_R_struct. Qed.
Close Scope R_scope.
(*****************************************************************************)
(* Proof that real is a fourcolor.model of real numbers. *)
(*****************************************************************************)
Require Import HOLLight_Real_With_N.terms.
Lemma real_add_of_num p q :
real_of_num (p + q) = real_add (real_of_num p) (real_of_num q).
Proof.
unfold real_of_num, real_add.
f_equal. rewrite treal_add_of_num. apply fun_ext; intro x.
apply prop_ext; intro h.
exists (treal_of_num p). exists (treal_of_num q). split. exact h. split.
rewrite axiom_24_aux. reflexivity. exists (treal_of_num p). reflexivity.
rewrite axiom_24_aux. reflexivity. exists (treal_of_num q). reflexivity.
destruct h as [p' [q' [h1 [h2 h3]]]].
rewrite axiom_24_aux in h2. 2: exists (treal_of_num p); reflexivity.
rewrite axiom_24_aux in h3. 2: exists (treal_of_num q); reflexivity.
rewrite h2, h3. exact h1.
Qed.
Definition real_sup : (real -> Prop) -> real.
Proof.
intro P. case (excluded_middle_informative (exists x, P x)); intro h.
case (excluded_middle_informative (exists M, forall x, (P x) -> real_le x M)); intro i.
set (Q := fun M => (forall x : real, P x -> real_le x M) /\
(forall M' : real, (forall x : real, P x -> real_le x M')
-> real_le M M')).
exact (ε Q). exact (real_of_num 0). exact (real_of_num 0).
Defined.
Definition real_struct : structure := {|
val := real;
le := real_le;
sup := real_sup;
add := real_add;
zero := real_of_num 0;
opp := real_neg;
mul := real_mul;
one := real_of_num 1;
inv := real_inv
|}.
Canonical real_struct.
Require Import HOLLight_Real_With_N.theorems.
Lemma real_sup_is_lub E :
has_sup E -> ub E (real_sup E) /\ (forall b, ub E b -> real_le (real_sup E) b).
Proof.
intros [i j]. unfold real_sup.
destruct (excluded_middle_informative (exists x : real, E x)).
destruct (excluded_middle_informative (exists M : real, forall x : real, E x -> real_le x M)).
set (Q := fun M : real =>
(forall x : real, E x -> real_le x M) /\
(forall M' : real, (forall x : real, E x -> real_le x M') -> real_le M M')).
assert (k: exists M : real, Q M). apply (thm_REAL_COMPLETE E (conj i j)).
generalize (ε_spec k); intros [l m]. auto. contradiction. contradiction.
Qed.
Lemma real_sup_upper_bound E : has_sup E -> ub E (real_sup E).
Proof. intro h. apply (proj1 (real_sup_is_lub E h)). Qed.
Lemma real_sup_total E x : has_sup E -> down E x \/ real_le (real_sup E) x.
Proof.
intro h. case (classic (down E x)); intro k. auto. right.
generalize (real_sup_is_lub E h); intros [i j]. apply j.
intros y hy.
unfold down in k. rewrite ex2_eq, not_exists_eq in k.
generalize (k y); intro k'. rewrite not_conj_eq, <- imp_eq_disj in k'.
apply thm_REAL_LT_IMP_LE. apply k'. apply hy.
Qed.
Lemma eq_real_struct: @eq real_struct = @Logic.eq real.
Proof.
apply fun_ext; intro x. apply fun_ext; intro y.
unfold eq. rewrite thm_REAL_LE_ANTISYM. reflexivity.
Qed.
Require Import Coq.micromega.Lia.
Lemma real_axioms : axioms real_struct.
Proof.
apply Axioms.
apply thm_REAL_LE_REFL.
intros x y z xy yz; apply (thm_REAL_LE_TRANS x y z (conj xy yz)).
apply real_sup_upper_bound.
apply real_sup_total.
intros x y z yz; rewrite thm_REAL_LE_LADD; exact yz.
intros x y. rewrite eq_real_struct. apply thm_REAL_ADD_SYM.
intros x y z. rewrite eq_real_struct. apply thm_REAL_ADD_ASSOC.
intro x. rewrite eq_real_struct. apply thm_REAL_ADD_LID.
intro x. rewrite eq_real_struct. rewrite thm_REAL_ADD_SYM. apply thm_REAL_ADD_LINV.
intros x y z hx yz. apply thm_REAL_LE_LMUL. auto.
intros x y. rewrite eq_real_struct. apply thm_REAL_MUL_SYM.
intros x y z. rewrite eq_real_struct. apply thm_REAL_MUL_ASSOC.
intros x y z. rewrite eq_real_struct. apply thm_REAL_ADD_LDISTRIB.
intro x. rewrite eq_real_struct. apply thm_REAL_MUL_LID.
intro x. rewrite eq_real_struct. rewrite thm_REAL_MUL_SYM. apply thm_REAL_MUL_LINV.
unfold one, zero. simpl. rewrite eq_real_struct, thm_REAL_OF_NUM_EQ. lia.
Qed.
Definition real_model : model := {|
model_structure := real_struct;
model_axioms := real_axioms;
|}.
Lemma eq_real_model:
@eq (model_structure real_model) = @Logic.eq (val (model_structure real_model)).
Proof. exact eq_real_struct. Qed.
Require Import fourcolor.reals.realcategorical.
Definition R_of_real := @Rmorph_to real_model R_model.
Definition real_of_R := @Rmorph_to R_model real_model.
Lemma R_of_real_of_R r : R_of_real (real_of_R r) = r.
Proof. rewrite <- eq_R_model. apply (@Rmorph_to_inv R_model real_model). Qed.
Lemma real_of_R_of_real r : real_of_R (R_of_real r) = r.
Proof. rewrite <- eq_real_model. apply (@Rmorph_to_inv real_model R_model). Qed.
(*****************************************************************************)
(* Mapping of HOL-Light reals to Coq reals. *)
(*****************************************************************************)
Definition mk_real : ((prod hreal hreal) -> Prop) -> R := fun x => R_of_real (mk_real x).
Definition dest_real : R -> (prod hreal hreal) -> Prop := fun x => dest_real (real_of_R x).
Lemma axiom_23 : forall (a : R), (mk_real (dest_real a)) = a.
Proof. intro a. unfold mk_real, dest_real. rewrite axiom_23. apply R_of_real_of_R. Qed.
Lemma axiom_24_aux : forall r, (exists x, r = treal_eq x) -> dest_real (mk_real r) = r.
Proof.
intros c [x h]. unfold dest_real, mk_real.
rewrite real_of_R_of_real, <- axiom_24.
exists x. exact h.
Qed.
Lemma axiom_24 : forall (r : (prod hreal hreal) -> Prop), ((fun s : (prod hreal hreal) -> Prop => exists x : prod hreal hreal, s = (treal_eq x)) r) = ((dest_real (mk_real r)) = r).
Proof.
intro c. unfold dest_real, mk_real. rewrite real_of_R_of_real, <- axiom_24.
reflexivity.
Qed.
Lemma real_of_R_morph : morphism real_of_R.
Proof. apply Rmorph_toP. Qed.
Lemma R_of_real_morph : morphism R_of_real.
Proof. apply Rmorph_toP. Qed.
Lemma le_morph_R x y : le x y = le (real_of_R x) (real_of_R y).
Proof.
generalize (morph_le real_of_R_morph x y); intros [h i]. apply prop_ext; auto.
Qed.
Lemma real_le_def : Rle = (fun x1 : R => fun y1 : R => @ε Prop (fun u : Prop => exists x1' : prod hreal hreal, exists y1' : prod hreal hreal, ((treal_le x1' y1') = u) /\ ((dest_real x1 x1') /\ (dest_real y1 y1')))).
Proof.
apply fun_ext; intro x. apply fun_ext; intro y.
unfold dest_real. rewrite le_morph_R.
generalize (real_of_R x); clear x; intro x.
generalize (real_of_R y); clear y; intro y.
reflexivity.
Qed.
Lemma add_morph_R x y : real_of_R (add x y) = (add (real_of_R x) (real_of_R y)).
Proof. rewrite <- eq_real_model. apply (morph_add real_of_R_morph). Qed.
Lemma add_eq x y : add x y = R_of_real (add (real_of_R x) (real_of_R y)).
Proof. rewrite <- add_morph_R, R_of_real_of_R. reflexivity. Qed.
Lemma real_add_def : Rplus = (fun x1 : R => fun y1 : R => mk_real (fun u : prod hreal hreal => exists x1' : prod hreal hreal, exists y1' : prod hreal hreal, (treal_eq (treal_add x1' y1') u) /\ ((dest_real x1 x1') /\ (dest_real y1 y1')))).
Proof.
apply fun_ext; intro x. apply fun_ext; intro y.
rewrite add_eq. unfold mk_real. apply f_equal. reflexivity.
Qed.
Lemma mul_morph_R x y : real_of_R (mul x y) = (mul (real_of_R x) (real_of_R y)).
Proof. rewrite <- eq_real_model. apply (morph_mul real_of_R_morph). Qed.
Lemma mul_eq x y : mul x y = R_of_real (mul (real_of_R x) (real_of_R y)).
Proof. rewrite <- mul_morph_R, R_of_real_of_R. reflexivity. Qed.
Lemma real_mul_def : Rmult = (fun x1 : R => fun y1 : R => mk_real (fun u : prod hreal hreal => exists x1' : prod hreal hreal, exists y1' : prod hreal hreal, (treal_eq (treal_mul x1' y1') u) /\ ((dest_real x1 x1') /\ (dest_real y1 y1')))).
Proof.
apply fun_ext; intro x. apply fun_ext; intro y.
rewrite mul_eq. unfold mk_real. apply f_equal. reflexivity.
Qed.
Lemma zero_morph_R : real_of_R 0%R = real_of_num 0.
Proof. rewrite <- eq_real_model. apply (morph_zero real_of_R_morph). Qed.
Lemma zero_eq : 0%R = R_of_real (real_of_num 0).
Proof. rewrite <- zero_morph_R, R_of_real_of_R. reflexivity. Qed.
Lemma inv_morph_R x : real_of_R (inv x) = inv (real_of_R x).
Proof.
case (classic (x = 0%R)); intro h.
subst x. unfold inv. simpl. rewrite Rinv_0, zero_eq, !real_of_R_of_real.
Set Printing All.
change (@Logic.eq (type real) (real_of_num 0) (real_inv (real_of_num 0))).
symmetry. apply thm_REAL_INV_0.
rewrite <- eq_real_model. apply (morph_inv real_of_R_morph).
rewrite eq_R_model. exact h.
Unset Printing All.
Qed.
Lemma inv_eq x : inv x = R_of_real (inv (real_of_R x)).
Proof. rewrite <- inv_morph_R, R_of_real_of_R. reflexivity. Qed.
Lemma real_inv_def : Rinv = (fun x : R => mk_real (fun u : prod hreal hreal => exists x' : prod hreal hreal, (treal_eq (treal_inv x') u) /\ (dest_real x x'))).
Proof. apply fun_ext; intro x. rewrite inv_eq. unfold mk_real. reflexivity. Qed.
Lemma neg_morph_R x : real_of_R (opp x) = opp (real_of_R x).
Proof. rewrite <- eq_real_model. apply (morph_opp real_of_R_morph). Qed.
Lemma neg_eq x : opp x = R_of_real (opp (real_of_R x)).
Proof. rewrite <- neg_morph_R, R_of_real_of_R. reflexivity. Qed.
Lemma real_neg_def : Ropp = (fun x1 : R => mk_real (fun u : prod hreal hreal => exists x1' : prod hreal hreal, (treal_eq (treal_neg x1') u) /\ (dest_real x1 x1'))).
Proof. apply fun_ext; intro x. rewrite neg_eq. unfold mk_real. reflexivity. Qed.
Lemma one_morph_R : real_of_R 1%R = real_of_num 1.
Proof. rewrite <- eq_real_model. apply (morph_one real_of_R_morph). Qed.
Lemma one_eq : 1%R = R_of_real (real_of_num 1).
Proof. rewrite <- one_morph_R, R_of_real_of_R. reflexivity. Qed.
Definition R_of_N n :=
match n with
| N0 => 0%R
| N.pos p => IPR p
end.
Require Import Coq.micromega.Lra.
Lemma R_of_N_succ n : R_of_N (N.succ n) = (R_of_N n + 1)%R.
Proof.
destruct n; simpl. unfold IPR. lra. rewrite Rplus_comm. apply succ_IPR.
Qed.
Lemma R_of_N_add p q : R_of_N (p + q)%N = (R_of_N p + R_of_N q)%R.
Proof.
destruct p; destruct q; simpl. lra. unfold IPR. lra. unfold IPR. lra.
apply plus_IPR.
Qed.
Lemma Npos_succ p : N.pos (Pos.succ p) = N.pos p + 1.
Proof. lia. Qed.
Lemma treal_eq_of_num_add m n :
treal_eq (treal_of_num (m + n))
= treal_eq (treal_add (treal_of_num m) (treal_of_num n)).
Proof.
apply eq_class_intro. apply treal_eq_sym. apply treal_eq_trans.
symmetry. apply thm_TREAL_OF_NUM_ADD.
Qed.
Lemma mk_real_treal_eq_add p q :
mk_real (treal_eq (treal_add (treal_of_num p) (treal_of_num q)))
= (mk_real (treal_eq (treal_of_num p)) + mk_real (treal_eq (treal_of_num q)))%R.
Proof.
rewrite add_eq. unfold mk_real. f_equal. rewrite !real_of_R_of_real.
rewrite <- treal_eq_of_num_add.
change (real_of_num (p + q) = add (real_of_num p) (real_of_num q)).
rewrite real_add_of_num. reflexivity.
Qed.
Lemma IPR_eq_mk_real p : IPR p = mk_real (treal_eq (treal_of_num (N.pos p))).
Proof.
pattern p; revert p; apply Pos.peano_ind.
apply one_eq.
intros p hp. rewrite succ_IPR, Rplus_comm.
assert (e: IPR 1 = mk_real (treal_eq (treal_of_num 1))). apply one_eq.
rewrite hp, e, Npos_succ, <- mk_real_treal_eq_add, <- treal_eq_of_num_add.
reflexivity.
Qed.
Lemma real_of_num_def : R_of_N = (fun m : N => mk_real (fun u : prod hreal hreal => treal_eq (treal_of_num m) u)).
Proof.
apply fun_ext; intro n.
change (R_of_N n = mk_real (treal_eq (treal_of_num n))).
destruct n; simpl. apply zero_eq. apply IPR_eq_mk_real.
Qed.
Lemma R_of_N0 : R_of_N 0 = 0%R.
Proof. reflexivity. Qed.
Lemma R_of_N1 : R_of_N 1 = 1%R.
Proof. reflexivity. Qed.
(*Fixpoint Rpower_nat r n : R :=
match n with
| 0 => 1
| S n => r * Rpower_nat r n
end.
Lemma real_pow_def : Rpower_nat = (@ε ((prod nat (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat nat))))))) -> R -> nat -> R) (fun real_pow' : (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat nat))))))) -> R -> nat -> R => forall _24085 : prod nat (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat nat)))))), (forall x : R, (real_pow' _24085 x (NUMERAL 0)) = (R_of_N (NUMERAL (BIT1 0)))) /\ (forall x : R, forall n : nat, (real_pow' _24085 x (S n)) = (Rmult x (real_pow' _24085 x n)))) (@pair nat (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat nat)))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat (prod nat (prod nat (prod nat nat))))) (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat (prod nat (prod nat nat)))) (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat (prod nat nat))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat nat)) (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 0)))))))) (@pair nat (prod nat nat) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 0)))))))) (@pair nat nat (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 0)))))))) (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 0)))))))))))))))).
Proof.
generalize (@pair nat (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat nat)))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat (prod nat (prod nat (prod nat nat))))) (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat (prod nat (prod nat nat)))) (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat (prod nat nat))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 0)))))))) (@pair nat (prod nat (prod nat nat)) (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 0)))))))) (@pair nat (prod nat nat) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 0)))))))) (@pair nat nat (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 0)))))))) (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 0))))))))))))))); generalize (@prod nat (prod nat (prod nat (prod nat (prod nat (prod nat (prod nat nat))))))); intros A a.
match goal with [|- _ = ε ?x _] => set (Q := x) end.
assert (i : exists q, Q q). exists (fun _ => Rpower_nat). split; simpl; intro x; reflexivity.
generalize (ε_spec i a). intros [h0 hs].
apply fun_ext; intro x. apply fun_ext; intro y.
induction y; simpl. rewrite h0. reflexivity. rewrite hs, IHy. reflexivity.
Qed.*)
Require Import Coq.Reals.RIneq.
Open Scope R_scope.
Lemma Rnot_le x y : (~ x <= y) = (x > y).
Proof.
apply prop_ext; intro h.
apply Rnot_le_gt. exact h.
apply Rgt_not_le. exact h.
Qed.
Lemma real_abs_def :
Rabs = (fun y0 : R => @COND R (Rle (R_of_N (NUMERAL 0)) y0) y0 (Ropp y0)).
Proof.
apply fun_ext; intro r. unfold Rabs. destruct (Rcase_abs r).
assert (h: (R_of_N (NUMERAL 0) <= r) = False). rewrite is_False, Rnot_le.
unfold NUMERAL. rewrite R_of_N0. exact r0.
rewrite h, COND_False. reflexivity.
assert (h: (R_of_N (NUMERAL 0) <= r) = True). rewrite is_True. apply Rge_le.
unfold NUMERAL. rewrite R_of_N0. exact r0.
rewrite h, COND_True. reflexivity.
Qed.
Lemma real_div_def : Rdiv = (fun y0 : R => fun y1 : R => Rmult y0 (Rinv y1)).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. reflexivity.
Qed.
Lemma real_sub_def : Rminus = (fun y0 : R => fun y1 : R => Rplus y0 (Ropp y1)).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. reflexivity.
Qed.
Lemma real_ge_def : Rge = (fun y0 : R => fun y1 : R => Rle y1 y0).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. apply prop_ext; intro h.
apply Rge_le. exact h. apply Rle_ge. exact h.
Qed.
Lemma real_gt_def : Rgt = (fun y0 : R => fun y1 : R => Rlt y1 y0).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. apply prop_ext; intro h.
apply Rgt_lt. exact h. apply Rlt_gt. exact h.
Qed.
Lemma real_lt_def : Rlt = (fun y0 : R => fun y1 : R => ~ (Rle y1 y0)).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. apply prop_ext; intro h.
rewrite Rnot_le. exact h. rewrite Rnot_le in h. exact h.
Qed.
Lemma real_max_def : Rmax = (fun y0 : R => fun y1 : R => @COND R (Rle y0 y1) y1 y0).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. unfold Rmax.
destruct (Rle_dec x y).
rewrite <- (is_True (x <= y)) in r. rewrite r, COND_True. reflexivity.
rewrite <- (is_False (x <= y)) in n. rewrite n, COND_False. reflexivity.
Qed.
Lemma real_min_def : Rmin = (fun y0 : R => fun y1 : R => @COND R (Rle y0 y1) y0 y1).
Proof.
apply fun_ext; intro x; apply fun_ext; intro y. unfold Rmin.
destruct (Rle_dec x y).
rewrite <- (is_True (x <= y)) in r. rewrite r, COND_True. reflexivity.
rewrite <- (is_False (x <= y)) in n. rewrite n, COND_False. reflexivity.
Qed.
(*****************************************************************************)
(* Mapping of integers. *)
(*****************************************************************************)
Definition Z' := {| type := Z; el := 0%Z |}.
Canonical Z'.
Definition int_of_real r := Z.pred (up r).
Lemma axiom_25 : forall (a : Z), (int_of_real (IZR a)) = a.
Proof.
intro k. unfold int_of_real. generalize (archimed (IZR k)).
generalize (up (IZR k)); intros l [h1 h2].
apply lt_IZR in h1. rewrite <- minus_IZR in h2. apply le_IZR in h2. lia.
Qed.
Definition integer : R -> Prop := fun _28588 : R => exists n : N, (Rabs _28588) = (R_of_N n).
Lemma minus_eq_minus x y : -x = y -> x = - y.
Proof. intro e. subst y. symmetry. apply Ropp_involutive. Qed.
Lemma integer_IZR r : integer r -> exists k, r = IZR k.
Proof.
intros [n h]. destruct (Rcase_abs r) as [i|i].
rewrite (Rabs_left _ i) in h. apply minus_eq_minus in h. subst r. clear i.
pattern n; revert n; apply N.peano_ind.
exists 0%Z. rewrite R_of_N0. ring.
intros n [k hk]. rewrite R_of_N_succ.
exists (k - 1)%Z. rewrite minus_IZR, <- hk. ring.
rewrite (Rabs_right _ i) in h. subst r. clear i.
pattern n; revert n; apply N.peano_ind.
exists 0%Z. rewrite R_of_N0. reflexivity.
intros n [k hk]. rewrite R_of_N_succ.
exists (k + 1)%Z. rewrite plus_IZR, <- hk. reflexivity.
Qed.
Definition Zabs (z:Z): N :=
match z with
| Z0 => N0
| Zpos p => N.pos p
| Zneg p => N.pos p
end.
Lemma pos_succ p : N.pos (Pos.succ p) = N.succ (N.pos p).
Proof. induction p; simpl; reflexivity. Qed.
Lemma IZR_pos_eq_R_of_N_pos p: IZR (Z.pos p) = R_of_N (N.pos p).
Proof.
pattern p; revert p; apply Pos.peano_ind.
rewrite R_of_N1. reflexivity.
intros p hp. rewrite Pos2Z.inj_succ, succ_IZR, pos_succ, R_of_N_succ, hp.
reflexivity.
Qed.
Lemma IZR_integer r : (exists k, r = IZR k) -> integer r.
Proof.
intros [k h]. subst r. exists (Zabs k). rewrite <- abs_IZR. destruct k; simpl.
rewrite <- R_of_N0. reflexivity. apply IZR_pos_eq_R_of_N_pos.
apply IZR_pos_eq_R_of_N_pos.
Qed.
Lemma axiom_26 : forall (r : R), ((fun x : R => integer x) r) = ((IZR (int_of_real r)) = r).
Proof.
intro r. apply prop_ext; intro h.
apply integer_IZR in h. destruct h as [k h]. subst r. apply f_equal.
apply axiom_25.
apply IZR_integer. exists (int_of_real r). symmetry. exact h.
Qed.
Definition Z_of_N (n:N): Z :=
match n with
| N0 => Z0
| N.pos p => Z.pos p
end.
Lemma Z_of_N_succ n : Z_of_N (N.succ n) = (Z_of_N n + 1)%Z.
Proof.
destruct n. reflexivity.
pattern p; revert p; apply Pos.peano_ind.
reflexivity.
intro p. simpl. lia.
Qed.
Require Import Coq.micromega.Lra Coq.Reals.R_Ifp.
Lemma up_IZR z : up (IZR z) = (z + 1)%Z.
Proof. symmetry; apply tech_up; rewrite plus_IZR; lra.
Qed.
Lemma up_shiftz r z : up (r + IZR z)%R = (up r + z)%Z.
Proof. assert (H := archimed r). symmetry; apply tech_up; rewrite plus_IZR; lra. Qed.
Lemma up0 : up 0 = 1%Z.
Proof. apply up_IZR. Qed.
Lemma up_succ r : up (r + 1) = (up r + 1)%Z.
Proof. apply up_shiftz. Qed.
Lemma int_of_num_def : Z_of_N = (fun _28789 : N => int_of_real (R_of_N _28789)).
Proof.
apply fun_ext; intro n; pattern n; revert n; apply N.peano_ind; unfold int_of_real.
rewrite R_of_N0, up0. reflexivity.
intro n. unfold int_of_real. rewrite Z_of_N_succ, R_of_N_succ, up_succ. lia.
Qed.
Close Scope R_scope.
(*****************************************************************************)
(* Sets. *)
(*****************************************************************************)
Definition IN {A : Type'} : A -> (A -> Prop) -> Prop := fun _32683 : A => fun _32684 : A -> Prop => _32684 _32683.
Definition EMPTY {A : Type'} : A -> Prop := fun x : A => False.
Definition INSERT {A : Type'} : A -> (A -> Prop) -> A -> Prop := fun _32739 : A => fun _32740 : A -> Prop => fun y : A => (@IN A y _32740) \/ (y = _32739).
Definition UNIV (A : Type') : A -> Prop := fun x : A => True.
Lemma UNIV_eq_INSERT A : UNIV A = INSERT (el A) (fun x => x <> el A).
Proof.
apply fun_ext; intro x. unfold INSERT. apply prop_ext; intro h.
2: exact Logic.I.
destruct (classic (x = el A)) as [i|i]. right. exact i. left. exact i.
Qed.
Lemma IN_el_not_el A: IN (el A) (fun x => x <> el A) = False.
Proof. rewrite is_False. intro h. apply h. reflexivity. Qed.
Definition Incl {A:Type'} (s s': A -> Prop) := forall x, IN x s -> IN x s'.
Lemma IN_set_eq_INSERT {A:Type'} (a:A) s:
IN a s -> s = INSERT a (fun x => IN x s /\ x <> a).
Proof.
intro h. apply fun_ext; intro x. apply prop_ext; unfold INSERT, IN; intro i.
destruct (classic (x = a)) as [j|j]; auto.
destruct i as [[i j]|i]. exact i. subst x. exact h.
Qed.
(*****************************************************************************)
(* Finite sets. *)
(*****************************************************************************)
Definition FINITE {A : Type'} : (A -> Prop) -> Prop := fun a : A -> Prop => forall FINITE' : (A -> Prop) -> Prop, (forall a' : A -> Prop, ((a' = (@EMPTY A)) \/ (exists x : A, exists s : A -> Prop, (a' = (@INSERT A x s)) /\ (FINITE' s))) -> FINITE' a') -> FINITE' a.
Inductive finite {A : Type'} : (A -> Prop) -> Prop :=
finite_EMPTY: finite EMPTY
| finite_INSERT a s : finite s -> finite (INSERT a s).
Lemma FINITE_eq_finite (A:Type') (s:A -> Prop) : FINITE s = finite s.
Proof.
apply prop_ext; intro h.
apply h. intros P [i|[x [s' [i j]]]]; rewrite i.
apply finite_EMPTY.
apply finite_INSERT. exact j.
induction h; intros P H; apply H.
left. reflexivity.
right. exists a. exists s. split. reflexivity. apply IHh. exact H.
Qed.
Require Import Coq.Lists.List.
Lemma finite_list_NoDup {A:Type'} (s:A -> Prop):
finite s = (exists l, NoDup l /\ s = fun x => In x l).
Proof.
apply prop_ext; intro H.
induction H.
exists nil. split.
apply NoDup_nil.
apply fun_ext; intro x. apply prop_ext; intro h; contradiction.
destruct IHfinite as [l [n e]]. subst s. destruct (classic (In a l)).
exists l. split. exact n. apply fun_ext; intro x. unfold INSERT.
apply prop_ext; intro h.
destruct h as [h|h]. exact h. subst x. exact H0. left. exact h.
exists (cons a l). split. apply NoDup_cons. exact H0. exact n.
apply fun_ext; intro x. unfold INSERT. apply prop_ext; simpl; intro h.
destruct h as [h|h]. right. exact h. left. symmetry; exact h.
destruct h as [h|h]. right. symmetry; exact h. left. exact h.
destruct H as [l e]. generalize e; clear e. generalize s; clear s.
induction l; intros s [n e].
subst s. apply finite_EMPTY.
set (s' := fun x => In x l). assert (j: s = INSERT a s').
apply fun_ext; intro x. rewrite e. unfold INSERT. simpl.
apply prop_ext; intro h; firstorder.
rewrite j. apply finite_INSERT. apply IHl. split.
inversion n. assumption. reflexivity.
Qed.
Definition elements {A:Type'} (s:A -> Prop) := ε (fun l => NoDup l /\ s = fun x => In x l).
Lemma elements_spec {A:Type'} {s:A -> Prop} (h:finite s):
NoDup (elements s) /\ s = fun x => In x (elements s).
Proof.
unfold elements.
match goal with [|- _ (ε ?x) /\ _] => set (Q := x) end.
assert (i: exists l, Q l). rewrite finite_list_NoDup in h.
destruct h as [l h]. exists l. exact h.
generalize (ε_spec i). firstorder.
Qed.
Lemma In_elements {A:Type'} {s: A -> Prop} : finite s -> forall x, In x (elements s) = IN x s.
Proof.
intro h. generalize (elements_spec h); intros [n e] x. rewrite e at 2.
reflexivity.
Qed.
Lemma elements_EMPTY (A:Type') : elements (@EMPTY A) = nil.
Proof.
generalize (elements_spec (@finite_EMPTY A)). intros [n e].
destruct (elements EMPTY). reflexivity.
apply False_rec. apply ext_fun with (x:=t) in e. unfold EMPTY in e.
rewrite e. simpl. left. reflexivity.
Qed.
Require Import Coq.Sorting.Permutation.
Lemma eq_mod_permut A (l: list A):
forall l', (forall x, In x l = In x l') -> NoDup l -> NoDup l' -> Permutation l l'.
Proof.
induction l; destruct l'.
intros. apply perm_nil.
intro e. generalize (e a). simpl. intro h. symmetry in h.
apply False_rec. rewrite <- h. left. reflexivity.
intro e. generalize (e a). simpl. intro h.
apply False_rec. rewrite <- h. left. reflexivity.
intros e n n'. inversion n; inversion n'; subst.
destruct (classic (a = a0)).
(* case a = a0 *)
subst a0. apply perm_skip. apply IHl.
intro x. apply prop_ext; intro h.
assert (i: In x (a::l)). right. exact h.
rewrite e in i. destruct i. subst x. contradiction. exact H.
assert (i: In x (a::l')). right. exact h.
rewrite <- e in i. destruct i. subst x. contradiction. exact H.
assumption.
assumption.
(* case a <> a0 *)
assert (i: In a (a0 :: l')). rewrite <- (e a). left. reflexivity.
apply in_split in i. destruct i as [l1 [l2 i]]. rewrite i.
rewrite <- Permutation_middle. apply perm_skip. apply IHl.
2: assumption.
2: apply NoDup_remove_1 with a; rewrite <- i; apply NoDup_cons; assumption.
intro x. apply prop_ext; intro h.
assert (j: In x (a::l)). right. exact h.
rewrite e, i in j. apply in_elt_inv in j. destruct j as [j|j].
subst x. contradiction. exact j.
assert (j: In x (l1 ++ a :: l2)). apply in_or_app. apply in_app_or in h.
destruct h as [h|h]. left. exact h. right. right. exact h.
rewrite <- i, <- e in j. destruct j as [j|j].
subst x. rewrite i in n'. apply NoDup_remove in n'. destruct n' as [h1 h2].
contradiction. exact j.
Qed.
Lemma elements_INSERT {A:Type'} (a:A) {s} :
finite s -> exists l', Permutation l' (a :: elements s) /\
elements (INSERT a s) = COND (IN a s) (elements s) l'.
Proof.
intro h. assert (h': finite (INSERT a s)). apply finite_INSERT. exact h.
destruct (prop_degen (IN a s)) as [e|e]; rewrite e.
exists (a :: elements s). split. reflexivity.
rewrite COND_True. f_equal. apply fun_ext; intro x.
apply prop_ext; firstorder. subst x. rewrite is_True in e. exact e.
exists (elements (INSERT a s)). split. 2: rewrite COND_False; reflexivity.
apply eq_mod_permut. intro x. rewrite (In_elements h').
unfold IN, INSERT, IN. simpl.
apply prop_ext; intros [i|i]. right. rewrite (In_elements h). exact i.
subst x. left. reflexivity. subst x. right. reflexivity.
rewrite (In_elements h) in i. left. exact i.
generalize (elements_spec h'); intros [n i]. exact n.
apply NoDup_cons. rewrite (In_elements h), <- is_False. exact e.
generalize (elements_spec h); intros [n i]. exact n.
Qed.
Definition permut_inv {A B:Type} (f:A -> B -> A) :=
forall a y x, f (f a y) x = f (f a x) y.
Lemma eq_fold_left_permut {A B} {f:A -> B -> A}: permut_inv f ->
forall l l', Permutation l l' -> forall a, fold_left f l a = fold_left f l' a.
Proof.
intro H. induction 1; intro a; simpl.
reflexivity.
apply IHPermutation. rewrite H. reflexivity.
transitivity (fold_left f l' a). apply IHPermutation1. apply IHPermutation2.
Qed.
Lemma fold_left_eq_permut {A B} {f:A -> B -> A}: permut_inv f ->
forall l a b, fold_left f l (f a b) = f (fold_left f l a) b.
Proof.
intro H. induction l as [|x l]; intros a b; simpl.
reflexivity.
rewrite <- IHl. f_equal. apply H.
Qed.
Definition ITSET {A B : Type'} : (B -> A -> A) -> (B -> Prop) -> A -> A := fun f : B -> A -> A => fun s : B -> Prop => fun a : A => @ε ((B -> Prop) -> A) (fun g : (B -> Prop) -> A => ((g (@EMPTY B)) = a) /\ (forall x : B, forall s : B -> Prop, (@FINITE B s) -> (g (@INSERT B x s)) = (@COND A (@IN B x s) (g s) (f x (g s))))) s.
Definition itset {A B:Type'} : (B -> A -> A) -> (B -> Prop) -> A -> A := fun f : B -> A -> A => let F := fun a b => f b a in fun s : B -> Prop => fun a : A => fold_left F (elements s) a.
Lemma itset_EMPTY {A B:Type'} (f:B -> A -> A) a: itset f EMPTY a = a.
Proof. unfold itset. rewrite elements_EMPTY. simpl. reflexivity. Qed.
Definition permut_inv' {A B:Type} (f:B -> A -> A) := forall a y x, f x (f y a) = f y (f x a).
Lemma itset_INSERT {A B:Type'} (f:B -> A -> A) a b s: permut_inv' f -> finite s ->
itset f (INSERT b s) a = COND (IN b s) (itset f s a) (f b (itset f s a)).
Proof.
intros H h. unfold itset. set (F := fun a b => f b a).
assert (H': permut_inv F). exact H.
destruct (elements_INSERT b h) as [l [p e]]. rewrite e.
destruct (prop_degen (IN b s)) as [i|i]; rewrite i.
rewrite !COND_True. reflexivity.
rewrite !COND_False. transitivity (fold_left F (b :: elements s) a).
apply (eq_fold_left_permut H). exact p.
apply (fold_left_eq_permut H').
Qed.
Lemma exists_ITSET {A B:Type'} {f:B -> A -> A}: permut_inv' f ->
forall a, let g := fun s => itset f s a in
g EMPTY = a /\
(forall (x : B) (s : B -> Prop),
FINITE s -> g (INSERT x s) = COND (IN x s) (g s) (f x (g s))).
Proof.
intros H a g. unfold g. split. apply itset_EMPTY. intros b s h.
apply itset_INSERT. exact H. rewrite <- FINITE_eq_finite. exact h.
Qed.
Lemma ITSET_eq_itset (A B:Type') (f:B -> A -> A):
permut_inv' f -> forall s, finite s -> forall a, ITSET f s a = itset f s a.
Proof.
intros H s h a.
unfold ITSET. match goal with [|- ε ?x _ = _] => set (Q := x) end.
assert (i: exists g, Q g). exists (fun s => itset f s a). apply exists_ITSET. exact H.
generalize (ε_spec i). intros [j k]. generalize s h. induction 1.
rewrite itset_EMPTY. exact j.
rewrite k. unfold reverse_coercion in IHh0. rewrite IHh0, itset_INSERT.
reflexivity. exact H. exact h0. rewrite FINITE_eq_finite. exact h0.
Qed.
Definition CARD {A : Type'} : (A -> Prop) -> N := fun s => @ITSET N A (fun x : A => fun n : N => N.succ n) s (NUMERAL 0).
Definition dimindex {A : Type'} : (A -> Prop) -> N := fun _ => @COND N (@FINITE A (UNIV A)) (@CARD A (UNIV A)) (NUMERAL (BIT1 0)).
Lemma Incl_finite {A:Type'} (s: A -> Prop) : finite s -> forall s', Incl s' s -> finite s'.
Proof.
induction 1.
intros s i. assert (e: s = EMPTY). apply fun_ext; intro x.
unfold EMPTY. rewrite is_False. intro h. apply (i x). exact h.
rewrite e. apply finite_EMPTY.
intros s' i. destruct (classic (Incl s' s)) as [h|h].
apply IHfinite. exact h.
assert (j: IN a s'). unfold Incl in h. rewrite not_forall_eq in h.
destruct h as [x h]. rewrite imp_eq_disj, not_disj_eq in h.
destruct h as [h1 h2]. apply NNPP in h1. generalize (i x h1).
intros [j|j]. contradiction. subst x. exact h1.
apply IN_set_eq_INSERT in j. rewrite j. apply finite_INSERT.
apply IHfinite. intros x [h1 h2]. generalize (i x h1). intros [k|k].
exact k. contradiction.
Qed.
Lemma dimindex_UNIV_gt_0 A : 0 < dimindex (UNIV A).
Proof.
assert (p1: permut_inv' (fun (_ : A) (n : N) => N.succ n)). unfold permut_inv'. reflexivity.
unfold dimindex. case (prop_degen (FINITE (UNIV A))); intro h; rewrite h.
assert (p2: finite (UNIV A)). rewrite <- FINITE_eq_finite, h. exact Logic.I.
assert (p3: finite (fun x : A => x <> el A)).
apply (Incl_finite (UNIV A)). exact p2. intros x _. exact Logic.I.
rewrite COND_True. unfold CARD.
rewrite ITSET_eq_itset, UNIV_eq_INSERT, itset_INSERT, IN_el_not_el, COND_False; auto.
lia.
rewrite COND_False. unfold NUMERAL, BIT1. lia.
Qed.
Lemma CARD_ge_1 {A : Type'} : FINITE (UNIV A) -> 1 <= CARD (UNIV A).
Proof.
rewrite FINITE_eq_finite. intro p2.
assert (p1: permut_inv' (fun (_ : A) (n : N) => N.succ n)). unfold permut_inv'. reflexivity.
assert (p3: finite (fun x : A => x <> el A)).
apply (Incl_finite (UNIV A)). exact p2. intros x _. exact Logic.I.
rewrite UNIV_eq_INSERT. unfold CARD.
rewrite ITSET_eq_itset. 2: exact p1. 2: rewrite UNIV_eq_INSERT in p2; exact p2.
rewrite itset_INSERT, IN_el_not_el, COND_False. lia. exact p1. exact p3.
Qed.
(*****************************************************************************)
(* Cart.finite_image: natural numbers between 1 and the cardinal of A,
if A is finite, and 1 otherwise. *)
(*****************************************************************************)
Definition GSPEC {A : Type'} : (A -> Prop) -> A -> Prop := fun _32695 : A -> Prop => _32695.
Definition SETSPEC {_83031 : Type'} : _83031 -> Prop -> _83031 -> Prop := fun _32700 : _83031 => fun _32701 : Prop => fun _32702 : _83031 => _32701 /\ (_32700 = _32702).
Definition dotdot : N -> N -> N -> Prop := fun _69692 : N => fun _69693 : N => @GSPEC N (fun GEN_PVAR_229 : N => exists x : N, @SETSPEC N GEN_PVAR_229 ((N.le _69692 x) /\ (N.le x _69693)) x).
Definition finite_image_pred (A:Type') x :=
@IN N x (dotdot (NUMERAL (BIT1 0)) (@dimindex A (UNIV A))).
Lemma finite_image_pred1 (A:Type') : finite_image_pred A 1.
Proof.
unfold finite_image_pred, IN, dotdot, GSPEC, SETSPEC, NUMERAL, BIT1, BIT0.
exists 1. generalize (dimindex_UNIV_gt_0 A). lia.
Qed.
Definition finite_image : Type' -> Type' :=
fun A => subtype (finite_image_pred1 A).
Definition finite_index : forall {A : Type'}, N -> finite_image A :=
fun A => mk (finite_image_pred1 A).
Definition dest_finite_image : forall {A : Type'}, (finite_image A) -> N :=
fun A => dest (finite_image_pred1 A).
Lemma axiom_27 : forall {A : Type'} (a : finite_image A), (@finite_index A (@dest_finite_image A a)) = a.
Proof. intros A a. apply mk_dest. Qed.
Lemma axiom_28 : forall {A : Type'} (r : N), ((fun x : N => @IN N x (dotdot (NUMERAL (BIT1 0)) (@dimindex A (UNIV A)))) r) = ((@dest_finite_image A (@finite_index A r)) = r).
Proof. intros A r. apply dest_mk. Qed.
(*****************************************************************************)
(* Cart.cart A B is finite_image B -> A. *)
(*****************************************************************************)
Definition cart A B := finite_image B -> A.
Definition mk_cart : forall {A B : Type'}, ((finite_image B) -> A) -> cart A B :=
fun A B f => f.
Definition dest_cart : forall {A B : Type'}, (cart A B) -> (finite_image B) -> A :=
fun A B f => f.
Lemma axiom_29 : forall {A B : Type'} (a : cart A B), (@mk_cart A B (@dest_cart A B a)) = a.
Proof. reflexivity. Qed.
Lemma axiom_30 : forall {A B : Type'} (r : (finite_image B) -> A), ((fun f : (finite_image B) -> A => True) r) = ((@dest_cart A B (@mk_cart A B r)) = r).
Proof. intros A B r. apply prop_ext; intros _. reflexivity. exact Logic.I. Qed.
(*****************************************************************************)
(* Cart.finite_sum *)
(*****************************************************************************)
Definition finite_sum_pred (A B: Type') x := @IN N x (dotdot (NUMERAL (BIT1 0)) (N.add (@dimindex A (UNIV A)) (@dimindex B (UNIV B)))).
Lemma finite_sum_pred1 (A B:Type') : finite_sum_pred A B 1.
Proof.
unfold finite_sum_pred, IN, dotdot, GSPEC, SETSPEC, NUMERAL, BIT1, BIT0.
exists 1. generalize (dimindex_UNIV_gt_0 A) (dimindex_UNIV_gt_0 B). lia.
Qed.
Definition finite_sum : Type' -> Type' -> Type' :=
fun A B => subtype (finite_sum_pred1 A B).
Definition mk_finite_sum : forall {A B : Type'}, N -> finite_sum A B :=
fun A B => mk (finite_sum_pred1 A B).
Definition dest_finite_sum : forall {A B : Type'}, (finite_sum A B) -> N :=
fun A B => dest (finite_sum_pred1 A B).
Lemma axiom_31 : forall {A B : Type'} (a : finite_sum A B), (@mk_finite_sum A B (@dest_finite_sum A B a)) = a.
Proof. intros A a. apply mk_dest. Qed.
Lemma axiom_32 : forall {A B : Type'} (r : N), ((fun x : N => @IN N x (dotdot (NUMERAL (BIT1 0)) (N.add (@dimindex A (UNIV A)) (@dimindex B (UNIV B))))) r) = ((@dest_finite_sum A B (@mk_finite_sum A B r)) = r).
Proof. intros A r. apply dest_mk. Qed.
(*****************************************************************************)
(* Cart.finite_diff *)
(*****************************************************************************)
Definition finite_diff_pred (A B: Type') x := @IN N x (dotdot (NUMERAL (BIT1 0)) (@COND N (N.lt (@dimindex B (UNIV B)) (@dimindex A (UNIV A))) (N.sub (@dimindex A (UNIV A)) (@dimindex B (UNIV B))) (NUMERAL (BIT1 0)))).
Lemma finite_diff_pred1 (A B:Type') : finite_diff_pred A B 1.
Proof.
unfold finite_diff_pred, IN, dotdot, GSPEC, SETSPEC, NUMERAL, BIT1, BIT0.
exists 1. generalize (dimindex_UNIV_gt_0 A) (dimindex_UNIV_gt_0 B); intros.
case (prop_degen (dimindex (UNIV B) < dimindex (UNIV A))); intro h; rewrite h.
rewrite COND_True. rewrite is_True in h. lia.
rewrite COND_False. rewrite is_False in h. lia.
Qed.
Definition finite_diff : Type' -> Type' -> Type' :=
fun A B => subtype (finite_diff_pred1 A B).
Definition mk_finite_diff : forall {A B : Type'}, N -> finite_diff A B :=
fun A B => mk (finite_diff_pred1 A B).
Definition dest_finite_diff : forall {A B : Type'}, (finite_diff A B) -> N :=
fun A B => dest (finite_diff_pred1 A B).