forked from AbsInt/CompCert
-
Notifications
You must be signed in to change notification settings - Fork 2
/
ConstpropOpproof.v
747 lines (704 loc) · 29.4 KB
/
ConstpropOpproof.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
(* *********************************************************************)
(* *)
(* The Compcert verified compiler *)
(* *)
(* Xavier Leroy, INRIA Paris *)
(* *)
(* Copyright Institut National de Recherche en Informatique et en *)
(* Automatique. All rights reserved. This file is distributed *)
(* under the terms of the INRIA Non-Commercial License Agreement. *)
(* *)
(* *********************************************************************)
(** Correctness proof for operator strength reduction. *)
Require Import Coqlib Compopts.
Require Import Integers Floats Values Memory Globalenvs Events.
Require Import Op Registers RTL ValueDomain.
Require Import ConstpropOp.
Section STRENGTH_REDUCTION.
Variable bc: block_classification.
Variable ge: genv.
Hypothesis GENV: genv_match bc ge.
Variable sp: block.
Hypothesis STACK: bc sp = BCstack.
Variable ae: AE.t.
Variable e: regset.
Variable m: mem.
Hypothesis MATCH: ematch bc e ae.
Lemma match_G:
forall r id ofs,
AE.get r ae = Ptr(Gl id ofs) -> Val.lessdef e#r (Genv.symbol_address ge id ofs).
Proof.
intros. apply vmatch_ptr_gl with bc; auto. rewrite <- H. apply MATCH.
Qed.
Lemma match_S:
forall r ofs,
AE.get r ae = Ptr(Stk ofs) -> Val.lessdef e#r (Vptr sp ofs).
Proof.
intros. apply vmatch_ptr_stk with bc; auto. rewrite <- H. apply MATCH.
Qed.
Ltac InvApproxRegs :=
match goal with
| [ H: _ :: _ = _ :: _ |- _ ] =>
injection H; clear H; intros; InvApproxRegs
| [ H: ?v = AE.get ?r ae |- _ ] =>
generalize (MATCH r); rewrite <- H; clear H; intro; InvApproxRegs
| _ => idtac
end.
Ltac SimplVM :=
match goal with
| [ H: vmatch _ ?v (I ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vint n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (L ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vlong n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (F ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vfloat n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (FS ?n) |- _ ] =>
let E := fresh in
assert (E: v = Vsingle n) by (inversion H; auto);
rewrite E in *; clear H; SimplVM
| [ H: vmatch _ ?v (Ptr(Gl ?id ?ofs)) |- _ ] =>
let E := fresh in
assert (E: Val.lessdef v (Genv.symbol_address ge id ofs)) by (eapply vmatch_ptr_gl; eauto);
clear H; SimplVM
| [ H: vmatch _ ?v (Ptr(Stk ?ofs)) |- _ ] =>
let E := fresh in
assert (E: Val.lessdef v (Vptr sp ofs)) by (eapply vmatch_ptr_stk; eauto);
clear H; SimplVM
| _ => idtac
end.
Lemma const_for_result_correct:
forall a op v,
const_for_result a = Some op ->
vmatch bc v a ->
exists v', eval_operation ge (Vptr sp Ptrofs.zero) op nil m = Some v' /\ Val.lessdef v v'.
Proof.
unfold const_for_result. generalize Archi.ptr64; intros ptr64; intros.
destruct a; inv H; SimplVM.
- (* integer *)
exists (Vint n); auto.
- (* integer or undef *)
exists (Vint n); split; auto. inv H0; auto.
- (* long *)
destruct ptr64; inv H2. exists (Vlong n); auto.
- (* float *)
destruct (Compopts.generate_float_constants tt); inv H2. exists (Vfloat f); auto.
- (* single *)
destruct (Compopts.generate_float_constants tt); inv H2. exists (Vsingle f); auto.
- (* pointer *)
destruct p; try discriminate; SimplVM.
+ (* global *)
inv H2. exists (Genv.symbol_address ge id ofs); auto.
+ (* stack *)
inv H2. exists (Vptr sp ofs); split; auto. simpl. rewrite Ptrofs.add_zero_l; auto.
Qed.
Lemma cond_strength_reduction_correct:
forall cond args vl,
vl = map (fun r => AE.get r ae) args ->
let (cond', args') := cond_strength_reduction cond args vl in
eval_condition cond' e##args' m = eval_condition cond e##args m.
Proof.
intros until vl. unfold cond_strength_reduction.
case (cond_strength_reduction_match cond args vl); simpl; intros; InvApproxRegs; SimplVM.
- apply Val.swap_cmp_bool.
- auto.
- apply Val.swap_cmpu_bool.
- auto.
- apply Val.swap_cmpl_bool.
- auto.
- apply Val.swap_cmplu_bool.
- auto.
- auto.
Qed.
Lemma make_cmp_base_correct:
forall c args vl,
vl = map (fun r => AE.get r ae) args ->
let (op', args') := make_cmp_base c args vl in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op' e##args' m = Some v
/\ Val.lessdef (Val.of_optbool (eval_condition c e##args m)) v.
Proof.
intros. unfold make_cmp_base.
generalize (cond_strength_reduction_correct c args vl H).
destruct (cond_strength_reduction c args vl) as [c' args']. intros EQ.
econstructor; split. simpl; eauto. rewrite EQ. auto.
Qed.
Lemma make_cmp_correct:
forall c args vl,
vl = map (fun r => AE.get r ae) args ->
let (op', args') := make_cmp c args vl in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op' e##args' m = Some v
/\ Val.lessdef (Val.of_optbool (eval_condition c e##args m)) v.
Proof.
intros c args vl.
assert (Y: forall r, vincl (AE.get r ae) (Uns Ptop 1) = true ->
e#r = Vundef \/ e#r = Vint Int.zero \/ e#r = Vint Int.one).
{ intros. apply vmatch_Uns_1 with bc Ptop. eapply vmatch_ge. eapply vincl_ge; eauto. apply MATCH. }
unfold make_cmp. case (make_cmp_match c args vl); intros.
- unfold make_cmp_imm_eq.
destruct (Int.eq_dec n Int.one && vincl v1 (Uns Ptop 1)) eqn:E1.
+ simpl in H; inv H. InvBooleans. subst n.
exists (e#r1); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
+ destruct (Int.eq_dec n Int.zero && vincl v1 (Uns Ptop 1)) eqn:E0.
* simpl in H; inv H. InvBooleans. subst n.
exists (Val.xor e#r1 (Vint Int.one)); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
* apply make_cmp_base_correct; auto.
- unfold make_cmp_imm_ne.
destruct (Int.eq_dec n Int.zero && vincl v1 (Uns Ptop 1)) eqn:E0.
+ simpl in H; inv H. InvBooleans. subst n.
exists (e#r1); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
+ destruct (Int.eq_dec n Int.one && vincl v1 (Uns Ptop 1)) eqn:E1.
* simpl in H; inv H. InvBooleans. subst n.
exists (Val.xor e#r1 (Vint Int.one)); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
* apply make_cmp_base_correct; auto.
- unfold make_cmp_imm_eq.
destruct (Int.eq_dec n Int.one && vincl v1 (Uns Ptop 1)) eqn:E1.
+ simpl in H; inv H. InvBooleans. subst n.
exists (e#r1); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
+ destruct (Int.eq_dec n Int.zero && vincl v1 (Uns Ptop 1)) eqn:E0.
* simpl in H; inv H. InvBooleans. subst n.
exists (Val.xor e#r1 (Vint Int.one)); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
* apply make_cmp_base_correct; auto.
- unfold make_cmp_imm_ne.
destruct (Int.eq_dec n Int.zero && vincl v1 (Uns Ptop 1)) eqn:E0.
+ simpl in H; inv H. InvBooleans. subst n.
exists (e#r1); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
+ destruct (Int.eq_dec n Int.one && vincl v1 (Uns Ptop 1)) eqn:E1.
* simpl in H; inv H. InvBooleans. subst n.
exists (Val.xor e#r1 (Vint Int.one)); split; auto. simpl.
exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
* apply make_cmp_base_correct; auto.
- apply make_cmp_base_correct; auto.
Qed.
Lemma make_addimm_correct:
forall n r,
let (op, args) := make_addimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.add e#r (Vint n)) v.
Proof.
intros. unfold make_addimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros.
subst. exists (e#r); split; auto.
destruct (e#r); simpl; auto; rewrite ?Int.add_zero, ?Ptrofs.add_zero; auto.
destruct Archi.ptr64; auto.
exists (Val.add e#r (Vint n)); split; auto.
Qed.
Lemma make_shlimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_shlimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.shl e#r1 (Vint n)) v.
Proof.
intros; unfold make_shlimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto. rewrite Int.shl_zero. auto.
destruct (Int.ltu n Int.iwordsize).
econstructor; split. simpl. eauto. auto.
econstructor; split. simpl. eauto. rewrite H; auto.
Qed.
Lemma make_shrimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_shrimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.shr e#r1 (Vint n)) v.
Proof.
intros; unfold make_shrimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto. rewrite Int.shr_zero. auto.
destruct (Int.ltu n Int.iwordsize).
econstructor; split. simpl. eauto. auto.
econstructor; split. simpl. eauto. rewrite H; auto.
Qed.
Lemma make_shruimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_shruimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.shru e#r1 (Vint n)) v.
Proof.
intros; unfold make_shruimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto. rewrite Int.shru_zero. auto.
destruct (Int.ltu n Int.iwordsize).
econstructor; split. simpl. eauto. auto.
econstructor; split. simpl. eauto. rewrite H; auto.
Qed.
Lemma make_mulimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_mulimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.mul e#r1 (Vint n)) v.
Proof.
intros; unfold make_mulimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (Vint Int.zero); split; auto. destruct (e#r1); simpl; auto. rewrite Int.mul_zero; auto.
predSpec Int.eq Int.eq_spec n Int.one; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto. rewrite Int.mul_one; auto.
destruct (Int.is_power2 n) eqn:?; intros.
rewrite (Val.mul_pow2 e#r1 _ _ Heqo). econstructor; split. simpl; eauto. auto.
econstructor; split; eauto. simpl. rewrite H; auto.
Qed.
Lemma make_divimm_correct:
forall n r1 r2 v,
Val.divs e#r1 e#r2 = Some v ->
e#r2 = Vint n ->
let (op, args) := make_divimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some w /\ Val.lessdef v w.
Proof.
intros; unfold make_divimm.
predSpec Int.eq Int.eq_spec n Int.one; intros. subst. rewrite H0 in H.
destruct (e#r1) eqn:?;
try (rewrite Val.divs_one in H; exists (Vint i); split; simpl; try rewrite Heqv0; auto);
inv H; auto.
destruct (Int.is_power2 n) eqn:?.
destruct (Int.ltu i (Int.repr 31)) eqn:?.
exists v; split; auto. simpl. eapply Val.divs_pow2; eauto. congruence.
exists v; auto.
exists v; auto.
Qed.
Lemma make_divuimm_correct:
forall n r1 r2 v,
Val.divu e#r1 e#r2 = Some v ->
e#r2 = Vint n ->
let (op, args) := make_divuimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some w /\ Val.lessdef v w.
Proof.
intros; unfold make_divuimm.
predSpec Int.eq Int.eq_spec n Int.one; intros. subst. rewrite H0 in H.
destruct (e#r1) eqn:?;
try (rewrite Val.divu_one in H; exists (Vint i); split; simpl; try rewrite Heqv0; auto);
inv H; auto.
destruct (Int.is_power2 n) eqn:?.
econstructor; split. simpl; eauto.
rewrite H0 in H. erewrite Val.divu_pow2 by eauto. auto.
exists v; auto.
Qed.
Lemma make_moduimm_correct:
forall n r1 r2 v,
Val.modu e#r1 e#r2 = Some v ->
e#r2 = Vint n ->
let (op, args) := make_moduimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some w /\ Val.lessdef v w.
Proof.
intros; unfold make_moduimm.
destruct (Int.is_power2 n) eqn:?.
exists v; split; auto. simpl. decEq. eapply Val.modu_pow2; eauto. congruence.
exists v; auto.
Qed.
Lemma make_andimm_correct:
forall n r x,
vmatch bc e#r x ->
let (op, args) := make_andimm n r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.and e#r (Vint n)) v.
Proof.
intros; unfold make_andimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros.
subst n. exists (Vint Int.zero); split; auto. destruct (e#r); simpl; auto. rewrite Int.and_zero; auto.
predSpec Int.eq Int.eq_spec n Int.mone; intros.
subst n. exists (e#r); split; auto. destruct (e#r); simpl; auto. rewrite Int.and_mone; auto.
destruct (match x with Uns _ k => Int.eq (Int.zero_ext k (Int.not n)) Int.zero
| _ => false end) eqn:UNS.
destruct x; try congruence.
exists (e#r); split; auto.
inv H; auto. simpl. replace (Int.and i n) with i; auto.
generalize (Int.eq_spec (Int.zero_ext n0 (Int.not n)) Int.zero); rewrite UNS; intro EQ.
Int.bit_solve. destruct (zlt i0 n0).
replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)).
rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto.
rewrite <- EQ. rewrite Int.bits_zero_ext by lia. rewrite zlt_true by auto.
rewrite Int.bits_not by auto. apply negb_involutive.
rewrite H6 by auto. auto.
econstructor; split; eauto. auto.
Qed.
Lemma make_orimm_correct:
forall n r,
let (op, args) := make_orimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.or e#r (Vint n)) v.
Proof.
intros; unfold make_orimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros.
subst n. exists (e#r); split; auto. destruct (e#r); simpl; auto. rewrite Int.or_zero; auto.
predSpec Int.eq Int.eq_spec n Int.mone; intros.
subst n. exists (Vint Int.mone); split; auto. destruct (e#r); simpl; auto. rewrite Int.or_mone; auto.
econstructor; split; eauto. auto.
Qed.
Lemma make_xorimm_correct:
forall n r,
let (op, args) := make_xorimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.xor e#r (Vint n)) v.
Proof.
intros; unfold make_xorimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros.
subst n. exists (e#r); split; auto. destruct (e#r); simpl; auto. rewrite Int.xor_zero; auto.
predSpec Int.eq Int.eq_spec n Int.mone; intros.
subst n. exists (Val.notint e#r); split; auto.
econstructor; split; eauto. auto.
Qed.
Lemma make_addlimm_correct:
forall n r,
let (op, args) := make_addlimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.addl e#r (Vlong n)) v.
Proof.
intros. unfold make_addlimm.
predSpec Int64.eq Int64.eq_spec n Int64.zero; intros.
subst. exists (e#r); split; auto.
destruct (e#r); simpl; auto; rewrite ? Int64.add_zero, ? Ptrofs.add_zero; auto.
destruct Archi.ptr64; auto.
exists (Val.addl e#r (Vlong n)); split; auto.
Qed.
Lemma make_shllimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_shllimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.shll e#r1 (Vint n)) v.
Proof.
intros; unfold make_shllimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto.
unfold Int64.shl'. rewrite Z.shiftl_0_r, Int64.repr_unsigned. auto.
destruct (Int.ltu n Int64.iwordsize').
econstructor; split. simpl. eauto. auto.
econstructor; split. simpl. eauto. rewrite H; auto.
Qed.
Lemma make_shrlimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_shrlimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.shrl e#r1 (Vint n)) v.
Proof.
intros; unfold make_shrlimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto.
unfold Int64.shr'. rewrite Z.shiftr_0_r, Int64.repr_signed. auto.
destruct (Int.ltu n Int64.iwordsize').
econstructor; split. simpl. eauto. auto.
econstructor; split. simpl. eauto. rewrite H; auto.
Qed.
Lemma make_shrluimm_correct:
forall n r1 r2,
e#r2 = Vint n ->
let (op, args) := make_shrluimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.shrlu e#r1 (Vint n)) v.
Proof.
intros; unfold make_shrluimm.
predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto.
unfold Int64.shru'. rewrite Z.shiftr_0_r, Int64.repr_unsigned. auto.
destruct (Int.ltu n Int64.iwordsize').
econstructor; split. simpl. eauto. auto.
econstructor; split. simpl. eauto. rewrite H; auto.
Qed.
Lemma make_mullimm_correct:
forall n r1 r2,
e#r2 = Vlong n ->
let (op, args) := make_mullimm n r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.mull e#r1 (Vlong n)) v.
Proof.
intros; unfold make_mullimm.
predSpec Int64.eq Int64.eq_spec n Int64.zero; intros. subst.
exists (Vlong Int64.zero); split; auto. destruct (e#r1); simpl; auto. rewrite Int64.mul_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.one; intros. subst.
exists (e#r1); split; auto. destruct (e#r1); simpl; auto. rewrite Int64.mul_one; auto.
destruct (Int64.is_power2' n) eqn:?; intros.
exists (Val.shll e#r1 (Vint i)); split; auto.
destruct (e#r1); simpl; auto.
erewrite Int64.is_power2'_range by eauto.
erewrite Int64.mul_pow2' by eauto. auto.
econstructor; split; eauto. simpl; rewrite H; auto.
Qed.
Lemma make_divlimm_correct:
forall n r1 r2 v,
Val.divls e#r1 e#r2 = Some v ->
e#r2 = Vlong n ->
let (op, args) := make_divlimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some w /\ Val.lessdef v w.
Proof.
intros; unfold make_divlimm.
destruct (Int64.is_power2' n) eqn:?. destruct (Int.ltu i (Int.repr 63)) eqn:?.
rewrite H0 in H. econstructor; split. simpl; eauto. eapply Val.divls_pow2; eauto. auto.
exists v; auto.
exists v; auto.
Qed.
Lemma make_divluimm_correct:
forall n r1 r2 v,
Val.divlu e#r1 e#r2 = Some v ->
e#r2 = Vlong n ->
let (op, args) := make_divluimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some w /\ Val.lessdef v w.
Proof.
intros; unfold make_divluimm.
destruct (Int64.is_power2' n) eqn:?.
econstructor; split. simpl; eauto.
rewrite H0 in H. destruct (e#r1); inv H. destruct (Int64.eq n Int64.zero); inv H2.
simpl.
erewrite Int64.is_power2'_range by eauto.
erewrite Int64.divu_pow2' by eauto. auto.
exists v; auto.
Qed.
Lemma make_modluimm_correct:
forall n r1 r2 v,
Val.modlu e#r1 e#r2 = Some v ->
e#r2 = Vlong n ->
let (op, args) := make_modluimm n r1 r2 in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some w /\ Val.lessdef v w.
Proof.
intros; unfold make_modluimm.
destruct (Int64.is_power2 n) eqn:?.
exists v; split; auto. simpl. decEq.
rewrite H0 in H. destruct (e#r1); inv H. destruct (Int64.eq n Int64.zero); inv H2.
simpl. erewrite Int64.modu_and by eauto. auto.
exists v; auto.
Qed.
Lemma make_andlimm_correct:
forall n r x,
let (op, args) := make_andlimm n r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.andl e#r (Vlong n)) v.
Proof.
intros; unfold make_andlimm.
predSpec Int64.eq Int64.eq_spec n Int64.zero; intros.
subst n. exists (Vlong Int64.zero); split; auto. destruct (e#r); simpl; auto. rewrite Int64.and_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.mone; intros.
subst n. exists (e#r); split; auto. destruct (e#r); simpl; auto. rewrite Int64.and_mone; auto.
econstructor; split; eauto. auto.
Qed.
Lemma make_orlimm_correct:
forall n r,
let (op, args) := make_orlimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.orl e#r (Vlong n)) v.
Proof.
intros; unfold make_orlimm.
predSpec Int64.eq Int64.eq_spec n Int64.zero; intros.
subst n. exists (e#r); split; auto. destruct (e#r); simpl; auto. rewrite Int64.or_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.mone; intros.
subst n. exists (Vlong Int64.mone); split; auto. destruct (e#r); simpl; auto. rewrite Int64.or_mone; auto.
econstructor; split; eauto. auto.
Qed.
Lemma make_xorlimm_correct:
forall n r,
let (op, args) := make_xorlimm n r in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.xorl e#r (Vlong n)) v.
Proof.
intros; unfold make_xorlimm.
predSpec Int64.eq Int64.eq_spec n Int64.zero; intros.
subst n. exists (e#r); split; auto. destruct (e#r); simpl; auto. rewrite Int64.xor_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.mone; intros.
subst n. exists (Val.notl e#r); split; auto.
econstructor; split; eauto. auto.
Qed.
Lemma make_mulfimm_correct:
forall n r1 r2,
e#r2 = Vfloat n ->
let (op, args) := make_mulfimm n r1 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.mulf e#r1 e#r2) v.
Proof.
intros; unfold make_mulfimm.
destruct (Float.eq_dec n (Float.of_int (Int.repr 2))); intros.
simpl. econstructor; split. eauto. rewrite H; subst n.
destruct (e#r1); simpl; auto. rewrite Float.mul2_add; auto.
simpl. econstructor; split; eauto.
Qed.
Lemma make_mulfimm_correct_2:
forall n r1 r2,
e#r1 = Vfloat n ->
let (op, args) := make_mulfimm n r2 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.mulf e#r1 e#r2) v.
Proof.
intros; unfold make_mulfimm.
destruct (Float.eq_dec n (Float.of_int (Int.repr 2))); intros.
simpl. econstructor; split. eauto. rewrite H; subst n.
destruct (e#r2); simpl; auto. rewrite Float.mul2_add; auto.
rewrite Float.mul_commut; auto.
simpl. econstructor; split; eauto.
Qed.
Lemma make_mulfsimm_correct:
forall n r1 r2,
e#r2 = Vsingle n ->
let (op, args) := make_mulfsimm n r1 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.mulfs e#r1 e#r2) v.
Proof.
intros; unfold make_mulfsimm.
destruct (Float32.eq_dec n (Float32.of_int (Int.repr 2))); intros.
simpl. econstructor; split. eauto. rewrite H; subst n.
destruct (e#r1); simpl; auto. rewrite Float32.mul2_add; auto.
simpl. econstructor; split; eauto.
Qed.
Lemma make_mulfsimm_correct_2:
forall n r1 r2,
e#r1 = Vsingle n ->
let (op, args) := make_mulfsimm n r2 r1 r2 in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.mulfs e#r1 e#r2) v.
Proof.
intros; unfold make_mulfsimm.
destruct (Float32.eq_dec n (Float32.of_int (Int.repr 2))); intros.
simpl. econstructor; split. eauto. rewrite H; subst n.
destruct (e#r2); simpl; auto. rewrite Float32.mul2_add; auto.
rewrite Float32.mul_commut; auto.
simpl. econstructor; split; eauto.
Qed.
Lemma make_cast8signed_correct:
forall r x,
vmatch bc e#r x ->
let (op, args) := make_cast8signed r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.sign_ext 8 e#r) v.
Proof.
intros; unfold make_cast8signed. destruct (vincl x (Sgn Ptop 8)) eqn:INCL.
exists e#r; split; auto.
assert (V: vmatch bc e#r (Sgn Ptop 8)).
{ eapply vmatch_ge; eauto. apply vincl_ge; auto. }
inv V; simpl; auto. rewrite is_sgn_sign_ext in H4 by auto. rewrite H4; auto.
econstructor; split; simpl; eauto.
Qed.
Lemma make_cast16signed_correct:
forall r x,
vmatch bc e#r x ->
let (op, args) := make_cast16signed r x in
exists v, eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v /\ Val.lessdef (Val.sign_ext 16 e#r) v.
Proof.
intros; unfold make_cast16signed. destruct (vincl x (Sgn Ptop 16)) eqn:INCL.
exists e#r; split; auto.
assert (V: vmatch bc e#r (Sgn Ptop 16)).
{ eapply vmatch_ge; eauto. apply vincl_ge; auto. }
inv V; simpl; auto. rewrite is_sgn_sign_ext in H4 by auto. rewrite H4; auto.
econstructor; split; simpl; eauto.
Qed.
Lemma op_strength_reduction_correct:
forall op args vl v,
vl = map (fun r => AE.get r ae) args ->
eval_operation ge (Vptr sp Ptrofs.zero) op e##args m = Some v ->
let (op', args') := op_strength_reduction op args vl in
exists w, eval_operation ge (Vptr sp Ptrofs.zero) op' e##args' m = Some w /\ Val.lessdef v w.
Proof.
intros until v; unfold op_strength_reduction;
case (op_strength_reduction_match op args vl); simpl; intros.
- (* cast8signed *)
InvApproxRegs; SimplVM; inv H0. apply make_cast8signed_correct; auto.
- (* cast16signed *)
InvApproxRegs; SimplVM; inv H0. apply make_cast16signed_correct; auto.
- (* add 1 *)
rewrite Val.add_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_addimm_correct; auto.
- (* add 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_addimm_correct; auto.
- (* sub *)
InvApproxRegs; SimplVM; inv H0. rewrite Val.sub_add_opp. apply make_addimm_correct; auto.
- (* mul 1 *)
rewrite Val.mul_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_mulimm_correct; auto.
- (* mul 2*)
InvApproxRegs; SimplVM; inv H0. apply make_mulimm_correct; auto.
- (* divs *)
assert (e#r2 = Vint n2). clear H0. InvApproxRegs; SimplVM; auto.
apply make_divimm_correct; auto.
- (* divu *)
assert (e#r2 = Vint n2). clear H0. InvApproxRegs; SimplVM; auto.
apply make_divuimm_correct; auto.
- (* modu *)
assert (e#r2 = Vint n2). clear H0. InvApproxRegs; SimplVM; auto.
apply make_moduimm_correct; auto.
- (* and 1 *)
rewrite Val.and_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_andimm_correct; auto.
- (* and 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_andimm_correct; auto.
- (* andimm *)
inv H; inv H0. apply make_andimm_correct; auto.
- (* or 1 *)
rewrite Val.or_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_orimm_correct; auto.
- (* or 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_orimm_correct; auto.
- (* xor 1 *)
rewrite Val.xor_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_xorimm_correct; auto.
- (* xor 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_xorimm_correct; auto.
- (* shl *)
InvApproxRegs; SimplVM; inv H0. apply make_shlimm_correct; auto.
- (* shr *)
InvApproxRegs; SimplVM; inv H0. apply make_shrimm_correct; auto.
- (* shru *)
InvApproxRegs; SimplVM; inv H0. apply make_shruimm_correct; auto.
- (* addl 1 *)
rewrite Val.addl_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_addlimm_correct; auto.
- (* addl 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_addlimm_correct; auto.
- (* subl *)
InvApproxRegs; SimplVM; inv H0.
replace (Val.subl e#r1 (Vlong n2)) with (Val.addl e#r1 (Vlong (Int64.neg n2))).
apply make_addlimm_correct; auto.
unfold Val.addl, Val.subl. destruct Archi.ptr64 eqn:SF, e#r1; auto.
rewrite Int64.sub_add_opp; auto.
rewrite Ptrofs.sub_add_opp. do 2 f_equal. auto with ptrofs.
rewrite Int64.sub_add_opp; auto.
- (* mull 1 *)
rewrite Val.mull_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_mullimm_correct; auto.
- (* mull 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_mullimm_correct; auto.
- (* divl *)
assert (e#r2 = Vlong n2). clear H0. InvApproxRegs; SimplVM; auto.
apply make_divlimm_correct; auto.
- (* divlu *)
assert (e#r2 = Vlong n2). clear H0. InvApproxRegs; SimplVM; auto.
apply make_divluimm_correct; auto.
- (* modlu *)
assert (e#r2 = Vlong n2). clear H0. InvApproxRegs; SimplVM; auto.
apply make_modluimm_correct; auto.
- (* andl 1 *)
rewrite Val.andl_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_andlimm_correct; auto.
- (* andl 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_andlimm_correct; auto.
- (* andlimm *)
inv H; inv H0. apply make_andlimm_correct; auto.
- (* orl 1 *)
rewrite Val.orl_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_orlimm_correct; auto.
- (* orl 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_orlimm_correct; auto.
- (* xorl 1 *)
rewrite Val.xorl_commut in H0. InvApproxRegs; SimplVM; inv H0. apply make_xorlimm_correct; auto.
- (* xorl 2 *)
InvApproxRegs; SimplVM; inv H0. apply make_xorlimm_correct; auto.
- (* shll *)
InvApproxRegs; SimplVM; inv H0. apply make_shllimm_correct; auto.
- (* shrl *)
InvApproxRegs; SimplVM; inv H0. apply make_shrlimm_correct; auto.
- (* shrlu *)
InvApproxRegs; SimplVM; inv H0. apply make_shrluimm_correct; auto.
- (* cond *)
inv H0. apply make_cmp_correct; auto.
- (* mulf 1 *)
InvApproxRegs; SimplVM; inv H0. rewrite <- H2. apply make_mulfimm_correct; auto.
- (* mulf 2 *)
InvApproxRegs; SimplVM; inv H0. fold (Val.mulf (Vfloat n1) e#r2).
rewrite <- H2. apply make_mulfimm_correct_2; auto.
- (* mulfs 1 *)
InvApproxRegs; SimplVM; inv H0. rewrite <- H2. apply make_mulfsimm_correct; auto.
- (* mulfs 2 *)
InvApproxRegs; SimplVM; inv H0. fold (Val.mulfs (Vsingle n1) e#r2).
rewrite <- H2. apply make_mulfsimm_correct_2; auto.
- (* default *)
exists v; auto.
Qed.
Lemma addr_strength_reduction_correct:
forall addr args vl res,
vl = map (fun r => AE.get r ae) args ->
eval_addressing ge (Vptr sp Ptrofs.zero) addr e##args = Some res ->
let (addr', args') := addr_strength_reduction addr args vl in
exists res', eval_addressing ge (Vptr sp Ptrofs.zero) addr' e##args' = Some res' /\ Val.lessdef res res'.
Proof.
intros until res. unfold addr_strength_reduction.
destruct (addr_strength_reduction_match addr args vl); simpl;
intros VL EA; InvApproxRegs; SimplVM; try (inv EA).
- destruct (Archi.pic_code tt).
+ exists (Val.offset_ptr e#r1 n); auto.
+ simpl. rewrite Genv.shift_symbol_address. econstructor; split; eauto.
inv H0; simpl; auto.
- rewrite Ptrofs.add_zero_l. econstructor; split; eauto.
change (Vptr sp (Ptrofs.add n1 n)) with (Val.offset_ptr (Vptr sp n1) n).
inv H0; simpl; auto.
- exists res; auto.
Qed.
End STRENGTH_REDUCTION.