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
|
(in-package "ACL2")
;this book contains very basic expt stuff (i couldn't include expt.lisp in basic.lisp because of a circular dependency)
;todo:
;make a separate expt-proofs book
;there's a distinction between expt and expt-2 rules
;make consistent names: expt vs. expt2
;think about the rules to combine and split exponents
;generalize some of these rules to be about expt with any base (not just 2)
;remove this?
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(include-book "ground-zero")
(include-book "negative-syntaxp")
(local (include-book "predicate"))
(local (include-book "fp2"))
(local (include-book "numerator"))
(local (include-book "denominator"))
(local (include-book "integerp"))
(local (include-book "fl")) ;or use floor?
(local (include-book "arith2"))
(encapsulate
()
(local (include-book "arithmetic/top" :dir :system))
(defthm a16
(equal (expt (* a b) i)
(* (expt a i) (expt b i)))
:hints
(("Goal" :in-theory (enable distributivity-of-expt-over-*))))
;gen
;split off the non-integer case
;make an expt2-split?
;instead of i1 and i2, use i and j?
(defthmd expt-split
(implies (and (integerp i)
(integerp j)
(case-split (acl2-numberp r)) ;(integerp r)
(case-split (not (equal r 0)))
)
(equal (expt r (+ i j))
(* (expt r i) (expt r j))))
:hints (("Goal" :in-theory (enable expt)))
)
)
(theory-invariant
(not (and (active-runep '(:rewrite expt-split))
(active-runep '(:rewrite a15))))
:key expt-split-invariant)
(theory-invariant
(not (and (active-runep '(:rewrite expt-split))
(active-runep '(:definition expt))))
:key expt-split-invariant-2)
;see also a14
;generalize? use arith books?
(defthm expt-2-positive-rational-type
(and (rationalp (expt 2 i))
(< 0 (expt 2 i)))
:hints (("Goal" :in-theory (enable expt) ))
:rule-classes (:rewrite (:type-prescription :typed-term (expt 2 i))))
;like a14?
(defthm expt-2-positive-integer-type
(implies (<= 0 i)
(and (integerp (expt 2 i))
(< 0 (expt 2 i))))
:hints (("Goal" :in-theory (enable expt)))
:rule-classes (:type-prescription))
;someday our rules may be better, but right now, ours only talk about when when the base is 2
;(in-theory (disable (:TYPE-PRESCRIPTION EXPT)))
;the rewrite rule counterpart to expt-2-positive-integer-type
(defthm expt-2-integerp
(implies (<= 0 i)
(integerp (expt 2 i))))
(defthm expt-2-type-linear
(implies (<= 0 i)
(<= 1 (expt 2 i)))
:rule-classes ((:linear :trigger-terms ((expt 2 i)))))
;when you disable either of the two rules below, you might want to also disable expt-compare?
;took these rules out of :rewrite since we have expt-compare?
;are these bad :linear rules because they have free vars?
(encapsulate
()
(local (defthm expt-strong-monotone-1
(implies (and (integerp n)
(integerp k)
(> k 0))
(< (expt 2 n) (expt 2 (+ n k))))
:hints (("Goal" :in-theory (enable expt
)))
:rule-classes ()))
(defthmd expt-strong-monotone
(implies (and (integerp n)
(integerp m))
(equal (< (expt 2 n) (expt 2 m))
(< n m)))
:hints (("Goal" :use ((:instance expt-strong-monotone-1 (k (- m n)))
(:instance expt-strong-monotone-1 (k (- n m)) (n m))
)))))
;improve to handle i non-integer?
(defthm expt2-integer
(implies (case-split (integerp i))
(equal (integerp (expt 2 i))
(<= 0 i)))
:hints (("Goal" :use (:instance expt-strong-monotone (n i) (m 0)))))
;trying :match-free :all
;why disabled?
(defthmd expt-strong-monotone-linear
(implies (and (< n m)
(case-split (integerp n))
(case-split (integerp m))
)
(< (expt 2 n) (expt 2 m)))
:rule-classes ((:linear :match-free :all))
:hints (("Goal" :use expt-strong-monotone)))
;why disabled?
(defthmd expt-weak-monotone
(implies (and (integerp n)
(integerp m))
(equal (<= (expt 2 n) (expt 2 m))
(<= n m)))
:hints (("Goal" :use (expt-strong-monotone
(:instance expt-strong-monotone (m n) (n m))))))
;disabled because of the free var
;but is occasionally helpful
;make linear?
;BOZO rename params?
;trying :match-free :all
(defthmd expt-weak-monotone-linear
(implies (and (<= n m)
(case-split (integerp n))
(case-split (integerp m)))
(<= (expt 2 n) (expt 2 m)))
:rule-classes ((:linear :match-free :all))
:hints (("Goal" :use (expt-strong-monotone
(:instance expt-strong-monotone (m n) (n m))))))
;generalize? rewrite (< (expt 2 i) k) to a comparison of (expt 2 i) and the greatest power of 2 <= k
;is this what expt-compare does?
(defthmd expt-between-one-and-two
(implies (and (<= 1 (expt 2 i))
(< (expt 2 i) 2))
(equal (expt 2 i) 1))
:hints (("goal"
:in-theory (enable expt zip))
("subgoal *1/7" :use (:instance expt-weak-monotone (n (+ i 1)) (m 0)))))
;We could disable this if it causes problems (but it seems okay).
;should always use case-split n hyps that say exponents are integers
(defthm expt-with-i-non-integer
(implies (not (integerp i))
(equal (expt r i)
1))
:hints (("Goal" :in-theory (enable expt))))
(defthmd expt-minus-helper
(equal (expt r (* -1 i))
(/ (expt r i)))
:otf-flg t
:hints (("Goal" :cases ((integerp i) (and (not (integerp i)) (acl2-numberp i)))
:in-theory (enable expt))))
;BOZO disable or enable by default?
;Loops with expt-inverse. How do we want to handle this??
;I'd rather have the inverting done outside EXPT since most rules don't look inside EXPT.
;This rule can be said to scatter exponents...
(defthmd expt-minus
(implies (syntaxp (negative-syntaxp i))
(equal (expt r i)
(/ (expt r (* -1 i)))))
:hints (("Goal" :in-theory (enable expt-minus-helper
expt-split))))
(local (in-theory (enable expt-minus)))
;This can loop with expt-minus (see theory-invariant).
(defthmd expt-inverse
(equal (/ (expt 2 i))
(expt 2 (* -1 i))))
(theory-invariant
(not (and (active-runep '(:rewrite expt-minus))
(active-runep '(:rewrite expt-inverse))))
:key expt-minus-invariant)
;could prove a rule for (expt (* -1 r) i) ... or maybe we have a rule for expt when r is a product...
;rename to expt-gather? !
;Note that this rule isn't enough to gather exponents in every situation. For example, two factors, (expt 2 i)
;and (expt 2 j), won't be gathered if there are intervening factors between them.
;BOZO change param names
(defthmd a15
(implies (and (rationalp i)
(not (equal i 0))
(integerp j1)
(integerp j2))
(and (equal (* (expt i j1) (expt i j2))
(expt i (+ j1 j2)))
(equal (* (expt i j1) (* (expt i j2) x))
(* (expt i (+ j1 j2)) x))))
:hints (("Goal" :in-theory (enable expt-split))))
(defthm expt-r-0
(equal (expt r 0)
1)
:hints (("Goal" :in-theory (enable expt))))
(defthm expt-0-i
(implies (and (case-split (integerp i)) ;since expt with a non-integer index is 1
(case-split (not (equal 0 i))) ;since (expt 0 0) is 1
)
(equal (expt 0 i)
0))
:hints (("Goal" :in-theory (enable expt))))
;==== A scheme for preventing massively expensive calls to expt =======
#| When ACL2 encounters a function call with constant arguments, the simplifier just evaluates the function on
those arguments. However, calls of (expt r i) with huge i can be very expensive to compute. (I suppose calls
with huge r might be very expensive too, but r is almost always 2 in my work.) The scheme below prevents
(expt r i) from being evaluated when i is too large (but allows evaluation in the case of small i).
|#
(in-theory (disable (:executable-counterpart expt)))
(set-compile-fns t)
(defun expt-execute (r i) (expt r i))
;Allows expt calls with small exponents to be computed
;You can change 1000 to your own desired bound.
(defthm expt-execute-rewrite
(implies (and (syntaxp (and (quotep r) (quotep i) (< (abs (cadr i)) 1000))))
(equal (expt r i)
(expt-execute r i))))
#|
The rules below are not complete, I proved them as needed to simplify terms like:
(* x
(EXPT 2 1000001)
(/ (EXPT 2 1000000))
y)
Note that we could just compute (EXPT 2 1000001) and (EXPT 2 1000000) but that would be very expensive.
Perhaps we can make this into a complete theory, based on the observation that if a product contains two
factors of the form (expt 2 k) of (/ (expt 2 k)), where k is a constant, those factors will be brought
together because they are very close in the term order used order arguments to * (recall that unary-/ is
ignored when we decide how to order arguments to *).
|#
;this could be made more general (replace the lhs with its second arg...)
(defthm expt2-constants-collect-special-1
(implies (and (syntaxp (and (quotep i1) (quotep i2)))
(case-split (rationalp y))
(case-split (integerp i1))
(case-split (integerp i2)))
(equal (* (EXPT 2 i1)
(/ (EXPT 2 i2))
y)
(* (expt 2 (- i1 i2)) y)))
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'())))
)
(defthm expt2-constants-collect-special-2
(implies (and (syntaxp (and (quotep i1) (quotep i2)))
(case-split (integerp i1))
(case-split (integerp i2)))
(equal (* (EXPT 2 i1)
(/ (EXPT 2 i2))
)
(expt 2 (- i1 i2))))
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'())))
)
(defthm expt2-constants-collect-special-4
(implies (and (syntaxp (and (quotep i1) (quotep i2)))
(case-split (rationalp y))
(case-split (integerp i1))
(case-split (integerp i2)))
(equal (* (/ (EXPT 2 i2)) (EXPT 2 i1) y)
(* (expt 2 (- i1 i2)) y)))
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'()))))
(defthm expt2-constants-collect-special-5
(implies (and (syntaxp (and (quotep i1) (quotep i2)))
(case-split (integerp i1))
(case-split (integerp i2)))
(equal (* (/ (EXPT 2 i2)) (EXPT 2 i1))
(expt 2 (- i1 i2))))
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'()))))
;will this happen?
(defthm expt2-constants-collect-special-6
(implies (and (syntaxp (and (quotep i1) (quotep i2)))
(case-split (rationalp x))
(case-split (integerp i1))
(case-split (integerp i2)))
(equal (* (EXPT 2 i2) x (EXPT 2 i1))
(* (expt 2 (+ i1 i2)) x)))
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'()))))
;whoa this one is sort of different... (it rewrites an equality)
(defthm expt2-constants-collect-special-3
(implies (and (syntaxp (and (quotep i1) (quotep i2)))
(case-split (rationalp x))
(case-split (integerp i1))
(case-split (integerp i2)))
(equal (equal (* x (EXPT 2 i1)) (EXPT 2 i2))
(equal x (expt 2 (- i2 i1)))))
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'())))
)
;==================================================================
;expt-compare
;handle constants as args?
(defthm expt2-1-to-1
(implies (and (integerp i1)
(integerp i2))
(equal (equal (expt 2 i1) (expt 2 i2))
(equal i1 i2)))
:hints (("Goal"
:use ((:instance expt-strong-monotone (n i1) (m i2))
(:instance expt-strong-monotone (n i2) (m i1))))))
;could gen? move hyps to concl?
(defthm expt-even
(implies (and (< 0 i)
(case-split (integerp i))
)
(integerp (* 1/2 (expt 2 i))))
:hints (("goal" :in-theory (enable expt))))
;generalize rules like this with a power2-syntaxp (not power2p!) ?
;make conclusion an equality?
(defthm expt-quotient-integerp
(implies (and (<= j i)
(case-split (integerp i))
(case-split (integerp j))
)
(integerp (* (expt 2 i) (/ (expt 2 j)))))
:rule-classes (:rewrite :type-prescription)
:hints (("Goal" :in-theory (e/d (expt-split) ( expt-2-integerp))
:use (:instance expt-2-positive-integer-type (i (- i j))))))
(defthm expt-quotient-integerp-alt
(implies (and (<= j i)
(case-split (integerp i))
(case-split (integerp j))
)
(integerp (* (/ (expt 2 j)) (expt 2 i))))
:rule-classes (:rewrite :type-prescription))
;is there a 2 term version?
(defthm expt-prod-integer-3-terms
(implies (and (integerp n)
(<= 0 (+ i j))
(integerp i)
(integerp j)
)
(integerp (* (expt 2 i) (expt 2 j) n)))
:hints (("Goal" :in-theory (enable a15))))
;drop these?
;generalize to comparisons to any constant (any power of 2)?
;bad name?
(defthm expt2-inverse-integer
(implies (case-split (integerp i))
(equal (integerp (/ (expt 2 i)))
(<= i 0)))
:hints (("Goal" :in-theory (disable expt2-integer)
:use (:instance expt2-integer (i (- i))))))
;figure out a better solution to this problem
;perhaps say if a term is a power of 2, then it's an integer iff its expo is >=0
(defthm expt-prod-integer-3-terms-2
(implies (and (<= 0 (+ i (- j) (- l)))
(integerp i)
(integerp j)
(integerp l)
)
(integerp (* (expt 2 i) (/ (expt 2 j)) (/ (expt 2 l)))))
:hints (("Goal" :in-theory (set-difference-theories (enable a15 expt-inverse)
'(expt-minus))))
)
#| would be nice (use expt2-1-to-1)?
(defthm expt2-equal-1
(implies (integerp i)
(equal (EQUAL (EXPT 2 i) 1)
(equal i 0)))
; :rule-classes nil
:hints (("Goal" :in-theory (enable expt-split-rewrite)))
)
|#
(defthm expt2-inverse-even
(implies (case-split (integerp i))
(equal (integerp (* 1/2 (/ (expt 2 i))))
(<= (+ 1 i) 0)))
:otf-flg t
:hints (("Goal" :in-theory (set-difference-theories
(enable expt-split)
'(expt2-integer EXPT2-INVERSE-INTEGER))
:use (:instance expt2-integer (i (+ -1 (- i)))))))
;loops with a15?
; (expt (* 2 i)) was matching with (expt 2 0) (booo!) so I added the syntaxp hyp
(defthmd expt-2-split-product-index
(implies (and (syntaxp (not (quotep i)))
(case-split (rationalp r))
(case-split (integerp i)))
(equal (expt r (* 2 i))
(* (expt r i) (expt r i))))
:hints (("Goal" :in-theory (disable expt-split)
:use (:instance expt-split (i i) (j i)))))
;linear?
(defthm expt-bigger-than-i
(implies (integerp i)
(< i (expt 2 i)))
:hints (("Goal" :in-theory (enable expt)))
)
;BOZO this might loop with expt-split
;i'm not sure that this is a good rewrite anyway
(defthmd expt-compare-with-double
(implies (and (integerp x)
(integerp i))
(equal (< (* 2 x) (expt 2 i))
(< x (expt 2 (+ -1 i)))))
:hints (("Goal" :in-theory (enable expt-split))))
;leave this disabled?
(defthmd expt-2-reduce-leading-constant-gen
(implies (case-split (integerp (+ k d)))
(equal (expt 2 (+ k d))
(* (expt 2 (fl k)) (expt 2 (+ (mod k 1) d)))))
:hints (("Goal" :in-theory (set-difference-theories
(enable mod)
'(expt-split))
:use (:instance expt-split (r 2) (i (fl k)) (j (+ (mod k 1) d))))))
;handles the case when k isn't even an integer!
;loops with a15! add theory invariant....
(defthmd expt-2-reduce-leading-constant
(implies (and (syntaxp (and (quotep k)
(or (>= (cadr k) 1) (< (cadr k) 0))))
(case-split (integerp (+ k d)))
)
(equal (expt 2 (+ k d))
(* (expt 2 (fl k)) (expt 2 (+ (mod k 1) d)))))
:hints (("Goal" :in-theory (set-difference-theories
(enable)
'(expt-split))
:use (expt-2-reduce-leading-constant-gen
(:instance expt-split (r 2) (i (fl k)) (j (+ (mod k 1) d)))))))
; BOZO better than a15?
(defthmd expt-combine
(implies (and (case-split (rationalp r))
(case-split (not (equal r 0)))
(case-split (integerp i1))
(case-split (integerp i2)))
(and (equal (* (expt r i1) (expt r i2))
(expt r (+ i1 i2)))
(equal (* (expt r i1) (* (expt r i2) x))
(* (expt r (+ i1 i2)) x))))
:hints (("goal" :in-theory (enable a15))))
;remove since we have expt-compare?
(defthm expt-with-small-n
(implies (<= n 0)
(<= (expt 2 n) 1))
:hints (("Goal" :use (:instance expt-weak-monotone (m 0))))
:rule-classes (:linear))
#|
(include-book
"factor-2")
;which way do we want to do this?
;disable later?
;add a "can have a 2 multiplied in" hyp to this series?
(defthm expt-2-combine-like-is
(implies (and (syntaxp (should-have-a-2-factor-multiplied-in i))
(integerp i))
(equal (* (expt 2 i) (expt 2 i))
(expt 2 (* 2 i))))
:hints (("Goal" :in-theory (disable expt-split)
:use (:instance expt-split (r 2) (i i) (j i)))))
(defthm expt-2-combine-like-is-3-and-4-of-6
(implies (and (syntaxp (should-have-a-2-factor-multiplied-in i))
(integerp i)
(rationalp a)
(rationalp b)
(rationalp c)
(rationalp d)
)
(equal (* a b (expt 2 i) (expt 2 i) c d )
(* a b c d (expt 2 (* 2 i)))))
:hints (("Goal" :in-theory (disable expt-split)
:use (:instance expt-split (r 2) (i i) (j i)))))
(defthm expt-2-combine-like-is-4-and-5-of-6
(implies (and (syntaxp (should-have-a-2-factor-multiplied-in i))
(integerp i)
(rationalp a)
(rationalp b)
(rationalp c)
(rationalp d)
)
(equal (* a b c (expt 2 i) (expt 2 i) d )
(* a b c d (expt 2 (* 2 i)))))
:hints (("Goal" :in-theory (disable expt-split)
:use (:instance expt-split (r 2) (i i) (j i)))))
(defthm expt-2-combine-like-is-inverted
(implies (and (syntaxp (should-have-a-2-factor-multiplied-in i))
(integerp i))
(equal (* (/ (EXPT 2 i))
(/ (EXPT 2 i)))
(/ (expt 2 (* 2 i)))))
:hints (("Goal" :in-theory (disable
expt-2-combine-like-is
expt-split)
:use (:instance expt-split (r 2) (i (* 1/2 i)) (j (* 1/2 i))))))
|#
#|
(defthm expt-prod-integer-4-terms
(implies (and (integerp i)
(integerp j)
(integerp l)
(<= 0 (+ i (- j) l))
(integerp n))
(integerp (* (expt 2 i) (/ (expt 2 j)) (expt 2 l) n)))
:hints (("Goal" :in-theory (set-difference-theories (enable a15 expt-inverse)
'(expt-minus))))
)
would be nice (use expt2-1-to-1)?
(defthm expt2-equal-1
(implies (integerp i)
(equal (EQUAL (EXPT 2 i) 1)
(equal i 0)))
; :rule-classes nil
:hints (("Goal" :in-theory (enable expt-split)))
)
;remove?
;actually, maybe this is good whether we are scattering or gathering...
;bad name?
;in general, are there rules which are good for scattering and for gathering?
(defthm expt-simp
(implies (integerp x)
(equal (* 2 (EXPT 2 (+ -1 x)))
(expt 2 x)))
:hints (("Goal" :use (:instance a15 (i 2) (j1 1) (j2 (+ -1 x))))))
(defthmd expt-next
(implies (and (integerp i1)
(integerp i2)
(< (expt 2 i1) (expt 2 i2)))
(<= (expt 2 i1) (expt 2 (+ -1 i2)))))
|#
(local (include-book "even-odd"))
;move? make fw-chaining rule?
(defthmd even-not-equal-odd
(implies (and (evenp x)
(oddp y))
(not (equal x y)))
:hints (("Goal" :in-theory (enable oddp))))
;this is sort of strange...
(defthm expt-2-is-not-odd
(implies (and (evenp x)
(< 0 i) ;drop?
(integerp i)
)
(equal (equal (expt 2 i)
(+ 1 x))
nil))
:hints (("Goal" :in-theory (enable evenp oddp even-not-equal-odd))) ;shouldn't have to enable oddp
:otf-flg t)
(defthm a14
(and
(implies (and (integerp i)
(<= 0 i)
(<= 0 j))
(and (integerp (expt i j))
(<= 0 (expt i j))))
(implies (and (rationalp i)
(not (equal i 0)))
(not (equal (expt i j) 0))))
:hints
(("Goal" :in-theory (enable expt)))
:rule-classes
((:type-prescription
:corollary
(implies (and (integerp i)
(<= 0 i)
(<= 0 j))
(and (integerp (expt i j))
(<= 0 (expt i j)))))
(:type-prescription
:corollary
(implies (and (rationalp i)
(not (equal i 0)))
(not (equal (expt i j) 0))))))
#|
;this will get rewritten away?
(defthm expt-in-product-linear
(implies (and (<= 0 i)
(<= 0 x)
(case-split (rationalp x))
)
(<= x (* x (expt 2 i))))
:rule-classes (:linear)
)
;this will get rewritten away?
(defthm expt-in-product-linear-2
(implies (and (case-split (<= i 0))
(case-split (<= 0 x))
(case-split (rationalp x))
)
(<= (* x (expt 2 i)) x))
:rule-classes (:linear)
)
|#
;crap. This unifies with (EXPT '2 '0), which we see because we disable the executable counterpart of expt to
;prevent massively expensive calls.
;loops with a15!!
;add theory invar!
;does this already exist?
(defthmd expt-with-product-exponent
(implies (and (syntaxp (not (quotep i)))
(case-split (integerp i)))
(equal (expt 2 (* 2 i))
(* (expt 2 i) (expt 2 i))))
:hints (("Goal" :in-theory (enable a15))))
;yuck
;perhaps use only expt-2-positive-integer-type;
;don't need this if natp is enabled?
(defthmd natp-expt
(implies (natp n)
(and (integerp (expt 2 n))
(< 0 (expt 2 n))))
:rule-classes (:type-prescription :rewrite))
#|
these deal with arbitrary bases (not just 2):
(local (include-book
"../../../arithmetic-2/meta/expt"))
(local
(include-book
"../arithmetic/product"))
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(>= x 0))))
;allows both a and b to be non-integers:
(defthm expt-non-negative
(implies (and (<= 0 a)
(<= 0 b)
(case-split (rationalp a))
)
(<= 0 (expt a b))))
(defthm expt-integerp
(implies (and (natp a)
(<= 0 b)
)
(integerp (expt a b))))
|#
;there are funny little rules...
(defthm expt-exceeds-another-by-more-than-1
(implies (and (<= 0 i)
(<= 0 j)
(integerp i)
(integerp j)
)
(implies (< (+ 1 i) j)
(< (+ 1 (expt 2 i)) (expt 2 j))))
:hints (("Goal" :in-theory (enable expt-split)
:use (:instance expt-strong-monotone (n (+ 1 i)) (m j)))))
(defthm expt-exceeds-2
(IMPLIES (AND (< i j)
(<= 0 i)
(<= 0 j)
(INTEGERP i)
(INTEGERP j)
)
(<= (+ 1 (EXPT 2 i)) (EXPT 2 j)))
:rule-classes (:rewrite :linear)
:hints (("Goal" :use (:instance expt-strong-monotone (n i) (m j)))))
#|
Are there some rules (such as this one) which we want enabled for both scaterring and gathering exponents?
(defthm expt-hack
(equal (* (expt 2 n) (expt 2 (* -1 n))) 1)
:hints (("Goal" :in-theory (e/d () (EXPT-minus)))))
(defthm expt-hack-2
(equal (* (expt 2 (* -1 n)) (expt 2 n)) 1)
:hints (("Goal" :in-theory (e/d () (EXPT-minus)))))
|#
(defthm expt-with-i-non-integer-special
(implies (not (integerp i))
(equal (EXPT 2 (+ -1 i))
(if (acl2-numberp i)
1
1/2))))
|
