-
Notifications
You must be signed in to change notification settings - Fork 226
/
Asmgenproof1.v
1585 lines (1491 loc) · 60.9 KB
/
Asmgenproof1.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
(* *********************************************************************)
(* *)
(* The Compcert verified compiler *)
(* *)
(* Xavier Leroy, INRIA Paris-Rocquencourt *)
(* *)
(* 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 ARM code generation: auxiliary results. *)
Require Import Coqlib.
Require Import Errors.
Require Import Maps.
Require Import AST.
Require Import Zbits.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Op.
Require Import Locations.
Require Import Mach.
Require Import Compopts.
Require Import Asm.
Require Import Asmgen.
Require Import Conventions.
Require Import Asmgenproof0.
Local Transparent Archi.ptr64.
(** Useful properties of the R14 registers. *)
Lemma ireg_of_not_R14:
forall m r, ireg_of m = OK r -> IR r <> IR IR14.
Proof.
intros. erewrite <- ireg_of_eq; eauto with asmgen.
Qed.
Global Hint Resolve ireg_of_not_R14: asmgen.
Lemma ireg_of_not_R14':
forall m r, ireg_of m = OK r -> r <> IR14.
Proof.
intros. generalize (ireg_of_not_R14 _ _ H). congruence.
Qed.
Global Hint Resolve ireg_of_not_R14': asmgen.
(** [undef_flags] and [nextinstr_nf] *)
Lemma nextinstr_nf_pc:
forall rs, (nextinstr_nf rs)#PC = Val.offset_ptr rs#PC Ptrofs.one.
Proof.
intros. reflexivity.
Qed.
Definition if_preg (r: preg) : bool :=
match r with
| IR _ => true
| FR _ => true
| CR _ => false
| PC => false
end.
Lemma data_if_preg: forall r, data_preg r = true -> if_preg r = true.
Proof.
intros. destruct r; reflexivity || discriminate.
Qed.
Lemma if_preg_not_PC: forall r, if_preg r = true -> r <> PC.
Proof.
intros; red; intros; subst; discriminate.
Qed.
Global Hint Resolve data_if_preg if_preg_not_PC: asmgen.
Lemma nextinstr_nf_inv:
forall r rs, if_preg r = true -> (nextinstr_nf rs)#r = rs#r.
Proof.
intros. destruct r; reflexivity || discriminate.
Qed.
Lemma nextinstr_nf_inv1:
forall r rs, data_preg r = true -> (nextinstr_nf rs)#r = rs#r.
Proof.
intros. destruct r; reflexivity || discriminate.
Qed.
(** Useful simplification tactic *)
Ltac Simplif :=
((rewrite nextinstr_inv by eauto with asmgen)
|| (rewrite nextinstr_inv1 by eauto with asmgen)
|| (rewrite nextinstr_nf_inv by eauto with asmgen)
|| (rewrite Pregmap.gss)
|| (rewrite nextinstr_pc)
|| (rewrite nextinstr_nf_pc)
|| (rewrite Pregmap.gso by eauto with asmgen)); auto with asmgen.
Ltac Simpl := repeat Simplif.
(** * Correctness of ARM constructor functions *)
Section CONSTRUCTORS.
Variable ge: genv.
Variable fn: function.
(** Decomposition of an integer constant *)
Lemma decompose_int_arm_or:
forall N n p x, List.fold_left Int.or (decompose_int_arm N n p) x = Int.or x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.or_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl (Int.repr 3) p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.or_assoc. decEq. rewrite <- Int.and_or_distrib.
rewrite Int.or_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_arm_xor:
forall N n p x, List.fold_left Int.xor (decompose_int_arm N n p) x = Int.xor x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.xor_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl (Int.repr 3) p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.xor_assoc. decEq. rewrite <- Int.and_xor_distrib.
rewrite Int.xor_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_arm_add:
forall N n p x, List.fold_left Int.add (decompose_int_arm N n p) x = Int.add x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.add_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl (Int.repr 3) p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.add_assoc. decEq. rewrite Int.add_and.
rewrite Int.or_not_self. apply Int.and_mone. apply Int.and_not_self.
Qed.
Remark decompose_int_arm_nil:
forall N n p, decompose_int_arm N n p = nil -> n = Int.zero.
Proof.
intros. generalize (decompose_int_arm_or N n p Int.zero). rewrite H. simpl.
rewrite Int.or_commut; rewrite Int.or_zero; auto.
Qed.
Lemma decompose_int_thumb_or:
forall N n p x, List.fold_left Int.or (decompose_int_thumb N n p) x = Int.or x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.or_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl Int.one p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.or_assoc. decEq. rewrite <- Int.and_or_distrib.
rewrite Int.or_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_thumb_xor:
forall N n p x, List.fold_left Int.xor (decompose_int_thumb N n p) x = Int.xor x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.xor_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl Int.one p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.xor_assoc. decEq. rewrite <- Int.and_xor_distrib.
rewrite Int.xor_not_self. apply Int.and_mone.
Qed.
Lemma decompose_int_thumb_add:
forall N n p x, List.fold_left Int.add (decompose_int_thumb N n p) x = Int.add x n.
Proof.
induction N; intros; simpl.
predSpec Int.eq Int.eq_spec n Int.zero; simpl.
subst n. rewrite Int.add_zero. auto.
auto.
predSpec Int.eq Int.eq_spec (Int.and n (Int.shl Int.one p)) Int.zero.
auto.
simpl. rewrite IHN. rewrite Int.add_assoc. decEq. rewrite Int.add_and.
rewrite Int.or_not_self. apply Int.and_mone. apply Int.and_not_self.
Qed.
Remark decompose_int_thumb_nil:
forall N n p, decompose_int_thumb N n p = nil -> n = Int.zero.
Proof.
intros. generalize (decompose_int_thumb_or N n p Int.zero). rewrite H. simpl.
rewrite Int.or_commut; rewrite Int.or_zero; auto.
Qed.
Lemma decompose_int_general:
forall (f: val -> int -> val) (g: int -> int -> int),
(forall v1 n2 n3, f (f v1 n2) n3 = f v1 (g n2 n3)) ->
(forall n1 n2 n3, g (g n1 n2) n3 = g n1 (g n2 n3)) ->
(forall n, g Int.zero n = n) ->
(forall N n p x, List.fold_left g (decompose_int_arm N n p) x = g x n) ->
(forall N n p x, List.fold_left g (decompose_int_thumb N n p) x = g x n) ->
forall n v,
List.fold_left f (decompose_int n) v = f v n.
Proof.
intros f g DISTR ASSOC ZERO DECOMP1 DECOMP2.
assert (A: forall l x y, g x (fold_left g l y) = fold_left g l (g x y)).
induction l; intros; simpl. auto. rewrite IHl. decEq. rewrite ASSOC; auto.
assert (B: forall l v n, fold_left f l (f v n) = f v (fold_left g l n)).
induction l; intros; simpl.
auto.
rewrite IHl. rewrite DISTR. decEq. decEq. auto.
intros. unfold decompose_int, decompose_int_base.
destruct (thumb tt); [destruct (is_immed_arith_thumb_special n)|].
- reflexivity.
- destruct (decompose_int_thumb 24%nat n Int.zero) eqn:DB.
+ simpl. exploit decompose_int_thumb_nil; eauto. congruence.
+ simpl. rewrite B. decEq.
generalize (DECOMP2 24%nat n Int.zero Int.zero).
rewrite DB; simpl. rewrite ! ZERO. auto.
- destruct (decompose_int_arm 12%nat n Int.zero) eqn:DB.
+ simpl. exploit decompose_int_arm_nil; eauto. congruence.
+ simpl. rewrite B. decEq.
generalize (DECOMP1 12%nat n Int.zero Int.zero).
rewrite DB; simpl. rewrite ! ZERO. auto.
Qed.
Lemma decompose_int_or:
forall n v,
List.fold_left (fun v i => Val.or v (Vint i)) (decompose_int n) v = Val.or v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.or v (Vint n)) (g := Int.or).
intros. rewrite Val.or_assoc. auto.
apply Int.or_assoc.
intros. rewrite Int.or_commut. apply Int.or_zero.
apply decompose_int_arm_or. apply decompose_int_thumb_or.
Qed.
Lemma decompose_int_bic:
forall n v,
List.fold_left (fun v i => Val.and v (Vint (Int.not i))) (decompose_int n) v = Val.and v (Vint (Int.not n)).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.and v (Vint (Int.not n))) (g := Int.or).
intros. rewrite Val.and_assoc. simpl. decEq. decEq. rewrite Int.not_or_and_not. auto.
apply Int.or_assoc.
intros. rewrite Int.or_commut. apply Int.or_zero.
apply decompose_int_arm_or. apply decompose_int_thumb_or.
Qed.
Lemma decompose_int_xor:
forall n v,
List.fold_left (fun v i => Val.xor v (Vint i)) (decompose_int n) v = Val.xor v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.xor v (Vint n)) (g := Int.xor).
intros. rewrite Val.xor_assoc. auto.
apply Int.xor_assoc.
intros. rewrite Int.xor_commut. apply Int.xor_zero.
apply decompose_int_arm_xor. apply decompose_int_thumb_xor.
Qed.
Lemma decompose_int_add:
forall n v,
List.fold_left (fun v i => Val.add v (Vint i)) (decompose_int n) v = Val.add v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.add v (Vint n)) (g := Int.add).
intros. rewrite Val.add_assoc. auto.
apply Int.add_assoc.
intros. rewrite Int.add_commut. apply Int.add_zero.
apply decompose_int_arm_add. apply decompose_int_thumb_add.
Qed.
Lemma decompose_int_sub:
forall n v,
List.fold_left (fun v i => Val.sub v (Vint i)) (decompose_int n) v = Val.sub v (Vint n).
Proof.
intros. apply decompose_int_general with (f := fun v n => Val.sub v (Vint n)) (g := Int.add).
intros. repeat rewrite Val.sub_add_opp. rewrite Val.add_assoc. decEq. simpl. decEq.
rewrite Int.neg_add_distr; auto.
apply Int.add_assoc.
intros. rewrite Int.add_commut. apply Int.add_zero.
apply decompose_int_arm_add. apply decompose_int_thumb_add.
Qed.
Lemma iterate_op_correct:
forall op1 op2 (f: val -> int -> val) (rs: regset) (r: ireg) m v0 n k,
(forall (rs:regset) n,
exec_instr ge fn (op2 (SOimm n)) rs m =
Next (nextinstr_nf (rs#r <- (f (rs#r) n))) m) ->
(forall n,
exec_instr ge fn (op1 (SOimm n)) rs m =
Next (nextinstr_nf (rs#r <- (f v0 n))) m) ->
exists rs',
exec_straight ge fn (iterate_op op1 op2 (decompose_int n) k) rs m k rs' m
/\ rs'#r = List.fold_left f (decompose_int n) v0
/\ forall r': preg, r' <> r -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros until k; intros SEM2 SEM1.
unfold iterate_op.
destruct (decompose_int n) as [ | i tl] eqn:DI.
unfold decompose_int in DI. destruct (decompose_int_base n); congruence.
revert k. pattern tl. apply List.rev_ind.
(* base case *)
intros; simpl. econstructor.
split. apply exec_straight_one. rewrite SEM1. reflexivity. reflexivity.
intuition Simpl.
(* inductive case *)
intros.
rewrite List.map_app. simpl. rewrite app_ass. simpl.
destruct (H (op2 (SOimm x) :: k)) as [rs' [A [B C]]].
econstructor.
split. eapply exec_straight_trans. eexact A. apply exec_straight_one.
rewrite SEM2. reflexivity. reflexivity.
split. rewrite fold_left_app; simpl. Simpl. rewrite B. auto.
intros; Simpl.
Qed.
(** Loading a constant. *)
Lemma loadimm_correct:
forall r n k rs m,
exists rs',
exec_straight ge fn (loadimm r n k) rs m k rs' m
/\ rs'#r = Vint n
/\ forall r': preg, r' <> r -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold loadimm.
set (l1 := length (decompose_int n)).
set (l2 := length (decompose_int (Int.not n))).
destruct (Nat.leb l1 1%nat).
{ (* single mov *)
econstructor; split. apply exec_straight_one. simpl; reflexivity. auto.
split; intros; Simpl. }
destruct (Nat.leb l2 1%nat).
{ (* single movn *)
econstructor; split. apply exec_straight_one.
simpl. rewrite Int.not_involutive. reflexivity. auto.
split; intros; Simpl. }
destruct Archi.thumb2_support.
{ (* movw / movt *)
unfold loadimm_word. destruct (Int.eq (Int.shru n (Int.repr 16)) Int.zero).
econstructor; split.
apply exec_straight_one. simpl; eauto. auto. split; intros; Simpl.
econstructor; split.
eapply exec_straight_two. simpl; reflexivity. simpl; reflexivity. auto. auto.
split; intros; Simpl. simpl. f_equal. rewrite Int.zero_ext_and by lia.
rewrite Int.and_assoc. change 65535 with (two_p 16 - 1). rewrite Int.and_idem.
apply Int.same_bits_eq; intros.
rewrite Int.bits_or, Int.bits_and, Int.bits_shl, Int.testbit_repr by auto.
rewrite Ztestbit_two_p_m1 by lia. change (Int.unsigned (Int.repr 16)) with 16.
destruct (zlt i 16).
rewrite andb_true_r, orb_false_r; auto.
rewrite andb_false_r; simpl. rewrite Int.bits_shru by lia.
change (Int.unsigned (Int.repr 16)) with 16. rewrite zlt_true by lia. f_equal; lia.
}
destruct (Nat.leb l1 l2).
{ (* mov - orr* *)
replace (Vint n) with (List.fold_left (fun v i => Val.or v (Vint i)) (decompose_int n) Vzero).
apply iterate_op_correct.
auto.
intros; simpl. rewrite Int.or_commut; rewrite Int.or_zero; auto.
rewrite decompose_int_or. simpl. rewrite Int.or_commut; rewrite Int.or_zero; auto.
}
{ (* mvn - bic* *)
replace (Vint n) with (List.fold_left (fun v i => Val.and v (Vint (Int.not i))) (decompose_int (Int.not n)) (Vint Int.mone)).
apply iterate_op_correct.
auto.
intros. simpl. rewrite Int.and_commut; rewrite Int.and_mone; auto.
rewrite decompose_int_bic. simpl. rewrite Int.not_involutive. rewrite Int.and_commut. rewrite Int.and_mone; auto.
}
Qed.
(** Add integer immediate. *)
Lemma addimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (addimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.add rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold addimm.
destruct (Int.ltu (Int.repr (-256)) n).
(* sub *)
econstructor; split. apply exec_straight_one; simpl; auto.
split; intros; Simpl. apply Val.sub_opp_add.
destruct (Nat.leb (length (decompose_int n)) (length (decompose_int (Int.neg n)))).
(* add - add* *)
replace (Val.add (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.add v (Vint i)) (decompose_int n) (rs r2)).
apply iterate_op_correct.
auto.
auto.
apply decompose_int_add.
(* sub - sub* *)
replace (Val.add (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.sub v (Vint i)) (decompose_int (Int.neg n)) (rs r2)).
apply iterate_op_correct.
auto.
auto.
rewrite decompose_int_sub. apply Val.sub_opp_add.
Qed.
(* And integer immediate *)
Lemma andimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (andimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.and rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold andimm. destruct (is_immed_arith n).
(* andi *)
exists (nextinstr_nf (rs#r1 <- (Val.and rs#r2 (Vint n)))).
split. apply exec_straight_one; auto. split; intros; Simpl.
(* bic - bic* *)
replace (Val.and (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.and v (Vint (Int.not i))) (decompose_int (Int.not n)) (rs r2)).
apply iterate_op_correct.
auto. auto.
rewrite decompose_int_bic. rewrite Int.not_involutive. auto.
Qed.
(** Reverse sub immediate *)
Lemma rsubimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (rsubimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.sub (Vint n) rs#r2
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold rsubimm.
(* rsb - add* *)
replace (Val.sub (Vint n) (rs r2))
with (List.fold_left (fun v i => Val.add v (Vint i)) (decompose_int n) (Val.neg (rs r2))).
apply iterate_op_correct.
auto.
intros. simpl. destruct (rs r2); auto. simpl. rewrite Int.sub_add_opp.
rewrite Int.add_commut; auto.
rewrite decompose_int_add.
destruct (rs r2); simpl; auto. rewrite Int.sub_add_opp. rewrite Int.add_commut; auto.
Qed.
(** Or immediate *)
Lemma orimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (orimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.or rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold orimm.
(* ori - ori* *)
replace (Val.or (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.or v (Vint i)) (decompose_int n) (rs r2)).
apply iterate_op_correct.
auto.
auto.
apply decompose_int_or.
Qed.
(** Xor immediate *)
Lemma xorimm_correct:
forall r1 r2 n k rs m,
exists rs',
exec_straight ge fn (xorimm r1 r2 n k) rs m k rs' m
/\ rs'#r1 = Val.xor rs#r2 (Vint n)
/\ forall r': preg, r' <> r1 -> if_preg r' = true -> rs'#r' = rs#r'.
Proof.
intros. unfold xorimm.
(* xori - xori* *)
replace (Val.xor (rs r2) (Vint n))
with (List.fold_left (fun v i => Val.xor v (Vint i)) (decompose_int n) (rs r2)).
apply iterate_op_correct.
auto.
auto.
apply decompose_int_xor.
Qed.
(** Indexed memory loads. *)
Lemma indexed_memory_access_correct:
forall (P: regset -> Prop) (mk_instr: ireg -> int -> instruction)
(mk_immed: int -> int) (base: ireg) n k (rs: regset) m m',
(forall (r1: ireg) (rs1: regset) n1 k,
Val.add rs1#r1 (Vint n1) = Val.add rs#base (Vint n) ->
(forall (r: preg), if_preg r = true -> r <> IR14 -> rs1 r = rs r) ->
exists rs',
exec_straight ge fn (mk_instr r1 n1 :: k) rs1 m k rs' m' /\ P rs') ->
exists rs',
exec_straight ge fn
(indexed_memory_access mk_instr mk_immed base n k) rs m
k rs' m'
/\ P rs'.
Proof.
intros until m'; intros SEM.
unfold indexed_memory_access.
destruct (Int.eq n (mk_immed n)).
- apply SEM; auto.
- destruct (addimm_correct IR14 base (Int.sub n (mk_immed n)) (mk_instr IR14 (mk_immed n) :: k) rs m)
as (rs1 & A & B & C).
destruct (SEM IR14 rs1 (mk_immed n) k) as (rs2 & D & E).
rewrite B. rewrite Val.add_assoc. f_equal. simpl.
rewrite Int.sub_add_opp. rewrite Int.add_assoc.
rewrite (Int.add_commut (Int.neg (mk_immed n))).
rewrite Int.add_neg_zero. rewrite Int.add_zero. auto.
auto with asmgen.
exists rs2; split; auto. eapply exec_straight_trans; eauto.
Qed.
Lemma loadind_int_correct:
forall (base: ireg) ofs dst (rs: regset) m v k,
Mem.loadv Mint32 m (Val.offset_ptr rs#base ofs) = Some v ->
exists rs',
exec_straight ge fn (loadind_int base ofs dst k) rs m k rs' m
/\ rs'#dst = v
/\ forall r, if_preg r = true -> r <> IR14 -> r <> dst -> rs'#r = rs#r.
Proof.
intros; unfold loadind_int.
assert (Val.offset_ptr (rs base) ofs = Val.add (rs base) (Vint (Ptrofs.to_int ofs))).
{ destruct (rs base); try discriminate. simpl. f_equal; f_equal. symmetry; auto with ptrofs. }
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H1, <- H0, H. eauto. auto.
split; intros; Simpl.
Qed.
Lemma loadind_correct:
forall (base: ireg) ofs ty dst k c (rs: regset) m v,
loadind base ofs ty dst k = OK c ->
Mem.loadv (chunk_of_type ty) m (Val.offset_ptr rs#base ofs) = Some v ->
exists rs',
exec_straight ge fn c rs m k rs' m
/\ rs'#(preg_of dst) = v
/\ forall r, if_preg r = true -> r <> IR14 -> r <> preg_of dst -> rs'#r = rs#r.
Proof.
unfold loadind; intros.
assert (Val.offset_ptr (rs base) ofs = Val.add (rs base) (Vint (Ptrofs.to_int ofs))).
{ destruct (rs base); try discriminate. simpl. f_equal; f_equal. symmetry; auto with ptrofs. }
destruct ty; destruct (preg_of dst); inv H; simpl in H0.
- (* int *)
apply loadind_int_correct; auto.
- (* float *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
- (* single *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
- (* any32 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
- (* any64 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_load. rewrite H, <- H1, H0. eauto. auto.
split; intros; Simpl.
Qed.
(** Indexed memory stores. *)
Lemma storeind_correct:
forall (base: ireg) ofs ty src k c (rs: regset) m m',
storeind src base ofs ty k = OK c ->
Mem.storev (chunk_of_type ty) m (Val.offset_ptr rs#base ofs) (rs#(preg_of src)) = Some m' ->
exists rs',
exec_straight ge fn c rs m k rs' m'
/\ forall r, if_preg r = true -> r <> IR14 -> rs'#r = rs#r.
Proof.
unfold storeind; intros.
assert (DATA: data_preg (preg_of src) = true) by eauto with asmgen.
assert (Val.offset_ptr (rs base) ofs = Val.add (rs base) (Vint (Ptrofs.to_int ofs))).
{ destruct (rs base); try discriminate. simpl. f_equal; f_equal. symmetry; auto with ptrofs. }
destruct ty; destruct (preg_of src); inv H; simpl in H0.
- (* int *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* float *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* single *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* any32 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
- (* any64 *)
apply indexed_memory_access_correct; intros.
econstructor; split.
apply exec_straight_one. simpl. unfold exec_store. rewrite H, <- H1, H2, H0 by auto with asmgen; eauto. auto.
intros; Simpl.
Qed.
(** Saving the link register *)
Lemma save_lr_correct:
forall ofs k (rs: regset) m m',
Mem.storev Mint32 m (Val.offset_ptr rs#IR13 ofs) (rs#IR14) = Some m' ->
exists rs',
exec_straight ge fn (save_lr ofs k) rs m k rs' m'
/\ (forall r, if_preg r = true -> r <> IR12 -> rs'#r = rs#r)
/\ (save_lr_preserves_R12 ofs = true -> rs'#IR12 = rs#IR12).
Proof.
intros; unfold save_lr, save_lr_preserves_R12.
set (n := Ptrofs.to_int ofs). set (n1 := mk_immed_mem_word n).
assert (EQ: Val.offset_ptr rs#IR13 ofs = Val.add rs#IR13 (Vint n)).
{ destruct rs#IR13; try discriminate. simpl. f_equal; f_equal. unfold n; symmetry; auto with ptrofs. }
destruct (Int.eq n n1).
- econstructor; split. apply exec_straight_one. simpl; unfold exec_store. rewrite <- EQ, H; reflexivity. auto.
split. intros; Simpl. intros; Simpl.
- destruct (addimm_correct IR12 IR13 (Int.sub n n1) (Pstr IR14 IR12 (SOimm n1) :: k) rs m)
as (rs1 & A & B & C).
econstructor; split.
eapply exec_straight_trans. eexact A.
apply exec_straight_one. simpl; unfold exec_store.
rewrite B. rewrite Val.add_assoc. simpl.
rewrite Int.sub_add_opp. rewrite Int.add_assoc.
rewrite (Int.add_commut (Int.neg n1)).
rewrite Int.add_neg_zero. rewrite Int.add_zero.
rewrite <- EQ. rewrite C by eauto with asmgen. rewrite H. reflexivity.
auto.
split. intros; Simpl. congruence.
Qed.
(** Translation of shift immediates *)
Lemma transl_shift_correct:
forall s (r: ireg) (rs: regset),
eval_shift_op (transl_shift s r) rs = eval_shift s (rs#r).
Proof.
intros. destruct s; simpl; auto.
Qed.
(** Translation of conditions *)
Lemma compare_int_spec:
forall rs v1 v2 m,
let rs1 := nextinstr (compare_int rs v1 v2 m) in
rs1#CN = Val.negative (Val.sub v1 v2)
/\ rs1#CZ = Val.cmpu (Mem.valid_pointer m) Ceq v1 v2
/\ rs1#CC = Val.cmpu (Mem.valid_pointer m) Cge v1 v2
/\ rs1#CV = Val.sub_overflow v1 v2.
Proof.
intros. unfold rs1. intuition.
Qed.
Lemma compare_int_inv:
forall rs v1 v2 m,
let rs1 := nextinstr (compare_int rs v1 v2 m) in
forall r', data_preg r' = true -> rs1#r' = rs#r'.
Proof.
intros. unfold rs1, compare_int.
repeat Simplif.
Qed.
Lemma int_signed_eq:
forall x y, Int.eq x y = zeq (Int.signed x) (Int.signed y).
Proof.
intros. unfold Int.eq. unfold proj_sumbool.
destruct (zeq (Int.unsigned x) (Int.unsigned y));
destruct (zeq (Int.signed x) (Int.signed y)); auto.
elim n. unfold Int.signed. rewrite e; auto.
elim n. apply Int.eqm_small_eq; auto with ints.
eapply Int.eqm_trans. apply Int.eqm_sym. apply Int.eqm_signed_unsigned.
rewrite e. apply Int.eqm_signed_unsigned.
Qed.
Lemma int_not_lt:
forall x y, negb (Int.lt y x) = (Int.lt x y || Int.eq x y).
Proof.
intros. unfold Int.lt. rewrite int_signed_eq. unfold proj_sumbool.
destruct (zlt (Int.signed y) (Int.signed x)).
rewrite zlt_false. rewrite zeq_false. auto. lia. lia.
destruct (zeq (Int.signed x) (Int.signed y)).
rewrite zlt_false. auto. lia.
rewrite zlt_true. auto. lia.
Qed.
Lemma int_lt_not:
forall x y, Int.lt y x = negb (Int.lt x y) && negb (Int.eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- int_not_lt. rewrite negb_involutive. auto.
Qed.
Lemma int_not_ltu:
forall x y, negb (Int.ltu y x) = (Int.ltu x y || Int.eq x y).
Proof.
intros. unfold Int.ltu, Int.eq.
destruct (zlt (Int.unsigned y) (Int.unsigned x)).
rewrite zlt_false. rewrite zeq_false. auto. lia. lia.
destruct (zeq (Int.unsigned x) (Int.unsigned y)).
rewrite zlt_false. auto. lia.
rewrite zlt_true. auto. lia.
Qed.
Lemma int_ltu_not:
forall x y, Int.ltu y x = negb (Int.ltu x y) && negb (Int.eq x y).
Proof.
intros. rewrite <- negb_orb. rewrite <- int_not_ltu. rewrite negb_involutive. auto.
Qed.
Lemma cond_for_signed_cmp_correct:
forall c v1 v2 rs m b,
Val.cmp_bool c v1 v2 = Some b ->
eval_testcond (cond_for_signed_cmp c)
(nextinstr (compare_int rs v1 v2 m)) = Some b.
Proof.
intros. generalize (compare_int_spec rs v1 v2 m).
set (rs' := nextinstr (compare_int rs v1 v2 m)).
intros [A [B [C D]]].
destruct v1; destruct v2; simpl in H; inv H.
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
simpl. unfold Val.cmp, Val.cmpu.
rewrite Int.lt_sub_overflow.
destruct c; simpl.
destruct (Int.eq i i0); auto.
destruct (Int.eq i i0); auto.
destruct (Int.lt i i0); auto.
rewrite int_not_lt. destruct (Int.lt i i0); simpl; destruct (Int.eq i i0); auto.
rewrite (int_lt_not i i0). destruct (Int.lt i i0); destruct (Int.eq i i0); reflexivity.
destruct (Int.lt i i0); reflexivity.
Qed.
Lemma cond_for_unsigned_cmp_correct:
forall c v1 v2 rs m b,
Val.cmpu_bool (Mem.valid_pointer m) c v1 v2 = Some b ->
eval_testcond (cond_for_unsigned_cmp c)
(nextinstr (compare_int rs v1 v2 m)) = Some b.
Proof.
intros. generalize (compare_int_spec rs v1 v2 m).
set (rs' := nextinstr (compare_int rs v1 v2 m)).
intros [A [B [C D]]].
unfold eval_testcond. rewrite B; rewrite C. unfold Val.cmpu, Val.cmp.
destruct v1; destruct v2; simpl in H; inv H.
(* int int *)
destruct c; simpl; auto.
destruct (Int.eq i i0); reflexivity.
destruct (Int.eq i i0); auto.
destruct (Int.ltu i i0); auto.
rewrite (int_not_ltu i i0). destruct (Int.ltu i i0); destruct (Int.eq i i0); auto.
rewrite (int_ltu_not i i0). destruct (Int.ltu i i0); destruct (Int.eq i i0); reflexivity.
destruct (Int.ltu i i0); reflexivity.
(* int ptr *)
destruct (Int.eq i Int.zero &&
(Mem.valid_pointer m b0 (Ptrofs.unsigned i0) || Mem.valid_pointer m b0 (Ptrofs.unsigned i0 - 1))) eqn:?; try discriminate.
destruct c; simpl in *; inv H1.
rewrite Heqb1; reflexivity.
rewrite Heqb1; reflexivity.
(* ptr int *)
destruct (Int.eq i0 Int.zero &&
(Mem.valid_pointer m b0 (Ptrofs.unsigned i) || Mem.valid_pointer m b0 (Ptrofs.unsigned i - 1))) eqn:?; try discriminate.
destruct c; simpl in *; inv H1.
rewrite Heqb1; reflexivity.
rewrite Heqb1; reflexivity.
(* ptr ptr *)
simpl.
fold (Mem.weak_valid_pointer m b0 (Ptrofs.unsigned i)) in *.
fold (Mem.weak_valid_pointer m b1 (Ptrofs.unsigned i0)) in *.
destruct (eq_block b0 b1).
destruct (Mem.weak_valid_pointer m b0 (Ptrofs.unsigned i) &&
Mem.weak_valid_pointer m b1 (Ptrofs.unsigned i0)); inversion H1.
destruct c; simpl; auto.
destruct (Ptrofs.eq i i0); reflexivity.
destruct (Ptrofs.eq i i0); auto.
destruct (Ptrofs.ltu i i0); auto.
rewrite (Ptrofs.not_ltu i i0). destruct (Ptrofs.ltu i i0); simpl; destruct (Ptrofs.eq i i0); auto.
rewrite (Ptrofs.ltu_not i i0). destruct (Ptrofs.ltu i i0); destruct (Ptrofs.eq i i0); reflexivity.
destruct (Ptrofs.ltu i i0); reflexivity.
destruct (Mem.valid_pointer m b0 (Ptrofs.unsigned i) &&
Mem.valid_pointer m b1 (Ptrofs.unsigned i0)); try discriminate.
destruct c; simpl in *; inv H1; reflexivity.
Qed.
Lemma compare_float_spec:
forall rs f1 f2,
let rs1 := nextinstr (compare_float rs (Vfloat f1) (Vfloat f2)) in
rs1#CN = Val.of_bool (Float.cmp Clt f1 f2)
/\ rs1#CZ = Val.of_bool (Float.cmp Ceq f1 f2)
/\ rs1#CC = Val.of_bool (negb (Float.cmp Clt f1 f2))
/\ rs1#CV = Val.of_bool (negb (Float.cmp Ceq f1 f2 || Float.cmp Clt f1 f2 || Float.cmp Cgt f1 f2)).
Proof.
intros. intuition.
Qed.
Lemma compare_float_inv:
forall rs v1 v2,
let rs1 := nextinstr (compare_float rs v1 v2) in
forall r', data_preg r' = true -> rs1#r' = rs#r'.
Proof.
intros. unfold rs1, compare_float.
assert (nextinstr (rs#CN <- Vundef #CZ <- Vundef #CC <- Vundef #CV <- Vundef) r' = rs r').
{ repeat Simplif. }
destruct v1; destruct v2; auto.
repeat Simplif.
Qed.
Lemma compare_float_nextpc:
forall rs v1 v2,
nextinstr (compare_float rs v1 v2) PC = Val.offset_ptr (rs PC) Ptrofs.one.
Proof.
intros. unfold compare_float. destruct v1; destruct v2; reflexivity.
Qed.
Lemma cond_for_float_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_cmp c)
(nextinstr (compare_float rs (Vfloat n1) (Vfloat n2))) =
Some(Float.cmp c n1 n2).
Proof.
intros.
generalize (compare_float_spec rs n1 n2).
set (rs' := nextinstr (compare_float rs (Vfloat n1) (Vfloat n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float.cmp_ne_eq. destruct (Float.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float.cmp Clt n1 n2); auto.
(* le *)
rewrite Float.cmp_le_lt_eq.
destruct (Float.cmp Clt n1 n2); destruct (Float.cmp Ceq n1 n2); auto.
(* gt *)
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
exfalso; eapply Float.cmp_gt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
(* ge *)
rewrite Float.cmp_ge_gt_eq.
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
Qed.
Lemma cond_for_float_not_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_not_cmp c)
(nextinstr (compare_float rs (Vfloat n1) (Vfloat n2)))=
Some(negb(Float.cmp c n1 n2)).
Proof.
intros.
generalize (compare_float_spec rs n1 n2).
set (rs' := nextinstr (compare_float rs (Vfloat n1) (Vfloat n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float.cmp_ne_eq. destruct (Float.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float.cmp Clt n1 n2); auto.
(* le *)
rewrite Float.cmp_le_lt_eq.
destruct (Float.cmp Clt n1 n2) eqn:LT; destruct (Float.cmp Ceq n1 n2) eqn:EQ; auto.
(* gt *)
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
exfalso; eapply Float.cmp_gt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
(* ge *)
rewrite Float.cmp_ge_gt_eq.
destruct (Float.cmp Ceq n1 n2) eqn:EQ;
destruct (Float.cmp Clt n1 n2) eqn:LT;
destruct (Float.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_eq_false; eauto.
exfalso; eapply Float.cmp_lt_gt_false; eauto.
Qed.
Lemma compare_float32_spec:
forall rs f1 f2,
let rs1 := nextinstr (compare_float32 rs (Vsingle f1) (Vsingle f2)) in
rs1#CN = Val.of_bool (Float32.cmp Clt f1 f2)
/\ rs1#CZ = Val.of_bool (Float32.cmp Ceq f1 f2)
/\ rs1#CC = Val.of_bool (negb (Float32.cmp Clt f1 f2))
/\ rs1#CV = Val.of_bool (negb (Float32.cmp Ceq f1 f2 || Float32.cmp Clt f1 f2 || Float32.cmp Cgt f1 f2)).
Proof.
intros. intuition.
Qed.
Lemma compare_float32_inv:
forall rs v1 v2,
let rs1 := nextinstr (compare_float32 rs v1 v2) in
forall r', data_preg r' = true -> rs1#r' = rs#r'.
Proof.
intros. unfold rs1, compare_float32.
assert (nextinstr (rs#CN <- Vundef #CZ <- Vundef #CC <- Vundef #CV <- Vundef) r' = rs r').
{ repeat Simplif. }
destruct v1; destruct v2; auto.
repeat Simplif.
Qed.
Lemma compare_float32_nextpc:
forall rs v1 v2,
nextinstr (compare_float32 rs v1 v2) PC = Val.offset_ptr (rs PC) Ptrofs.one.
Proof.
intros. unfold compare_float32. destruct v1; destruct v2; reflexivity.
Qed.
Lemma cond_for_float32_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_cmp c)
(nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2))) =
Some(Float32.cmp c n1 n2).
Proof.
intros.
generalize (compare_float32_spec rs n1 n2).
set (rs' := nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float32.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float32.cmp_ne_eq. destruct (Float32.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float32.cmp Clt n1 n2); auto.
(* le *)
rewrite Float32.cmp_le_lt_eq.
destruct (Float32.cmp Clt n1 n2); destruct (Float32.cmp Ceq n1 n2); auto.
(* gt *)
destruct (Float32.cmp Ceq n1 n2) eqn:EQ;
destruct (Float32.cmp Clt n1 n2) eqn:LT;
destruct (Float32.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
exfalso; eapply Float32.cmp_gt_eq_false; eauto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
(* ge *)
rewrite Float32.cmp_ge_gt_eq.
destruct (Float32.cmp Ceq n1 n2) eqn:EQ;
destruct (Float32.cmp Clt n1 n2) eqn:LT;
destruct (Float32.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float32.cmp_lt_eq_false; eauto.
exfalso; eapply Float32.cmp_lt_eq_false; eauto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
Qed.
Lemma cond_for_float32_not_cmp_correct:
forall c n1 n2 rs,
eval_testcond (cond_for_float_not_cmp c)
(nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2)))=
Some(negb(Float32.cmp c n1 n2)).
Proof.
intros.
generalize (compare_float32_spec rs n1 n2).
set (rs' := nextinstr (compare_float32 rs (Vsingle n1) (Vsingle n2))).
intros [A [B [C D]]].
unfold eval_testcond. rewrite A; rewrite B; rewrite C; rewrite D.
destruct c; simpl.
(* eq *)
destruct (Float32.cmp Ceq n1 n2); auto.
(* ne *)
rewrite Float32.cmp_ne_eq. destruct (Float32.cmp Ceq n1 n2); auto.
(* lt *)
destruct (Float32.cmp Clt n1 n2); auto.
(* le *)
rewrite Float32.cmp_le_lt_eq.
destruct (Float32.cmp Clt n1 n2) eqn:LT; destruct (Float32.cmp Ceq n1 n2) eqn:EQ; auto.
(* gt *)
destruct (Float32.cmp Ceq n1 n2) eqn:EQ;
destruct (Float32.cmp Clt n1 n2) eqn:LT;
destruct (Float32.cmp Cgt n1 n2) eqn:GT; auto.
exfalso; eapply Float32.cmp_lt_gt_false; eauto.
exfalso; eapply Float32.cmp_gt_eq_false; eauto.