-
Notifications
You must be signed in to change notification settings - Fork 0
/
proof_slot_bag_oloc.v
1091 lines (978 loc) · 37.5 KB
/
proof_slot_bag_oloc.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
From iris.base_logic Require Export lib.invariants lib.ghost_map.
From smr.program_logic Require Import atomic.
From smr Require Import helpers.
From smr.lang Require Import proofmode notation.
From iris.prelude Require Import options.
From smr.base_logic Require Export lib.mono_list.
From smr Require Import spec_slot_bag_oloc code_slot_bag_oloc.
Set Printing Projections.
Class slot_bag_olocG Σ := SlotBagoLocG {
slot_bag_oloc_ghost_mapG :> ghost_mapG Σ loc (bool * (option blk));
slot_bag_oloc_mono_listG :> mono_listG loc Σ;
}.
Definition slot_bag_olocΣ : gFunctors := #[ ghost_mapΣ loc (bool * (option blk)); mono_listΣ loc].
Global Instance subG_slot_bag_olocΣ {Σ} :
subG slot_bag_olocΣ Σ → slot_bag_olocG Σ.
Proof. solve_inG. Qed.
Section slot_bag.
Context `{!heapGS Σ, !slot_bag_olocG Σ}.
Notation iProp := (iProp Σ).
Implicit Types
(γsb γm γxs : gname)
(xs : list loc)
(sbvmap : gmap loc (bool * (option blk))).
(* domain relation *)
Definition domain_of xs sbvmap :=
∀ slot, slot ∈ xs ↔ is_Some (sbvmap !! slot).
(* physical contents *)
Definition slot_phys slot (b : bool) (v : option blk) : iProp :=
(slot +ₗ slotActive) ↦{# if b then (1/2)%Qp else 1%Qp } #b ∗
(slot +ₗ slotValue) ↦{# if b then (1/2)%Qp else 1%Qp } #(oblk_to_lit v) ∗
⌜if (negb b) then v = None else True⌝.
Fixpoint phys_list_rec xs : iProp :=
match xs with
| [] => True
| slot :: xs =>
(slot +ₗ slotNext) ↦□ #(oloc_to_lit (head xs)) ∗
phys_list_rec xs
end.
Definition phys_list xs : iProp := phys_list_rec (reverse xs).
Definition phys_map xs sbvmap : iProp :=
phys_list xs ∗
([∗ map] slot ↦ bv ∈ sbvmap, slot_phys slot (bv.1) (bv.2)) ∗
⌜domain_of xs sbvmap⌝.
Definition ghost_slot_bag γm sbvmap : iProp :=
ghost_map_auth γm 1 sbvmap ∗
([∗ map] slot ↦ bv ∈ sbvmap,
if decide (sbvmap !! slot = Some (false, None))
then slot ↪[γm] (false, None) else True).
(* slot and bag definitions *)
Definition Slot γsb (slot : loc) (idx : nat) (v : option blk) : iProp :=
∃ γm γxs,
⌜γsb = encode (γm, γxs)⌝ ∗
(slot +ₗ slotActive) ↦{# 1/2} #true ∗
(slot +ₗ slotValue) ↦{# 1/2} #(oblk_to_lit v) ∗
slot ↪[γm] (true, v) ∗
mono_list_idx_own γxs idx slot.
Definition SlotBag γsb (slotBag : loc) sbvmap xs : iProp :=
∃ γm γxs,
⌜γsb = encode (γm, γxs)⌝ ∗
(slotBag +ₗ slotBagHead) ↦ #(oloc_to_lit (last xs)) ∗
phys_map xs sbvmap ∗
ghost_slot_bag γm sbvmap ∗
mono_list_auth_own γxs 1 xs.
Definition SlotList γsb xs : iProp :=
∃ γm γxs,
⌜γsb = encode (γm, γxs)⌝ ∗
mono_list_lb_own γxs xs.
Fixpoint seq_bag_list (hd : base_lit) (vs : list (option blk)) : iProp :=
match vs with
| [] => ⌜hd = NULL⌝
| v :: vs =>
∃ (l : loc) (hd' : base_lit), ⌜hd = LitLoc l⌝ ∗
(l +ₗ seqNodeValue) ↦ #(oblk_to_lit v) ∗
(l +ₗ seqNodeNext) ↦ #hd' ∗
seq_bag_list hd' vs
end.
Definition SeqBag (seqBag : loc) (vs : list (option blk)) : iProp :=
∃ (hd : base_lit),
(seqBag +ₗ seqHead) ↦ #hd ∗
seq_bag_list hd vs.
(* Typeclass Instances *)
Global Instance Slot_TimeLess γsb slot idx v : Timeless (Slot γsb slot idx v).
Proof. apply _. Qed.
Global Instance ghost_slot_bag_timeless γm sbvmap : Timeless (ghost_slot_bag γm sbvmap).
Proof.
apply bi.sep_timeless; first apply _.
apply big_sepM_timeless. intros.
case_decide; apply _.
Qed.
Global Instance phys_list_rec_timeless xs : Timeless (phys_list_rec xs).
Proof. induction xs; apply _. Qed.
Global Instance phys_list_rec_persistent xs : Persistent (phys_list_rec xs).
Proof. induction xs; apply _. Qed.
Global Instance phys_list_persistent xs : Persistent (phys_list xs).
Proof. apply _. Qed.
Global Instance phys_list_timeless xs : Timeless (phys_list xs).
Proof. apply _. Qed.
Global Instance SlotBag_TimeLess γsb slotbag sbvmap slist : Timeless (SlotBag γsb slotbag sbvmap slist).
Proof. apply _. Qed.
Global Instance SlotList_Persistent γsb slist : Persistent (SlotList γsb slist).
Proof. apply _. Qed.
(* domain lemmas *)
Lemma elem_of_domain_neg xs sbvmap slot :
(slot ∈ xs ↔ is_Some (sbvmap !! slot)) ↔
(slot ∉ xs ↔ sbvmap !! slot = None).
Proof.
split; split; unfold not; intros.
- destruct (sbvmap !! slot); naive_solver.
- rewrite H H0 in H1. by destruct H1.
- destruct (sbvmap !! slot); naive_solver.
- destruct (decide (slot ∈ xs)); auto.
apply H in n. rewrite n in H0. by destruct H0.
Qed.
Lemma domain_of_snoc xs sbvmap slot :
domain_of (xs ++ [slot]) sbvmap → slot ∉ xs →
domain_of xs (delete slot sbvmap).
Proof.
intros. intros slot'.
destruct (decide (slot = slot')).
- subst. apply elem_of_domain_neg. by simpl_map.
- specialize (H slot'). simpl_map.
rewrite <- H, elem_of_snoc. naive_solver.
Qed.
Lemma domain_of_insert xs sbvmap slot st :
domain_of xs sbvmap →
domain_of (xs ++ [slot]) (<[slot:=st]> sbvmap).
Proof.
intros. intros slot'.
destruct (decide (slot = slot')).
- subst. simpl_map. rewrite elem_of_snoc. naive_solver.
- specialize (H slot'). simpl_map.
rewrite <- H, elem_of_snoc. naive_solver.
Qed.
Lemma domain_of_undelete xs sbvmap slot :
domain_of xs (delete slot sbvmap) →
is_Some (sbvmap !! slot) →
domain_of (xs ++ [slot]) sbvmap.
Proof.
intros. intros slot'.
destruct (decide (slot = slot')).
- subst. rewrite elem_of_snoc. naive_solver.
- specialize (H slot'). simpl_map.
rewrite <- H, elem_of_snoc. naive_solver.
Qed.
Lemma domain_of_update xs sbvmap slot st :
domain_of xs sbvmap →
is_Some (sbvmap !! slot) →
domain_of xs (<[slot:=st]> sbvmap).
Proof.
intros. intros slot'.
apply H in H0. destruct (decide (slot = slot')).
- subst. simpl_map. naive_solver.
- by simpl_map.
Qed.
(* phys list/map lemmas *)
Lemma phys_list_snoc xs slot :
phys_list (xs ++ [slot]) ⊣⊢
(slot +ₗ slotNext) ↦□ #(oloc_to_lit (last xs)) ∗
phys_list xs.
Proof.
unfold phys_list. rewrite reverse_snoc. simpl.
by rewrite head_reverse.
Qed.
Lemma phys_list_prefix xs xs' :
xs' `prefix_of` xs → phys_list xs -∗ phys_list xs'.
Proof.
assert (∀ xs xs',
xs' `suffix_of` xs → phys_list_rec xs -∗ phys_list_rec xs');
last first.
{ rewrite prefix_suffix_reverse. apply H. }
clear xs xs'.
intros. induction xs; iIntros "Lphys".
{ apply suffix_nil_inv in H. by subst. }
destruct xs'; auto.
apply suffix_cons_inv in H as [H|H].
- by rewrite H.
- iDestruct "Lphys" as "[Lhd Lphys]".
iApply (IHxs with "[Lphys]"); auto.
Qed.
Lemma phys_list_agree xs xs' :
last xs = last xs' → phys_list xs -∗ phys_list xs' -∗ ⌜xs = xs'⌝.
Proof.
revert xs'.
induction xs using rev_ind; iIntros (xs' Lhd) "#Lphys #Lphys'";
destruct (rev_des xs') as [|[x' [l' H']]]; subst; auto.
{ by rewrite last_snoc in Lhd. }
{ by rewrite last_snoc in Lhd. }
do 2 rewrite last_snoc in Lhd. injection Lhd as ->.
iDestruct (phys_list_snoc with "Lphys") as "[Lhdx Lphysx]".
iDestruct (phys_list_snoc with "Lphys'") as "[Lhdx' Lphysx']".
iDestruct (mapsto_agree with "Lhdx Lhdx'") as %[= ?].
iDestruct (IHxs with "Lphysx Lphysx'") as %->; auto.
Qed.
Lemma phys_list_NoDup xs :
phys_list xs -∗ ⌜NoDup xs⌝.
Proof.
induction xs using rev_ind; iIntros "#Lphys".
{ iPureIntro. by apply NoDup_nil. }
iDestruct (phys_list_snoc with "Lphys") as "[_ Lphys']".
iDestruct (IHxs with "Lphys'") as %LND.
destruct (decide (x ∈ xs)); last first.
{ iPureIntro. by apply NoDup_snoc. }
(* if a ∉ xs, derive contradiction due to cycle *)
apply elem_of_list_lookup in e as [i Hia].
iDestruct (phys_list_prefix xs (take (S i) xs) with "Lphys'") as
"#Lphysd". { apply take_prefix. }
iDestruct (phys_list_agree (xs++[x]) (take (S i) xs) with "[] []") as "%"; auto.
{ rewrite take_last; auto. rewrite Hia. apply last_snoc. }
assert (length (xs ++ [x]) > length (take (S i) xs)).
{ rewrite take_length snoc_length. lia. }
rewrite H in H0; lia.
Qed.
Lemma phys_map_snoc slot xs sbvmap :
phys_map (xs ++ [slot]) sbvmap -∗
phys_list (xs ++ [slot]) ∗
phys_map xs (delete slot sbvmap) ∗
∃ act v, slot_phys slot act v ∗ ⌜sbvmap !! slot = Some (act, v)⌝.
Proof.
iIntros "Mphys";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
iDestruct (phys_list_NoDup with "Lphys") as %LND.
iDestruct (phys_list_snoc with "Lphys") as "[_ Lphys']".
assert (is_Some (sbvmap !! slot)) as [[b v] H].
{ apply Mdom. apply elem_of_snoc; by left. }
iDestruct (big_sepM_delete _ sbvmap slot (b, v) with "Ms")
as "[Sphys Ms]"; auto.
iFrame. repeat iSplit; auto.
iPureIntro. apply domain_of_snoc; auto.
by apply NoDup_snoc in LND as [_ LND].
Qed.
Lemma phys_map_undelete slot xs act v sbvmap :
sbvmap !! slot = Some (act, v) →
domain_of xs sbvmap →
phys_list xs -∗
slot_phys slot act v -∗
([∗ map] k↦y ∈ delete slot sbvmap, slot_phys k y.1 y.2) -∗
phys_map xs sbvmap.
Proof.
iIntros "%Ml %Mdom Lphys Sphys Ms". iFrame. iSplit; auto.
iApply big_sepM_delete; first apply Ml. iFrame.
Qed.
Lemma phys_map_insert slot xs act v sbvmap :
sbvmap !! slot = None →
phys_map xs sbvmap -∗
slot_phys slot act v -∗
(slot +ₗ slotNext) ↦□ #(oloc_to_lit (last xs)) -∗
phys_map (xs ++ [slot]) (<[slot:=(act, v)]> sbvmap).
Proof.
iIntros "% Mphys Snxt Sphys";
iDestruct "Mphys" as "(Lphys & Ms & %Mdom)".
unfold phys_map. simpl.
repeat iSplit; auto.
- iApply phys_list_snoc. iFrame.
- iApply (big_sepM_delete _ _ slot).
{ apply lookup_insert. }
rewrite delete_insert_delete delete_notin; auto. iFrame.
- iPureIntro. by apply domain_of_insert.
Qed.
(* slot lemmas *)
Lemma slot_lookup γsb γm γxs xs l idx v :
γsb = prod_countable.(encode) (γm, γxs) →
mono_list_auth_own γxs 1 xs -∗
Slot γsb l idx v -∗
⌜xs !! idx = Some l⌝.
Proof.
iIntros (H) "●L S".
iDestruct "S" as (??) "(%&_&_&_&◯i)".
encode_agree H0.
iApply (mono_list_auth_idx_lookup with "●L ◯i").
Qed.
(* ghost slot bag lemmas *)
Lemma ghost_slot_bag_lookup γm sbvmap slot v :
ghost_slot_bag γm sbvmap -∗
slot ↪[γm] v -∗
⌜sbvmap !! slot = Some v⌝.
Proof.
iIntros "[●Mm _] ●Ms".
iApply (ghost_map_lookup with "●Mm ●Ms").
Qed.
Lemma ghost_slot_bag_insert v γm sbvmap slot :
sbvmap !! slot = None →
ghost_slot_bag γm sbvmap ==∗
ghost_slot_bag γm (<[slot:=v]> sbvmap) ∗
if decide (v = (false, None)) then True else slot ↪[γm] v.
Proof.
iIntros "% [●Mm ●S]".
iMod (ghost_map_insert slot v with "●Mm") as "[●Mm ●Ms]"; auto.
iModIntro. iFrame.
destruct (decide (v = (false, None))).
- iSplit; auto. iApply big_sepM_insert; auto.
rewrite lookup_insert; subst; simpl. simplify_option_eq. iFrame.
iApply big_sepM_mono; last auto; simpl.
iIntros (k x Hkx) "Dec".
destruct (decide (k = slot)); subst. { by rewrite H in Hkx. }
by rewrite lookup_insert_ne.
- iFrame. iApply big_sepM_insert; auto.
rewrite lookup_insert decide_False; last naive_solver.
iSplit; auto. iApply big_sepM_mono; last auto; simpl.
iIntros (k x Hkx) "Dec".
destruct (decide (k = slot)); subst. { by rewrite H in Hkx. }
by rewrite lookup_insert_ne.
Qed.
Lemma ghost_slot_bag_update v' γm sbvmap slot v :
sbvmap !! slot = Some v →
(if decide (v = (false, None)) then True else slot ↪[γm] v) -∗
ghost_slot_bag γm sbvmap ==∗
ghost_slot_bag γm (<[slot:=v']> sbvmap) ∗
if decide (v' = (false, None)) then True else slot ↪[γm] v'.
Proof.
iIntros "%H Decv [●Mm ●S]".
destruct (decide (v = (false, None))).
- (* ghost slot is inside the bag *)
subst.
iPoseProof (big_sepM_delete _ _ slot with "●S") as "[●Ms ●S]".
{ apply H. }
rewrite decide_True; auto.
iMod (ghost_map_update v' with "●Mm ●Ms") as "[●Mm ●Ms]"; auto.
iModIntro. iFrame.
destruct (decide (v' = (false, None))).
+ (* put ghost slot back into the bag *)
subst.
iPoseProof (big_sepM_insert _ (delete slot sbvmap) slot
with "[●Ms ●S]") as "●S".
{ by rewrite lookup_delete. }
{ iFrame. by rewrite decide_True. }
simpl. rewrite insert_delete_insert. iFrame.
rewrite insert_id; last apply H.
rewrite insert_id; last apply H. auto.
+ (* leave ghost slot outside the bag *)
iFrame. rewrite <- insert_delete_insert.
iApply big_sepM_insert.
{ by rewrite lookup_delete. }
rewrite lookup_insert.
rewrite decide_False; last naive_solver. iFrame.
iApply big_sepM_mono; last auto; simpl.
iIntros (k x Hkx) "Dec".
destruct (decide (k = slot)); subst. { by rewrite lookup_delete in Hkx. }
rewrite lookup_insert_ne; auto.
by rewrite lookup_delete_ne.
- (* ghost slot is outside the bag *)
iMod (ghost_map_update with "●Mm Decv") as "[●Mm ●Ms]"; auto.
iModIntro. iFrame.
rewrite <- insert_delete_insert.
destruct (decide (v' = (false, None))).
+ (* put ghost slot into the bag *)
subst. iSplit; auto.
iApply big_sepM_insert. { by rewrite lookup_delete. }
rewrite decide_True; last by rewrite lookup_insert. iFrame.
iPoseProof (big_sepM_delete with "●S") as "[_ ●S]";
first apply H.
iApply big_sepM_mono; last auto; simpl.
iIntros (k x Hkx) "Dec".
destruct (decide (k = slot)); subst. { by rewrite lookup_delete in Hkx. }
rewrite lookup_insert_ne; auto.
by rewrite lookup_delete_ne.
+ (* leave ghost slot outside the bag *)
iFrame. iApply big_sepM_insert. { by rewrite lookup_delete. }
rewrite decide_False; last first.
{ rewrite lookup_insert. naive_solver. }
iSplit; auto.
iPoseProof (big_sepM_delete with "●S") as "[_ ●S]";
first apply H.
iApply big_sepM_mono; last auto; simpl.
iIntros (k x Hkx) "Dec".
destruct (decide (k = slot)); subst. { by rewrite lookup_delete in Hkx. }
rewrite lookup_insert_ne; auto.
by rewrite lookup_delete_ne.
Qed.
(* lookup spec *)
Lemma SlotBag_lookup :
SlotBag_lookup' Slot SlotBag.
Proof.
iIntros (???????) "B S".
iDestruct "B" as (??) "(%Hγ & Bhd & Mphys & ●Mm & ●L)".
iDestruct "S" as (???) "(Sact & Sv & ●Ms & ◯i)";
encode_agree Hγ.
iSplit.
- iApply (mono_list_auth_idx_lookup with "●L ◯i").
- iApply (ghost_slot_bag_lookup with "●Mm ●Ms").
Qed.
Lemma SlotBag_prefix :
SlotBag_prefix' SlotBag SlotList.
Proof.
iIntros (?????) "B S".
iDestruct "B" as (??) "(%Hγ & Bhd & Mphys & ●Mm & ●L)".
iDestruct "S" as (???) "◯L";
encode_agree Hγ.
by iDestruct (mono_list_auth_lb_valid with "●L ◯L") as %[_ ?].
Qed.
Lemma SlotBag_NoDup :
SlotBag_NoDup' SlotBag.
Proof.
iIntros (????) "B";
iDestruct "B" as (??) "(%Hγ & Bhd & Mphys & ●Mm & ●L)".
iDestruct "Mphys" as "(Lphys & _ & _)".
by iApply phys_list_NoDup.
Qed.
(* slot acquire specs *)
Lemma phys_map_no_slot_xs xs sbvmap slot v :
phys_map xs sbvmap -∗
(slot +ₗ 2%nat) ↦ #v -∗
⌜slot ∉ xs⌝.
Proof.
iIntros "Mphys Sv";
iDestruct "Mphys" as "(Lphys & Ms & %Mdom)".
destruct (decide (slot ∈ xs)); auto.
apply Mdom in e as [bv e].
iDestruct (big_sepM_lookup with "Ms") as "Sphys"; [apply e|].
iDestruct "Sphys" as "(_ & Sv' & _)".
iDestruct (mapsto_valid_2 with "Sv Sv'") as %[H ->].
destruct bv. by destruct b.
Qed.
Lemma phys_map_no_slot_sbvmap xs sbvmap slot v :
phys_map xs sbvmap -∗
(slot +ₗ 2%nat) ↦ #v -∗
⌜sbvmap !! slot = None⌝.
Proof.
iIntros "Mphys Sv";
iDestruct "Mphys" as "(Lphys & Ms & %Mdom)".
destruct (sbvmap !! slot) eqn:e; auto.
iDestruct (big_sepM_lookup with "Ms") as "Sphys"; [apply e|].
iDestruct "Sphys" as "(_ & Sv' & _)".
iDestruct (mapsto_valid_2 with "Sv Sv'") as %[H ->].
destruct p. by destruct b.
Qed.
Lemma slot_bag_new_spec :
slot_bag_new_spec' SlotBag.
Proof.
iIntros (E Φ). iModIntro.
iIntros "_ HΦ".
wp_lam. wp_alloc slotbag as "sb↦" "†sb".
wp_pures. rewrite loc_add_0 array_singleton. wp_store.
iMod (ghost_map_alloc_empty) as (γm) "●m".
iMod (mono_list_own_alloc []) as (γxs) "[●xs _]".
remember (encode (γm, γxs)) as γsb eqn:Hγsb.
iAssert (SlotBag γsb slotbag ∅ []) with "[●m ●xs sb↦]" as "SlotBag".
{ repeat iExists _. rewrite loc_add_0. iFrame "∗%".
unfold phys_map, phys_list, phys_list_rec. rewrite !big_sepM_empty. iSplit; [|done].
simpl. iPureIntro. split_and!; [done..|]. unfold domain_of.
intro s. split; intro Empty.
- inversion Empty.
- rewrite lookup_empty in Empty. destruct Empty as [? EQ]. inversion EQ. }
iApply "HΦ". by iFrame.
Qed.
Lemma slot_bag_push_slot_loop_spec :
⊢ ∀ γsb (slotBag slot : loc) E,
(∃ p, (slot +ₗ slotNext) ↦ p) -∗
(slot +ₗ slotActive) ↦ #true -∗
(slot +ₗ slotValue) ↦ #NULL -∗
<<< ∀∀ sbvmap xs, ▷ SlotBag γsb slotBag sbvmap xs >>>
slot_bag_push_slot_loop #slotBag #slot @ E,∅,∅
<<< let idx := length xs in
let sbvmap' := <[slot := (true, None)]> sbvmap in
SlotBag γsb slotBag sbvmap' (xs ++ [slot]) ∗
Slot γsb slot idx None ∗
⌜sbvmap !! slot = None⌝,
RET #() >>>.
Proof.
iLöb as "IH".
iIntros (γsb slotBag slot E) "[%sp Snxt] Sact Sv % AU".
wp_rec. wp_pures. wp_bind (! _)%E.
iMod "AU" as (sbvmap xs) "[>B [Abort _]]";
iDestruct "B" as (γm γxs) "(%Enc & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
wp_load. iMod ("Abort" with "[Bhd ●Mm ●L Ms]") as "AU".
{ iNext; iFrame. iExists γm, γxs. iFrame; auto. }
clear -γsb. iModIntro. wp_pures. wp_store.
wp_pures. wp_bind (CmpXchg _ _ _)%E.
iMod "AU" as (sbvmap' xs') "[>B AU]";
iDestruct "B" as (γm γxs) "(%Enc & Bhd' & Mphys' & ●Mm & ●L)".
destruct (decide (last xs = last xs')) as [EQ|NE]; subst.
- (* success *)
rewrite -EQ.
iDestruct "AU" as "[_ Commit]".
wp_cmpxchg_suc.
(* make resources to commit *)
iDestruct (phys_map_no_slot_sbvmap with "[-Sv] Sv") as %Nosbv; auto.
iDestruct "Mphys'" as "(#Lphys' & Ms' & %Mdom')".
iDestruct (phys_list_agree with "Lphys Lphys'") as "%"; subst; auto.
iMod (mapsto_persist with "Snxt") as "#Snxt".
iMod (ghost_slot_bag_insert (true, None) with "●Mm")
as "[●Mm ●Ms]"; first apply Nosbv; simpl.
iMod (mono_list_auth_own_update (xs' ++ [slot]) with "●L")
as "[●L ◯L]". { apply prefix_app_cut. }
iDestruct "Sact" as "[Sact1 Sact2]".
iDestruct "Sv" as "[Sv1 Sv2]".
(* commit *)
iMod ("Commit" with "[-]") as "HΦ";
last (iModIntro; wp_pures; by iApply "HΦ").
iFrame. iSplitR "●Ms ◯L".
+ iExists γm, γxs. iFrame. iSplit; auto.
rewrite last_snoc. iFrame.
iApply (phys_map_insert with "[Ms'] [Sact1 Sv1]"); auto.
* iFrame. iSplit; auto.
* iFrame.
+ iSplit; auto. iExists γm, γxs. iSplit; auto. iFrame.
iApply (mono_list_idx_own_get with "◯L").
apply snoc_lookup.
- (* fail, loop *)
iDestruct "AU" as "[Abort _]".
wp_cmpxchg_fail.
iMod ("Abort" with "[Bhd' Mphys' ●Mm ●L]") as "AU".
{ iNext; iFrame. iExists γm, γxs. iFrame; auto. }
iModIntro. wp_pures.
iApply ("IH" with "[Snxt] [Sact] [Sv] [AU]"); auto.
Qed.
Lemma slot_bag_try_acquire_inactive_slot_spec (slotBag : loc) γsb E :
⊢ <<< ∀∀ γm γxs xs (sbvmap : gmap loc (bool * option blk)),
⌜γsb = encode (γm, γxs)⌝ ∗
(slotBag +ₗ slotBagHead) ↦ #(oloc_to_lit (last xs)) ∗
phys_map xs sbvmap ∗
ghost_slot_bag γm sbvmap ∗
mono_list_auth_own γxs 1 xs >>>
slot_bag_try_acquire_inactive_slot #slotBag @ E,∅,∅
<<< ∃∃ (q : option loc),
(slotBag +ₗ slotBagHead) ↦ #(oloc_to_lit (last xs)) ∗
mono_list_auth_own γxs 1 xs ∗
match q with
| None =>
phys_map xs sbvmap ∗
ghost_slot_bag γm sbvmap
| Some slot =>
⌜sbvmap !! slot = Some (false, (None))⌝ ∗
let sbvmap' := <[slot := (true, (None))]> sbvmap in
phys_map xs sbvmap' ∗
ghost_slot_bag γm sbvmap' ∗
(∃ idx, Slot γsb slot idx (None))
end,
RET #(oloc_to_lit q) >>>.
Proof.
iIntros "% AU".
wp_rec. wp_pures. wp_bind (! _)%E.
iMod "AU" as (γm γxs xs sbvmap) "[(%Hγ & Bhd & Mphys & ●Mm & ●L) [Abort _]]";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
iDestruct (mono_list_lb_own_get with "●L") as "#◯L".
wp_load. iMod ("Abort" with "[-]") as "AU"; iFrame; auto.
clear -Hγ. iModIntro. wp_pures.
iLöb as "IH" forall (xs) "Lphys ◯L".
(* fail loop if head is null *)
destruct (rev_des xs).
{ rewrite H. clear H xs.
wp_pures. wp_rec. wp_pures. iMod "AU" as (?? xs sbvmap) "[[%Hγ' P] [_ Commit]]".
iMod ("Commit" $! None with "[P]") as "HΦ"; last by iApply "HΦ".
iDestruct "P" as "(P1&P2&P3&P4)"; iFrame. }
destruct H as [x [xs' H]]. rewrite H. clear H xs. rename xs' into xs.
rewrite last_snoc. wp_pures.
(* load head *)
iDestruct (phys_list_snoc with "Lphys") as "[Lhd Lphysx]".
wp_rec. wp_load. wp_pures.
(* cas *)
wp_bind (CmpXchg _ _ _)%E.
iMod "AU" as (?? xs' sbvmap') "[(%Hγ' & Bhd' & Mphys' & ●Mm & ●L) AU]";
iDestruct "Mphys'" as "(#Lphys' & Ms' & %Mdom')";
encode_agree Hγ'.
iDestruct (mono_list_auth_lb_valid with "●L ◯L") as
%[_ Lpf].
iAssert ⌜is_Some (sbvmap' !! x)⌝%I as ((act, v)) "%Ml".
{ iPureIntro. apply Mdom'.
eapply elem_of_prefix; last apply Lpf.
apply elem_of_snoc; by left. }
iDestruct (big_sepM_delete _ sbvmap' x with "Ms'") as "[Sphys Ms']";
first apply Ml.
iDestruct "Sphys" as "(Sact & Sv & %Sav)".
destruct act; simpl in *.
- (* slot active, cas fail *)
iDestruct "AU" as "[Abort _]".
wp_cmpxchg_fail.
iMod ("Abort" with "[-]") as "AU".
{ iFrame. iSplit; auto.
iApply (phys_map_undelete with "[] [Sact Sv]"); auto.
1: apply Ml. iFrame. }
iModIntro. wp_pures.
iApply ("IH" with "[AU]"); auto.
iApply (mono_list_lb_own_le with "◯L").
apply prefix_app_cut.
- (* slot inactive, cas success *)
iDestruct "AU" as "[_ Commit]". subst.
wp_cmpxchg_suc.
iMod (ghost_slot_bag_update (true, None)
with "[] ●Mm") as "[●Mm ●Ms]"; eauto.
iDestruct "Sact" as "[Sact1 Sact2]".
iDestruct "Sv" as "[Sv1 Sv2]".
iMod ("Commit" $! (Some x) with "[-]") as "HΦ";
last (iModIntro; wp_pures; by iApply "HΦ").
simpl. iFrame. iSplit; auto.
iSplitL "Sact1 Sv1 Ms'".
+ iApply (phys_map_undelete x xs' true (None) _
with "[] [Sact1 Sv1]"); auto.
* apply lookup_insert.
* by apply domain_of_update.
* iFrame.
* by rewrite delete_insert_delete.
+ iExists (length xs), γm, γxs.
iFrame. iSplit; auto.
iApply mono_list_idx_own_get; auto.
apply snoc_lookup.
Qed.
(* slot specs *)
Lemma slot_bag_acquire_slot_spec :
slot_bag_acquire_slot_spec' Slot SlotBag.
Proof.
iIntros (γsb slotBag E) "% AU".
wp_rec. wp_bind (slot_bag_try_acquire_inactive_slot _)%E.
iApply (slot_bag_try_acquire_inactive_slot_spec slotBag γsb).
iAuIntro. unfold atomic_acc.
iMod "AU" as (sbvmap xs) "[>B AU]";
iDestruct "B" as (γm γxs) "(%Enc & Bhd & Mphys & ●Mm & ●L)".
iModIntro. iExists γm, γxs, xs, sbvmap. iFrame.
iSplit; auto.
iSplit.
{ (* prove abort *)
iDestruct "AU" as "[Abort _]".
iIntros "(Bhd & Mphys & ●Mm & ●L)".
iMod ("Abort" with "[-]"); auto.
iFrame. iExists γm, γxs; auto.
}
iIntros (q) "(Bhd & ●L & Scase)".
destruct q.
{ (* acquire success *)
iDestruct "AU" as "[_ Commit]".
iDestruct "Scase" as "(%Ml & Mphys & ●Mm & [%idx S])".
iDestruct (slot_lookup with "●L S") as "%"; eauto.
iMod ("Commit" with "[-]") as "HΦ";
last (iModIntro; iIntros "_"; wp_pures; by iApply "HΦ").
iFrame. iSplit; eauto.
iExists γm, γxs. iFrame "∗%".
}
(* acquire fail *)
iDestruct "AU" as "[Abort _]".
iDestruct "Scase" as "[Mphys ●Mm]".
iMod ("Abort" with "[-]") as "AU".
{ iFrame. iNext. iExists γm, γxs; iFrame; auto. }
iModIntro. iIntros "_". simpl. wp_pures. clear.
(* new slot *)
unfold slot_new. wp_pures. wp_alloc slot as "S" "?".
rewrite 3!array_cons 2!loc_add_assoc.
iDestruct "S" as "(Snxt & Sact & Sv & _)".
wp_pures. rewrite loc_add_0.
wp_store. wp_pures. wp_store. wp_pures. wp_store. wp_let.
(* push the new slot *)
wp_bind (slot_bag_push_slot_loop _ _)%E.
iApply (slot_bag_push_slot_loop_spec $! γsb with
"[Snxt] [Sact] [Sv]"); auto; auto.
iAuIntro. unfold atomic_acc.
iMod "AU" as (sbvmap xs) "[B AU]". iModIntro.
iExists sbvmap, xs. iFrame.
iSplit.
- (* abort *)
iDestruct "AU" as "[Abort _]".
iIntros "B". iMod ("Abort" with "B") as "AU".
iModIntro. iFrame.
- (* commit *)
iDestruct "AU" as "[_ Commit]".
iIntros "[B [S %]]".
iMod ("Commit" with "[B S]") as "HΦ"; iFrame; auto.
iModIntro. iIntros "_". wp_pures. by iApply "HΦ".
Qed.
Lemma slot_set_spec :
slot_set_spec' Slot SlotBag.
Proof.
iIntros (γsb slotBag slot idx oldv v E) ">S %Φ AU".
wp_rec. wp_pures.
iMod "AU" as (sbvmap xs) "[>B [_ Commit]]".
iDestruct (SlotBag_lookup with "B S") as "%".
iDestruct "S" as (γm γxs) "(%Hγ & Sact & Sv & ●Ms & ◯i)".
iDestruct "B" as (??) "(%Hγ' & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)";
encode_agree Hγ.
iDestruct (ghost_slot_bag_lookup with "●Mm ●Ms") as "%Hmap".
iMod (ghost_slot_bag_update (true, v) with "[●Ms] [●Mm]") as "[●Mm ●Ms]"; eauto with iFrame.
iDestruct (big_sepM_delete with "Ms") as "[Sphys Ms]"; first apply Hmap; simpl.
iDestruct "Sphys" as "[Sact2 [Sv2 %]]".
iCombine "Sv" "Sv2" as "Sv". wp_store.
(* make resources to commit *)
iDestruct "Sv" as "[Sv Sv2]".
iDestruct (big_sepM_insert
_ (delete slot sbvmap) slot (true, v) with "[Ms Sact2 Sv2]") as
"Ms".
{ by rewrite lookup_delete. }
{ iFrame. iFrame. }
rewrite insert_delete_insert.
(* commit *)
iMod ("Commit" with "[-]") as "HΦ"; last (iModIntro; auto; by iApply "HΦ").
iFrame. iSplit; auto. iSplitL "●L ●Mm"; iExists γm, γxs; iFrame; auto.
repeat iSplit; auto; iPureIntro.
by apply domain_of_update.
Qed.
Lemma slot_unset_spec :
slot_unset_spec' Slot SlotBag.
Proof.
iIntros (γsb slotBag slot idx v E) ">S %Φ AU".
wp_rec. wp_pures.
iMod "AU" as (sbvmap xs) "[>B [_ Commit]]".
iDestruct (SlotBag_lookup with "B S") as "%".
iDestruct "S" as (γm γxs) "(%Hγ & Sact & Sv & ●Ms & ◯i)".
iDestruct "B" as (??) "(%Hγ' & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)";
encode_agree Hγ.
iDestruct (ghost_slot_bag_lookup with "●Mm ●Ms") as "%Hmap".
iMod (ghost_slot_bag_update (true, None) with "[●Ms] [●Mm]") as "[●Mm ●Ms]"; eauto with iFrame.
iDestruct (big_sepM_delete with "Ms") as "[Sphys Ms]"; first apply Hmap; simpl.
iDestruct "Sphys" as "[Sact2 [Sv2 %]]".
iCombine "Sv" "Sv2" as "Sv". wp_store.
(* make resources to commit *)
iDestruct "Sv" as "[Sv Sv2]".
iDestruct (big_sepM_insert
_ (delete slot sbvmap) slot (true, None) with "[Ms Sact2 Sv2]") as
"Ms".
{ by rewrite lookup_delete. }
{ iFrame. iFrame. }
rewrite insert_delete_insert.
(* commit *)
iMod ("Commit" with "[-]") as "HΦ"; last (iModIntro; auto; by iApply "HΦ").
iFrame. iSplit; auto.
iSplitL "●L ●Mm"; iExists γm, γxs; iFrame; auto.
repeat iSplit; auto; iPureIntro. by apply domain_of_update.
Qed.
Lemma slot_drop_spec :
slot_drop_spec' Slot SlotBag.
Proof.
iIntros (γsb slotBag slot idx E) "Slot %Φ AU".
wp_lam. wp_op.
iMod "AU" as (sbvmap slist) "[>SlotBag [_ Commit]]".
iDestruct (SlotBag_lookup with "SlotBag Slot") as "%Hsb".
iDestruct "Slot" as (γm γxs) "(%Hγ & Sact & Sv & ●Ms & ◯i)".
iDestruct "SlotBag" as (??) "(%Hγ' & Bhd & Mphys & ●Mm & ●L)".
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
encode_agree Hγ.
destruct Hsb as [Hslist Hslot].
iMod (ghost_slot_bag_update (false, None) with "[●Ms] [●Mm]") as "[●Mm ●Ms]"; eauto with iFrame; simpl.
iDestruct (big_sepM_delete with "Ms") as "[Sphys Ms]"; first apply Hslot; simpl.
iDestruct "Sphys" as "[Sact2 [Sv2 %]]".
iCombine "Sact" "Sact2" as "Sact". wp_store.
(* make resources to commit *)
iDestruct (big_sepM_insert
_ (delete slot sbvmap) slot (false, None) with "[Ms Sact Sv Sv2]") as
"Ms".
{ by rewrite lookup_delete. }
{ by do 2 iFrame. }
rewrite insert_delete_insert.
(* commit *)
iMod ("Commit" with "[-]") as "HΦ"; last (iModIntro; auto; by iApply "HΦ").
iFrame. iSplit; first by done.
iExists γm, γxs; iFrame; auto.
repeat iSplit; auto; iPureIntro. by apply domain_of_update.
Qed.
(* snapshot specs *)
Lemma seq_bag_new_spec :
seq_bag_new_spec' SeqBag.
Proof.
iIntros (E Φ _) "HΦ".
wp_rec. wp_alloc seqBag as "SBhd" "?". wp_pures.
rewrite array_singleton loc_add_0.
wp_store.
iModIntro. iApply "HΦ". iExists _.
rewrite loc_add_0. by iFrame.
Qed.
Lemma seq_bag_add_spec :
seq_bag_add_spec' SeqBag.
Proof.
iIntros (E seqBag vs v Φ) "SB HΦ";
iDestruct "SB" as (hd) "(SBhd & SBphys)".
wp_rec. wp_pures. wp_load. wp_pures.
wp_alloc seqNode as "N" "?". wp_pures.
rewrite 2!array_cons. iDestruct "N" as "[Nnxt [Nv _]]".
rewrite 2!loc_add_0.
wp_store. wp_pures. wp_store. wp_pures. rewrite loc_add_0. wp_store.
iModIntro. iApply "HΦ". iExists seqNode; iFrame.
rewrite loc_add_0. iFrame.
iExists seqNode, hd. by iFrame.
Qed.
Lemma slot_bag_read_head_spec :
slot_bag_read_head_spec' SlotBag SlotList.
Proof.
iIntros (????? Φ) "B HΦ".
iDestruct "B" as (γm γxs) "(%Hγ & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
iDestruct (mono_list_lb_own_get with "●L") as "#◯L".
wp_load.
iApply "HΦ". iModIntro. iSplit; (iExists _,_; iFrame "∗#%").
Qed.
Lemma slot_read_active_spec :
slot_read_active_spec' SlotBag.
Proof.
iIntros (??????? Hidx Φ) "!> B HΦ".
iDestruct "B" as (γm γxs) "(%Hγ & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
have [[act v] Ml] : is_Some (sbvmap !! slot).
{ apply Mdom. by eapply elem_of_list_lookup_2. }
iDestruct (big_sepM_lookup_acc _ _ _ _ Ml with "Ms") as "[Sphys Ms]".
iDestruct "Sphys" as "(Sact & Sv & %Sav)".
wp_load.
iSpecialize ("Ms" with "[$Sact $Sv //]").
iApply "HΦ".
iModIntro. iSplit; [done|].
iExists γm, γxs. iFrame. repeat iSplit; auto.
Qed.
Lemma slot_read_value_spec :
slot_read_value_spec' SlotBag.
Proof.
iIntros (??????? Hidx Φ) "!> B HΦ".
iDestruct "B" as (γm γxs) "(%Hγ & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
have [[act v] Ml] : is_Some (sbvmap !! slot).
{ apply Mdom. by eapply elem_of_list_lookup_2. }
iDestruct (big_sepM_lookup_acc _ _ _ _ Ml with "Ms") as "[Sphys Ms]".
iDestruct "Sphys" as "(Sact & Sv & %Sav)".
wp_load.
iSpecialize ("Ms" with "[$Sact $Sv //]").
iApply "HΦ".
iModIntro. iSplit; [done|].
iExists γm, γxs. iFrame. repeat iSplit; auto.
Qed.
Lemma phys_list_lookup slist idx slot :
slist !! idx = Some slot →
phys_list slist -∗
(slot +ₗ slotNext) ↦□ #(oloc_to_lit (last (take idx slist))).
Proof.
unfold phys_list.
iIntros (Hidx) "#LPhys".
iInduction slist as [|slot' slist'] "IH" using rev_ind forall (idx slot Hidx); first done.
rewrite reverse_app /=. iDestruct "LPhys" as "[slot↦ Lphys']". fold (phys_list slist').
rewrite head_reverse.
case (decide (idx = length slist')) as [->|NE].
{ rewrite snoc_lookup in Hidx. injection Hidx as ->.
rewrite take_app. done. }
have {}Hidx : slist' !! idx = Some slot.
{ apply lookup_lt_Some in Hidx as Hlen. rewrite app_length /= in Hlen.
by rewrite lookup_app_l in Hidx; last lia. }
apply lookup_lt_Some in Hidx as Hlen.
rewrite take_app_le; last lia.
by iApply "IH".
Qed.
Lemma slot_read_next_spec :
slot_read_next_spec' SlotBag.
Proof.
iIntros (??????? Hidx Φ) "!> B HΦ".
iDestruct "B" as (γm γxs) "(%Hγ & Bhd & Mphys & ●Mm & ●L)";
iDestruct "Mphys" as "(#Lphys & Ms & %Mdom)".
have [[act v] Ml] : is_Some (sbvmap !! slot).
{ apply Mdom. by eapply elem_of_list_lookup_2. }
iDestruct (big_sepM_lookup_acc _ _ _ _ Ml with "Ms") as "[Sphys Ms]".
iDestruct "Sphys" as "(Sact & Sv & %Sav)". simpl.
iDestruct (phys_list_lookup _ _ _ Hidx with "Lphys") as "?".
wp_load.
iModIntro. iApply "HΦ".
iSpecialize ("Ms" with "[$Sact $Sv //]").
iExists γm, γxs. iFrame. repeat iSplit; auto.
Qed.
Lemma seq_bag_contains_loop_spec :
∀ hd vs (v : option blk) E,
{{{ seq_bag_list hd vs }}}
seq_bag_contains_loop #hd #(oblk_to_lit v) @ E
{{{ RET #(bool_decide (v ∈ vs));
seq_bag_list hd vs }}}.
Proof.
iIntros (hd vs v E Φ) "SBphys HΦ".
iLöb as "IH" forall (hd vs).
wp_rec. wp_pures.
destruct vs as [|b vs].
{ (* search fail *)
simpl. iDestruct "SBphys" as %->. wp_pures.
by iApply "HΦ". }
simpl. iDestruct "SBphys" as (l hd') "(% & SBv & SBnxt & SBphys')".
subst. wp_pures. wp_load.
wp_pures.
destruct (decide (b = v)); subst.
{ (* search success *)
rewrite bool_decide_eq_true_2; last first.
{ apply elem_of_cons; by left. }
wp_pures. destruct v as [v|].
- wp_pures. rewrite bool_decide_eq_true_2; auto; wp_pures.
iModIntro. iApply "HΦ". by repeat (iExists _; iFrame).
- wp_pures. iApply "HΦ". by repeat (iExists _; iFrame).
}
replace (bool_decide (b = v)) with false; last first.
{ rewrite bool_decide_eq_false_2; done. }
wp_pures. wp_load.
iApply ("IH" $! hd' vs with "[SBphys']"); auto.
iNext. iIntros "SBphys".
replace (bool_decide (v ∈ vs)) with (bool_decide (v ∈ b::vs)).
- iApply "HΦ". iExists l, hd'; by iFrame.
- destruct (decide (v ∈ vs)).
+ rewrite bool_decide_eq_true_2. 2: apply elem_of_cons; by right.
rewrite bool_decide_eq_true_2; auto.
+ rewrite bool_decide_eq_false_2. 2: apply not_elem_of_cons; auto.
rewrite bool_decide_eq_false_2; auto.
Qed.
Lemma seq_bag_contains_spec :
seq_bag_contains_spec' SeqBag.
Proof.
iIntros (seqBag vs v E Φ) "SB HΦ";
iDestruct "SB" as (hd) "(SBhd & SBphys)".
wp_rec. wp_pures. wp_load. wp_let.
iApply (seq_bag_contains_loop_spec hd vs v with "SBphys"); auto.
iNext. iIntros "SBphys". iApply "HΦ". iExists hd; by iFrame.
Qed.