aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/ZArith/BinInt.v
blob: ad3781832a4339fbc09b1a20ddc011d084cd4e6f (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
(* -*- coding: utf-8 -*- *)
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(***********************************************************)
(** * Binary Integers *)
(** Initial author: Pierre Crégut, CNET, Lannion, France *)
(***********************************************************)

Require Export BinPos Pnat.
Require Import BinNat Plus Mult.

Inductive Z : Set :=
  | Z0 : Z
  | Zpos : positive -> Z
  | Zneg : positive -> Z.


(** Automatically open scope positive_scope for the constructors of Z *)
Delimit Scope Z_scope with Z.
Bind Scope Z_scope with Z.
Arguments Scope Zpos [positive_scope].
Arguments Scope Zneg [positive_scope].

(*************************************)
(** * Basic operations *)

(** ** Subtraction of positive into Z *)

Definition Zdouble_plus_one (x:Z) :=
  match x with
    | Z0 => Zpos 1
    | Zpos p => Zpos p~1
    | Zneg p => Zneg (Pdouble_minus_one p)
  end.

Definition Zdouble_minus_one (x:Z) :=
  match x with
    | Z0 => Zneg 1
    | Zneg p => Zneg p~1
    | Zpos p => Zpos (Pdouble_minus_one p)
  end.

Definition Zdouble (x:Z) :=
  match x with
    | Z0 => Z0
    | Zpos p => Zpos p~0
    | Zneg p => Zneg p~0
  end.

Open Local Scope positive_scope.

Fixpoint ZPminus (x y:positive) {struct y} : Z :=
  match x, y with
    | p~1, q~1 => Zdouble (ZPminus p q)
    | p~1, q~0 => Zdouble_plus_one (ZPminus p q)
    | p~1, 1 => Zpos p~0
    | p~0, q~1 => Zdouble_minus_one (ZPminus p q)
    | p~0, q~0 => Zdouble (ZPminus p q)
    | p~0, 1 => Zpos (Pdouble_minus_one p)
    | 1, q~1 => Zneg q~0
    | 1, q~0 => Zneg (Pdouble_minus_one q)
    | 1, 1 => Z0
  end.

Close Local Scope positive_scope.

(** ** Addition on integers *)

Definition Zplus (x y:Z) :=
  match x, y with
    | Z0, y => y
    | Zpos x', Z0 => Zpos x'
    | Zneg x', Z0 => Zneg x'
    | Zpos x', Zpos y' => Zpos (x' + y')
    | Zpos x', Zneg y' =>
      match (x' ?= y')%positive Eq with
	| Eq => Z0
	| Lt => Zneg (y' - x')
	| Gt => Zpos (x' - y')
      end
    | Zneg x', Zpos y' =>
      match (x' ?= y')%positive Eq with
	| Eq => Z0
	| Lt => Zpos (y' - x')
	| Gt => Zneg (x' - y')
      end
    | Zneg x', Zneg y' => Zneg (x' + y')
  end.

Infix "+" := Zplus : Z_scope.

(** ** Opposite *)

Definition Zopp (x:Z) :=
  match x with
    | Z0 => Z0
    | Zpos x => Zneg x
    | Zneg x => Zpos x
  end.

Notation "- x" := (Zopp x) : Z_scope.

(** ** Successor on integers *)

Definition Zsucc (x:Z) := (x + Zpos 1)%Z.

(** ** Predecessor on integers *)

Definition Zpred (x:Z) := (x + Zneg 1)%Z.

(** ** Subtraction on integers *)

Definition Zminus (m n:Z) := (m + - n)%Z.

Infix "-" := Zminus : Z_scope.

(** ** Multiplication on integers *)

Definition Zmult (x y:Z) :=
  match x, y with
    | Z0, _ => Z0
    | _, Z0 => Z0
    | Zpos x', Zpos y' => Zpos (x' * y')
    | Zpos x', Zneg y' => Zneg (x' * y')
    | Zneg x', Zpos y' => Zneg (x' * y')
    | Zneg x', Zneg y' => Zpos (x' * y')
  end.

Infix "*" := Zmult : Z_scope.

(** ** Comparison of integers *)

Definition Zcompare (x y:Z) :=
  match x, y with
    | Z0, Z0 => Eq
    | Z0, Zpos y' => Lt
    | Z0, Zneg y' => Gt
    | Zpos x', Z0 => Gt
    | Zpos x', Zpos y' => (x' ?= y')%positive Eq
    | Zpos x', Zneg y' => Gt
    | Zneg x', Z0 => Lt
    | Zneg x', Zpos y' => Lt
    | Zneg x', Zneg y' => CompOpp ((x' ?= y')%positive Eq)
  end.

Infix "?=" := Zcompare (at level 70, no associativity) : Z_scope.

Ltac elim_compare com1 com2 :=
  case (Dcompare (com1 ?= com2)%Z);
    [ idtac | let x := fresh "H" in
      (intro x; case x; clear x) ].

(** ** Sign function *)

Definition Zsgn (z:Z) : Z :=
  match z with
    | Z0 => Z0
    | Zpos p => Zpos 1
    | Zneg p => Zneg 1
  end.

(** ** Direct, easier to handle variants of successor and addition *)

Definition Zsucc' (x:Z) :=
  match x with
    | Z0 => Zpos 1
    | Zpos x' => Zpos (Psucc x')
    | Zneg x' => ZPminus 1 x'
  end.

Definition Zpred' (x:Z) :=
  match x with
    | Z0 => Zneg 1
    | Zpos x' => ZPminus x' 1
    | Zneg x' => Zneg (Psucc x')
  end.

Definition Zplus' (x y:Z) :=
  match x, y with
    | Z0, y => y
    | x, Z0 => x
    | Zpos x', Zpos y' => Zpos (x' + y')
    | Zpos x', Zneg y' => ZPminus x' y'
    | Zneg x', Zpos y' => ZPminus y' x'
    | Zneg x', Zneg y' => Zneg (x' + y')
  end.

Open Local Scope Z_scope.

(**********************************************************************)
(** ** Inductive specification of Z *)

Theorem Zind :
  forall P:Z -> Prop,
    P Z0 ->
    (forall x:Z, P x -> P (Zsucc' x)) ->
    (forall x:Z, P x -> P (Zpred' x)) -> forall n:Z, P n.
Proof.
  intros P H0 Hs Hp z; destruct z.
  assumption.
  apply Pind with (P := fun p => P (Zpos p)).
    change (P (Zsucc' Z0)); apply Hs; apply H0.
    intro n; exact (Hs (Zpos n)).
  apply Pind with (P := fun p => P (Zneg p)).
    change (P (Zpred' Z0)); apply Hp; apply H0.
    intro n; exact (Hp (Zneg n)).
Qed.

(**********************************************************************)
(** * Misc properties about binary integer operations *)

(**********************************************************************)
(** ** Properties of opposite on binary integer numbers *)

Theorem Zopp_0 : Zopp Z0 = Z0.
Proof.
  reflexivity.
Qed.

Theorem Zopp_neg : forall p:positive, - Zneg p = Zpos p.
Proof.
  reflexivity.
Qed.

(** [opp] is involutive *)

Theorem Zopp_involutive : forall n:Z, - - n = n.
Proof.
  intro x; destruct x; reflexivity.
Qed.

(** Injectivity of the opposite *)

Theorem Zopp_inj : forall n m:Z, - n = - m -> n = m.
Proof.
  intros x y; case x; case y; simpl; intros;
    [ trivial
      | discriminate H
      | discriminate H
      | discriminate H
      | simplify_eq H; intro E; rewrite E; trivial
      | discriminate H
      | discriminate H
      | discriminate H
      | simplify_eq H; intro E; rewrite E; trivial ].
Qed.

(**********************************************************************)
(** ** Other properties of binary integer numbers *)

Lemma ZL0 : 2%nat = (1 + 1)%nat.
Proof.
  reflexivity.
Qed.

(**********************************************************************)
(** * Properties of the addition on integers *)

(** ** Zero is left neutral for addition *)

Theorem Zplus_0_l : forall n:Z, Z0 + n = n.
Proof.
  intro x; destruct x; reflexivity.
Qed.

(** ** Zero is right neutral for addition *)

Theorem Zplus_0_r : forall n:Z, n + Z0 = n.
Proof.
  intro x; destruct x; reflexivity.
Qed.

(** ** Addition is commutative *)

Theorem Zplus_comm : forall n m:Z, n + m = m + n.
Proof.
  induction n as [|p|p]; intros [|q|q]; simpl; try reflexivity.
  rewrite Pplus_comm; reflexivity.
  rewrite ZC4. now case Pcompare_spec.
  rewrite ZC4; now case Pcompare_spec.
  rewrite Pplus_comm; reflexivity.
Qed.

(** ** Opposite distributes over addition *)

Theorem Zopp_plus_distr : forall n m:Z, - (n + m) = - n + - m.
Proof.
  intro x; destruct x as [| p| p]; intro y; destruct y as [| q| q];
    simpl; reflexivity || destruct ((p ?= q)%positive Eq);
      reflexivity.
Qed.

Theorem Zopp_succ : forall n:Z, Zopp (Zsucc n) = Zpred (Zopp n).
Proof.
intro; unfold Zsucc; now rewrite Zopp_plus_distr.
Qed.

(** ** Opposite is inverse for addition *)

Theorem Zplus_opp_r : forall n:Z, n + - n = Z0.
Proof.
  intro x; destruct x as [| p| p]; simpl;
    [ reflexivity
      | rewrite (Pcompare_refl p); reflexivity
      | rewrite (Pcompare_refl p); reflexivity ].
Qed.

Theorem Zplus_opp_l : forall n:Z, - n + n = Z0.
Proof.
  intro; rewrite Zplus_comm; apply Zplus_opp_r.
Qed.

Hint Local Resolve Zplus_0_l Zplus_0_r.

(** ** Addition is associative *)

Lemma weak_assoc :
  forall (p q:positive) (n:Z), Zpos p + (Zpos q + n) = Zpos p + Zpos q + n.
Proof.
 intros x y [|z|z]; simpl; trivial.
 now rewrite Pplus_assoc.
 case (Pcompare_spec y z); intros E0.
 (* y = z *)
 subst.
 assert (H := Plt_plus_r z x). rewrite Pplus_comm in H. apply ZC2 in H.
 now rewrite H, Pplus_minus_eq.
 (* y < z *)
 assert (Hz : (z = (z-y)+y)%positive).
  rewrite Pplus_comm, Pplus_minus_lt; trivial.
 pattern z at 4. rewrite Hz, Pplus_compare_mono_r.
 case Pcompare_spec; intros E1; trivial; f_equal.
 symmetry. rewrite Pplus_comm. apply Pminus_plus_distr.
 rewrite Hz, Pplus_comm. now apply Pplus_lt_mono_r.
 apply Pminus_minus_distr; trivial.
 (* z < y *)
 assert (LT : (z < x + y)%positive).
  rewrite Pplus_comm. apply Plt_trans with y; trivial using Plt_plus_r.
 apply ZC2 in LT. rewrite LT. f_equal.
 now apply Pplus_minus_assoc.
Qed.

Theorem Zplus_assoc : forall n m p:Z, n + (m + p) = n + m + p.
Proof.
 intros [|x|x] [|y|y] [|z|z]; trivial.
 apply weak_assoc.
 apply weak_assoc.
 now rewrite !Zplus_0_r.
 rewrite 2 (Zplus_comm _ (Zpos z)), 2 weak_assoc.
  f_equal; apply Zplus_comm.
 apply Zopp_inj. rewrite !Zopp_plus_distr, !Zopp_neg.
  rewrite 2 (Zplus_comm (-Zpos x)), 2 (Zplus_comm _ (Zpos z)).
  now rewrite weak_assoc.
 now rewrite !Zplus_0_r.
 rewrite 2 (Zplus_comm (Zneg x)), 2 (Zplus_comm _ (Zpos z)).
  now rewrite weak_assoc.
 apply Zopp_inj. rewrite !Zopp_plus_distr, !Zopp_neg.
 rewrite 2 (Zplus_comm _ (Zpos z)), 2 weak_assoc.
  f_equal; apply Zplus_comm.
 apply Zopp_inj. rewrite !Zopp_plus_distr, !Zopp_neg.
  apply weak_assoc.
 apply Zopp_inj. rewrite !Zopp_plus_distr, !Zopp_neg.
  apply weak_assoc.
Qed.

Lemma Zplus_assoc_reverse : forall n m p:Z, n + m + p = n + (m + p).
Proof.
  intros; symmetry ; apply Zplus_assoc.
Qed.

(** ** Associativity mixed with commutativity *)

Theorem Zplus_permute : forall n m p:Z, n + (m + p) = m + (n + p).
Proof.
  intros n m p; rewrite Zplus_comm; rewrite <- Zplus_assoc;
    rewrite (Zplus_comm p n); trivial with arith.
Qed.

(** ** Addition simplifies *)

Theorem Zplus_reg_l : forall n m p:Z, n + m = n + p -> m = p.
  intros n m p H; cut (- n + (n + m) = - n + (n + p));
    [ do 2 rewrite Zplus_assoc; rewrite (Zplus_comm (- n) n);
      rewrite Zplus_opp_r; simpl; trivial with arith
      | rewrite H; trivial with arith ].
Qed.

(** ** Addition and successor permutes *)

Lemma Zplus_succ_l : forall n m:Z, Zsucc n + m = Zsucc (n + m).
Proof.
  intros x y; unfold Zsucc; rewrite (Zplus_comm (x + y));
    rewrite Zplus_assoc; rewrite (Zplus_comm (Zpos 1));
      trivial with arith.
Qed.

Lemma Zplus_succ_r_reverse : forall n m:Z, Zsucc (n + m) = n + Zsucc m.
Proof.
  intros n m; unfold Zsucc; rewrite Zplus_assoc; trivial with arith.
Qed.

Notation Zplus_succ_r := Zplus_succ_r_reverse (only parsing).

Lemma Zplus_succ_comm : forall n m:Z, Zsucc n + m = n + Zsucc m.
Proof.
  unfold Zsucc; intros n m; rewrite <- Zplus_assoc;
    rewrite (Zplus_comm (Zpos 1)); trivial with arith.
Qed.

(** ** Misc properties, usually redundant or non natural *)

Lemma Zplus_0_r_reverse : forall n:Z, n = n + Z0.
Proof.
  symmetry ; apply Zplus_0_r.
Qed.

Lemma Zplus_0_simpl_l : forall n m:Z, n + Z0 = m -> n = m.
Proof.
  intros n m; rewrite Zplus_0_r; intro; assumption.
Qed.

Lemma Zplus_0_simpl_l_reverse : forall n m:Z, n = m + Z0 -> n = m.
Proof.
  intros n m; rewrite Zplus_0_r; intro; assumption.
Qed.

Lemma Zplus_eq_compat : forall n m p q:Z, n = m -> p = q -> n + p = m + q.
Proof.
  intros; rewrite H; rewrite H0; reflexivity.
Qed.

Lemma Zplus_opp_expand : forall n m p:Z, n + - m = n + - p + (p + - m).
Proof.
  intros x y z.
  rewrite <- (Zplus_assoc x).
  rewrite (Zplus_assoc (- z)).
  rewrite Zplus_opp_l.
  reflexivity.
Qed.

(************************************************************************)
(** * Properties of successor and predecessor on binary integer numbers *)

Theorem Zsucc_discr : forall n:Z, n <> Zsucc n.
Proof.
  intros n; cut (Z0 <> Zpos 1);
    [ unfold not; intros H1 H2; apply H1; apply (Zplus_reg_l n);
      rewrite Zplus_0_r; exact H2
      | discriminate ].
Qed.

Theorem Zpos_succ_morphism :
  forall p:positive, Zpos (Psucc p) = Zsucc (Zpos p).
Proof.
  intro; rewrite Pplus_one_succ_r; unfold Zsucc; simpl;
    trivial with arith.
Qed.

(** successor and predecessor are inverse functions *)

Theorem Zsucc_pred : forall n:Z, n = Zsucc (Zpred n).
Proof.
  intros n; unfold Zsucc, Zpred; rewrite <- Zplus_assoc; simpl;
    rewrite Zplus_0_r; trivial with arith.
Qed.

Hint Immediate Zsucc_pred: zarith.

Theorem Zpred_succ : forall n:Z, n = Zpred (Zsucc n).
Proof.
  intros m; unfold Zpred, Zsucc; rewrite <- Zplus_assoc; simpl;
    rewrite Zplus_comm; auto with arith.
Qed.

Theorem Zsucc_inj : forall n m:Z, Zsucc n = Zsucc m -> n = m.
Proof.
  intros n m H.
  change (Zneg 1 + Zpos 1 + n = Zneg 1 + Zpos 1 + m);
    do 2 rewrite <- Zplus_assoc; do 2 rewrite (Zplus_comm (Zpos 1));
      unfold Zsucc in H; rewrite H; trivial with arith.
Qed.

(*************************************************************************)
(** **  Properties of the direct definition of successor and predecessor *)

Theorem Zsucc_succ' : forall n:Z, Zsucc n = Zsucc' n.
Proof.
destruct n as [| p | p]; simpl.
reflexivity.
now rewrite Pplus_one_succ_r.
now destruct p as [q | q |].
Qed.

Theorem Zpred_pred' : forall n:Z, Zpred n = Zpred' n.
Proof.
destruct n as [| p | p]; simpl.
reflexivity.
now destruct p as [q | q |].
now rewrite Pplus_one_succ_r.
Qed.

Theorem Zsucc'_inj : forall n m:Z, Zsucc' n = Zsucc' m -> n = m.
Proof.
intros n m; do 2 rewrite <- Zsucc_succ'; now apply Zsucc_inj.
Qed.

Theorem Zsucc'_pred' : forall n:Z, Zsucc' (Zpred' n) = n.
Proof.
intro; rewrite <- Zsucc_succ'; rewrite <- Zpred_pred';
symmetry; apply Zsucc_pred.
Qed.

Theorem Zpred'_succ' : forall n:Z, Zpred' (Zsucc' n) = n.
Proof.
intro; apply Zsucc'_inj; now rewrite Zsucc'_pred'.
Qed.

Theorem Zpred'_inj : forall n m:Z, Zpred' n = Zpred' m -> n = m.
Proof.
intros n m H.
rewrite <- (Zsucc'_pred' n); rewrite <- (Zsucc'_pred' m); now rewrite H.
Qed.

Theorem Zsucc'_discr : forall n:Z, n <> Zsucc' n.
Proof.
  intro x; destruct x; simpl.
  discriminate.
  injection; apply Psucc_discr.
  destruct p; simpl.
    discriminate.
    intro H; symmetry  in H; injection H; apply double_moins_un_xO_discr.
    discriminate.
Qed.

(** Misc properties, usually redundant or non natural *)

Lemma Zsucc_eq_compat : forall n m:Z, n = m -> Zsucc n = Zsucc m.
Proof.
  intros n m H; rewrite H; reflexivity.
Qed.

Lemma Zsucc_inj_contrapositive : forall n m:Z, n <> m -> Zsucc n <> Zsucc m.
Proof.
  unfold not; intros n m H1 H2; apply H1; apply Zsucc_inj; assumption.
Qed.

(**********************************************************************)
(** * Properties of subtraction on binary integer numbers *)

(** ** [minus] and [Z0] *)

Lemma Zminus_0_r : forall n:Z, n - Z0 = n.
Proof.
  intro; unfold Zminus; simpl; rewrite Zplus_0_r;
    trivial with arith.
Qed.

Lemma Zminus_0_l_reverse : forall n:Z, n = n - Z0.
Proof.
  intro; symmetry ; apply Zminus_0_r.
Qed.

Lemma Zminus_diag : forall n:Z, n - n = Z0.
Proof.
  intro; unfold Zminus; rewrite Zplus_opp_r; trivial with arith.
Qed.

Lemma Zminus_diag_reverse : forall n:Z, Z0 = n - n.
Proof.
  intro; symmetry ; apply Zminus_diag.
Qed.


(** ** Relating [minus] with [plus] and [Zsucc] *)

Lemma Zminus_plus_distr : forall n m p:Z, n - (m + p) = n - m - p.
Proof.
intros; unfold Zminus; rewrite Zopp_plus_distr; apply Zplus_assoc.
Qed.

Lemma Zminus_succ_l : forall n m:Z, Zsucc (n - m) = Zsucc n - m.
Proof.
  intros n m; unfold Zminus, Zsucc; rewrite (Zplus_comm n (- m));
    rewrite <- Zplus_assoc; apply Zplus_comm.
Qed.

Lemma Zminus_succ_r : forall n m:Z, n - (Zsucc m) = Zpred (n - m).
Proof.
intros; unfold Zsucc; now rewrite Zminus_plus_distr.
Qed.

Lemma Zplus_minus_eq : forall n m p:Z, n = m + p -> p = n - m.
Proof.
  intros n m p H; unfold Zminus; apply (Zplus_reg_l m);
    rewrite (Zplus_comm m (n + - m)); rewrite <- Zplus_assoc;
      rewrite Zplus_opp_l; rewrite Zplus_0_r; rewrite H;
	trivial with arith.
Qed.

Lemma Zminus_plus : forall n m:Z, n + m - n = m.
Proof.
  intros n m; unfold Zminus; rewrite (Zplus_comm n m);
    rewrite <- Zplus_assoc; rewrite Zplus_opp_r; apply Zplus_0_r.
Qed.

Lemma Zplus_minus : forall n m:Z, n + (m - n) = m.
Proof.
  unfold Zminus; intros n m; rewrite Zplus_permute; rewrite Zplus_opp_r;
    apply Zplus_0_r.
Qed.

Lemma Zminus_plus_simpl_l : forall n m p:Z, p + n - (p + m) = n - m.
Proof.
  intros n m p; unfold Zminus; rewrite Zopp_plus_distr;
    rewrite Zplus_assoc; rewrite (Zplus_comm p); rewrite <- (Zplus_assoc n p);
      rewrite Zplus_opp_r; rewrite Zplus_0_r; trivial with arith.
Qed.

Lemma Zminus_plus_simpl_l_reverse : forall n m p:Z, n - m = p + n - (p + m).
Proof.
  intros; symmetry ; apply Zminus_plus_simpl_l.
Qed.

Lemma Zminus_plus_simpl_r : forall n m p:Z, n + p - (m + p) = n - m.
Proof.
  intros x y n.
  unfold Zminus.
  rewrite Zopp_plus_distr.
  rewrite (Zplus_comm (- y) (- n)).
  rewrite Zplus_assoc.
  rewrite <- (Zplus_assoc x n (- n)).
  rewrite (Zplus_opp_r n).
  rewrite <- Zplus_0_r_reverse.
  reflexivity.
Qed.

Lemma Zpos_minus_morphism : forall a b:positive, Pcompare a b Eq = Lt ->
  Zpos (b-a) = Zpos b - Zpos a.
Proof.
  intros.
  simpl.
  change Eq with (CompOpp Eq).
  rewrite <- Pcompare_antisym.
  rewrite H; simpl; auto.
Qed.

(** ** Misc redundant properties *)

Lemma Zeq_minus : forall n m:Z, n = m -> n - m = Z0.
Proof.
  intros x y H; rewrite H; symmetry ; apply Zminus_diag_reverse.
Qed.

Lemma Zminus_eq : forall n m:Z, n - m = Z0 -> n = m.
Proof.
  intros x y H; rewrite <- (Zplus_minus y x); rewrite H; apply Zplus_0_r.
Qed.


(**********************************************************************)
(** * Properties of multiplication on binary integer numbers *)

Theorem Zpos_mult_morphism :
  forall p q:positive, Zpos (p*q) = Zpos p * Zpos q.
Proof.
  auto.
Qed.

(** ** One is neutral for multiplication *)

Theorem Zmult_1_l : forall n:Z, Zpos 1 * n = n.
Proof.
  intro x; destruct x; reflexivity.
Qed.

Theorem Zmult_1_r : forall n:Z, n * Zpos 1 = n.
Proof.
  intro x; destruct x; simpl; try rewrite Pmult_1_r; reflexivity.
Qed.

(** ** Zero property of multiplication *)

Theorem Zmult_0_l : forall n:Z, Z0 * n = Z0.
Proof.
  intro x; destruct x; reflexivity.
Qed.

Theorem Zmult_0_r : forall n:Z, n * Z0 = Z0.
Proof.
  intro x; destruct x; reflexivity.
Qed.

Hint Local Resolve Zmult_0_l Zmult_0_r.

Lemma Zmult_0_r_reverse : forall n:Z, Z0 = n * Z0.
Proof.
  intro x; destruct x; reflexivity.
Qed.

(** ** Commutativity of multiplication *)

Theorem Zmult_comm : forall n m:Z, n * m = m * n.
Proof.
  intros x y; destruct x as [| p| p]; destruct y as [| q| q]; simpl;
    try rewrite (Pmult_comm p q); reflexivity.
Qed.

(** ** Associativity of multiplication *)

Theorem Zmult_assoc : forall n m p:Z, n * (m * p) = n * m * p.
Proof.
  intros x y z; destruct x; destruct y; destruct z; simpl;
    try rewrite Pmult_assoc; reflexivity.
Qed.

Lemma Zmult_assoc_reverse : forall n m p:Z, n * m * p = n * (m * p).
Proof.
  intros n m p; rewrite Zmult_assoc; trivial with arith.
Qed.

(** ** Associativity mixed with commutativity *)

Theorem Zmult_permute : forall n m p:Z, n * (m * p) = m * (n * p).
Proof.
  intros x y z; rewrite (Zmult_assoc y x z); rewrite (Zmult_comm y x).
  apply Zmult_assoc.
Qed.

(** ** Z is integral *)

Theorem Zmult_integral_l : forall n m:Z, n <> Z0 -> m * n = Z0 -> m = Z0.
Proof.
  intros x y; destruct x as [| p| p].
  intro H; absurd (Z0 = Z0); trivial.
  intros _ H; destruct y as [| q| q]; reflexivity || discriminate.
  intros _ H; destruct y as [| q| q]; reflexivity || discriminate.
Qed.


Theorem Zmult_integral : forall n m:Z, n * m = Z0 -> n = Z0 \/ m = Z0.
Proof.
  intros x y; destruct x; destruct y; auto; simpl; intro H;
    discriminate H.
Qed.


Lemma Zmult_1_inversion_l :
  forall n m:Z, n * m = Zpos 1 -> n = Zpos 1 \/ n = Zneg 1.
Proof.
  intros x y; destruct x as [| p| p]; intro; [ discriminate | left | right ];
    (destruct y as [| q| q]; try discriminate; simpl in H; injection H; clear H;
      intro H; rewrite Pmult_1_inversion_l with (1 := H);
	reflexivity).
Qed.

(** ** Multiplication and Doubling *)

Lemma Zdouble_mult : forall z, Zdouble z = (Zpos 2) * z.
Proof.
  reflexivity.
Qed.

Lemma Zdouble_plus_one_mult : forall z,
  Zdouble_plus_one z = (Zpos 2) * z + (Zpos 1).
Proof.
  destruct z; simpl; auto with zarith.
Qed.

(** ** Multiplication and Opposite *)

Theorem Zopp_mult_distr_l : forall n m:Z, - (n * m) = - n * m.
Proof.
  intros x y; destruct x; destruct y; reflexivity.
Qed.

Theorem Zopp_mult_distr_r : forall n m:Z, - (n * m) = n * - m.
Proof.
  intros x y; rewrite (Zmult_comm x y); rewrite Zopp_mult_distr_l;
    apply Zmult_comm.
Qed.

Lemma Zopp_mult_distr_l_reverse : forall n m:Z, - n * m = - (n * m).
Proof.
  intros x y; symmetry ; apply Zopp_mult_distr_l.
Qed.

Theorem Zmult_opp_comm : forall n m:Z, - n * m = n * - m.
Proof.
  intros x y; rewrite Zopp_mult_distr_l_reverse; rewrite Zopp_mult_distr_r;
    trivial with arith.
Qed.

Theorem Zmult_opp_opp : forall n m:Z, - n * - m = n * m.
Proof.
  intros x y; destruct x; destruct y; reflexivity.
Qed.

Theorem Zopp_eq_mult_neg_1 : forall n:Z, - n = n * Zneg 1.
Proof.
  intro x; induction x; intros; rewrite Zmult_comm; auto with arith.
Qed.

(** ** Distributivity of multiplication over addition *)

Lemma weak_Zmult_plus_distr_r :
  forall (p:positive) (n m:Z), Zpos p * (n + m) = Zpos p * n + Zpos p * m.
Proof.
 intros x [ |y|y] [ |z|z]; simpl; trivial; f_equal;
  apply Pmult_plus_distr_l || rewrite Pmult_compare_mono_l;
  case_eq ((y ?= z) Eq)%positive; intros H; trivial;
   rewrite Pmult_minus_distr_l; trivial; now apply ZC1.
Qed.

Theorem Zmult_plus_distr_r : forall n m p:Z, n * (m + p) = n * m + n * p.
Proof.
 intros [|x|x] y z. trivial.
 apply weak_Zmult_plus_distr_r.
 apply Zopp_inj; rewrite Zopp_plus_distr, !Zopp_mult_distr_l, !Zopp_neg.
 apply weak_Zmult_plus_distr_r.
Qed.

Theorem Zmult_plus_distr_l : forall n m p:Z, (n + m) * p = n * p + m * p.
Proof.
  intros n m p; rewrite Zmult_comm; rewrite Zmult_plus_distr_r;
    do 2 rewrite (Zmult_comm p); trivial with arith.
Qed.

(** ** Distributivity of multiplication over subtraction *)

Lemma Zmult_minus_distr_r : forall n m p:Z, (n - m) * p = n * p - m * p.
Proof.
  intros x y z; unfold Zminus.
  rewrite <- Zopp_mult_distr_l_reverse.
  apply Zmult_plus_distr_l.
Qed.


Lemma Zmult_minus_distr_l : forall n m p:Z, p * (n - m) = p * n - p * m.
Proof.
  intros x y z; rewrite (Zmult_comm z (x - y)).
  rewrite (Zmult_comm z x).
  rewrite (Zmult_comm z y).
  apply Zmult_minus_distr_r.
Qed.

(** ** Simplification of multiplication for non-zero integers *)

Lemma Zmult_reg_l : forall n m p:Z, p <> Z0 -> p * n = p * m -> n = m.
Proof.
  intros x y z H H0.
  generalize (Zeq_minus _ _ H0).
  intro.
  apply Zminus_eq.
  rewrite <- Zmult_minus_distr_l in H1.
  clear H0; destruct (Zmult_integral _ _ H1).
  contradiction.
  trivial.
Qed.

Lemma Zmult_reg_r : forall n m p:Z, p <> Z0 -> n * p = m * p -> n = m.
Proof.
  intros x y z Hz.
  rewrite (Zmult_comm x z).
  rewrite (Zmult_comm y z).
  intro; apply Zmult_reg_l with z; assumption.
Qed.

(** ** Addition and multiplication by 2 *)

Lemma Zplus_diag_eq_mult_2 : forall n:Z, n + n = n * Zpos 2.
Proof.
  intros x; pattern x at 1 2; rewrite <- (Zmult_1_r x);
    rewrite <- Zmult_plus_distr_r; reflexivity.
Qed.

(** ** Multiplication and successor *)

Lemma Zmult_succ_r : forall n m:Z, n * Zsucc m = n * m + n.
Proof.
  intros n m; unfold Zsucc; rewrite Zmult_plus_distr_r;
    rewrite (Zmult_comm n (Zpos 1)); rewrite Zmult_1_l;
      trivial with arith.
Qed.

Lemma Zmult_succ_r_reverse : forall n m:Z, n * m + n = n * Zsucc m.
Proof.
  intros; symmetry ; apply Zmult_succ_r.
Qed.

Lemma Zmult_succ_l : forall n m:Z, Zsucc n * m = n * m + m.
Proof.
  intros n m; unfold Zsucc; rewrite Zmult_plus_distr_l;
    rewrite Zmult_1_l; trivial with arith.
Qed.

Lemma Zmult_succ_l_reverse : forall n m:Z, n * m + m = Zsucc n * m.
Proof.
  intros; symmetry; apply Zmult_succ_l.
Qed.



(** ** Misc redundant properties *)

Lemma Z_eq_mult : forall n m:Z, m = Z0 -> m * n = Z0.
Proof.
  intros x y H; rewrite H; auto with arith.
Qed.



(**********************************************************************)
(** * Relating binary positive numbers and binary integers *)

Lemma Zpos_eq : forall p q:positive, p = q -> Zpos p = Zpos q.
Proof.
  intros; f_equal; auto.
Qed.

Lemma Zpos_eq_rev : forall p q:positive, Zpos p = Zpos q -> p = q.
Proof.
  inversion 1; auto.
Qed.

Lemma Zpos_eq_iff : forall p q:positive, p = q <-> Zpos p = Zpos q.
Proof.
  split; [apply Zpos_eq|apply Zpos_eq_rev].
Qed.

Lemma Zpos_xI : forall p:positive, Zpos p~1 = Zpos 2 * Zpos p + Zpos 1.
Proof.
  intro; apply refl_equal.
Qed.

Lemma Zpos_xO : forall p:positive, Zpos p~0 = Zpos 2 * Zpos p.
Proof.
  intro; apply refl_equal.
Qed.

Lemma Zneg_xI : forall p:positive, Zneg p~1 = Zpos 2 * Zneg p - Zpos 1.
Proof.
  intro; apply refl_equal.
Qed.

Lemma Zneg_xO : forall p:positive, Zneg p~0 = Zpos 2 * Zneg p.
Proof.
  reflexivity.
Qed.

Lemma Zpos_plus_distr : forall p q:positive, Zpos (p + q) = Zpos p + Zpos q.
Proof.
  intros p p'; destruct p;
    [ destruct p' as [p0| p0| ]
      | destruct p' as [p0| p0| ]
      | destruct p' as [p| p| ] ]; reflexivity.
Qed.

Lemma Zneg_plus_distr : forall p q:positive, Zneg (p + q) = Zneg p + Zneg q.
Proof.
  intros p p'; destruct p;
    [ destruct p' as [p0| p0| ]
      | destruct p' as [p0| p0| ]
      | destruct p' as [p| p| ] ]; reflexivity.
Qed.

(**********************************************************************)
(** * Order relations *)

Definition Zlt (x y:Z) := (x ?= y) = Lt.
Definition Zgt (x y:Z) := (x ?= y) = Gt.
Definition Zle (x y:Z) := (x ?= y) <> Gt.
Definition Zge (x y:Z) := (x ?= y) <> Lt.
Definition Zne (x y:Z) := x <> y.

Infix "<=" := Zle : Z_scope.
Infix "<" := Zlt : Z_scope.
Infix ">=" := Zge : Z_scope.
Infix ">" := Zgt : Z_scope.

Notation "x <= y <= z" := (x <= y /\ y <= z) : Z_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : Z_scope.
Notation "x < y < z" := (x < y /\ y < z) : Z_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : Z_scope.

Lemma Zpos_lt : forall p q, Zlt (Zpos p) (Zpos q) <-> Plt p q.
Proof.
 intros. apply iff_refl.
Qed.

Lemma Zpos_le : forall p q, Zle (Zpos p) (Zpos q) <-> Ple p q.
Proof.
 intros. apply iff_refl.
Qed.

(**********************************************************************)
(** * Minimum and maximum *)

Definition Zmax (n m:Z) :=
  match n ?= m with
    | Eq | Gt => n
    | Lt => m
  end.

Definition Zmin (n m:Z) :=
  match n ?= m with
    | Eq | Lt => n
    | Gt => m
  end.

(**********************************************************************)
(** * Absolute value on integers *)

Definition Zabs_nat (x:Z) : nat :=
  match x with
    | Z0 => 0%nat
    | Zpos p => nat_of_P p
    | Zneg p => nat_of_P p
  end.

Definition Zabs (z:Z) : Z :=
  match z with
    | Z0 => Z0
    | Zpos p => Zpos p
    | Zneg p => Zpos p
  end.

(**********************************************************************)
(** * From [nat] to [Z] *)

Definition Z_of_nat (x:nat) :=
  match x with
    | O => Z0
    | S y => Zpos (P_of_succ_nat y)
  end.

Definition Zabs_N (z:Z) :=
  match z with
    | Z0 => 0%N
    | Zpos p => Npos p
    | Zneg p => Npos p
  end.

Definition Z_of_N (x:N) :=
  match x with
    | N0 => Z0
    | Npos p => Zpos p
  end.