summaryrefslogtreecommitdiff
path: root/tests/hol4/betree/betree_FunsTheory.sig
blob: 4127cba1bb37f28179e03ec1f37d0908ac20d09f (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
signature betree_FunsTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val betree_be_tree_apply_back_def : thm
    val betree_be_tree_apply_fwd_def : thm
    val betree_be_tree_delete_back_def : thm
    val betree_be_tree_delete_fwd_def : thm
    val betree_be_tree_insert_back_def : thm
    val betree_be_tree_insert_fwd_def : thm
    val betree_be_tree_lookup_back_def : thm
    val betree_be_tree_lookup_fwd_def : thm
    val betree_be_tree_new_fwd_def : thm
    val betree_be_tree_upsert_back_def : thm
    val betree_be_tree_upsert_fwd_def : thm
    val betree_fresh_node_id_back_def : thm
    val betree_fresh_node_id_fwd_def : thm
    val betree_internal_flush_back_def : thm
    val betree_internal_flush_fwd_def : thm
    val betree_internal_lookup_in_children_back_def : thm
    val betree_internal_lookup_in_children_fwd_def : thm
    val betree_leaf_split_back_def : thm
    val betree_leaf_split_fwd_def : thm
    val betree_list_hd_fwd_def : thm
    val betree_list_head_has_key_fwd_def : thm
    val betree_list_len_fwd_def : thm
    val betree_list_partition_at_pivot_fwd_def : thm
    val betree_list_pop_front_back_def : thm
    val betree_list_pop_front_fwd_def : thm
    val betree_list_push_front_fwd_back_def : thm
    val betree_list_split_at_fwd_def : thm
    val betree_load_internal_node_fwd_def : thm
    val betree_load_leaf_node_fwd_def : thm
    val betree_node_apply_back_def : thm
    val betree_node_apply_fwd_def : thm
    val betree_node_apply_messages_back_def : thm
    val betree_node_apply_messages_fwd_def : thm
    val betree_node_apply_messages_to_internal_fwd_back_def : thm
    val betree_node_apply_messages_to_leaf_fwd_back_def : thm
    val betree_node_apply_to_internal_fwd_back_def : thm
    val betree_node_apply_to_leaf_fwd_back_def : thm
    val betree_node_apply_upserts_back_def : thm
    val betree_node_apply_upserts_fwd_def : thm
    val betree_node_filter_messages_for_key_fwd_back_def : thm
    val betree_node_id_counter_fresh_id_back_def : thm
    val betree_node_id_counter_fresh_id_fwd_def : thm
    val betree_node_id_counter_new_fwd_def : thm
    val betree_node_lookup_back_def : thm
    val betree_node_lookup_first_message_after_key_back_def : thm
    val betree_node_lookup_first_message_after_key_fwd_def : thm
    val betree_node_lookup_first_message_for_key_back_def : thm
    val betree_node_lookup_first_message_for_key_fwd_def : thm
    val betree_node_lookup_fwd_def : thm
    val betree_node_lookup_in_bindings_fwd_def : thm
    val betree_node_lookup_mut_in_bindings_back_def : thm
    val betree_node_lookup_mut_in_bindings_fwd_def : thm
    val betree_store_internal_node_fwd_def : thm
    val betree_store_leaf_node_fwd_def : thm
    val betree_upsert_update_fwd_def : thm
    val main_fwd_def : thm
  
  val betree_Funs_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [betree_Opaque] Parent theory of "betree_Funs"
   
   [betree_be_tree_apply_back_def]  Definition
      
      ⊢ ∀self key msg st.
          betree_be_tree_apply_back self key msg st =
          do
            (n,nic) <-
              betree_node_apply_back self.betree_be_tree_root
                self.betree_be_tree_params self.betree_be_tree_node_id_cnt
                key msg st;
            Return
              (self with
               <|betree_be_tree_node_id_cnt := nic;
                 betree_be_tree_root := n|>)
          od
   
   [betree_be_tree_apply_fwd_def]  Definition
      
      ⊢ ∀self key msg st.
          betree_be_tree_apply_fwd self key msg st =
          do
            (st0,_) <-
              betree_node_apply_fwd self.betree_be_tree_root
                self.betree_be_tree_params self.betree_be_tree_node_id_cnt
                key msg st;
            monad_ignore_bind
              (betree_node_apply_back self.betree_be_tree_root
                 self.betree_be_tree_params self.betree_be_tree_node_id_cnt
                 key msg st) (Return (st0,()))
          od
   
   [betree_be_tree_delete_back_def]  Definition
      
      ⊢ ∀self key st.
          betree_be_tree_delete_back self key st =
          betree_be_tree_apply_back self key BetreeMessageDelete st
   
   [betree_be_tree_delete_fwd_def]  Definition
      
      ⊢ ∀self key st.
          betree_be_tree_delete_fwd self key st =
          do
            (st0,_) <-
              betree_be_tree_apply_fwd self key BetreeMessageDelete st;
            monad_ignore_bind
              (betree_be_tree_apply_back self key BetreeMessageDelete st)
              (Return (st0,()))
          od
   
   [betree_be_tree_insert_back_def]  Definition
      
      ⊢ ∀self key value st.
          betree_be_tree_insert_back self key value st =
          betree_be_tree_apply_back self key (BetreeMessageInsert value) st
   
   [betree_be_tree_insert_fwd_def]  Definition
      
      ⊢ ∀self key value st.
          betree_be_tree_insert_fwd self key value st =
          do
            (st0,_) <-
              betree_be_tree_apply_fwd self key (BetreeMessageInsert value)
                st;
            monad_ignore_bind
              (betree_be_tree_apply_back self key
                 (BetreeMessageInsert value) st) (Return (st0,()))
          od
   
   [betree_be_tree_lookup_back_def]  Definition
      
      ⊢ ∀self key st.
          betree_be_tree_lookup_back self key st =
          do
            n <- betree_node_lookup_back self.betree_be_tree_root key st;
            Return (self with betree_be_tree_root := n)
          od
   
   [betree_be_tree_lookup_fwd_def]  Definition
      
      ⊢ ∀self key st.
          betree_be_tree_lookup_fwd self key st =
          betree_node_lookup_fwd self.betree_be_tree_root key st
   
   [betree_be_tree_new_fwd_def]  Definition
      
      ⊢ ∀min_flush_size split_size st.
          betree_be_tree_new_fwd min_flush_size split_size st =
          do
            node_id_cnt <- betree_node_id_counter_new_fwd;
            id <- betree_node_id_counter_fresh_id_fwd node_id_cnt;
            (st0,_) <- betree_store_leaf_node_fwd id BetreeListNil st;
            node_id_cnt0 <-
              betree_node_id_counter_fresh_id_back node_id_cnt;
            Return
              (st0,
               <|betree_be_tree_params :=
                   <|betree_params_min_flush_size := min_flush_size;
                     betree_params_split_size := split_size|>;
                 betree_be_tree_node_id_cnt := node_id_cnt0;
                 betree_be_tree_root :=
                   BetreeNodeLeaf
                     <|betree_leaf_id := id;
                       betree_leaf_size := int_to_u64 0|> |>)
          od
   
   [betree_be_tree_upsert_back_def]  Definition
      
      ⊢ ∀self key upd st.
          betree_be_tree_upsert_back self key upd st =
          betree_be_tree_apply_back self key (BetreeMessageUpsert upd) st
   
   [betree_be_tree_upsert_fwd_def]  Definition
      
      ⊢ ∀self key upd st.
          betree_be_tree_upsert_fwd self key upd st =
          do
            (st0,_) <-
              betree_be_tree_apply_fwd self key (BetreeMessageUpsert upd)
                st;
            monad_ignore_bind
              (betree_be_tree_apply_back self key (BetreeMessageUpsert upd)
                 st) (Return (st0,()))
          od
   
   [betree_fresh_node_id_back_def]  Definition
      
      ⊢ ∀counter.
          betree_fresh_node_id_back counter =
          u64_add counter (int_to_u64 1)
   
   [betree_fresh_node_id_fwd_def]  Definition
      
      ⊢ ∀counter.
          betree_fresh_node_id_fwd counter =
          monad_ignore_bind (u64_add counter (int_to_u64 1))
            (Return counter)
   
   [betree_internal_flush_back_def]  Definition
      
      ⊢ ∀self params node_id_cnt content st.
          betree_internal_flush_back self params node_id_cnt content st =
          do
            p <-
              betree_list_partition_at_pivot_fwd content
                self.betree_internal_pivot;
            (msgs_left,msgs_right) <<- p;
            len_left <- betree_list_len_fwd msgs_left;
            if u64_ge len_left params.betree_params_min_flush_size then
              do
                (st0,_) <-
                  betree_node_apply_messages_fwd self.betree_internal_left
                    params node_id_cnt msgs_left st;
                (n,node_id_cnt0) <-
                  betree_node_apply_messages_back self.betree_internal_left
                    params node_id_cnt msgs_left st;
                len_right <- betree_list_len_fwd msgs_right;
                if
                  u64_ge len_right params.betree_params_min_flush_size
                then
                  do
                    (n0,node_id_cnt1) <-
                      betree_node_apply_messages_back
                        self.betree_internal_right params node_id_cnt0
                        msgs_right st0;
                    Return
                      (self with
                       <|betree_internal_left := n;
                         betree_internal_right := n0|>,node_id_cnt1)
                  od
                else
                  Return (self with betree_internal_left := n,node_id_cnt0)
              od
            else
              do
                (n,node_id_cnt0) <-
                  betree_node_apply_messages_back
                    self.betree_internal_right params node_id_cnt
                    msgs_right st;
                Return (self with betree_internal_right := n,node_id_cnt0)
              od
          od
   
   [betree_internal_flush_fwd_def]  Definition
      
      ⊢ ∀self params node_id_cnt content st.
          betree_internal_flush_fwd self params node_id_cnt content st =
          do
            p <-
              betree_list_partition_at_pivot_fwd content
                self.betree_internal_pivot;
            (msgs_left,msgs_right) <<- p;
            len_left <- betree_list_len_fwd msgs_left;
            if u64_ge len_left params.betree_params_min_flush_size then
              do
                (st0,_) <-
                  betree_node_apply_messages_fwd self.betree_internal_left
                    params node_id_cnt msgs_left st;
                (_,node_id_cnt0) <-
                  betree_node_apply_messages_back self.betree_internal_left
                    params node_id_cnt msgs_left st;
                len_right <- betree_list_len_fwd msgs_right;
                if
                  u64_ge len_right params.betree_params_min_flush_size
                then
                  do
                    (st1,_) <-
                      betree_node_apply_messages_fwd
                        self.betree_internal_right params node_id_cnt0
                        msgs_right st0;
                    monad_ignore_bind
                      (betree_node_apply_messages_back
                         self.betree_internal_right params node_id_cnt0
                         msgs_right st0) (Return (st1,BetreeListNil))
                  od
                else Return (st0,msgs_right)
              od
            else
              do
                (st0,_) <-
                  betree_node_apply_messages_fwd self.betree_internal_right
                    params node_id_cnt msgs_right st;
                monad_ignore_bind
                  (betree_node_apply_messages_back
                     self.betree_internal_right params node_id_cnt
                     msgs_right st) (Return (st0,msgs_left))
              od
          od
   
   [betree_internal_lookup_in_children_back_def]  Definition
      
      ⊢ ∀self key st.
          betree_internal_lookup_in_children_back self key st =
          if u64_lt key self.betree_internal_pivot then
            do
              n <- betree_node_lookup_back self.betree_internal_left key st;
              Return (self with betree_internal_left := n)
            od
          else
            do
              n <-
                betree_node_lookup_back self.betree_internal_right key st;
              Return (self with betree_internal_right := n)
            od
   
   [betree_internal_lookup_in_children_fwd_def]  Definition
      
      ⊢ ∀self key st.
          betree_internal_lookup_in_children_fwd self key st =
          if u64_lt key self.betree_internal_pivot then
            betree_node_lookup_fwd self.betree_internal_left key st
          else betree_node_lookup_fwd self.betree_internal_right key st
   
   [betree_leaf_split_back_def]  Definition
      
      ⊢ ∀self content params node_id_cnt st.
          betree_leaf_split_back self content params node_id_cnt st =
          do
            p <-
              betree_list_split_at_fwd content
                params.betree_params_split_size;
            (content0,content1) <<- p;
            monad_ignore_bind (betree_list_hd_fwd content1)
              do
                id0 <- betree_node_id_counter_fresh_id_fwd node_id_cnt;
                node_id_cnt0 <-
                  betree_node_id_counter_fresh_id_back node_id_cnt;
                id1 <- betree_node_id_counter_fresh_id_fwd node_id_cnt0;
                (st0,_) <- betree_store_leaf_node_fwd id0 content0 st;
                monad_ignore_bind
                  (betree_store_leaf_node_fwd id1 content1 st0)
                  (betree_node_id_counter_fresh_id_back node_id_cnt0)
              od
          od
   
   [betree_leaf_split_fwd_def]  Definition
      
      ⊢ ∀self content params node_id_cnt st.
          betree_leaf_split_fwd self content params node_id_cnt st =
          do
            p <-
              betree_list_split_at_fwd content
                params.betree_params_split_size;
            (content0,content1) <<- p;
            p0 <- betree_list_hd_fwd content1;
            (pivot,_) <<- p0;
            id0 <- betree_node_id_counter_fresh_id_fwd node_id_cnt;
            node_id_cnt0 <-
              betree_node_id_counter_fresh_id_back node_id_cnt;
            id1 <- betree_node_id_counter_fresh_id_fwd node_id_cnt0;
            (st0,_) <- betree_store_leaf_node_fwd id0 content0 st;
            (st1,_) <- betree_store_leaf_node_fwd id1 content1 st0;
            n <<-
              BetreeNodeLeaf
                <|betree_leaf_id := id0;
                  betree_leaf_size := params.betree_params_split_size|>;
            n0 <<-
              BetreeNodeLeaf
                <|betree_leaf_id := id1;
                  betree_leaf_size := params.betree_params_split_size|>;
            Return
              (st1,
               <|betree_internal_id := self.betree_leaf_id;
                 betree_internal_pivot := pivot; betree_internal_left := n;
                 betree_internal_right := n0|>)
          od
   
   [betree_list_hd_fwd_def]  Definition
      
      ⊢ ∀self.
          betree_list_hd_fwd self =
          case self of
            BetreeListCons hd l => Return hd
          | BetreeListNil => Fail Failure
   
   [betree_list_head_has_key_fwd_def]  Definition
      
      ⊢ ∀self key.
          betree_list_head_has_key_fwd self key =
          case self of
            BetreeListCons hd l => (let (i,_) = hd in Return (i = key))
          | BetreeListNil => Return F
   
   [betree_list_len_fwd_def]  Definition
      
      ⊢ ∀self.
          betree_list_len_fwd self =
          case self of
            BetreeListCons t tl =>
              do i <- betree_list_len_fwd tl; u64_add (int_to_u64 1) i od
          | BetreeListNil => Return (int_to_u64 0)
   
   [betree_list_partition_at_pivot_fwd_def]  Definition
      
      ⊢ ∀self pivot.
          betree_list_partition_at_pivot_fwd self pivot =
          case self of
            BetreeListCons hd tl =>
              (let
                 (i,t) = hd
               in
                 if u64_ge i pivot then
                   Return (BetreeListNil,BetreeListCons (i,t) tl)
                 else
                   do
                     p <- betree_list_partition_at_pivot_fwd tl pivot;
                     (ls0,ls1) <<- p;
                     l <<- ls0;
                     Return (BetreeListCons (i,t) l,ls1)
                   od)
          | BetreeListNil => Return (BetreeListNil,BetreeListNil)
   
   [betree_list_pop_front_back_def]  Definition
      
      ⊢ ∀self.
          betree_list_pop_front_back self =
          (let
             ls = mem_replace_fwd self BetreeListNil
           in
             case ls of
               BetreeListCons x tl => Return tl
             | BetreeListNil => Fail Failure)
   
   [betree_list_pop_front_fwd_def]  Definition
      
      ⊢ ∀self.
          betree_list_pop_front_fwd self =
          (let
             ls = mem_replace_fwd self BetreeListNil
           in
             case ls of
               BetreeListCons x tl => Return x
             | BetreeListNil => Fail Failure)
   
   [betree_list_push_front_fwd_back_def]  Definition
      
      ⊢ ∀self x.
          betree_list_push_front_fwd_back self x =
          (let
             tl = mem_replace_fwd self BetreeListNil;
             l = tl
           in
             Return (BetreeListCons x l))
   
   [betree_list_split_at_fwd_def]  Definition
      
      ⊢ ∀self n.
          betree_list_split_at_fwd self n =
          if n = int_to_u64 0 then Return (BetreeListNil,self)
          else
            case self of
              BetreeListCons hd tl =>
                do
                  i <- u64_sub n (int_to_u64 1);
                  p <- betree_list_split_at_fwd tl i;
                  (ls0,ls1) <<- p;
                  l <<- ls0;
                  Return (BetreeListCons hd l,ls1)
                od
            | BetreeListNil => Fail Failure
   
   [betree_load_internal_node_fwd_def]  Definition
      
      ⊢ ∀id st.
          betree_load_internal_node_fwd id st =
          betree_utils_load_internal_node_fwd id st
   
   [betree_load_leaf_node_fwd_def]  Definition
      
      ⊢ ∀id st.
          betree_load_leaf_node_fwd id st =
          betree_utils_load_leaf_node_fwd id st
   
   [betree_node_apply_back_def]  Definition
      
      ⊢ ∀self params node_id_cnt key new_msg st.
          betree_node_apply_back self params node_id_cnt key new_msg st =
          (let
             l = BetreeListNil
           in
             betree_node_apply_messages_back self params node_id_cnt
               (BetreeListCons (key,new_msg) l) st)
   
   [betree_node_apply_fwd_def]  Definition
      
      ⊢ ∀self params node_id_cnt key new_msg st.
          betree_node_apply_fwd self params node_id_cnt key new_msg st =
          (let
             l = BetreeListNil
           in
             do
               (st0,_) <-
                 betree_node_apply_messages_fwd self params node_id_cnt
                   (BetreeListCons (key,new_msg) l) st;
               monad_ignore_bind
                 (betree_node_apply_messages_back self params node_id_cnt
                    (BetreeListCons (key,new_msg) l) st) (Return (st0,()))
             od)
   
   [betree_node_apply_messages_back_def]  Definition
      
      ⊢ ∀self params node_id_cnt msgs st.
          betree_node_apply_messages_back self params node_id_cnt msgs st =
          case self of
            BetreeNodeInternal node =>
              do
                (st0,content) <-
                  betree_load_internal_node_fwd node.betree_internal_id st;
                content0 <-
                  betree_node_apply_messages_to_internal_fwd_back content
                    msgs;
                num_msgs <- betree_list_len_fwd content0;
                if u64_ge num_msgs params.betree_params_min_flush_size then
                  do
                    (st1,content1) <-
                      betree_internal_flush_fwd node params node_id_cnt
                        content0 st0;
                    (node0,node_id_cnt0) <-
                      betree_internal_flush_back node params node_id_cnt
                        content0 st0;
                    monad_ignore_bind
                      (betree_store_internal_node_fwd
                         node0.betree_internal_id content1 st1)
                      (Return (BetreeNodeInternal node0,node_id_cnt0))
                  od
                else
                  monad_ignore_bind
                    (betree_store_internal_node_fwd node.betree_internal_id
                       content0 st0)
                    (Return (BetreeNodeInternal node,node_id_cnt))
              od
          | BetreeNodeLeaf node' =>
            do
              (st0,content) <-
                betree_load_leaf_node_fwd node'.betree_leaf_id st;
              content0 <-
                betree_node_apply_messages_to_leaf_fwd_back content msgs;
              len <- betree_list_len_fwd content0;
              i <- u64_mul (int_to_u64 2) params.betree_params_split_size;
              if u64_ge len i then
                do
                  (st1,new_node) <-
                    betree_leaf_split_fwd node' content0 params node_id_cnt
                      st0;
                  monad_ignore_bind
                    (betree_store_leaf_node_fwd node'.betree_leaf_id
                       BetreeListNil st1)
                    do
                      node_id_cnt0 <-
                        betree_leaf_split_back node' content0 params
                          node_id_cnt st0;
                      Return (BetreeNodeInternal new_node,node_id_cnt0)
                    od
                od
              else
                monad_ignore_bind
                  (betree_store_leaf_node_fwd node'.betree_leaf_id content0
                     st0)
                  (Return
                     (BetreeNodeLeaf (node' with betree_leaf_size := len),
                      node_id_cnt))
            od
   
   [betree_node_apply_messages_fwd_def]  Definition
      
      ⊢ ∀self params node_id_cnt msgs st.
          betree_node_apply_messages_fwd self params node_id_cnt msgs st =
          case self of
            BetreeNodeInternal node =>
              do
                (st0,content) <-
                  betree_load_internal_node_fwd node.betree_internal_id st;
                content0 <-
                  betree_node_apply_messages_to_internal_fwd_back content
                    msgs;
                num_msgs <- betree_list_len_fwd content0;
                if u64_ge num_msgs params.betree_params_min_flush_size then
                  do
                    (st1,content1) <-
                      betree_internal_flush_fwd node params node_id_cnt
                        content0 st0;
                    (node0,_) <-
                      betree_internal_flush_back node params node_id_cnt
                        content0 st0;
                    (st2,_) <-
                      betree_store_internal_node_fwd
                        node0.betree_internal_id content1 st1;
                    Return (st2,())
                  od
                else
                  do
                    (st1,_) <-
                      betree_store_internal_node_fwd
                        node.betree_internal_id content0 st0;
                    Return (st1,())
                  od
              od
          | BetreeNodeLeaf node' =>
            do
              (st0,content) <-
                betree_load_leaf_node_fwd node'.betree_leaf_id st;
              content0 <-
                betree_node_apply_messages_to_leaf_fwd_back content msgs;
              len <- betree_list_len_fwd content0;
              i <- u64_mul (int_to_u64 2) params.betree_params_split_size;
              if u64_ge len i then
                do
                  (st1,_) <-
                    betree_leaf_split_fwd node' content0 params node_id_cnt
                      st0;
                  (st2,_) <-
                    betree_store_leaf_node_fwd node'.betree_leaf_id
                      BetreeListNil st1;
                  Return (st2,())
                od
              else
                do
                  (st1,_) <-
                    betree_store_leaf_node_fwd node'.betree_leaf_id
                      content0 st0;
                  Return (st1,())
                od
            od
   
   [betree_node_apply_messages_to_internal_fwd_back_def]  Definition
      
      ⊢ ∀msgs new_msgs.
          betree_node_apply_messages_to_internal_fwd_back msgs new_msgs =
          case new_msgs of
            BetreeListCons new_msg new_msgs_tl =>
              (let
                 (i,m) = new_msg
               in
                 do
                   msgs0 <- betree_node_apply_to_internal_fwd_back msgs i m;
                   betree_node_apply_messages_to_internal_fwd_back msgs0
                     new_msgs_tl
                 od)
          | BetreeListNil => Return msgs
   
   [betree_node_apply_messages_to_leaf_fwd_back_def]  Definition
      
      ⊢ ∀bindings new_msgs.
          betree_node_apply_messages_to_leaf_fwd_back bindings new_msgs =
          case new_msgs of
            BetreeListCons new_msg new_msgs_tl =>
              (let
                 (i,m) = new_msg
               in
                 do
                   bindings0 <-
                     betree_node_apply_to_leaf_fwd_back bindings i m;
                   betree_node_apply_messages_to_leaf_fwd_back bindings0
                     new_msgs_tl
                 od)
          | BetreeListNil => Return bindings
   
   [betree_node_apply_to_internal_fwd_back_def]  Definition
      
      ⊢ ∀msgs key new_msg.
          betree_node_apply_to_internal_fwd_back msgs key new_msg =
          do
            msgs0 <- betree_node_lookup_first_message_for_key_fwd key msgs;
            b <- betree_list_head_has_key_fwd msgs0 key;
            if b then
              case new_msg of
                BetreeMessageInsert i =>
                  do
                    msgs1 <-
                      betree_node_filter_messages_for_key_fwd_back key
                        msgs0;
                    msgs2 <-
                      betree_list_push_front_fwd_back msgs1
                        (key,BetreeMessageInsert i);
                    betree_node_lookup_first_message_for_key_back key msgs
                      msgs2
                  od
              | BetreeMessageDelete =>
                do
                  msgs1 <-
                    betree_node_filter_messages_for_key_fwd_back key msgs0;
                  msgs2 <-
                    betree_list_push_front_fwd_back msgs1
                      (key,BetreeMessageDelete);
                  betree_node_lookup_first_message_for_key_back key msgs
                    msgs2
                od
              | BetreeMessageUpsert s =>
                do
                  p <- betree_list_hd_fwd msgs0;
                  (_,m) <<- p;
                  case m of
                    BetreeMessageInsert prev =>
                      do
                        v <- betree_upsert_update_fwd (SOME prev) s;
                        msgs1 <- betree_list_pop_front_back msgs0;
                        msgs2 <-
                          betree_list_push_front_fwd_back msgs1
                            (key,BetreeMessageInsert v);
                        betree_node_lookup_first_message_for_key_back key
                          msgs msgs2
                      od
                  | BetreeMessageDelete =>
                    do
                      v <- betree_upsert_update_fwd NONE s;
                      msgs1 <- betree_list_pop_front_back msgs0;
                      msgs2 <-
                        betree_list_push_front_fwd_back msgs1
                          (key,BetreeMessageInsert v);
                      betree_node_lookup_first_message_for_key_back key
                        msgs msgs2
                    od
                  | BetreeMessageUpsert ufs =>
                    do
                      msgs1 <-
                        betree_node_lookup_first_message_after_key_fwd key
                          msgs0;
                      msgs2 <-
                        betree_list_push_front_fwd_back msgs1
                          (key,BetreeMessageUpsert s);
                      msgs3 <-
                        betree_node_lookup_first_message_after_key_back key
                          msgs0 msgs2;
                      betree_node_lookup_first_message_for_key_back key
                        msgs msgs3
                    od
                od
            else
              do
                msgs1 <-
                  betree_list_push_front_fwd_back msgs0 (key,new_msg);
                betree_node_lookup_first_message_for_key_back key msgs
                  msgs1
              od
          od
   
   [betree_node_apply_to_leaf_fwd_back_def]  Definition
      
      ⊢ ∀bindings key new_msg.
          betree_node_apply_to_leaf_fwd_back bindings key new_msg =
          do
            bindings0 <-
              betree_node_lookup_mut_in_bindings_fwd key bindings;
            b <- betree_list_head_has_key_fwd bindings0 key;
            if b then
              do
                hd <- betree_list_pop_front_fwd bindings0;
                case new_msg of
                  BetreeMessageInsert v =>
                    do
                      bindings1 <- betree_list_pop_front_back bindings0;
                      bindings2 <-
                        betree_list_push_front_fwd_back bindings1 (key,v);
                      betree_node_lookup_mut_in_bindings_back key bindings
                        bindings2
                    od
                | BetreeMessageDelete =>
                  do
                    bindings1 <- betree_list_pop_front_back bindings0;
                    betree_node_lookup_mut_in_bindings_back key bindings
                      bindings1
                  od
                | BetreeMessageUpsert s =>
                  (let
                     (_,i) = hd
                   in
                     do
                       v <- betree_upsert_update_fwd (SOME i) s;
                       bindings1 <- betree_list_pop_front_back bindings0;
                       bindings2 <-
                         betree_list_push_front_fwd_back bindings1 (key,v);
                       betree_node_lookup_mut_in_bindings_back key bindings
                         bindings2
                     od)
              od
            else
              case new_msg of
                BetreeMessageInsert v =>
                  do
                    bindings1 <-
                      betree_list_push_front_fwd_back bindings0 (key,v);
                    betree_node_lookup_mut_in_bindings_back key bindings
                      bindings1
                  od
              | BetreeMessageDelete =>
                betree_node_lookup_mut_in_bindings_back key bindings
                  bindings0
              | BetreeMessageUpsert s =>
                do
                  v <- betree_upsert_update_fwd NONE s;
                  bindings1 <-
                    betree_list_push_front_fwd_back bindings0 (key,v);
                  betree_node_lookup_mut_in_bindings_back key bindings
                    bindings1
                od
          od
   
   [betree_node_apply_upserts_back_def]  Definition
      
      ⊢ ∀msgs prev key st.
          betree_node_apply_upserts_back msgs prev key st =
          do
            b <- betree_list_head_has_key_fwd msgs key;
            if b then
              do
                msg <- betree_list_pop_front_fwd msgs;
                (_,m) <<- msg;
                case m of
                  BetreeMessageInsert i => Fail Failure
                | BetreeMessageDelete => Fail Failure
                | BetreeMessageUpsert s =>
                  do
                    v <- betree_upsert_update_fwd prev s;
                    msgs0 <- betree_list_pop_front_back msgs;
                    betree_node_apply_upserts_back msgs0 (SOME v) key st
                  od
              od
            else
              do
                (_,v) <- core_option_option_unwrap_fwd prev st;
                betree_list_push_front_fwd_back msgs
                  (key,BetreeMessageInsert v)
              od
          od
   
   [betree_node_apply_upserts_fwd_def]  Definition
      
      ⊢ ∀msgs prev key st.
          betree_node_apply_upserts_fwd msgs prev key st =
          do
            b <- betree_list_head_has_key_fwd msgs key;
            if b then
              do
                msg <- betree_list_pop_front_fwd msgs;
                (_,m) <<- msg;
                case m of
                  BetreeMessageInsert i => Fail Failure
                | BetreeMessageDelete => Fail Failure
                | BetreeMessageUpsert s =>
                  do
                    v <- betree_upsert_update_fwd prev s;
                    msgs0 <- betree_list_pop_front_back msgs;
                    betree_node_apply_upserts_fwd msgs0 (SOME v) key st
                  od
              od
            else
              do
                (st0,v) <- core_option_option_unwrap_fwd prev st;
                monad_ignore_bind
                  (betree_list_push_front_fwd_back msgs
                     (key,BetreeMessageInsert v)) (Return (st0,v))
              od
          od
   
   [betree_node_filter_messages_for_key_fwd_back_def]  Definition
      
      ⊢ ∀key msgs.
          betree_node_filter_messages_for_key_fwd_back key msgs =
          case msgs of
            BetreeListCons p l =>
              (let
                 (k,m) = p
               in
                 if k = key then
                   do
                     msgs0 <-
                       betree_list_pop_front_back (BetreeListCons (k,m) l);
                     betree_node_filter_messages_for_key_fwd_back key msgs0
                   od
                 else Return (BetreeListCons (k,m) l))
          | BetreeListNil => Return BetreeListNil
   
   [betree_node_id_counter_fresh_id_back_def]  Definition
      
      ⊢ ∀self.
          betree_node_id_counter_fresh_id_back self =
          do
            i <-
              u64_add self.betree_node_id_counter_next_node_id
                (int_to_u64 1);
            Return <|betree_node_id_counter_next_node_id := i|>
          od
   
   [betree_node_id_counter_fresh_id_fwd_def]  Definition
      
      ⊢ ∀self.
          betree_node_id_counter_fresh_id_fwd self =
          monad_ignore_bind
            (u64_add self.betree_node_id_counter_next_node_id
               (int_to_u64 1))
            (Return self.betree_node_id_counter_next_node_id)
   
   [betree_node_id_counter_new_fwd_def]  Definition
      
      ⊢ betree_node_id_counter_new_fwd =
        Return <|betree_node_id_counter_next_node_id := int_to_u64 0|>
   
   [betree_node_lookup_back_def]  Definition
      
      ⊢ ∀self key st.
          betree_node_lookup_back self key st =
          case self of
            BetreeNodeInternal node =>
              do
                (st0,msgs) <-
                  betree_load_internal_node_fwd node.betree_internal_id st;
                pending <-
                  betree_node_lookup_first_message_for_key_fwd key msgs;
                case pending of
                  BetreeListCons p l =>
                    (let
                       (k,msg) = p
                     in
                       if k ≠ key then
                         monad_ignore_bind
                           (betree_node_lookup_first_message_for_key_back
                              key msgs (BetreeListCons (k,msg) l))
                           do
                             node0 <-
                               betree_internal_lookup_in_children_back node
                                 key st0;
                             Return (BetreeNodeInternal node0)
                           od
                       else
                         case msg of
                           BetreeMessageInsert v =>
                             monad_ignore_bind
                               (betree_node_lookup_first_message_for_key_back
                                  key msgs
                                  (BetreeListCons (k,BetreeMessageInsert v)
                                     l)) (Return (BetreeNodeInternal node))
                         | BetreeMessageDelete =>
                           monad_ignore_bind
                             (betree_node_lookup_first_message_for_key_back
                                key msgs
                                (BetreeListCons (k,BetreeMessageDelete) l))
                             (Return (BetreeNodeInternal node))
                         | BetreeMessageUpsert ufs =>
                           do
                             (st1,v) <-
                               betree_internal_lookup_in_children_fwd node
                                 key st0;
                             (st2,_) <-
                               betree_node_apply_upserts_fwd
                                 (BetreeListCons
                                    (k,BetreeMessageUpsert ufs) l) v key
                                 st1;
                             node0 <-
                               betree_internal_lookup_in_children_back node
                                 key st0;
                             pending0 <-
                               betree_node_apply_upserts_back
                                 (BetreeListCons
                                    (k,BetreeMessageUpsert ufs) l) v key
                                 st1;
                             msgs0 <-
                               betree_node_lookup_first_message_for_key_back
                                 key msgs pending0;
                             monad_ignore_bind
                               (betree_store_internal_node_fwd
                                  node0.betree_internal_id msgs0 st2)
                               (Return (BetreeNodeInternal node0))
                           od)
                | BetreeListNil =>
                  monad_ignore_bind
                    (betree_node_lookup_first_message_for_key_back key msgs
                       BetreeListNil)
                    do
                      node0 <-
                        betree_internal_lookup_in_children_back node key
                          st0;
                      Return (BetreeNodeInternal node0)
                    od
              od
          | BetreeNodeLeaf node' =>
            do
              (_,bindings) <-
                betree_load_leaf_node_fwd node'.betree_leaf_id st;
              monad_ignore_bind
                (betree_node_lookup_in_bindings_fwd key bindings)
                (Return (BetreeNodeLeaf node'))
            od
   
   [betree_node_lookup_first_message_after_key_back_def]  Definition
      
      ⊢ ∀key msgs ret.
          betree_node_lookup_first_message_after_key_back key msgs ret =
          case msgs of
            BetreeListCons p next_msgs =>
              (let
                 (k,m) = p
               in
                 if k = key then
                   do
                     next_msgs0 <-
                       betree_node_lookup_first_message_after_key_back key
                         next_msgs ret;
                     Return (BetreeListCons (k,m) next_msgs0)
                   od
                 else Return ret)
          | BetreeListNil => Return ret
   
   [betree_node_lookup_first_message_after_key_fwd_def]  Definition
      
      ⊢ ∀key msgs.
          betree_node_lookup_first_message_after_key_fwd key msgs =
          case msgs of
            BetreeListCons p next_msgs =>
              (let
                 (k,m) = p
               in
                 if k = key then
                   betree_node_lookup_first_message_after_key_fwd key
                     next_msgs
                 else Return (BetreeListCons (k,m) next_msgs))
          | BetreeListNil => Return BetreeListNil
   
   [betree_node_lookup_first_message_for_key_back_def]  Definition
      
      ⊢ ∀key msgs ret.
          betree_node_lookup_first_message_for_key_back key msgs ret =
          case msgs of
            BetreeListCons x next_msgs =>
              (let
                 (i,m) = x
               in
                 if u64_ge i key then Return ret
                 else
                   do
                     next_msgs0 <-
                       betree_node_lookup_first_message_for_key_back key
                         next_msgs ret;
                     Return (BetreeListCons (i,m) next_msgs0)
                   od)
          | BetreeListNil => Return ret
   
   [betree_node_lookup_first_message_for_key_fwd_def]  Definition
      
      ⊢ ∀key msgs.
          betree_node_lookup_first_message_for_key_fwd key msgs =
          case msgs of
            BetreeListCons x next_msgs =>
              (let
                 (i,m) = x
               in
                 if u64_ge i key then
                   Return (BetreeListCons (i,m) next_msgs)
                 else
                   betree_node_lookup_first_message_for_key_fwd key
                     next_msgs)
          | BetreeListNil => Return BetreeListNil
   
   [betree_node_lookup_fwd_def]  Definition
      
      ⊢ ∀self key st.
          betree_node_lookup_fwd self key st =
          case self of
            BetreeNodeInternal node =>
              do
                (st0,msgs) <-
                  betree_load_internal_node_fwd node.betree_internal_id st;
                pending <-
                  betree_node_lookup_first_message_for_key_fwd key msgs;
                case pending of
                  BetreeListCons p l =>
                    (let
                       (k,msg) = p
                     in
                       if k ≠ key then
                         do
                           (st1,opt) <-
                             betree_internal_lookup_in_children_fwd node
                               key st0;
                           monad_ignore_bind
                             (betree_node_lookup_first_message_for_key_back
                                key msgs (BetreeListCons (k,msg) l))
                             (Return (st1,opt))
                         od
                       else
                         case msg of
                           BetreeMessageInsert v =>
                             monad_ignore_bind
                               (betree_node_lookup_first_message_for_key_back
                                  key msgs
                                  (BetreeListCons (k,BetreeMessageInsert v)
                                     l)) (Return (st0,SOME v))
                         | BetreeMessageDelete =>
                           monad_ignore_bind
                             (betree_node_lookup_first_message_for_key_back
                                key msgs
                                (BetreeListCons (k,BetreeMessageDelete) l))
                             (Return (st0,NONE))
                         | BetreeMessageUpsert ufs =>
                           do
                             (st1,v) <-
                               betree_internal_lookup_in_children_fwd node
                                 key st0;
                             (st2,v0) <-
                               betree_node_apply_upserts_fwd
                                 (BetreeListCons
                                    (k,BetreeMessageUpsert ufs) l) v key
                                 st1;
                             node0 <-
                               betree_internal_lookup_in_children_back node
                                 key st0;
                             pending0 <-
                               betree_node_apply_upserts_back
                                 (BetreeListCons
                                    (k,BetreeMessageUpsert ufs) l) v key
                                 st1;
                             msgs0 <-
                               betree_node_lookup_first_message_for_key_back
                                 key msgs pending0;
                             (st3,_) <-
                               betree_store_internal_node_fwd
                                 node0.betree_internal_id msgs0 st2;
                             Return (st3,SOME v0)
                           od)
                | BetreeListNil =>
                  do
                    (st1,opt) <-
                      betree_internal_lookup_in_children_fwd node key st0;
                    monad_ignore_bind
                      (betree_node_lookup_first_message_for_key_back key
                         msgs BetreeListNil) (Return (st1,opt))
                  od
              od
          | BetreeNodeLeaf node' =>
            do
              (st0,bindings) <-
                betree_load_leaf_node_fwd node'.betree_leaf_id st;
              opt <- betree_node_lookup_in_bindings_fwd key bindings;
              Return (st0,opt)
            od
   
   [betree_node_lookup_in_bindings_fwd_def]  Definition
      
      ⊢ ∀key bindings.
          betree_node_lookup_in_bindings_fwd key bindings =
          case bindings of
            BetreeListCons hd tl =>
              (let
                 (i,i0) = hd
               in
                 if i = key then Return (SOME i0)
                 else if u64_gt i key then Return NONE
                 else betree_node_lookup_in_bindings_fwd key tl)
          | BetreeListNil => Return NONE
   
   [betree_node_lookup_mut_in_bindings_back_def]  Definition
      
      ⊢ ∀key bindings ret.
          betree_node_lookup_mut_in_bindings_back key bindings ret =
          case bindings of
            BetreeListCons hd tl =>
              (let
                 (i,i0) = hd
               in
                 if u64_ge i key then Return ret
                 else
                   do
                     tl0 <-
                       betree_node_lookup_mut_in_bindings_back key tl ret;
                     Return (BetreeListCons (i,i0) tl0)
                   od)
          | BetreeListNil => Return ret
   
   [betree_node_lookup_mut_in_bindings_fwd_def]  Definition
      
      ⊢ ∀key bindings.
          betree_node_lookup_mut_in_bindings_fwd key bindings =
          case bindings of
            BetreeListCons hd tl =>
              (let
                 (i,i0) = hd
               in
                 if u64_ge i key then Return (BetreeListCons (i,i0) tl)
                 else betree_node_lookup_mut_in_bindings_fwd key tl)
          | BetreeListNil => Return BetreeListNil
   
   [betree_store_internal_node_fwd_def]  Definition
      
      ⊢ ∀id content st.
          betree_store_internal_node_fwd id content st =
          do
            (st0,_) <- betree_utils_store_internal_node_fwd id content st;
            Return (st0,())
          od
   
   [betree_store_leaf_node_fwd_def]  Definition
      
      ⊢ ∀id content st.
          betree_store_leaf_node_fwd id content st =
          do
            (st0,_) <- betree_utils_store_leaf_node_fwd id content st;
            Return (st0,())
          od
   
   [betree_upsert_update_fwd_def]  Definition
      
      ⊢ ∀prev st.
          betree_upsert_update_fwd prev st =
          case prev of
            NONE =>
              (case st of
                 BetreeUpsertFunStateAdd v => Return v
               | BetreeUpsertFunStateSub i => Return (int_to_u64 0))
          | SOME prev0 =>
            case st of
              BetreeUpsertFunStateAdd v =>
                do
                  margin <- u64_sub core_u64_max prev0;
                  if u64_ge margin v then u64_add prev0 v
                  else Return core_u64_max
                od
            | BetreeUpsertFunStateSub v' =>
              if u64_ge prev0 v' then u64_sub prev0 v'
              else Return (int_to_u64 0)
   
   [main_fwd_def]  Definition
      
      ⊢ main_fwd = Return ()
   
   
*)
end