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Datasets:
HyperCactus0
/
LeanTransitionCorpus

Tasks:
Text Generation
Tags:
lean
lean4
automated-theorem-proving
theorem-proving
formal-mathematics
mathematics
License:
Dataset card Data Studio Files
xet
 Community
1
HyperCactus0 commited on
Commit
3c1af7d
·
verified ·
1 Parent(s): d9462ab

Record 26 verified HPC batch manifests

Browse files
Files changed (26) hide show
  1. manifests/batches/mathlib/001250.json +0 -0
  2. manifests/batches/mathlib/001251.json +1079 -0
  3. manifests/batches/mathlib/001252.json +406 -0
  4. manifests/batches/mathlib/001253.json +1261 -0
  5. manifests/batches/mathlib/001256.json +0 -0
  6. manifests/batches/mathlib/001257.json +0 -0
  7. manifests/batches/mathlib/001259.json +330 -0
  8. manifests/batches/mathlib/001260.json +0 -0
  9. manifests/batches/mathlib/001261.json +0 -0
  10. manifests/batches/mathlib/001262.json +0 -0
  11. manifests/batches/mathlib/001263.json +599 -0
  12. manifests/batches/mathlib/001264.json +0 -0
  13. manifests/batches/mathlib/001265.json +0 -0
  14. manifests/batches/mathlib/001266.json +0 -0
  15. manifests/batches/mathlib/001267.json +0 -0
  16. manifests/batches/mathlib/001268.json +0 -0
  17. manifests/batches/mathlib/001269.json +605 -0
  18. manifests/batches/mathlib/001270.json +0 -0
  19. manifests/batches/mathlib/001271.json +1107 -0
  20. manifests/batches/mathlib/001272.json +0 -0
  21. manifests/batches/mathlib/001273.json +0 -0
  22. manifests/batches/mathlib/001274.json +382 -0
  23. manifests/batches/mathlib/001275.json +168 -0
  24. manifests/batches/mathlib/001276.json +239 -0
  25. manifests/batches/mathlib/001277.json +185 -0
  26. manifests/batches/mathlib/001280.json +0 -0
manifests/batches/mathlib/001250.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001251.json ADDED
@@ -0,0 +1,1079 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ "selection": {
433
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
434
+ "end": [
435
+ 152,
436
+ 25
437
+ ],
438
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
439
+ "full_name": "LinearMap.map_add\u2082",
440
+ "start": [
441
+ 151,
442
+ 1
443
+ ],
444
+ "traced_tactics": [],
445
+ "url": "https://github.com/leanprover-community/mathlib4"
446
+ },
447
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
448
+ },
449
+ {
450
+ "error": "theorem has no selected tactic transitions",
451
+ "full_name": "LinearMap.map_neg\u2082",
452
+ "id": "LinearMap.map_neg\u2082",
453
+ "reason": "import_failure",
454
+ "selection": {
455
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
456
+ "end": [
457
+ 144,
458
+ 23
459
+ ],
460
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
461
+ "full_name": "LinearMap.map_neg\u2082",
462
+ "start": [
463
+ 143,
464
+ 1
465
+ ],
466
+ "traced_tactics": [],
467
+ "url": "https://github.com/leanprover-community/mathlib4"
468
+ },
469
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
470
+ },
471
+ {
472
+ "error": "theorem has no selected tactic transitions",
473
+ "full_name": "LinearMap.map_smul\u2082",
474
+ "id": "LinearMap.map_smul\u2082",
475
+ "reason": "import_failure",
476
+ "selection": {
477
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
478
+ "end": [
479
+ 156,
480
+ 26
481
+ ],
482
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
483
+ "full_name": "LinearMap.map_smul\u2082",
484
+ "start": [
485
+ 155,
486
+ 1
487
+ ],
488
+ "traced_tactics": [],
489
+ "url": "https://github.com/leanprover-community/mathlib4"
490
+ },
491
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
492
+ },
493
+ {
494
+ "error": "theorem has no selected tactic transitions",
495
+ "full_name": "LinearMap.map_smul\u209b\u2097\u2082",
496
+ "id": "LinearMap.map_smul\u209b\u2097\u2082",
497
+ "reason": "import_failure",
498
+ "selection": {
499
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
500
+ "end": [
501
+ 160,
502
+ 28
503
+ ],
504
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
505
+ "full_name": "LinearMap.map_smul\u209b\u2097\u2082",
506
+ "start": [
507
+ 159,
508
+ 1
509
+ ],
510
+ "traced_tactics": [],
511
+ "url": "https://github.com/leanprover-community/mathlib4"
512
+ },
513
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
514
+ },
515
+ {
516
+ "error": "theorem has no selected tactic transitions",
517
+ "full_name": "LinearMap.map_sub\u2082",
518
+ "id": "LinearMap.map_sub\u2082",
519
+ "reason": "import_failure",
520
+ "selection": {
521
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
522
+ "end": [
523
+ 148,
524
+ 25
525
+ ],
526
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
527
+ "full_name": "LinearMap.map_sub\u2082",
528
+ "start": [
529
+ 147,
530
+ 1
531
+ ],
532
+ "traced_tactics": [],
533
+ "url": "https://github.com/leanprover-community/mathlib4"
534
+ },
535
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
536
+ },
537
+ {
538
+ "error": "theorem has no selected tactic transitions",
539
+ "full_name": "LinearMap.map_sum\u2082",
540
+ "id": "LinearMap.map_sum\u2082",
541
+ "reason": "import_failure",
542
+ "selection": {
543
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
544
+ "end": [
545
+ 165,
546
+ 32
547
+ ],
548
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
549
+ "full_name": "LinearMap.map_sum\u2082",
550
+ "start": [
551
+ 163,
552
+ 1
553
+ ],
554
+ "traced_tactics": [],
555
+ "url": "https://github.com/leanprover-community/mathlib4"
556
+ },
557
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
558
+ },
559
+ {
560
+ "error": "theorem has no selected tactic transitions",
561
+ "full_name": "LinearMap.map_zero\u2082",
562
+ "id": "LinearMap.map_zero\u2082",
563
+ "reason": "import_failure",
564
+ "selection": {
565
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
566
+ "end": [
567
+ 140,
568
+ 22
569
+ ],
570
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
571
+ "full_name": "LinearMap.map_zero\u2082",
572
+ "start": [
573
+ 139,
574
+ 1
575
+ ],
576
+ "traced_tactics": [],
577
+ "url": "https://github.com/leanprover-community/mathlib4"
578
+ },
579
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
580
+ },
581
+ {
582
+ "error": "theorem has no selected tactic transitions",
583
+ "full_name": "LinearMap.mk\u2082'_apply",
584
+ "id": "LinearMap.mk\u2082'_apply",
585
+ "reason": "import_failure",
586
+ "selection": {
587
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
588
+ "end": [
589
+ 93,
590
+ 69
591
+ ],
592
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
593
+ "full_name": "LinearMap.mk\u2082'_apply",
594
+ "start": [
595
+ 92,
596
+ 1
597
+ ],
598
+ "traced_tactics": [],
599
+ "url": "https://github.com/leanprover-community/mathlib4"
600
+ },
601
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
602
+ },
603
+ {
604
+ "error": "theorem has no selected tactic transitions",
605
+ "full_name": "LinearMap.mk\u2082_apply",
606
+ "id": "LinearMap.mk\u2082_apply",
607
+ "reason": "import_failure",
608
+ "selection": {
609
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
610
+ "end": [
611
+ 252,
612
+ 67
613
+ ],
614
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
615
+ "full_name": "LinearMap.mk\u2082_apply",
616
+ "start": [
617
+ 251,
618
+ 1
619
+ ],
620
+ "traced_tactics": [],
621
+ "url": "https://github.com/leanprover-community/mathlib4"
622
+ },
623
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
624
+ },
625
+ {
626
+ "error": "theorem has no selected tactic transitions",
627
+ "full_name": "LinearMap.restrictScalars\u2081\u2082_inj",
628
+ "id": "LinearMap.restrictScalars\u2081\u2082_inj",
629
+ "reason": "import_failure",
630
+ "selection": {
631
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
632
+ "end": [
633
+ 218,
634
+ 45
635
+ ],
636
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
637
+ "full_name": "LinearMap.restrictScalars\u2081\u2082_inj",
638
+ "start": [
639
+ 216,
640
+ 1
641
+ ],
642
+ "traced_tactics": [],
643
+ "url": "https://github.com/leanprover-community/mathlib4"
644
+ },
645
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
646
+ },
647
+ {
648
+ "error": "theorem has no selected tactic transitions",
649
+ "full_name": "LinearMap.restrictScalars\u2081\u2082_injective",
650
+ "id": "LinearMap.restrictScalars\u2081\u2082_injective",
651
+ "reason": "import_failure",
652
+ "selection": {
653
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
654
+ "end": [
655
+ 213,
656
+ 38
657
+ ],
658
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
659
+ "full_name": "LinearMap.restrictScalars\u2081\u2082_injective",
660
+ "start": [
661
+ 211,
662
+ 1
663
+ ],
664
+ "traced_tactics": [],
665
+ "url": "https://github.com/leanprover-community/mathlib4"
666
+ },
667
+ "source_file": "Mathlib/LinearAlgebra/BilinearMap.lean"
668
+ },
669
+ {
670
+ "error": "theorem has no selected tactic transitions",
671
+ "full_name": "LinearMap.aeval_eq_aeval_mod_charpoly",
672
+ "id": "LinearMap.aeval_eq_aeval_mod_charpoly",
673
+ "reason": "import_failure",
674
+ "selection": {
675
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
676
+ "end": [
677
+ 90,
678
+ 81
679
+ ],
680
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
681
+ "full_name": "LinearMap.aeval_eq_aeval_mod_charpoly",
682
+ "start": [
683
+ 89,
684
+ 1
685
+ ],
686
+ "traced_tactics": [],
687
+ "url": "https://github.com/leanprover-community/mathlib4"
688
+ },
689
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/Basic.lean"
690
+ },
691
+ {
692
+ "error": "theorem has no selected tactic transitions",
693
+ "full_name": "LinearMap.charpoly_def",
694
+ "id": "LinearMap.charpoly_def",
695
+ "reason": "import_failure",
696
+ "selection": {
697
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
698
+ "end": [
699
+ 48,
700
+ 6
701
+ ],
702
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
703
+ "full_name": "LinearMap.charpoly_def",
704
+ "start": [
705
+ 47,
706
+ 1
707
+ ],
708
+ "traced_tactics": [],
709
+ "url": "https://github.com/leanprover-community/mathlib4"
710
+ },
711
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/Basic.lean"
712
+ },
713
+ {
714
+ "error": "theorem has no selected tactic transitions",
715
+ "full_name": "LinearMap.charpoly_monic",
716
+ "id": "LinearMap.charpoly_monic",
717
+ "reason": "import_failure",
718
+ "selection": {
719
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
720
+ "end": [
721
+ 56,
722
+ 26
723
+ ],
724
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
725
+ "full_name": "LinearMap.charpoly_monic",
726
+ "start": [
727
+ 55,
728
+ 1
729
+ ],
730
+ "traced_tactics": [],
731
+ "url": "https://github.com/leanprover-community/mathlib4"
732
+ },
733
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/Basic.lean"
734
+ },
735
+ {
736
+ "error": "theorem has no selected tactic transitions",
737
+ "full_name": "LinearMap.isIntegral",
738
+ "id": "LinearMap.isIntegral",
739
+ "reason": "import_failure",
740
+ "selection": {
741
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
742
+ "end": [
743
+ 79,
744
+ 58
745
+ ],
746
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
747
+ "full_name": "LinearMap.isIntegral",
748
+ "start": [
749
+ 78,
750
+ 1
751
+ ],
752
+ "traced_tactics": [],
753
+ "url": "https://github.com/leanprover-community/mathlib4"
754
+ },
755
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/Basic.lean"
756
+ },
757
+ {
758
+ "error": "no internal trace for tactic 'rw [hP, mul_comm]'",
759
+ "full_name": "LinearMap.minpoly_coeff_zero_of_injective",
760
+ "id": "LinearMap.minpoly_coeff_zero_of_injective",
761
+ "reason": "import_failure",
762
+ "selection": {
763
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
764
+ "end": [
765
+ 117,
766
+ 61
767
+ ],
768
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
769
+ "full_name": "LinearMap.minpoly_coeff_zero_of_injective",
770
+ "start": [
771
+ 101,
772
+ 1
773
+ ],
774
+ "traced_tactics": [
775
+ {
776
+ "annotated_tactic": [
777
+ "\"intro h\"",
778
+ "[]"
779
+ ],
780
+ "state_after": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\n\u22a2 False",
781
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\n\u22a2 (minpoly R f).coeff 0 \u2260 0",
782
+ "tactic": "intro h"
783
+ },
784
+ {
785
+ "annotated_tactic": [
786
+ "\"obtain \\u27e8P, hP\\u27e9 := <a>X_dvd_iff<\\/a>.2 h\"",
787
+ "[{\"full_name\":\"Polynomial.X_dvd_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Div.lean\",\"def_pos\":[38,9],\"def_end_pos\":[38,18]}]"
788
+ ],
789
+ "state_after": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 False",
790
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\n\u22a2 False",
791
+ "tactic": "obtain \u27e8P, hP\u27e9 := X_dvd_iff.2 h"
792
+ },
793
+ {
794
+ "annotated_tactic": [
795
+ "\"have hdegP : P.degree < (<a>minpoly<\\/a> R f).<a>degree<\\/a> := by\\n rw [hP, <a>mul_comm<\\/a>]\\n refine <a>degree_lt_degree_mul_X<\\/a> fun h => ?_\\n rw [h, <a>mul_zero<\\/a>] at hP\\n exact <a>minpoly.ne_zero<\\/a> (<a>isIntegral<\\/a> f) hP\"",
796
+ "[{\"full_name\":\"minpoly\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[38,19],\"def_end_pos\":[38,26]},{\"full_name\":\"Polynomial.degree\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[52,5],\"def_end_pos\":[52,11]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Polynomial.degree_lt_degree_mul_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1189,9],\"def_end_pos\":[1189,31]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"minpoly.ne_zero\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,16]},{\"full_name\":\"LinearMap.isIntegral\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Charpoly\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,19]}]"
797
+ ],
798
+ "state_after": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\n\u22a2 False",
799
+ "state_before": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 False",
800
+ "tactic": "have hdegP : P.degree < (minpoly R f).degree := by\n rw [hP, mul_comm]\n refine degree_lt_degree_mul_X fun h => ?_\n rw [h, mul_zero] at hP\n exact minpoly.ne_zero (isIntegral f) hP"
801
+ },
802
+ {
803
+ "annotated_tactic": [
804
+ "\"have hPmonic : P.Monic := by\\n suffices (<a>minpoly<\\/a> R f).<a>Monic<\\/a> by\\n rwa [<a>Monic.def<\\/a>, hP, <a>mul_comm<\\/a>, <a>leadingCoeff_mul_X<\\/a>, \\u2190 <a>Monic.def<\\/a>] at this\\n exact <a>minpoly.monic<\\/a> (<a>isIntegral<\\/a> f)\"",
805
+ "[{\"full_name\":\"minpoly\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[38,19],\"def_end_pos\":[38,26]},{\"full_name\":\"Polynomial.Monic\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[77,5],\"def_end_pos\":[77,10]},{\"full_name\":\"Polynomial.Monic.def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Polynomial.leadingCoeff_mul_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1092,9],\"def_end_pos\":[1092,27]},{\"full_name\":\"Polynomial.Monic.def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]},{\"full_name\":\"minpoly.monic\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,14]},{\"full_name\":\"LinearMap.isIntegral\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Charpoly\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,19]}]"
806
+ ],
807
+ "state_after": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nhPmonic : P.Monic\n\u22a2 False",
808
+ "state_before": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\n\u22a2 False",
809
+ "tactic": "have hPmonic : P.Monic := by\n suffices (minpoly R f).Monic by\n rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, \u2190 Monic.def] at this\n exact minpoly.monic (isIntegral f)"
810
+ },
811
+ {
812
+ "annotated_tactic": [
813
+ "\"have hzero : <a>aeval<\\/a> f (<a>minpoly<\\/a> R f) = 0 := <a>minpoly.aeval<\\/a> _ _\"",
814
+ "[{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]},{\"full_name\":\"minpoly\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[38,19],\"def_end_pos\":[38,26]},{\"full_name\":\"minpoly.aeval\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,14]}]"
815
+ ],
816
+ "state_after": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nhPmonic : P.Monic\nhzero : (aeval f) (minpoly R f) = 0\n\u22a2 False",
817
+ "state_before": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nhPmonic : P.Monic\n\u22a2 False",
818
+ "tactic": "have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _"
819
+ },
820
+ {
821
+ "annotated_tactic": [
822
+ "\"simp only [hP, <a>mul_eq_comp<\\/a>, <a>ext_iff<\\/a>, hf, <a>aeval_X<\\/a>, <a>map_eq_zero_iff<\\/a>, <a>coe_comp<\\/a>, <a>AlgHom.map_mul<\\/a>,\\n <a>zero_apply<\\/a>, <a>Function.comp_apply<\\/a>] at hzero\"",
823
+ "[{\"full_name\":\"LinearMap.mul_eq_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,20]},{\"full_name\":\"LinearMap.ext_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[348,9],\"def_end_pos\":[348,16]},{\"full_name\":\"Polynomial.aeval_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[209,9],\"def_end_pos\":[209,16]},{\"full_name\":\"LinearMap.map_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[385,9],\"def_end_pos\":[385,24]},{\"full_name\":\"LinearMap.coe_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[557,9],\"def_end_pos\":[557,17]},{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"LinearMap.zero_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[871,9],\"def_end_pos\":[871,19]},{\"full_name\":\"Function.comp_apply\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[35,17],\"def_end_pos\":[35,36]}]"
824
+ ],
825
+ "state_after": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nhPmonic : P.Monic\nhzero : \u2200 (x : M), ((aeval f) P) x = 0\n\u22a2 False",
826
+ "state_before": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nhPmonic : P.Monic\nhzero : (aeval f) (minpoly R f) = 0\n\u22a2 False",
827
+ "tactic": "simp only [hP, mul_eq_comp, ext_iff, hf, aeval_X, map_eq_zero_iff, coe_comp, AlgHom.map_mul,\n zero_apply, Function.comp_apply] at hzero"
828
+ },
829
+ {
830
+ "annotated_tactic": [
831
+ "\"exact <a>not_le<\\/a>.2 hdegP (<a>minpoly.min<\\/a> _ _ hPmonic (<a>ext<\\/a> hzero))\"",
832
+ "[{\"full_name\":\"not_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[375,9],\"def_end_pos\":[375,15]},{\"full_name\":\"minpoly.min\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,12]},{\"full_name\":\"LinearMap.ext\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[244,9],\"def_end_pos\":[244,12]}]"
833
+ ],
834
+ "state_after": "no goals",
835
+ "state_before": "case intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nhPmonic : P.Monic\nhzero : \u2200 (x : M), ((aeval f) P) x = 0\n\u22a2 False",
836
+ "tactic": "exact not_le.2 hdegP (minpoly.min _ _ hPmonic (ext hzero))"
837
+ },
838
+ {
839
+ "annotated_tactic": [
840
+ "\"rw [hP, <a>mul_comm<\\/a>]\"",
841
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
842
+ ],
843
+ "state_after": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 P.degree < (P * X).degree",
844
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 P.degree < (minpoly R f).degree",
845
+ "tactic": "rw [hP, mul_comm]"
846
+ },
847
+ {
848
+ "annotated_tactic": [
849
+ "\"refine <a>degree_lt_degree_mul_X<\\/a> fun h => ?_\"",
850
+ "[{\"full_name\":\"Polynomial.degree_lt_degree_mul_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1189,9],\"def_end_pos\":[1189,31]}]"
851
+ ],
852
+ "state_after": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh\u271d : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nh : P = 0\n\u22a2 False",
853
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 P.degree < (P * X).degree",
854
+ "tactic": "refine degree_lt_degree_mul_X fun h => ?_"
855
+ },
856
+ {
857
+ "annotated_tactic": [
858
+ "\"rw [h, <a>mul_zero<\\/a>] at hP\"",
859
+ "[{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
860
+ ],
861
+ "state_after": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh\u271d : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = 0\nh : P = 0\n\u22a2 False",
862
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh\u271d : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nh : P = 0\n\u22a2 False",
863
+ "tactic": "rw [h, mul_zero] at hP"
864
+ },
865
+ {
866
+ "annotated_tactic": [
867
+ "\"exact <a>minpoly.ne_zero<\\/a> (<a>isIntegral<\\/a> f) hP\"",
868
+ "[{\"full_name\":\"minpoly.ne_zero\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,16]},{\"full_name\":\"LinearMap.isIntegral\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Charpoly\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,19]}]"
869
+ ],
870
+ "state_after": "no goals",
871
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh\u271d : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = 0\nh : P = 0\n\u22a2 False",
872
+ "tactic": "exact minpoly.ne_zero (isIntegral f) hP"
873
+ },
874
+ {
875
+ "annotated_tactic": [
876
+ "\"suffices (<a>minpoly<\\/a> R f).<a>Monic<\\/a> by\\n rwa [<a>Monic.def<\\/a>, hP, <a>mul_comm<\\/a>, <a>leadingCoeff_mul_X<\\/a>, \\u2190 <a>Monic.def<\\/a>] at this\"",
877
+ "[{\"full_name\":\"minpoly\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[38,19],\"def_end_pos\":[38,26]},{\"full_name\":\"Polynomial.Monic\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[77,5],\"def_end_pos\":[77,10]},{\"full_name\":\"Polynomial.Monic.def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Polynomial.leadingCoeff_mul_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1092,9],\"def_end_pos\":[1092,27]},{\"full_name\":\"Polynomial.Monic.def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]}]"
878
+ ],
879
+ "state_after": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\n\u22a2 (minpoly R f).Monic",
880
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\n\u22a2 P.Monic",
881
+ "tactic": "suffices (minpoly R f).Monic by\n rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, \u2190 Monic.def] at this"
882
+ },
883
+ {
884
+ "annotated_tactic": [
885
+ "\"exact <a>minpoly.monic<\\/a> (<a>isIntegral<\\/a> f)\"",
886
+ "[{\"full_name\":\"minpoly.monic\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,14]},{\"full_name\":\"LinearMap.isIntegral\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Charpoly\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,19]}]"
887
+ ],
888
+ "state_after": "no goals",
889
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\n\u22a2 (minpoly R f).Monic",
890
+ "tactic": "exact minpoly.monic (isIntegral f)"
891
+ },
892
+ {
893
+ "annotated_tactic": [
894
+ "\"rwa [<a>Monic.def<\\/a>, hP, <a>mul_comm<\\/a>, <a>leadingCoeff_mul_X<\\/a>, \\u2190 <a>Monic.def<\\/a>] at this\"",
895
+ "[{\"full_name\":\"Polynomial.Monic.def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Polynomial.leadingCoeff_mul_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1092,9],\"def_end_pos\":[1092,27]},{\"full_name\":\"Polynomial.Monic.def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]}]"
896
+ ],
897
+ "state_after": "no goals",
898
+ "state_before": "R : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u21d1f\nh : (minpoly R f).coeff 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : P.degree < (minpoly R f).degree\nthis : (minpoly R f).Monic\n\u22a2 P.Monic",
899
+ "tactic": "rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, \u2190 Monic.def] at this"
900
+ }
901
+ ],
902
+ "url": "https://github.com/leanprover-community/mathlib4"
903
+ },
904
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/Basic.lean"
905
+ },
906
+ {
907
+ "error": "theorem has no selected tactic transitions",
908
+ "full_name": "LinearMap.minpoly_dvd_charpoly",
909
+ "id": "LinearMap.minpoly_dvd_charpoly",
910
+ "reason": "import_failure",
911
+ "selection": {
912
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
913
+ "end": [
914
+ 84,
915
+ 42
916
+ ],
917
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
918
+ "full_name": "LinearMap.minpoly_dvd_charpoly",
919
+ "start": [
920
+ 82,
921
+ 1
922
+ ],
923
+ "traced_tactics": [],
924
+ "url": "https://github.com/leanprover-community/mathlib4"
925
+ },
926
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/Basic.lean"
927
+ }
928
+ ],
929
+ "input_proofs": 42,
930
+ "input_units": 2,
931
+ "proofs": 6,
932
+ "published_proofs": 5,
933
+ "published_transitions": 9,
934
+ "replay_failures": [],
935
+ "transitions": 18,
936
+ "validation_failures": [
937
+ {
938
+ "errors": [
939
+ "internal transition boundary mismatch",
940
+ "internal transition boundary mismatch"
941
+ ],
942
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
943
+ "proof_id": "mathlib:LinearMap.compl\u2081\u2082_inj",
944
+ "provenance": {
945
+ "selection": {
946
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
947
+ "end": [
948
+ 380,
949
+ 17
950
+ ],
951
+ "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean",
952
+ "full_name": "LinearMap.compl\u2081\u2082_inj",
953
+ "start": [
954
+ 368,
955
+ 1
956
+ ],
957
+ "traced_tactics": [
958
+ {
959
+ "annotated_tactic": [
960
+ "\"constructor <;> intro h\"",
961
+ "[]"
962
+ ],
963
+ "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\n\u22a2 f\u2081 = f\u2082\n\ncase mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081 = f\u2082\n\u22a2 f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'",
964
+ "state_before": "R : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\n\u22a2 f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g' \u2194 f\u2081 = f\u2082",
965
+ "tactic": "constructor <;> intro h"
966
+ },
967
+ {
968
+ "annotated_tactic": [
969
+ "\"ext x y\"",
970
+ "[]"
971
+ ],
972
+ "state_after": "case mp.h.h\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\nx : M\u2097\ny : N\u2097\n\u22a2 (f\u2081 x) y = (f\u2082 x) y",
973
+ "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\n\u22a2 f\u2081 = f\u2082",
974
+ "tactic": "ext x y"
975
+ },
976
+ {
977
+ "annotated_tactic": [
978
+ "\"cases' h\\u2097 x with x' hx\"",
979
+ "[]"
980
+ ],
981
+ "state_after": "case mp.h.h.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\nx : M\u2097\ny : N\u2097\nx' : Q\u2097\nhx : g x' = x\n\u22a2 (f\u2081 x) y = (f\u2082 x) y",
982
+ "state_before": "case mp.h.h\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\nx : M\u2097\ny : N\u2097\n\u22a2 (f\u2081 x) y = (f\u2082 x) y",
983
+ "tactic": "cases' h\u2097 x with x' hx"
984
+ },
985
+ {
986
+ "annotated_tactic": [
987
+ "\"subst hx\"",
988
+ "[]"
989
+ ],
990
+ "state_after": "case mp.h.h.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\ny : N\u2097\nx' : Q\u2097\n\u22a2 (f\u2081 (g x')) y = (f\u2082 (g x')) y",
991
+ "state_before": "case mp.h.h.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\nx : M\u2097\ny : N\u2097\nx' : Q\u2097\nhx : g x' = x\n\u22a2 (f\u2081 x) y = (f\u2082 x) y",
992
+ "tactic": "subst hx"
993
+ },
994
+ {
995
+ "annotated_tactic": [
996
+ "\"cases' h\\u1d63 y with y' hy\"",
997
+ "[]"
998
+ ],
999
+ "state_after": "case mp.h.h.intro.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\ny : N\u2097\nx' : Q\u2097\ny' : Q\u2097'\nhy : g' y' = y\n\u22a2 (f\u2081 (g x')) y = (f\u2082 (g x')) y",
1000
+ "state_before": "case mp.h.h.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\ny : N\u2097\nx' : Q\u2097\n\u22a2 (f\u2081 (g x')) y = (f\u2082 (g x')) y",
1001
+ "tactic": "cases' h\u1d63 y with y' hy"
1002
+ },
1003
+ {
1004
+ "annotated_tactic": [
1005
+ "\"subst hy\"",
1006
+ "[]"
1007
+ ],
1008
+ "state_after": "case mp.h.h.intro.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\nx' : Q\u2097\ny' : Q\u2097'\n\u22a2 (f\u2081 (g x')) (g' y') = (f\u2082 (g x')) (g' y')",
1009
+ "state_before": "case mp.h.h.intro.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\ny : N\u2097\nx' : Q\u2097\ny' : Q\u2097'\nhy : g' y' = y\n\u22a2 (f\u2081 (g x')) y = (f\u2082 (g x')) y",
1010
+ "tactic": "subst hy"
1011
+ },
1012
+ {
1013
+ "annotated_tactic": [
1014
+ "\"convert <a>LinearMap.congr_fun\\u2082<\\/a> h x' y' using 0\"",
1015
+ "[{\"full_name\":\"LinearMap.congr_fun\\u2082\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/BilinearMap.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,19]}]"
1016
+ ],
1017
+ "state_after": "no goals",
1018
+ "state_before": "case mp.h.h.intro.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'\nx' : Q\u2097\ny' : Q\u2097'\n\u22a2 (f\u2081 (g x')) (g' y') = (f\u2082 (g x')) (g' y')",
1019
+ "tactic": "convert LinearMap.congr_fun\u2082 h x' y' using 0"
1020
+ },
1021
+ {
1022
+ "annotated_tactic": [
1023
+ "\"subst h\"",
1024
+ "[]"
1025
+ ],
1026
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\n\u22a2 f\u2081.compl\u2081\u2082 g g' = f\u2081.compl\u2081\u2082 g g'",
1027
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\nh : f\u2081 = f\u2082\n\u22a2 f\u2081.compl\u2081\u2082 g g' = f\u2082.compl\u2081\u2082 g g'",
1028
+ "tactic": "subst h"
1029
+ },
1030
+ {
1031
+ "annotated_tactic": [
1032
+ "\"rfl\"",
1033
+ "[]"
1034
+ ],
1035
+ "state_after": "no goals",
1036
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u21d1g\nh\u1d63 : Function.Surjective \u21d1g'\n\u22a2 f\u2081.compl\u2081\u2082 g g' = f\u2081.compl\u2081\u2082 g g'",
1037
+ "tactic": "rfl"
1038
+ }
1039
+ ],
1040
+ "url": "https://github.com/leanprover-community/mathlib4"
1041
+ }
1042
+ },
1043
+ "theorem_name": "LinearMap.compl\u2081\u2082_inj"
1044
+ }
1045
+ ],
1046
+ "validation_rejected_proofs": 1,
1047
+ "zero_transition_proofs": 0
1048
+ },
1049
+ "shards": [
1050
+ {
1051
+ "path": "data/train/mathlib-001251-0000-d3829fd2c8de8b52.parquet",
1052
+ "sha256": "d3829fd2c8de8b5219edf16fd06fc167741ca156c813b33428d505c07c0a5869",
1053
+ "bytes": 88808,
1054
+ "rows": 6,
1055
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1056
+ },
1057
+ {
1058
+ "path": "data/internal_test/mathlib-001251-0000-db0233d6ae3395d2.parquet",
1059
+ "sha256": "db0233d6ae3395d25b4f5c87989b1149a0d71607c6bac2f2312328b76a6f3ab0",
1060
+ "bytes": 52135,
1061
+ "rows": 3,
1062
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1063
+ }
1064
+ ],
1065
+ "proof_ids": [
1066
+ "mathlib:LinearMap.aeval_self_charpoly",
1067
+ "mathlib:LinearMap.compl\u2081\u2082_id_id",
1068
+ "mathlib:LinearMap.compl\u2082_id",
1069
+ "mathlib:LinearMap.flip_inj",
1070
+ "mathlib:LinearMap.pow_eq_aeval_mod_charpoly"
1071
+ ],
1072
+ "proof_keys": [
1073
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/BilinearMap.lean\",null,null,\"LinearMap.compl\u2081\u2082_id_id\"]",
1074
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/BilinearMap.lean\",null,null,\"LinearMap.compl\u2082_id\"]",
1075
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/BilinearMap.lean\",null,null,\"LinearMap.flip_inj\"]",
1076
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Charpoly/Basic.lean\",null,null,\"LinearMap.aeval_self_charpoly\"]",
1077
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Charpoly/Basic.lean\",null,null,\"LinearMap.pow_eq_aeval_mod_charpoly\"]"
1078
+ ]
1079
+ }
manifests/batches/mathlib/001252.json ADDED
@@ -0,0 +1,406 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001252",
4
+ "batch_fingerprint": "00620c73e778ea7888417e5efb9937bd90ccd85509edd34c5dc4f163c24b1fe9",
5
+ "evidence": {
6
+ "batch_fingerprint": "00620c73e778ea7888417e5efb9937bd90ccd85509edd34c5dc4f163c24b1fe9",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "no internal trace for tactic 'rw [RingHom.mapMatrix_apply, RingHom.mapMatrix_apply, \u2190 Matrix.map_mul,\\n reindexLinearEquiv_mul R R, Basis.toMatrix_mul_toMatrix_flip,\\n reindexLinearEquiv_one, \u2190 RingHom.mapMatrix_apply, RingHom.map_one]'",
11
+ "full_name": "LinearMap.charpoly_toMatrix",
12
+ "id": "LinearMap.charpoly_toMatrix",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 87,
18
+ 26
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean",
21
+ "full_name": "LinearMap.charpoly_toMatrix",
22
+ "start": [
23
+ 48,
24
+ 1
25
+ ],
26
+ "traced_tactics": [
27
+ {
28
+ "annotated_tactic": [
29
+ "\"let A := <a>toMatrix<\\/a> b b f\"",
30
+ "[{\"full_name\":\"LinearMap.toMatrix\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/ToLin.lean\",\"def_pos\":[548,5],\"def_end_pos\":[548,23]}]"
31
+ ],
32
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
33
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
34
+ "tactic": "let A := toMatrix b b f"
35
+ },
36
+ {
37
+ "annotated_tactic": [
38
+ "\"let b' := <a>chooseBasis<\\/a> R M\"",
39
+ "[{\"full_name\":\"Module.Free.chooseBasis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Basic.lean\",\"def_pos\":[85,19],\"def_end_pos\":[85,30]}]"
40
+ ],
41
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
42
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
43
+ "tactic": "let b' := chooseBasis R M"
44
+ },
45
+ {
46
+ "annotated_tactic": [
47
+ "\"let \\u03b9' := <a>ChooseBasisIndex<\\/a> R M\"",
48
+ "[{\"full_name\":\"Module.Free.ChooseBasisIndex\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Basic.lean\",\"def_pos\":[75,5],\"def_end_pos\":[75,21]}]"
49
+ ],
50
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
51
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
52
+ "tactic": "let \u03b9' := ChooseBasisIndex R M"
53
+ },
54
+ {
55
+ "annotated_tactic": [
56
+ "\"let A' := <a>toMatrix<\\/a> b' b' f\"",
57
+ "[{\"full_name\":\"LinearMap.toMatrix\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/ToLin.lean\",\"def_pos\":[548,5],\"def_end_pos\":[548,23]}]"
58
+ ],
59
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
60
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
61
+ "tactic": "let A' := toMatrix b' b' f"
62
+ },
63
+ {
64
+ "annotated_tactic": [
65
+ "\"let e := <a>Basis.indexEquiv<\\/a> b b'\"",
66
+ "[{\"full_name\":\"Basis.indexEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/StrongRankCondition.lean\",\"def_pos\":[88,5],\"def_end_pos\":[88,21]}]"
67
+ ],
68
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
69
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
70
+ "tactic": "let e := Basis.indexEquiv b b'"
71
+ },
72
+ {
73
+ "annotated_tactic": [
74
+ "\"let \\u03c6 := <a>reindexLinearEquiv<\\/a> R R e e\"",
75
+ "[{\"full_name\":\"Matrix.reindexLinearEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,23]}]"
76
+ ],
77
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
78
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
79
+ "tactic": "let \u03c6 := reindexLinearEquiv R R e e"
80
+ },
81
+ {
82
+ "annotated_tactic": [
83
+ "\"let \\u03c6\\u2081 := <a>reindexLinearEquiv<\\/a> R R e (<a>Equiv.refl<\\/a> \\u03b9')\"",
84
+ "[{\"full_name\":\"Matrix.reindexLinearEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,23]},{\"full_name\":\"Equiv.refl\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[155,23],\"def_end_pos\":[155,27]}]"
85
+ ],
86
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
87
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
88
+ "tactic": "let \u03c6\u2081 := reindexLinearEquiv R R e (Equiv.refl \u03b9')"
89
+ },
90
+ {
91
+ "annotated_tactic": [
92
+ "\"let \\u03c6\\u2082 := <a>reindexLinearEquiv<\\/a> R R (<a>Equiv.refl<\\/a> \\u03b9') (<a>Equiv.refl<\\/a> \\u03b9')\"",
93
+ "[{\"full_name\":\"Matrix.reindexLinearEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,23]},{\"full_name\":\"Equiv.refl\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[155,23],\"def_end_pos\":[155,27]},{\"full_name\":\"Equiv.refl\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[155,23],\"def_end_pos\":[155,27]}]"
94
+ ],
95
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
96
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
97
+ "tactic": "let \u03c6\u2082 := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')"
98
+ },
99
+ {
100
+ "annotated_tactic": [
101
+ "\"let \\u03c6\\u2083 := <a>reindexLinearEquiv<\\/a> R R (<a>Equiv.refl<\\/a> \\u03b9') e\"",
102
+ "[{\"full_name\":\"Matrix.reindexLinearEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,23]},{\"full_name\":\"Equiv.refl\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[155,23],\"def_end_pos\":[155,27]}]"
103
+ ],
104
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
105
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
106
+ "tactic": "let \u03c6\u2083 := reindexLinearEquiv R R (Equiv.refl \u03b9') e"
107
+ },
108
+ {
109
+ "annotated_tactic": [
110
+ "\"let P := b.toMatrix b'\"",
111
+ "[]"
112
+ ],
113
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
114
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
115
+ "tactic": "let P := b.toMatrix b'"
116
+ },
117
+ {
118
+ "annotated_tactic": [
119
+ "\"let Q := b'.toMatrix b\"",
120
+ "[]"
121
+ ],
122
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
123
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
124
+ "tactic": "let Q := b'.toMatrix b"
125
+ },
126
+ {
127
+ "annotated_tactic": [
128
+ "\"have hPQ : C.mapMatrix (\\u03c6\\u2081 P) * C.mapMatrix (\\u03c6\\u2083 Q) = 1 := by\\n rw [<a>RingHom.mapMatrix_apply<\\/a>, <a>RingHom.mapMatrix_apply<\\/a>, \\u2190 <a>Matrix.map_mul<\\/a>,\\n <a>reindexLinearEquiv_mul<\\/a> R R, <a>Basis.toMatrix_mul_toMatrix_flip<\\/a>,\\n <a>reindexLinearEquiv_one<\\/a>, \\u2190 <a>RingHom.mapMatrix_apply<\\/a>, <a>RingHom.map_one<\\/a>]\"",
129
+ "[{\"full_name\":\"RingHom.mapMatrix_apply\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1551,3],\"def_end_pos\":[1551,8]},{\"full_name\":\"RingHom.mapMatrix_apply\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1551,3],\"def_end_pos\":[1551,8]},{\"full_name\":\"Matrix.map_mul\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1170,9],\"def_end_pos\":[1170,16]},{\"full_name\":\"Matrix.reindexLinearEquiv_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,31]},{\"full_name\":\"Basis.toMatrix_mul_toMatrix_flip\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Basis.lean\",\"def_pos\":[267,9],\"def_end_pos\":[267,41]},{\"full_name\":\"Matrix.reindexLinearEquiv_one\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,31]},{\"full_name\":\"RingHom.mapMatrix_apply\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1551,3],\"def_end_pos\":[1551,8]},{\"full_name\":\"RingHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[551,19],\"def_end_pos\":[551,26]}]"
130
+ ],
131
+ "state_after": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
132
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
133
+ "tactic": "have hPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1 := by\n rw [RingHom.mapMatrix_apply, RingHom.mapMatrix_apply, \u2190 Matrix.map_mul,\n reindexLinearEquiv_mul R R, Basis.toMatrix_mul_toMatrix_flip,\n reindexLinearEquiv_one, \u2190 RingHom.mapMatrix_apply, RingHom.map_one]"
134
+ },
135
+ {
136
+ "annotated_tactic": [
137
+ "\"calc\\n A.charpoly = (<a>reindex<\\/a> e e A).<a>charpoly<\\/a> := (<a>charpoly_reindex<\\/a> _ _).<a>symm<\\/a>\\n _ = <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix (\\u03c6 A)) := <a>rfl<\\/a>\\n _ = <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix (\\u03c6 (P * A' * Q))) := by\\n rw [<a>basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix<\\/a>]\\n _ = <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix (\\u03c6\\u2081 P * \\u03c6\\u2082 A' * \\u03c6\\u2083 Q)) := by\\n rw [<a>reindexLinearEquiv_mul<\\/a>, <a>reindexLinearEquiv_mul<\\/a>]\\n _ = <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix (\\u03c6\\u2081 P) * C.mapMatrix A' * C.mapMatrix (\\u03c6\\u2083 Q)) := by simp [\\u03c6\\u2082]\\n _ = <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> * C.mapMatrix (\\u03c6\\u2081 P) * C.mapMatrix (\\u03c6\\u2083 Q) -\\n C.mapMatrix (\\u03c6\\u2081 P) * C.mapMatrix A' * C.mapMatrix (\\u03c6\\u2083 Q)) := by\\n rw [<a>Matrix.mul_assoc<\\/a> ((<a>scalar<\\/a> \\u03b9') <a>X<\\/a>), hPQ, <a>Matrix.mul_one<\\/a>]\\n _ = <a>det<\\/a> (C.mapMatrix (\\u03c6\\u2081 P) * <a>scalar<\\/a> \\u03b9' <a>X<\\/a> * C.mapMatrix (\\u03c6\\u2083 Q) -\\n C.mapMatrix (\\u03c6\\u2081 P) * C.mapMatrix A' * C.mapMatrix (\\u03c6\\u2083 Q)) := by\\n rw [<a>scalar_commute<\\/a> _ <a>commute_X<\\/a>]\\n _ = <a>det<\\/a> (C.mapMatrix (\\u03c6\\u2081 P) * (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix A') * C.mapMatrix (\\u03c6\\u2083 Q)) := by\\n rw [\\u2190 <a>Matrix.sub_mul<\\/a>, \\u2190 <a>Matrix.mul_sub<\\/a>]\\n _ = <a>det<\\/a> (C.mapMatrix (\\u03c6\\u2081 P)) * <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix A') * <a>det<\\/a> (C.mapMatrix (\\u03c6\\u2083 Q)) :=\\n by rw [<a>det_mul<\\/a>, <a>det_mul<\\/a>]\\n _ = <a>det<\\/a> (C.mapMatrix (\\u03c6\\u2081 P)) * <a>det<\\/a> (C.mapMatrix (\\u03c6\\u2083 Q)) * <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix A') :=\\n by ring\\n _ = <a>det<\\/a> (<a>scalar<\\/a> \\u03b9' <a>X<\\/a> - C.mapMatrix A') := by\\n rw [\\u2190 <a>det_mul<\\/a>, hPQ, <a>det_one<\\/a>, <a>one_mul<\\/a>]\\n _ = f.charpoly := <a>rfl<\\/a>\"",
138
+ "[{\"full_name\":\"Matrix.reindex\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[2769,5],\"def_end_pos\":[2769,12]},{\"full_name\":\"Matrix.charpoly\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Charpoly\\/Basic.lean\",\"def_pos\":[99,5],\"def_end_pos\":[99,13]},{\"full_name\":\"Matrix.charpoly_reindex\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Charpoly\\/Basic.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,25]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Basis.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,65]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.reindexLinearEquiv_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,31]},{\"full_name\":\"Matrix.reindexLinearEquiv_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,31]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.mul_assoc\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1201,19],\"def_end_pos\":[1201,28]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.mul_one\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1157,19],\"def_end_pos\":[1157,26]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.scalar_commute\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1307,9],\"def_end_pos\":[1307,23]},{\"full_name\":\"Polynomial.commute_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[628,9],\"def_end_pos\":[628,18]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.sub_mul\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1237,19],\"def_end_pos\":[1237,26]},{\"full_name\":\"Matrix.mul_sub\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1242,19],\"def_end_pos\":[1242,26]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.det_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,16]},{\"full_name\":\"Matrix.det_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,16]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.det_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,16]},{\"full_name\":\"Matrix.det_one\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[91,9],\"def_end_pos\":[91,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
139
+ ],
140
+ "state_after": "no goals",
141
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((toMatrix b b) f).charpoly = f.charpoly",
142
+ "tactic": "calc\n A.charpoly = (reindex e e A).charpoly := (charpoly_reindex _ _).symm\n _ = det (scalar \u03b9' X - C.mapMatrix (\u03c6 A)) := rfl\n _ = det (scalar \u03b9' X - C.mapMatrix (\u03c6 (P * A' * Q))) := by\n rw [basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix]\n _ = det (scalar \u03b9' X - C.mapMatrix (\u03c6\u2081 P * \u03c6\u2082 A' * \u03c6\u2083 Q)) := by\n rw [reindexLinearEquiv_mul, reindexLinearEquiv_mul]\n _ = det (scalar \u03b9' X - C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)) := by simp [\u03c6\u2082]\n _ = det (scalar \u03b9' X * C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) -\n C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)) := by\n rw [Matrix.mul_assoc ((scalar \u03b9') X), hPQ, Matrix.mul_one]\n _ = det (C.mapMatrix (\u03c6\u2081 P) * scalar \u03b9' X * C.mapMatrix (\u03c6\u2083 Q) -\n C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)) := by\n rw [scalar_commute _ commute_X]\n _ = det (C.mapMatrix (\u03c6\u2081 P) * (scalar \u03b9' X - C.mapMatrix A') * C.mapMatrix (\u03c6\u2083 Q)) := by\n rw [\u2190 Matrix.sub_mul, \u2190 Matrix.mul_sub]\n _ = det (C.mapMatrix (\u03c6\u2081 P)) * det (scalar \u03b9' X - C.mapMatrix A') * det (C.mapMatrix (\u03c6\u2083 Q)) :=\n by rw [det_mul, det_mul]\n _ = det (C.mapMatrix (\u03c6\u2081 P)) * det (C.mapMatrix (\u03c6\u2083 Q)) * det (scalar \u03b9' X - C.mapMatrix A') :=\n by ring\n _ = det (scalar \u03b9' X - C.mapMatrix A') := by\n rw [\u2190 det_mul, hPQ, det_one, one_mul]\n _ = f.charpoly := rfl"
143
+ },
144
+ {
145
+ "annotated_tactic": [
146
+ "\"rw [<a>RingHom.mapMatrix_apply<\\/a>, <a>RingHom.mapMatrix_apply<\\/a>, \\u2190 <a>Matrix.map_mul<\\/a>,\\n <a>reindexLinearEquiv_mul<\\/a> R R, <a>Basis.toMatrix_mul_toMatrix_flip<\\/a>,\\n <a>reindexLinearEquiv_one<\\/a>, \\u2190 <a>RingHom.mapMatrix_apply<\\/a>, <a>RingHom.map_one<\\/a>]\"",
147
+ "[{\"full_name\":\"RingHom.mapMatrix_apply\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1551,3],\"def_end_pos\":[1551,8]},{\"full_name\":\"RingHom.mapMatrix_apply\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1551,3],\"def_end_pos\":[1551,8]},{\"full_name\":\"Matrix.map_mul\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1170,9],\"def_end_pos\":[1170,16]},{\"full_name\":\"Matrix.reindexLinearEquiv_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,31]},{\"full_name\":\"Basis.toMatrix_mul_toMatrix_flip\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Basis.lean\",\"def_pos\":[267,9],\"def_end_pos\":[267,41]},{\"full_name\":\"Matrix.reindexLinearEquiv_one\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,31]},{\"full_name\":\"RingHom.mapMatrix_apply\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1551,3],\"def_end_pos\":[1551,8]},{\"full_name\":\"RingHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[551,19],\"def_end_pos\":[551,26]}]"
148
+ ],
149
+ "state_after": "no goals",
150
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\n\u22a2 C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1",
151
+ "tactic": "rw [RingHom.mapMatrix_apply, RingHom.mapMatrix_apply, \u2190 Matrix.map_mul,\n reindexLinearEquiv_mul R R, Basis.toMatrix_mul_toMatrix_flip,\n reindexLinearEquiv_one, \u2190 RingHom.mapMatrix_apply, RingHom.map_one]"
152
+ },
153
+ {
154
+ "annotated_tactic": [
155
+ "\"rw [<a>basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix<\\/a>]\"",
156
+ "[{\"full_name\":\"basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Basis.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,65]}]"
157
+ ],
158
+ "state_after": "no goals",
159
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((scalar \u03b9') X - C.mapMatrix (\u03c6 A)).det = ((scalar \u03b9') X - C.mapMatrix (\u03c6 (P * A' * Q))).det",
160
+ "tactic": "rw [basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix]"
161
+ },
162
+ {
163
+ "annotated_tactic": [
164
+ "\"rw [<a>reindexLinearEquiv_mul<\\/a>, <a>reindexLinearEquiv_mul<\\/a>]\"",
165
+ "[{\"full_name\":\"Matrix.reindexLinearEquiv_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,31]},{\"full_name\":\"Matrix.reindexLinearEquiv_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Reindex.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,31]}]"
166
+ ],
167
+ "state_after": "no goals",
168
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((scalar \u03b9') X - C.mapMatrix (\u03c6 (P * A' * Q))).det = ((scalar \u03b9') X - C.mapMatrix (\u03c6\u2081 P * \u03c6\u2082 A' * \u03c6\u2083 Q)).det",
169
+ "tactic": "rw [reindexLinearEquiv_mul, reindexLinearEquiv_mul]"
170
+ },
171
+ {
172
+ "annotated_tactic": [
173
+ "\"simp [\\u03c6\\u2082]\"",
174
+ "[]"
175
+ ],
176
+ "state_after": "no goals",
177
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((scalar \u03b9') X - C.mapMatrix (\u03c6\u2081 P * \u03c6\u2082 A' * \u03c6\u2083 Q)).det =\n ((scalar \u03b9') X - C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)).det",
178
+ "tactic": "simp [\u03c6\u2082]"
179
+ },
180
+ {
181
+ "annotated_tactic": [
182
+ "\"rw [<a>Matrix.mul_assoc<\\/a> ((<a>scalar<\\/a> \\u03b9') <a>X<\\/a>), hPQ, <a>Matrix.mul_one<\\/a>]\"",
183
+ "[{\"full_name\":\"Matrix.mul_assoc\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1201,19],\"def_end_pos\":[1201,28]},{\"full_name\":\"Matrix.scalar\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1283,5],\"def_end_pos\":[1283,11]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Matrix.mul_one\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1157,19],\"def_end_pos\":[1157,26]}]"
184
+ ],
185
+ "state_after": "no goals",
186
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((scalar \u03b9') X - C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)).det =\n ((scalar \u03b9') X * C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) -\n C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)).det",
187
+ "tactic": "rw [Matrix.mul_assoc ((scalar \u03b9') X), hPQ, Matrix.mul_one]"
188
+ },
189
+ {
190
+ "annotated_tactic": [
191
+ "\"rw [<a>scalar_commute<\\/a> _ <a>commute_X<\\/a>]\"",
192
+ "[{\"full_name\":\"Matrix.scalar_commute\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1307,9],\"def_end_pos\":[1307,23]},{\"full_name\":\"Polynomial.commute_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[628,9],\"def_end_pos\":[628,18]}]"
193
+ ],
194
+ "state_after": "no goals",
195
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 ((scalar \u03b9') X * C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) -\n C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)).det =\n (C.mapMatrix (\u03c6\u2081 P) * (scalar \u03b9') X * C.mapMatrix (\u03c6\u2083 Q) -\n C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)).det",
196
+ "tactic": "rw [scalar_commute _ commute_X]"
197
+ },
198
+ {
199
+ "annotated_tactic": [
200
+ "\"rw [\\u2190 <a>Matrix.sub_mul<\\/a>, \\u2190 <a>Matrix.mul_sub<\\/a>]\"",
201
+ "[{\"full_name\":\"Matrix.sub_mul\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1237,19],\"def_end_pos\":[1237,26]},{\"full_name\":\"Matrix.mul_sub\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1242,19],\"def_end_pos\":[1242,26]}]"
202
+ ],
203
+ "state_after": "no goals",
204
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 (C.mapMatrix (\u03c6\u2081 P) * (scalar \u03b9') X * C.mapMatrix (\u03c6\u2083 Q) -\n C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix A' * C.mapMatrix (\u03c6\u2083 Q)).det =\n (C.mapMatrix (\u03c6\u2081 P) * ((scalar \u03b9') X - C.mapMatrix A') * C.mapMatrix (\u03c6\u2083 Q)).det",
205
+ "tactic": "rw [\u2190 Matrix.sub_mul, \u2190 Matrix.mul_sub]"
206
+ },
207
+ {
208
+ "annotated_tactic": [
209
+ "\"rw [<a>det_mul<\\/a>, <a>det_mul<\\/a>]\"",
210
+ "[{\"full_name\":\"Matrix.det_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,16]},{\"full_name\":\"Matrix.det_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,16]}]"
211
+ ],
212
+ "state_after": "no goals",
213
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 (C.mapMatrix (\u03c6\u2081 P) * ((scalar \u03b9') X - C.mapMatrix A') * C.mapMatrix (\u03c6\u2083 Q)).det =\n (C.mapMatrix (\u03c6\u2081 P)).det * ((scalar \u03b9') X - C.mapMatrix A').det * (C.mapMatrix (\u03c6\u2083 Q)).det",
214
+ "tactic": "rw [det_mul, det_mul]"
215
+ },
216
+ {
217
+ "annotated_tactic": [
218
+ "\"ring\"",
219
+ "[]"
220
+ ],
221
+ "state_after": "no goals",
222
+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 (C.mapMatrix (\u03c6\u2081 P)).det * ((scalar \u03b9') X - C.mapMatrix A').det * (C.mapMatrix (\u03c6\u2083 Q)).det =\n (C.mapMatrix (\u03c6\u2081 P)).det * (C.mapMatrix (\u03c6\u2083 Q)).det * ((scalar \u03b9') X - C.mapMatrix A').det",
223
+ "tactic": "ring"
224
+ },
225
+ {
226
+ "annotated_tactic": [
227
+ "\"rw [\\u2190 <a>det_mul<\\/a>, hPQ, <a>det_one<\\/a>, <a>one_mul<\\/a>]\"",
228
+ "[{\"full_name\":\"Matrix.det_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,16]},{\"full_name\":\"Matrix.det_one\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[91,9],\"def_end_pos\":[91,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
229
+ ],
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+ "state_after": "no goals",
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+ "state_before": "R : Type u_1\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : Nontrivial R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module.Free R M\ninst\u271d\u00b9\u2070 : Module.Finite R M\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module.Finite R M\u2081\ninst\u271d\u2076 : Module.Free R M\u2081\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module.Finite R M\u2082\ninst\u271d\u00b2 : Module.Free R M\u2082\nf : M \u2192\u2097[R] M\n\u03b9 : Type w\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R M\nA : Matrix \u03b9 \u03b9 R := (toMatrix b b) f\nb' : Basis (ChooseBasisIndex R M) R M := chooseBasis R M\n\u03b9' : Type u_2 := ChooseBasisIndex R M\nA' : Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := (toMatrix b' b') f\ne : \u03b9 \u2243 ChooseBasisIndex R M := b.indexEquiv b'\n\u03c6 : Matrix \u03b9 \u03b9 R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) (ChooseBasisIndex R M) R := reindexLinearEquiv R R e e\n\u03c6\u2081 : Matrix \u03b9 \u03b9' R \u2243\u2097[R] Matrix (ChooseBasisIndex R M) \u03b9' R := reindexLinearEquiv R R e (Equiv.refl \u03b9')\n\u03c6\u2082 : Matrix \u03b9' \u03b9' R \u2243\u2097[R] Matrix \u03b9' \u03b9' R := reindexLinearEquiv R R (Equiv.refl \u03b9') (Equiv.refl \u03b9')\n\u03c6\u2083 : Matrix \u03b9' \u03b9 R \u2243\u2097[R] Matrix \u03b9' (ChooseBasisIndex R M) R := reindexLinearEquiv R R (Equiv.refl \u03b9') e\nP : Matrix \u03b9 (ChooseBasisIndex R M) R := b.toMatrix \u21d1b'\nQ : Matrix (ChooseBasisIndex R M) \u03b9 R := b'.toMatrix \u21d1b\nhPQ : C.mapMatrix (\u03c6\u2081 P) * C.mapMatrix (\u03c6\u2083 Q) = 1\n\u22a2 (C.mapMatrix (\u03c6\u2081 P)).det * (C.mapMatrix (\u03c6\u2083 Q)).det * ((scalar \u03b9') X - C.mapMatrix A').det =\n ((scalar \u03b9') X - C.mapMatrix A').det",
232
+ "tactic": "rw [\u2190 det_mul, hPQ, det_one, one_mul]"
233
+ }
234
+ ],
235
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236
+ },
237
+ "source_file": "Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean"
238
+ },
239
+ {
240
+ "error": "theorem has no selected tactic transitions",
241
+ "full_name": "CliffordAlgebra.ofBaseChangeAux_\u03b9",
242
+ "id": "CliffordAlgebra.ofBaseChangeAux_\u03b9",
243
+ "reason": "import_failure",
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+ },
259
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean"
260
+ },
261
+ {
262
+ "error": "theorem has no selected tactic transitions",
263
+ "full_name": "CliffordAlgebra.ofBaseChange_toBaseChange",
264
+ "id": "CliffordAlgebra.ofBaseChange_toBaseChange",
265
+ "reason": "import_failure",
266
+ "selection": {
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+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
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+ "full_name": "CliffordAlgebra.ofBaseChange_toBaseChange",
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+ "start": [
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+ "traced_tactics": [],
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+ },
281
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean"
282
+ },
283
+ {
284
+ "error": "theorem has no selected tactic transitions",
285
+ "full_name": "CliffordAlgebra.toBaseChange_involute",
286
+ "id": "CliffordAlgebra.toBaseChange_involute",
287
+ "reason": "import_failure",
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+ },
303
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean"
304
+ },
305
+ {
306
+ "error": "theorem has no selected tactic transitions",
307
+ "full_name": "CliffordAlgebra.toBaseChange_ofBaseChange",
308
+ "id": "CliffordAlgebra.toBaseChange_ofBaseChange",
309
+ "reason": "import_failure",
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+ "selection": {
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+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean",
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
325
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean"
326
+ },
327
+ {
328
+ "error": "theorem has no selected tactic transitions",
329
+ "full_name": "CliffordAlgebra.toBaseChange_\u03b9",
330
+ "id": "CliffordAlgebra.toBaseChange_\u03b9",
331
+ "reason": "import_failure",
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+ "selection": {
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+ "full_name": "CliffordAlgebra.toBaseChange_\u03b9",
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+ "start": [
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean"
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+ }
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+ ],
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+ "input_units": 2,
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+ "proofs": 9,
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+ "published_proofs": 9,
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+ "replay_failures": [],
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+ "mathlib:CliffordAlgebra.ofBaseChange_tmul_one",
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+ "mathlib:CliffordAlgebra.ofBaseChange_tmul_\u03b9",
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+ "mathlib:CliffordAlgebra.toBaseChange_comp_involute",
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+ "mathlib:CliffordAlgebra.toBaseChange_comp_ofBaseChange",
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+ "mathlib:CliffordAlgebra.toBaseChange_comp_reverseOp",
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+ "mathlib:CliffordAlgebra.toBaseChange_reverse",
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+ "mathlib:LinearEquiv.charpoly_conj",
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+ "mathlib:LinearMap.charpoly_prodMap"
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+ ],
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+ "proof_keys": [
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.ofBaseChange_comp_toBaseChange\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.ofBaseChange_tmul_one\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.ofBaseChange_tmul_\u03b9\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.toBaseChange_comp_involute\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.toBaseChange_comp_ofBaseChange\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.toBaseChange_comp_reverseOp\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean\",null,null,\"CliffordAlgebra.toBaseChange_reverse\"]"
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+ ]
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+ }
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+ "contamination_matches": [],
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+ {
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+ "error": "theorem has no selected tactic transitions",
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+ "full_name": "CliffordAlgebra.equivOfIsometry_symm",
12
+ "id": "CliffordAlgebra.equivOfIsometry_symm",
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+ "reason": "import_failure",
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+ },
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+ },
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+ {
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+ "error": "no internal trace for tactic 'have of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) := by\\n ext\\n simp [of]\\n erw [LinearMap.codRestrict_apply]'",
33
+ "full_name": "CliffordAlgebra.induction",
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+ "reason": "import_failure",
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+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
43
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+ "start": [
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+ "traced_tactics": [
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+ {
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+ "annotated_tactic": [
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+ "\"let s : <a>Subalgebra<\\/a> R (<a>CliffordAlgebra<\\/a> Q) :=\\n { carrier := C\\n mul_mem' := @mul\\n add_mem' := @add\\n algebraMap_mem' := algebraMap }\"",
52
+ "[{\"full_name\":\"Subalgebra\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[22,11],\"def_end_pos\":[22,21]},{\"full_name\":\"CliffordAlgebra\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[69,5],\"def_end_pos\":[69,20]}]"
53
+ ],
54
+ "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\n\u22a2 C a",
55
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\n\u22a2 C a",
56
+ "tactic": "let s : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C\n mul_mem' := @mul\n add_mem' := @add\n algebraMap_mem' := algebraMap }"
57
+ },
58
+ {
59
+ "annotated_tactic": [
60
+ "\"letI h : <a>AddCommMonoid<\\/a> s := <a>inferInstanceAs<\\/a> (<a>AddCommMonoid<\\/a> (<a>Subalgebra.toSubmodule<\\/a> s))\"",
61
+ "[{\"full_name\":\"AddCommMonoid\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[735,7],\"def_end_pos\":[735,20]},{\"full_name\":\"inferInstanceAs\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[113,8],\"def_end_pos\":[113,23]},{\"full_name\":\"AddCommMonoid\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[735,7],\"def_end_pos\":[735,20]},{\"full_name\":\"Subalgebra.toSubmodule\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[280,5],\"def_end_pos\":[280,16]}]"
62
+ ],
63
+ "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\n\u22a2 C a",
64
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\n\u22a2 C a",
65
+ "tactic": "letI h : AddCommMonoid s := inferInstanceAs (AddCommMonoid (Subalgebra.toSubmodule s))"
66
+ },
67
+ {
68
+ "annotated_tactic": [
69
+ "\"let of : { f : M \\u2192\\u2097[R] s \\/\\/ \\u2200 m, f m * f m = <a>_root_.algebraMap<\\/a> _ _ (Q m) } :=\\n \\u27e8(<a>CliffordAlgebra.\\u03b9<\\/a> Q).<a>codRestrict<\\/a> (<a>Subalgebra.toSubmodule<\\/a> s) \\u03b9,\\n fun m => <a>Subtype.eq<\\/a> <| <a>\\u03b9_sq_scalar<\\/a> Q m\\u27e9\"",
70
+ "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"CliffordAlgebra.\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[111,5],\"def_end_pos\":[111,6]},{\"full_name\":\"LinearMap.codRestrict\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[147,5],\"def_end_pos\":[147,16]},{\"full_name\":\"Subalgebra.toSubmodule\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[280,5],\"def_end_pos\":[280,16]},{\"full_name\":\"Subtype.eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1110,19],\"def_end_pos\":[1110,21]},{\"full_name\":\"CliffordAlgebra.\\u03b9_sq_scalar\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[117,9],\"def_end_pos\":[117,20]}]"
71
+ ],
72
+ "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\n\u22a2 C a",
73
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\n\u22a2 C a",
74
+ "tactic": "let of : { f : M \u2192\u2097[R] s // \u2200 m, f m * f m = _root_.algebraMap _ _ (Q m) } :=\n \u27e8(CliffordAlgebra.\u03b9 Q).codRestrict (Subalgebra.toSubmodule s) \u03b9,\n fun m => Subtype.eq <| \u03b9_sq_scalar Q m\u27e9"
75
+ },
76
+ {
77
+ "annotated_tactic": [
78
+ "\"have of_id : <a>AlgHom.id<\\/a> R (<a>CliffordAlgebra<\\/a> Q) = s.val.comp (<a>lift<\\/a> Q of) := by\\n ext\\n simp [of]\\n -- Porting note: `simp` can't apply this\\n erw [<a>LinearMap.codRestrict_apply<\\/a>]\"",
79
+ "[{\"full_name\":\"AlgHom.id\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[293,15],\"def_end_pos\":[293,17]},{\"full_name\":\"CliffordAlgebra\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[69,5],\"def_end_pos\":[69,20]},{\"full_name\":\"CliffordAlgebra.lift\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[137,5],\"def_end_pos\":[137,9]},{\"full_name\":\"LinearMap.codRestrict_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,26]}]"
80
+ ],
81
+ "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nof_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)\n\u22a2 C a",
82
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\n\u22a2 C a",
83
+ "tactic": "have of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) := by\n ext\n simp [of]\n erw [LinearMap.codRestrict_apply]"
84
+ },
85
+ {
86
+ "annotated_tactic": [
87
+ "\"rw [\\u2190 <a>AlgHom.id_apply<\\/a> (R := R) a, of_id]\"",
88
+ "[{\"full_name\":\"AlgHom.id_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,17]}]"
89
+ ],
90
+ "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nof_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)\n\u22a2 C ((s.val.comp ((lift Q) of)) a)",
91
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nof_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)\n\u22a2 C a",
92
+ "tactic": "rw [\u2190 AlgHom.id_apply (R := R) a, of_id]"
93
+ },
94
+ {
95
+ "annotated_tactic": [
96
+ "\"exact <a>Subtype.prop<\\/a> (<a>lift<\\/a> Q of a)\"",
97
+ "[{\"full_name\":\"Subtype.prop\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,13]},{\"full_name\":\"CliffordAlgebra.lift\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[137,5],\"def_end_pos\":[137,9]}]"
98
+ ],
99
+ "state_after": "no goals",
100
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nof_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)\n\u22a2 C ((s.val.comp ((lift Q) of)) a)",
101
+ "tactic": "exact Subtype.prop (lift Q of a)"
102
+ },
103
+ {
104
+ "annotated_tactic": [
105
+ "\"ext\"",
106
+ "[]"
107
+ ],
108
+ "state_after": "case a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nx\u271d : M\n\u22a2 ((AlgHom.id R (CliffordAlgebra Q)).toLinearMap \u2218\u2097 CliffordAlgebra.\u03b9 Q) x\u271d =\n ((s.val.comp ((lift Q) of)).toLinearMap \u2218\u2097 CliffordAlgebra.\u03b9 Q) x\u271d",
109
+ "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\n\u22a2 AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)",
110
+ "tactic": "ext"
111
+ },
112
+ {
113
+ "annotated_tactic": [
114
+ "\"simp [of]\"",
115
+ "[]"
116
+ ],
117
+ "state_after": "case a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nx\u271d : M\n\u22a2 (CliffordAlgebra.\u03b9 Q) x\u271d = \u2191((LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9) x\u271d)",
118
+ "state_before": "case a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nx\u271d : M\n\u22a2 ((AlgHom.id R (CliffordAlgebra Q)).toLinearMap \u2218\u2097 CliffordAlgebra.\u03b9 Q) x\u271d =\n ((s.val.comp ((lift Q) of)).toLinearMap \u2218\u2097 CliffordAlgebra.\u03b9 Q) x\u271d",
119
+ "tactic": "simp [of]"
120
+ },
121
+ {
122
+ "annotated_tactic": [
123
+ "\"erw [<a>LinearMap.codRestrict_apply<\\/a>]\"",
124
+ "[{\"full_name\":\"LinearMap.codRestrict_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,26]}]"
125
+ ],
126
+ "state_after": "no goals",
127
+ "state_before": "case a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nalgebraMap : \u2200 (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)\n\u03b9 : \u2200 (x : M), C ((CliffordAlgebra.\u03b9 Q) x)\nmul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nadd : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C, mul_mem' := mul, one_mem' := \u22ef, add_mem' := add, zero_mem' := \u22ef, algebraMap_mem' := algebraMap }\nh : AddCommMonoid \u21a5s := inferInstanceAs (AddCommMonoid \u21a5(Subalgebra.toSubmodule s))\nof : { f // \u2200 (m : M), f m * f m = (_root_.algebraMap R \u21a5s) (Q m) } :=\n \u27e8LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9, \u22ef\u27e9\nx\u271d : M\n\u22a2 (CliffordAlgebra.\u03b9 Q) x\u271d = \u2191((LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.\u03b9 Q) \u03b9) x\u271d)",
128
+ "tactic": "erw [LinearMap.codRestrict_apply]"
129
+ }
130
+ ],
131
+ "url": "https://github.com/leanprover-community/mathlib4"
132
+ },
133
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
134
+ },
135
+ {
136
+ "error": "theorem has no selected tactic transitions",
137
+ "full_name": "CliffordAlgebra.lift_\u03b9_apply",
138
+ "id": "CliffordAlgebra.lift_\u03b9_apply",
139
+ "reason": "import_failure",
140
+ "selection": {
141
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
142
+ "end": [
143
+ 171,
144
+ 49
145
+ ],
146
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
147
+ "full_name": "CliffordAlgebra.lift_\u03b9_apply",
148
+ "start": [
149
+ 169,
150
+ 1
151
+ ],
152
+ "traced_tactics": [],
153
+ "url": "https://github.com/leanprover-community/mathlib4"
154
+ },
155
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
156
+ },
157
+ {
158
+ "error": "theorem has no selected tactic transitions",
159
+ "full_name": "CliffordAlgebra.map_apply_\u03b9",
160
+ "id": "CliffordAlgebra.map_apply_\u03b9",
161
+ "reason": "import_failure",
162
+ "selection": {
163
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
164
+ "end": [
165
+ 328,
166
+ 21
167
+ ],
168
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
169
+ "full_name": "CliffordAlgebra.map_apply_\u03b9",
170
+ "start": [
171
+ 327,
172
+ 1
173
+ ],
174
+ "traced_tactics": [],
175
+ "url": "https://github.com/leanprover-community/mathlib4"
176
+ },
177
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
178
+ },
179
+ {
180
+ "error": "theorem has no selected tactic transitions",
181
+ "full_name": "CliffordAlgebra.map_comp_\u03b9",
182
+ "id": "CliffordAlgebra.map_comp_\u03b9",
183
+ "reason": "import_failure",
184
+ "selection": {
185
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
186
+ "end": [
187
+ 323,
188
+ 18
189
+ ],
190
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
191
+ "full_name": "CliffordAlgebra.map_comp_\u03b9",
192
+ "start": [
193
+ 321,
194
+ 1
195
+ ],
196
+ "traced_tactics": [],
197
+ "url": "https://github.com/leanprover-community/mathlib4"
198
+ },
199
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
200
+ },
201
+ {
202
+ "error": "theorem has no selected tactic transitions",
203
+ "full_name": "CliffordAlgebra.\u03b9_comp_lift",
204
+ "id": "CliffordAlgebra.\u03b9_comp_lift",
205
+ "reason": "import_failure",
206
+ "selection": {
207
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
208
+ "end": [
209
+ 165,
210
+ 61
211
+ ],
212
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
213
+ "full_name": "CliffordAlgebra.\u03b9_comp_lift",
214
+ "start": [
215
+ 163,
216
+ 1
217
+ ],
218
+ "traced_tactics": [],
219
+ "url": "https://github.com/leanprover-community/mathlib4"
220
+ },
221
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
222
+ },
223
+ {
224
+ "error": "theorem has no selected tactic transitions",
225
+ "full_name": "CliffordAlgebra.\u03b9_mul_\u03b9_add_swap",
226
+ "id": "CliffordAlgebra.\u03b9_mul_\u03b9_add_swap",
227
+ "reason": "import_failure",
228
+ "selection": {
229
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
230
+ "end": [
231
+ 262,
232
+ 68
233
+ ],
234
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
235
+ "full_name": "CliffordAlgebra.\u03b9_mul_\u03b9_add_swap",
236
+ "start": [
237
+ 260,
238
+ 1
239
+ ],
240
+ "traced_tactics": [],
241
+ "url": "https://github.com/leanprover-community/mathlib4"
242
+ },
243
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
244
+ },
245
+ {
246
+ "error": "theorem has no selected tactic transitions",
247
+ "full_name": "CliffordAlgebra.\u03b9_mul_\u03b9_comm",
248
+ "id": "CliffordAlgebra.\u03b9_mul_\u03b9_comm",
249
+ "reason": "import_failure",
250
+ "selection": {
251
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
252
+ "end": [
253
+ 267,
254
+ 42
255
+ ],
256
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
257
+ "full_name": "CliffordAlgebra.\u03b9_mul_\u03b9_comm",
258
+ "start": [
259
+ 265,
260
+ 1
261
+ ],
262
+ "traced_tactics": [],
263
+ "url": "https://github.com/leanprover-community/mathlib4"
264
+ },
265
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
266
+ },
267
+ {
268
+ "error": "theorem has no selected tactic transitions",
269
+ "full_name": "CliffordAlgebra.\u03b9_mul_\u03b9_comm_of_isOrtho",
270
+ "id": "CliffordAlgebra.\u03b9_mul_\u03b9_comm_of_isOrtho",
271
+ "reason": "import_failure",
272
+ "selection": {
273
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
274
+ "end": [
275
+ 279,
276
+ 62
277
+ ],
278
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
279
+ "full_name": "CliffordAlgebra.\u03b9_mul_\u03b9_comm_of_isOrtho",
280
+ "start": [
281
+ 277,
282
+ 1
283
+ ],
284
+ "traced_tactics": [],
285
+ "url": "https://github.com/leanprover-community/mathlib4"
286
+ },
287
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
288
+ },
289
+ {
290
+ "error": "theorem has no selected tactic transitions",
291
+ "full_name": "CliffordAlgebra.\u03b9_range_map_map",
292
+ "id": "CliffordAlgebra.\u03b9_range_map_map",
293
+ "reason": "import_failure",
294
+ "selection": {
295
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
296
+ "end": [
297
+ 348,
298
+ 58
299
+ ],
300
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
301
+ "full_name": "CliffordAlgebra.\u03b9_range_map_map",
302
+ "start": [
303
+ 346,
304
+ 1
305
+ ],
306
+ "traced_tactics": [],
307
+ "url": "https://github.com/leanprover-community/mathlib4"
308
+ },
309
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean"
310
+ },
311
+ {
312
+ "error": "no internal trace for tactic 'exact AlgHom.commutes _ _'",
313
+ "full_name": "CliffordAlgebra.involute_eq_of_mem_even",
314
+ "id": "CliffordAlgebra.involute_eq_of_mem_even",
315
+ "reason": "import_failure",
316
+ "selection": {
317
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
318
+ "end": [
319
+ 342,
320
+ 68
321
+ ],
322
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
323
+ "full_name": "CliffordAlgebra.involute_eq_of_mem_even",
324
+ "start": [
325
+ 336,
326
+ 1
327
+ ],
328
+ "traced_tactics": [
329
+ {
330
+ "annotated_tactic": [
331
+ "\"induction x, h using <a>even_induction<\\/a> with\\n | algebraMap r => exact <a>AlgHom.commutes<\\/a> _ _\\n | add x y _hx _hy ihx ihy =>\\n rw [<a>map_add<\\/a>, ihx, ihy]\\n | \\u03b9_mul_\\u03b9_mul m\\u2081 m\\u2082 x _hx ihx =>\\n rw [<a>map_mul<\\/a>, <a>map_mul<\\/a>, <a>involute_\\u03b9<\\/a>, <a>involute_\\u03b9<\\/a>, ihx, <a>neg_mul_neg<\\/a>]\"",
332
+ "[{\"full_name\":\"CliffordAlgebra.even_induction\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Grading.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,23]},{\"full_name\":\"AlgHom.commutes\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,17]},{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"neg_mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[327,9],\"def_end_pos\":[327,20]}]"
333
+ ],
334
+ "state_after": "no goals",
335
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\nh : x \u2208 evenOdd Q 0\n\u22a2 involute x = x",
336
+ "tactic": "induction x, h using even_induction with\n| algebraMap r => exact AlgHom.commutes _ _\n| add x y _hx _hy ihx ihy =>\n rw [map_add, ihx, ihy]\n| \u03b9_mul_\u03b9_mul m\u2081 m\u2082 x _hx ihx =>\n rw [map_mul, map_mul, involute_\u03b9, involute_\u03b9, ihx, neg_mul_neg]"
337
+ },
338
+ {
339
+ "annotated_tactic": [
340
+ "\"exact <a>AlgHom.commutes<\\/a> _ _\"",
341
+ "[{\"full_name\":\"AlgHom.commutes\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,17]}]"
342
+ ],
343
+ "state_after": "no goals",
344
+ "state_before": "case algebraMap\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\nr : R\n\u22a2 involute ((algebraMap R (CliffordAlgebra Q)) r) = (algebraMap R (CliffordAlgebra Q)) r",
345
+ "tactic": "exact AlgHom.commutes _ _"
346
+ },
347
+ {
348
+ "annotated_tactic": [
349
+ "\"rw [<a>map_add<\\/a>, ihx, ihy]\"",
350
+ "[{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]}]"
351
+ ],
352
+ "state_after": "no goals",
353
+ "state_before": "case add\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx\u271d x y : CliffordAlgebra Q\n_hx : x \u2208 evenOdd Q 0\n_hy : y \u2208 evenOdd Q 0\nihx : involute x = x\nihy : involute y = y\n\u22a2 involute (x + y) = x + y",
354
+ "tactic": "rw [map_add, ihx, ihy]"
355
+ },
356
+ {
357
+ "annotated_tactic": [
358
+ "\"rw [<a>map_mul<\\/a>, <a>map_mul<\\/a>, <a>involute_\\u03b9<\\/a>, <a>involute_\\u03b9<\\/a>, ihx, <a>neg_mul_neg<\\/a>]\"",
359
+ "[{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"neg_mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[327,9],\"def_end_pos\":[327,20]}]"
360
+ ],
361
+ "state_after": "no goals",
362
+ "state_before": "case \u03b9_mul_\u03b9_mul\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx\u271d : CliffordAlgebra Q\nm\u2081 m\u2082 : M\nx : CliffordAlgebra Q\n_hx : x \u2208 evenOdd Q 0\nihx : involute x = x\n\u22a2 involute ((\u03b9 Q) m\u2081 * (\u03b9 Q) m\u2082 * x) = (\u03b9 Q) m\u2081 * (\u03b9 Q) m\u2082 * x",
363
+ "tactic": "rw [map_mul, map_mul, involute_\u03b9, involute_\u03b9, ihx, neg_mul_neg]"
364
+ }
365
+ ],
366
+ "url": "https://github.com/leanprover-community/mathlib4"
367
+ },
368
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
369
+ },
370
+ {
371
+ "error": "no internal trace for tactic 'exact involute_\u03b9 _'",
372
+ "full_name": "CliffordAlgebra.involute_eq_of_mem_odd",
373
+ "id": "CliffordAlgebra.involute_eq_of_mem_odd",
374
+ "reason": "import_failure",
375
+ "selection": {
376
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
377
+ "end": [
378
+ 351,
379
+ 77
380
+ ],
381
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
382
+ "full_name": "CliffordAlgebra.involute_eq_of_mem_odd",
383
+ "start": [
384
+ 345,
385
+ 1
386
+ ],
387
+ "traced_tactics": [
388
+ {
389
+ "annotated_tactic": [
390
+ "\"induction x, h using <a>odd_induction<\\/a> with\\n | \\u03b9 m => exact <a>involute_\\u03b9<\\/a> _\\n | add x y _hx _hy ihx ihy =>\\n rw [<a>map_add<\\/a>, ihx, ihy, <a>neg_add<\\/a>]\\n | \\u03b9_mul_\\u03b9_mul m\\u2081 m\\u2082 x _hx ihx =>\\n rw [<a>map_mul<\\/a>, <a>map_mul<\\/a>, <a>involute_\\u03b9<\\/a>, <a>involute_\\u03b9<\\/a>, ihx, <a>neg_mul_neg<\\/a>, <a>mul_neg<\\/a>]\"",
391
+ "[{\"full_name\":\"CliffordAlgebra.odd_induction\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Grading.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,22]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]},{\"full_name\":\"neg_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[725,15],\"def_end_pos\":[725,22]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"neg_mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[327,9],\"def_end_pos\":[327,20]},{\"full_name\":\"mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
392
+ ],
393
+ "state_after": "no goals",
394
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\nh : x \u2208 evenOdd Q 1\n\u22a2 involute x = -x",
395
+ "tactic": "induction x, h using odd_induction with\n| \u03b9 m => exact involute_\u03b9 _\n| add x y _hx _hy ihx ihy =>\n rw [map_add, ihx, ihy, neg_add]\n| \u03b9_mul_\u03b9_mul m\u2081 m\u2082 x _hx ihx =>\n rw [map_mul, map_mul, involute_\u03b9, involute_\u03b9, ihx, neg_mul_neg, mul_neg]"
396
+ },
397
+ {
398
+ "annotated_tactic": [
399
+ "\"exact <a>involute_\\u03b9<\\/a> _\"",
400
+ "[{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]}]"
401
+ ],
402
+ "state_after": "no goals",
403
+ "state_before": "case \u03b9\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\nm : M\n\u22a2 involute ((\u03b9 Q) m) = -(\u03b9 Q) m",
404
+ "tactic": "exact involute_\u03b9 _"
405
+ },
406
+ {
407
+ "annotated_tactic": [
408
+ "\"rw [<a>map_add<\\/a>, ihx, ihy, <a>neg_add<\\/a>]\"",
409
+ "[{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]},{\"full_name\":\"neg_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[725,15],\"def_end_pos\":[725,22]}]"
410
+ ],
411
+ "state_after": "no goals",
412
+ "state_before": "case add\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx\u271d x y : CliffordAlgebra Q\n_hx : x \u2208 evenOdd Q 1\n_hy : y \u2208 evenOdd Q 1\nihx : involute x = -x\nihy : involute y = -y\n\u22a2 involute (x + y) = -(x + y)",
413
+ "tactic": "rw [map_add, ihx, ihy, neg_add]"
414
+ },
415
+ {
416
+ "annotated_tactic": [
417
+ "\"rw [<a>map_mul<\\/a>, <a>map_mul<\\/a>, <a>involute_\\u03b9<\\/a>, <a>involute_\\u03b9<\\/a>, ihx, <a>neg_mul_neg<\\/a>, <a>mul_neg<\\/a>]\"",
418
+ "[{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"CliffordAlgebra.involute_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,19]},{\"full_name\":\"neg_mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[327,9],\"def_end_pos\":[327,20]},{\"full_name\":\"mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
419
+ ],
420
+ "state_after": "no goals",
421
+ "state_before": "case \u03b9_mul_\u03b9_mul\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx\u271d : CliffordAlgebra Q\nm\u2081 m\u2082 : M\nx : CliffordAlgebra Q\n_hx : x \u2208 evenOdd Q 1\nihx : involute x = -x\n\u22a2 involute ((\u03b9 Q) m\u2081 * (\u03b9 Q) m\u2082 * x) = -((\u03b9 Q) m\u2081 * (\u03b9 Q) m\u2082 * x)",
422
+ "tactic": "rw [map_mul, map_mul, involute_\u03b9, involute_\u03b9, ihx, neg_mul_neg, mul_neg]"
423
+ }
424
+ ],
425
+ "url": "https://github.com/leanprover-community/mathlib4"
426
+ },
427
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
428
+ },
429
+ {
430
+ "error": "theorem has no selected tactic transitions",
431
+ "full_name": "CliffordAlgebra.involute_involute",
432
+ "id": "CliffordAlgebra.involute_involute",
433
+ "reason": "import_failure",
434
+ "selection": {
435
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
436
+ "end": [
437
+ 65,
438
+ 22
439
+ ],
440
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
441
+ "full_name": "CliffordAlgebra.involute_involute",
442
+ "start": [
443
+ 64,
444
+ 1
445
+ ],
446
+ "traced_tactics": [],
447
+ "url": "https://github.com/leanprover-community/mathlib4"
448
+ },
449
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
450
+ },
451
+ {
452
+ "error": "theorem has no selected tactic transitions",
453
+ "full_name": "CliffordAlgebra.involute_involutive",
454
+ "id": "CliffordAlgebra.involute_involutive",
455
+ "reason": "import_failure",
456
+ "selection": {
457
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
458
+ "end": [
459
+ 60,
460
+ 42
461
+ ],
462
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
463
+ "full_name": "CliffordAlgebra.involute_involutive",
464
+ "start": [
465
+ 59,
466
+ 1
467
+ ],
468
+ "traced_tactics": [],
469
+ "url": "https://github.com/leanprover-community/mathlib4"
470
+ },
471
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
472
+ },
473
+ {
474
+ "error": "theorem has no selected tactic transitions",
475
+ "full_name": "CliffordAlgebra.involute_mem_evenOdd_iff",
476
+ "id": "CliffordAlgebra.involute_mem_evenOdd_iff",
477
+ "reason": "import_failure",
478
+ "selection": {
479
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
480
+ "end": [
481
+ 318,
482
+ 52
483
+ ],
484
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
485
+ "full_name": "CliffordAlgebra.involute_mem_evenOdd_iff",
486
+ "start": [
487
+ 316,
488
+ 1
489
+ ],
490
+ "traced_tactics": [],
491
+ "url": "https://github.com/leanprover-community/mathlib4"
492
+ },
493
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
494
+ },
495
+ {
496
+ "error": "theorem has no selected tactic transitions",
497
+ "full_name": "CliffordAlgebra.involute_\u03b9",
498
+ "id": "CliffordAlgebra.involute_\u03b9",
499
+ "reason": "import_failure",
500
+ "selection": {
501
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
502
+ "end": [
503
+ 51,
504
+ 21
505
+ ],
506
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
507
+ "full_name": "CliffordAlgebra.involute_\u03b9",
508
+ "start": [
509
+ 50,
510
+ 1
511
+ ],
512
+ "traced_tactics": [],
513
+ "url": "https://github.com/leanprover-community/mathlib4"
514
+ },
515
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
516
+ },
517
+ {
518
+ "error": "theorem has no selected tactic transitions",
519
+ "full_name": "CliffordAlgebra.op_reverse",
520
+ "id": "CliffordAlgebra.op_reverse",
521
+ "reason": "import_failure",
522
+ "selection": {
523
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
524
+ "end": [
525
+ 108,
526
+ 89
527
+ ],
528
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
529
+ "full_name": "CliffordAlgebra.op_reverse",
530
+ "start": [
531
+ 108,
532
+ 9
533
+ ],
534
+ "traced_tactics": [],
535
+ "url": "https://github.com/leanprover-community/mathlib4"
536
+ },
537
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
538
+ },
539
+ {
540
+ "error": "theorem has no selected tactic transitions",
541
+ "full_name": "CliffordAlgebra.reverse.commutes",
542
+ "id": "CliffordAlgebra.reverse.commutes",
543
+ "reason": "import_failure",
544
+ "selection": {
545
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
546
+ "end": [
547
+ 117,
548
+ 39
549
+ ],
550
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
551
+ "full_name": "CliffordAlgebra.reverse.commutes",
552
+ "start": [
553
+ 115,
554
+ 1
555
+ ],
556
+ "traced_tactics": [],
557
+ "url": "https://github.com/leanprover-community/mathlib4"
558
+ },
559
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
560
+ },
561
+ {
562
+ "error": "theorem has no selected tactic transitions",
563
+ "full_name": "CliffordAlgebra.reverse.map_mul",
564
+ "id": "CliffordAlgebra.reverse.map_mul",
565
+ "reason": "import_failure",
566
+ "selection": {
567
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
568
+ "end": [
569
+ 128,
570
+ 39
571
+ ],
572
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
573
+ "full_name": "CliffordAlgebra.reverse.map_mul",
574
+ "start": [
575
+ 126,
576
+ 1
577
+ ],
578
+ "traced_tactics": [],
579
+ "url": "https://github.com/leanprover-community/mathlib4"
580
+ },
581
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
582
+ },
583
+ {
584
+ "error": "theorem has no selected tactic transitions",
585
+ "full_name": "CliffordAlgebra.reverse.map_one",
586
+ "id": "CliffordAlgebra.reverse.map_one",
587
+ "reason": "import_failure",
588
+ "selection": {
589
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
590
+ "end": [
591
+ 122,
592
+ 33
593
+ ],
594
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
595
+ "full_name": "CliffordAlgebra.reverse.map_one",
596
+ "start": [
597
+ 121,
598
+ 1
599
+ ],
600
+ "traced_tactics": [],
601
+ "url": "https://github.com/leanprover-community/mathlib4"
602
+ },
603
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
604
+ },
605
+ {
606
+ "error": "theorem has no selected tactic transitions",
607
+ "full_name": "CliffordAlgebra.reverseOpEquiv_opComm",
608
+ "id": "CliffordAlgebra.reverseOpEquiv_opComm",
609
+ "reason": "import_failure",
610
+ "selection": {
611
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
612
+ "end": [
613
+ 98,
614
+ 75
615
+ ],
616
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
617
+ "full_name": "CliffordAlgebra.reverseOpEquiv_opComm",
618
+ "start": [
619
+ 97,
620
+ 1
621
+ ],
622
+ "traced_tactics": [],
623
+ "url": "https://github.com/leanprover-community/mathlib4"
624
+ },
625
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
626
+ },
627
+ {
628
+ "error": "theorem has no selected tactic transitions",
629
+ "full_name": "CliffordAlgebra.reverseOp_\u03b9",
630
+ "id": "CliffordAlgebra.reverseOp_\u03b9",
631
+ "reason": "import_failure",
632
+ "selection": {
633
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
634
+ "end": [
635
+ 87,
636
+ 83
637
+ ],
638
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
639
+ "full_name": "CliffordAlgebra.reverseOp_\u03b9",
640
+ "start": [
641
+ 87,
642
+ 1
643
+ ],
644
+ "traced_tactics": [],
645
+ "url": "https://github.com/leanprover-community/mathlib4"
646
+ },
647
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
648
+ },
649
+ {
650
+ "error": "no internal trace for tactic 'simp'",
651
+ "full_name": "CliffordAlgebra.reverse_comp_involute",
652
+ "id": "CliffordAlgebra.reverse_comp_involute",
653
+ "reason": "import_failure",
654
+ "selection": {
655
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
656
+ "end": [
657
+ 161,
658
+ 73
659
+ ],
660
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
661
+ "full_name": "CliffordAlgebra.reverse_comp_involute",
662
+ "start": [
663
+ 152,
664
+ 1
665
+ ],
666
+ "traced_tactics": [
667
+ {
668
+ "annotated_tactic": [
669
+ "\"ext x\"",
670
+ "[]"
671
+ ],
672
+ "state_after": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n\u22a2 (reverse \u2218\u2097 involute.toLinearMap) x = (involute.toLinearMap \u2218\u2097 reverse) x",
673
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 reverse \u2218\u2097 involute.toLinearMap = involute.toLinearMap \u2218\u2097 reverse",
674
+ "tactic": "ext x"
675
+ },
676
+ {
677
+ "annotated_tactic": [
678
+ "\"simp only [<a>LinearMap.comp_apply<\\/a>, <a>AlgHom.toLinearMap_apply<\\/a>]\"",
679
+ "[{\"full_name\":\"LinearMap.comp_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,19]},{\"full_name\":\"AlgHom.toLinearMap_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[356,9],\"def_end_pos\":[356,26]}]"
680
+ ],
681
+ "state_after": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n\u22a2 reverse (involute x) = involute (reverse x)",
682
+ "state_before": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n\u22a2 (reverse \u2218\u2097 involute.toLinearMap) x = (involute.toLinearMap \u2218\u2097 reverse) x",
683
+ "tactic": "simp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply]"
684
+ },
685
+ {
686
+ "annotated_tactic": [
687
+ "\"induction x using <a>CliffordAlgebra.induction<\\/a> with\\n | algebraMap => simp\\n | \\u03b9 => simp\\n | mul a b ha hb => simp only [ha, hb, <a>reverse.map_mul<\\/a>, <a>AlgHom.map_mul<\\/a>]\\n | add a b ha hb => simp only [ha, hb, reverse.map_add, <a>AlgHom.map_add<\\/a>]\"",
688
+ "[{\"full_name\":\"CliffordAlgebra.induction\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Basic.lean\",\"def_pos\":[206,9],\"def_end_pos\":[206,18]},{\"full_name\":\"CliffordAlgebra.reverse.map_mul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/CliffordAlgebra\\/Conjugation.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,24]},{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"AlgHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[238,19],\"def_end_pos\":[238,26]}]"
689
+ ],
690
+ "state_after": "no goals",
691
+ "state_before": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n\u22a2 reverse (involute x) = involute (reverse x)",
692
+ "tactic": "induction x using CliffordAlgebra.induction with\n| algebraMap => simp\n| \u03b9 => simp\n| mul a b ha hb => simp only [ha, hb, reverse.map_mul, AlgHom.map_mul]\n| add a b ha hb => simp only [ha, hb, reverse.map_add, AlgHom.map_add]"
693
+ },
694
+ {
695
+ "annotated_tactic": [
696
+ "\"simp\"",
697
+ "[]"
698
+ ],
699
+ "state_after": "no goals",
700
+ "state_before": "case h.algebraMap\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nr\u271d : R\n\u22a2 reverse (involute ((algebraMap R (CliffordAlgebra Q)) r\u271d)) =\n involute (reverse ((algebraMap R (CliffordAlgebra Q)) r\u271d))",
701
+ "tactic": "simp"
702
+ },
703
+ {
704
+ "annotated_tactic": [
705
+ "\"simp\"",
706
+ "[]"
707
+ ],
708
+ "state_after": "no goals",
709
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710
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711
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712
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713
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714
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715
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716
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717
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718
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719
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720
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722
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723
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725
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726
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727
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728
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729
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730
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731
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732
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733
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734
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735
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736
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737
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738
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739
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740
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741
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745
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746
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750
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751
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752
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753
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754
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755
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756
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757
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758
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759
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760
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761
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762
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763
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767
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768
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769
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770
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771
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772
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773
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774
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775
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776
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777
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778
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779
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780
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781
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782
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783
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784
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785
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788
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789
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790
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791
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792
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793
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794
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795
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796
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797
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798
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799
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800
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801
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802
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803
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804
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805
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806
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807
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808
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811
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812
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813
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814
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815
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816
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817
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818
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819
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820
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821
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
822
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823
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824
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825
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826
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827
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828
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829
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830
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833
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834
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835
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836
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837
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839
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840
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841
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842
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843
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844
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845
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846
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847
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848
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849
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850
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851
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853
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855
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856
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857
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858
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859
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860
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861
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862
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863
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864
+ },
865
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
866
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867
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868
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869
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870
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871
+ "reason": "import_failure",
872
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873
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874
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875
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876
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877
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878
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879
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880
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881
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882
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883
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884
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885
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886
+ },
887
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
888
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889
+ {
890
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891
+ "full_name": "CliffordAlgebra.submodule_map_reverse_eq_comap",
892
+ "id": "CliffordAlgebra.submodule_map_reverse_eq_comap",
893
+ "reason": "import_failure",
894
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895
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896
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897
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899
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900
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901
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902
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903
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904
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905
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906
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907
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908
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909
+ "source_file": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean"
910
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911
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912
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913
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914
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915
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916
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917
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918
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921
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922
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923
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924
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925
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926
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927
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928
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929
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930
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931
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932
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933
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934
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935
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936
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937
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938
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939
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940
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943
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944
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945
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946
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947
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949
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950
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951
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952
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953
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954
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955
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956
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957
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958
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959
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960
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961
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962
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963
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964
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965
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966
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967
+ "internal transition boundary mismatch",
968
+ "internal transition boundary mismatch"
969
+ ],
970
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971
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972
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973
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974
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978
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979
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980
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981
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982
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983
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984
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985
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986
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987
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988
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989
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990
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991
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992
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993
+ "tactic": "simp only [AlgHom.commutes]"
994
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995
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996
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997
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998
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999
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1000
+ "state_after": "no goals",
1001
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1002
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1003
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1004
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1005
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1006
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1007
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1008
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1009
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1010
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1011
+ "tactic": "simp only [map_mul, hx, hy]"
1012
+ },
1013
+ {
1014
+ "annotated_tactic": [
1015
+ "\"simp only [<a>map_add<\\/a>, hx, hy]\"",
1016
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1017
+ ],
1018
+ "state_after": "no goals",
1019
+ "state_before": "R : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081\u271d : QuadraticForm R M\u2081\nQ\u2082\u271d : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nf : Q\u2081 \u2192q\u1d62 Q\u2082\nhf : Function.Surjective \u21d1f\nx\u271d\u00b3 x\u271d\u00b2 : CliffordAlgebra Q\u2082\nx\u271d\u00b9 : \u2203 a, (map f) a = x\u271d\u00b3\nx\u271d : \u2203 a, (map f) a = x\u271d\u00b2\nx : CliffordAlgebra Q\u2081\nhx : (map f) x = x\u271d\u00b3\ny : CliffordAlgebra Q\u2081\nhy : (map f) y = x\u271d\u00b2\n\u22a2 (map f) (x + y) = x\u271d\u00b3 + x\u271d\u00b2",
1020
+ "tactic": "simp only [map_add, hx, hy]"
1021
+ }
1022
+ ],
1023
+ "url": "https://github.com/leanprover-community/mathlib4"
1024
+ }
1025
+ },
1026
+ "theorem_name": "CliffordAlgebra.map_surjective"
1027
+ },
1028
+ {
1029
+ "errors": [
1030
+ "internal transition boundary mismatch",
1031
+ "internal transition boundary mismatch"
1032
+ ],
1033
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean",
1034
+ "proof_id": "mathlib:CliffordAlgebra.mul_add_swap_eq_polar_of_forall_mul_self_eq",
1035
+ "provenance": {
1036
+ "selection": {
1037
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1038
+ "end": [
1039
+ 242,
1040
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1041
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1042
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1043
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+ "errors": [
1093
+ "internal transition boundary mismatch"
1094
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1095
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
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1105
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+ "tactic": "simp [pow_succ, involute_prod_map_\u03b9 xs]"
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+ }
1129
+ ],
1130
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+ }
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1134
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+ "errors": [
1137
+ "internal transition boundary mismatch"
1138
+ ],
1139
+ "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean",
1140
+ "proof_id": "mathlib:CliffordAlgebra.reverse_prod_map_\u03b9",
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+ "start": [
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+ "annotated_tactic": [
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1160
+ "state_after": "no goals",
1161
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 reverse (List.map \u21d1(\u03b9 Q) []).prod = (List.map \u21d1(\u03b9 Q) []).reverse.prod",
1162
+ "tactic": "simp"
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+ },
1164
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1165
+ "annotated_tactic": [
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+ "\"simp [reverse_prod_map_\\u03b9 xs]\"",
1167
+ "[]"
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1169
+ "state_after": "no goals",
1170
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : M\nxs : List M\n\u22a2 reverse (List.map (\u21d1(\u03b9 Q)) (x :: xs)).prod = (List.map (\u21d1(\u03b9 Q)) (x :: xs)).reverse.prod",
1171
+ "tactic": "simp [reverse_prod_map_\u03b9 xs]"
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+ }
1173
+ ],
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+ "url": "https://github.com/leanprover-community/mathlib4"
1175
+ }
1176
+ },
1177
+ "theorem_name": "CliffordAlgebra.reverse_prod_map_\u03b9"
1178
+ }
1179
+ ],
1180
+ "validation_rejected_proofs": 4,
1181
+ "zero_transition_proofs": 0
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+ },
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+ "shards": [
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+ {
1185
+ "path": "data/train/mathlib-001253-0000-26bbef70a1b1079d.parquet",
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+ "rows": 38,
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+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1190
+ },
1191
+ {
1192
+ "path": "data/internal_test/mathlib-001253-0000-ec8d8ad6aa7307bd.parquet",
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+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1197
+ }
1198
+ ],
1199
+ "proof_ids": [
1200
+ "mathlib:CliffordAlgebra.comp_\u03b9_sq_scalar",
1201
+ "mathlib:CliffordAlgebra.equivOfIsometry_refl",
1202
+ "mathlib:CliffordAlgebra.equivOfIsometry_trans",
1203
+ "mathlib:CliffordAlgebra.evenOdd_comap_involute",
1204
+ "mathlib:CliffordAlgebra.evenOdd_comap_reverse",
1205
+ "mathlib:CliffordAlgebra.evenOdd_map_involute",
1206
+ "mathlib:CliffordAlgebra.evenOdd_map_reverse",
1207
+ "mathlib:CliffordAlgebra.forall_mul_self_eq_iff",
1208
+ "mathlib:CliffordAlgebra.hom_ext",
1209
+ "mathlib:CliffordAlgebra.involute_comp_involute",
1210
+ "mathlib:CliffordAlgebra.leftInverse_map_of_leftInverse",
1211
+ "mathlib:CliffordAlgebra.lift_comp_\u03b9",
1212
+ "mathlib:CliffordAlgebra.lift_unique",
1213
+ "mathlib:CliffordAlgebra.map_comp_map",
1214
+ "mathlib:CliffordAlgebra.map_id",
1215
+ "mathlib:CliffordAlgebra.mul_\u03b9_mul_\u03b9_of_isOrtho",
1216
+ "mathlib:CliffordAlgebra.reverse_\u03b9",
1217
+ "mathlib:CliffordAlgebra.submodule_comap_mul_reverse",
1218
+ "mathlib:CliffordAlgebra.submodule_comap_pow_reverse",
1219
+ "mathlib:CliffordAlgebra.submodule_map_mul_reverse",
1220
+ "mathlib:CliffordAlgebra.submodule_map_pow_reverse",
1221
+ "mathlib:CliffordAlgebra.\u03b9_mul_\u03b9_add_swap_of_isOrtho",
1222
+ "mathlib:CliffordAlgebra.\u03b9_mul_\u03b9_mul_of_isOrtho",
1223
+ "mathlib:CliffordAlgebra.\u03b9_mul_\u03b9_mul_\u03b9",
1224
+ "mathlib:CliffordAlgebra.\u03b9_range_comap_involute",
1225
+ "mathlib:CliffordAlgebra.\u03b9_range_comap_reverse",
1226
+ "mathlib:CliffordAlgebra.\u03b9_range_map_lift",
1227
+ "mathlib:CliffordAlgebra.\u03b9_range_map_reverse",
1228
+ "mathlib:TensorAlgebra.toClifford_\u03b9"
1229
+ ],
1230
+ "proof_keys": [
1231
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1232
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1233
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1234
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1235
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean\",null,null,\"CliffordAlgebra.hom_ext\"]",
1236
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean\",null,null,\"CliffordAlgebra.leftInverse_map_of_leftInverse\"]",
1237
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean\",null,null,\"CliffordAlgebra.lift_comp_\u03b9\"]",
1238
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean\",null,null,\"CliffordAlgebra.lift_unique\"]",
1239
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1240
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1241
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1242
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1243
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1244
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1245
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1246
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean\",null,null,\"TensorAlgebra.toClifford_\u03b9\"]",
1247
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.evenOdd_comap_involute\"]",
1248
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.evenOdd_comap_reverse\"]",
1249
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1250
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.evenOdd_map_reverse\"]",
1251
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.involute_comp_involute\"]",
1252
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1253
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1254
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1255
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1256
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1257
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.\u03b9_range_comap_involute\"]",
1258
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.\u03b9_range_comap_reverse\"]",
1259
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean\",null,null,\"CliffordAlgebra.\u03b9_range_map_reverse\"]"
1260
+ ]
1261
+ }
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The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001259.json ADDED
@@ -0,0 +1,330 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001259",
4
+ "batch_fingerprint": "e22ed47770150a63eab0adac94e62b9ee4a9d0ff8592519a0aba6149fdb57488",
5
+ "evidence": {
6
+ "batch_fingerprint": "e22ed47770150a63eab0adac94e62b9ee4a9d0ff8592519a0aba6149fdb57488",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "contractLeft_apply",
12
+ "id": "contractLeft_apply",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 70,
18
+ 6
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/Contraction.lean",
21
+ "full_name": "contractLeft_apply",
22
+ "start": [
23
+ 69,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/LinearAlgebra/Contraction.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "contractRight_apply",
34
+ "id": "contractRight_apply",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 75,
40
+ 6
41
+ ],
42
+ "file_path": "Mathlib/LinearAlgebra/Contraction.lean",
43
+ "full_name": "contractRight_apply",
44
+ "start": [
45
+ 74,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/LinearAlgebra/Contraction.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "dualTensorHomEquivOfBasis_toLinearMap",
56
+ "id": "dualTensorHomEquivOfBasis_toLinearMap",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
60
+ "end": [
61
+ 181,
62
+ 6
63
+ ],
64
+ "file_path": "Mathlib/LinearAlgebra/Contraction.lean",
65
+ "full_name": "dualTensorHomEquivOfBasis_toLinearMap",
66
+ "start": [
67
+ 178,
68
+ 1
69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/LinearAlgebra/Contraction.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "dualTensorHom_apply",
78
+ "id": "dualTensorHom_apply",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
82
+ "end": [
83
+ 81,
84
+ 6
85
+ ],
86
+ "file_path": "Mathlib/LinearAlgebra/Contraction.lean",
87
+ "full_name": "dualTensorHom_apply",
88
+ "start": [
89
+ 79,
90
+ 1
91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/LinearAlgebra/Contraction.lean"
96
+ },
97
+ {
98
+ "error": "no internal trace for tactic 'classical let e := congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\\nhave h : Function.Surjective e.toLinearMap := e.surjective\\nrefine (cancel_right h).1 ?_\\next f q m\\ndsimp [e, lTensorHomEquivHomLTensor]\\nsimp only [lTensorHomEquivHomLTensor, dualTensorHomEquiv, compr\u2082_apply, mk_apply, coe_comp,\\n LinearEquiv.coe_toLinearMap, Function.comp_apply, map_tmul, LinearEquiv.coe_coe,\\n dualTensorHomEquivOfBasis_apply, LinearEquiv.trans_apply, congr_tmul, LinearEquiv.refl_apply,\\n dualTensorHomEquivOfBasis_symm_cancel_left, leftComm_tmul, dualTensorHom_apply, tmul_smul]'",
99
+ "full_name": "lTensorHomEquivHomLTensor_toLinearMap",
100
+ "id": "lTensorHomEquivHomLTensor_toLinearMap",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 256,
106
+ 95
107
+ ],
108
+ "file_path": "Mathlib/LinearAlgebra/Contraction.lean",
109
+ "full_name": "lTensorHomEquivHomLTensor_toLinearMap",
110
+ "start": [
111
+ 245,
112
+ 1
113
+ ],
114
+ "traced_tactics": [
115
+ {
116
+ "annotated_tactic": [
117
+ "\"classical -- Porting note: missing decidable for choosing basis\\n let e := <a>congr<\\/a> (<a>LinearEquiv.refl<\\/a> R P) (<a>dualTensorHomEquiv<\\/a> R M Q)\\n have h : <a>Function.Surjective<\\/a> e.toLinearMap := e.surjective\\n refine (<a>cancel_right<\\/a> h).1 ?_\\n ext f q m\\n dsimp [e, <a>lTensorHomEquivHomLTensor<\\/a>]\\n simp only [<a>lTensorHomEquivHomLTensor<\\/a>, <a>dualTensorHomEquiv<\\/a>, <a>compr\\u2082_apply<\\/a>, <a>mk_apply<\\/a>, <a>coe_comp<\\/a>,\\n <a>LinearEquiv.coe_toLinearMap<\\/a>, <a>Function.comp_apply<\\/a>, <a>map_tmul<\\/a>, <a>LinearEquiv.coe_coe<\\/a>,\\n <a>dualTensorHomEquivOfBasis_apply<\\/a>, <a>LinearEquiv.trans_apply<\\/a>, <a>congr_tmul<\\/a>, <a>LinearEquiv.refl_apply<\\/a>,\\n <a>dualTensorHomEquivOfBasis_symm_cancel_left<\\/a>, <a>leftComm_tmul<\\/a>, <a>dualTensorHom_apply<\\/a>, <a>tmul_smul<\\/a>]\"",
118
+ "[{\"full_name\":\"TensorProduct.congr\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[1025,5],\"def_end_pos\":[1025,10]},{\"full_name\":\"LinearEquiv.refl\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[269,5],\"def_end_pos\":[269,9]},{\"full_name\":\"dualTensorHomEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[204,19],\"def_end_pos\":[204,37]},{\"full_name\":\"Function.Surjective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[133,5],\"def_end_pos\":[133,15]},{\"full_name\":\"LinearMap.cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[588,9],\"def_end_pos\":[588,21]},{\"full_name\":\"lTensorHomEquivHomLTensor\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[229,19],\"def_end_pos\":[229,44]},{\"full_name\":\"lTensorHomEquivHomLTensor\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[229,19],\"def_end_pos\":[229,44]},{\"full_name\":\"dualTensorHomEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[204,19],\"def_end_pos\":[204,37]},{\"full_name\":\"LinearMap.compr\\u2082_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/BilinearMap.lean\",\"def_pos\":[390,9],\"def_end_pos\":[390,21]},{\"full_name\":\"TensorProduct.mk_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[451,9],\"def_end_pos\":[451,17]},{\"full_name\":\"LinearMap.coe_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[557,9],\"def_end_pos\":[557,17]},{\"full_name\":\"LinearEquiv.coe_toLinearMap\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,24]},{\"full_name\":\"Function.comp_apply\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[35,17],\"def_end_pos\":[35,36]},{\"full_name\":\"TensorProduct.map_tmul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[808,9],\"def_end_pos\":[808,17]},{\"full_name\":\"LinearEquiv.coe_coe\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[222,9],\"def_end_pos\":[222,16]},{\"full_name\":\"dualTensorHomEquivOfBasis_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[172,9],\"def_end_pos\":[172,40]},{\"full_name\":\"LinearEquiv.trans_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[366,9],\"def_end_pos\":[366,20]},{\"full_name\":\"TensorProduct.congr_tmul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[1032,9],\"def_end_pos\":[1032,19]},{\"full_name\":\"LinearEquiv.refl_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[276,9],\"def_end_pos\":[276,19]},{\"full_name\":\"dualTensorHomEquivOfBasis_symm_cancel_left\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,51]},{\"full_name\":\"TensorProduct.leftComm_tmul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[1080,9],\"def_end_pos\":[1080,22]},{\"full_name\":\"dualTensorHom_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,28]},{\"full_name\":\"TensorProduct.tmul_smul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[373,9],\"def_end_pos\":[373,18]}]"
119
+ ],
120
+ "state_after": "no goals",
121
+ "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor R M P Q",
122
+ "tactic": "classical let e := congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nhave h : Function.Surjective e.toLinearMap := e.surjective\nrefine (cancel_right h).1 ?_\next f q m\ndsimp [e, lTensorHomEquivHomLTensor]\nsimp only [lTensorHomEquivHomLTensor, dualTensorHomEquiv, compr\u2082_apply, mk_apply, coe_comp,\n LinearEquiv.coe_toLinearMap, Function.comp_apply, map_tmul, LinearEquiv.coe_coe,\n dualTensorHomEquivOfBasis_apply, LinearEquiv.trans_apply, congr_tmul, LinearEquiv.refl_apply,\n dualTensorHomEquivOfBasis_symm_cancel_left, leftComm_tmul, dualTensorHom_apply, tmul_smul]"
123
+ },
124
+ {
125
+ "annotated_tactic": [
126
+ "\"let e := <a>congr<\\/a> (<a>LinearEquiv.refl<\\/a> R P) (<a>dualTensorHomEquiv<\\/a> R M Q)\"",
127
+ "[{\"full_name\":\"TensorProduct.congr\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[1025,5],\"def_end_pos\":[1025,10]},{\"full_name\":\"LinearEquiv.refl\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[269,5],\"def_end_pos\":[269,9]},{\"full_name\":\"dualTensorHomEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[204,19],\"def_end_pos\":[204,37]}]"
128
+ ],
129
+ "state_after": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor R M P Q",
130
+ "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor R M P Q",
131
+ "tactic": "let e := congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)"
132
+ },
133
+ {
134
+ "annotated_tactic": [
135
+ "\"have h : <a>Function.Surjective<\\/a> e.toLinearMap := e.surjective\"",
136
+ "[{\"full_name\":\"Function.Surjective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[133,5],\"def_end_pos\":[133,15]}]"
137
+ ],
138
+ "state_after": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor R M P Q",
139
+ "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor R M P Q",
140
+ "tactic": "have h : Function.Surjective e.toLinearMap := e.surjective"
141
+ },
142
+ {
143
+ "annotated_tactic": [
144
+ "\"refine (<a>cancel_right<\\/a> h).1 ?_\"",
145
+ "[{\"full_name\":\"LinearMap.cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[588,9],\"def_end_pos\":[588,21]}]"
146
+ ],
147
+ "state_after": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) \u2218\u2097 \u2191e = lTensorHomToHomLTensor R M P Q \u2218\u2097 \u2191e",
148
+ "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor R M P Q",
149
+ "tactic": "refine (cancel_right h).1 ?_"
150
+ },
151
+ {
152
+ "annotated_tactic": [
153
+ "\"ext f q m\"",
154
+ "[]"
155
+ ],
156
+ "state_after": "case H.h.H.h.h.h\n\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\nf : P\nq : Dual R M\nm : Q\nx\u271d : M\n\u22a2 ((((TensorProduct.mk R (Dual R M) Q).compr\u2082\n (((TensorProduct.mk R P (Dual R M \u2297[R] Q)).compr\u2082 (\u2191(lTensorHomEquivHomLTensor R M P Q) \u2218\u2097 \u2191e)) f))\n q)\n m)\n x\u271d =\n ((((TensorProduct.mk R (Dual R M) Q).compr\u2082\n (((TensorProduct.mk R P (Dual R M \u2297[R] Q)).compr\u2082 (lTensorHomToHomLTensor R M P Q \u2218\u2097 \u2191e)) f))\n q)\n m)\n x\u271d",
157
+ "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) \u2218\u2097 \u2191e = lTensorHomToHomLTensor R M P Q \u2218\u2097 \u2191e",
158
+ "tactic": "ext f q m"
159
+ },
160
+ {
161
+ "annotated_tactic": [
162
+ "\"dsimp [e, <a>lTensorHomEquivHomLTensor<\\/a>]\"",
163
+ "[{\"full_name\":\"lTensorHomEquivHomLTensor\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[229,19],\"def_end_pos\":[229,44]}]"
164
+ ],
165
+ "state_after": "case H.h.H.h.h.h\n\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\nf : P\nq : Dual R M\nm : Q\nx\u271d : M\n\u22a2 ((dualTensorHomEquivOfBasis (Free.chooseBasis R M))\n ((leftComm R P (Dual R M) Q)\n (f \u2297\u209c[R]\n (dualTensorHomEquivOfBasis (Free.chooseBasis R M)).symm\n ((dualTensorHomEquivOfBasis (Free.chooseBasis R M)) (q \u2297\u209c[R] m)))))\n x\u271d =\n f \u2297\u209c[R] ((dualTensorHomEquivOfBasis (Free.chooseBasis R M)) (q \u2297\u209c[R] m)) x\u271d",
166
+ "state_before": "case H.h.H.h.h.h\n\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\nf : P\nq : Dual R M\nm : Q\nx\u271d : M\n\u22a2 ((((TensorProduct.mk R (Dual R M) Q).compr\u2082\n (((TensorProduct.mk R P (Dual R M \u2297[R] Q)).compr\u2082 (\u2191(lTensorHomEquivHomLTensor R M P Q) \u2218\u2097 \u2191e)) f))\n q)\n m)\n x\u271d =\n ((((TensorProduct.mk R (Dual R M) Q).compr\u2082\n (((TensorProduct.mk R P (Dual R M \u2297[R] Q)).compr\u2082 (lTensorHomToHomLTensor R M P Q \u2218\u2097 \u2191e)) f))\n q)\n m)\n x\u271d",
167
+ "tactic": "dsimp [e, lTensorHomEquivHomLTensor]"
168
+ },
169
+ {
170
+ "annotated_tactic": [
171
+ "\"simp only [<a>lTensorHomEquivHomLTensor<\\/a>, <a>dualTensorHomEquiv<\\/a>, <a>compr\\u2082_apply<\\/a>, <a>mk_apply<\\/a>, <a>coe_comp<\\/a>,\\n <a>LinearEquiv.coe_toLinearMap<\\/a>, <a>Function.comp_apply<\\/a>, <a>map_tmul<\\/a>, <a>LinearEquiv.coe_coe<\\/a>,\\n <a>dualTensorHomEquivOfBasis_apply<\\/a>, <a>LinearEquiv.trans_apply<\\/a>, <a>congr_tmul<\\/a>, <a>LinearEquiv.refl_apply<\\/a>,\\n <a>dualTensorHomEquivOfBasis_symm_cancel_left<\\/a>, <a>leftComm_tmul<\\/a>, <a>dualTensorHom_apply<\\/a>, <a>tmul_smul<\\/a>]\"",
172
+ "[{\"full_name\":\"lTensorHomEquivHomLTensor\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[229,19],\"def_end_pos\":[229,44]},{\"full_name\":\"dualTensorHomEquiv\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[204,19],\"def_end_pos\":[204,37]},{\"full_name\":\"LinearMap.compr\\u2082_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/BilinearMap.lean\",\"def_pos\":[390,9],\"def_end_pos\":[390,21]},{\"full_name\":\"TensorProduct.mk_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[451,9],\"def_end_pos\":[451,17]},{\"full_name\":\"LinearMap.coe_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[557,9],\"def_end_pos\":[557,17]},{\"full_name\":\"LinearEquiv.coe_toLinearMap\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,24]},{\"full_name\":\"Function.comp_apply\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[35,17],\"def_end_pos\":[35,36]},{\"full_name\":\"TensorProduct.map_tmul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[808,9],\"def_end_pos\":[808,17]},{\"full_name\":\"LinearEquiv.coe_coe\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[222,9],\"def_end_pos\":[222,16]},{\"full_name\":\"dualTensorHomEquivOfBasis_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[172,9],\"def_end_pos\":[172,40]},{\"full_name\":\"LinearEquiv.trans_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[366,9],\"def_end_pos\":[366,20]},{\"full_name\":\"TensorProduct.congr_tmul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[1032,9],\"def_end_pos\":[1032,19]},{\"full_name\":\"LinearEquiv.refl_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[276,9],\"def_end_pos\":[276,19]},{\"full_name\":\"dualTensorHomEquivOfBasis_symm_cancel_left\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,51]},{\"full_name\":\"TensorProduct.leftComm_tmul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[1080,9],\"def_end_pos\":[1080,22]},{\"full_name\":\"dualTensorHom_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Contraction.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,28]},{\"full_name\":\"TensorProduct.tmul_smul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/TensorProduct\\/Basic.lean\",\"def_pos\":[373,9],\"def_end_pos\":[373,18]}]"
173
+ ],
174
+ "state_after": "no goals",
175
+ "state_before": "case H.h.H.h.h.h\n\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : P \u2297[R] Dual R M \u2297[R] Q \u2243\u2097[R] P \u2297[R] (M \u2192\u2097[R] Q) :=\n TensorProduct.congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)\nh : Function.Surjective \u21d1\u2191e\nf : P\nq : Dual R M\nm : Q\nx\u271d : M\n\u22a2 ((dualTensorHomEquivOfBasis (Free.chooseBasis R M))\n ((leftComm R P (Dual R M) Q)\n (f \u2297\u209c[R]\n (dualTensorHomEquivOfBasis (Free.chooseBasis R M)).symm\n ((dualTensorHomEquivOfBasis (Free.chooseBasis R M)) (q \u2297\u209c[R] m)))))\n x\u271d =\n f \u2297\u209c[R] ((dualTensorHomEquivOfBasis (Free.chooseBasis R M)) (q \u2297\u209c[R] m)) x\u271d",
176
+ "tactic": "simp only [lTensorHomEquivHomLTensor, dualTensorHomEquiv, compr\u2082_apply, mk_apply, coe_comp,\n LinearEquiv.coe_toLinearMap, Function.comp_apply, map_tmul, LinearEquiv.coe_coe,\n dualTensorHomEquivOfBasis_apply, LinearEquiv.trans_apply, congr_tmul, LinearEquiv.refl_apply,\n dualTensorHomEquivOfBasis_symm_cancel_left, leftComm_tmul, dualTensorHom_apply, tmul_smul]"
177
+ }
178
+ ],
179
+ "url": "https://github.com/leanprover-community/mathlib4"
180
+ },
181
+ "source_file": "Mathlib/LinearAlgebra/Contraction.lean"
182
+ },
183
+ {
184
+ "error": "theorem has no selected tactic transitions",
185
+ "full_name": "cross_apply",
186
+ "id": "cross_apply",
187
+ "reason": "import_failure",
188
+ "selection": {
189
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
190
+ "end": [
191
+ 66,
192
+ 91
193
+ ],
194
+ "file_path": "Mathlib/LinearAlgebra/CrossProduct.lean",
195
+ "full_name": "cross_apply",
196
+ "start": [
197
+ 65,
198
+ 1
199
+ ],
200
+ "traced_tactics": [],
201
+ "url": "https://github.com/leanprover-community/mathlib4"
202
+ },
203
+ "source_file": "Mathlib/LinearAlgebra/CrossProduct.lean"
204
+ },
205
+ {
206
+ "error": "theorem has no selected tactic transitions",
207
+ "full_name": "cross_cross",
208
+ "id": "cross_cross",
209
+ "reason": "import_failure",
210
+ "selection": {
211
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
212
+ "end": [
213
+ 166,
214
+ 16
215
+ ],
216
+ "file_path": "Mathlib/LinearAlgebra/CrossProduct.lean",
217
+ "full_name": "cross_cross",
218
+ "start": [
219
+ 165,
220
+ 1
221
+ ],
222
+ "traced_tactics": [],
223
+ "url": "https://github.com/leanprover-community/mathlib4"
224
+ },
225
+ "source_file": "Mathlib/LinearAlgebra/CrossProduct.lean"
226
+ },
227
+ {
228
+ "error": "theorem has no selected tactic transitions",
229
+ "full_name": "jacobi_cross",
230
+ "id": "jacobi_cross",
231
+ "reason": "import_failure",
232
+ "selection": {
233
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
234
+ "end": [
235
+ 172,
236
+ 19
237
+ ],
238
+ "file_path": "Mathlib/LinearAlgebra/CrossProduct.lean",
239
+ "full_name": "jacobi_cross",
240
+ "start": [
241
+ 171,
242
+ 1
243
+ ],
244
+ "traced_tactics": [],
245
+ "url": "https://github.com/leanprover-community/mathlib4"
246
+ },
247
+ "source_file": "Mathlib/LinearAlgebra/CrossProduct.lean"
248
+ }
249
+ ],
250
+ "input_proofs": 29,
251
+ "input_units": 2,
252
+ "proofs": 21,
253
+ "published_proofs": 21,
254
+ "published_transitions": 39,
255
+ "replay_failures": [],
256
+ "transitions": 39,
257
+ "validation_failures": [],
258
+ "validation_rejected_proofs": 0,
259
+ "zero_transition_proofs": 0
260
+ },
261
+ "shards": [
262
+ {
263
+ "path": "data/train/mathlib-001259-0000-158b2caf6dfad65f.parquet",
264
+ "sha256": "158b2caf6dfad65f340391b854a388e08483467bc40f473e6ae71b655bd02381",
265
+ "bytes": 333333,
266
+ "rows": 26,
267
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
268
+ },
269
+ {
270
+ "path": "data/dev/mathlib-001259-0000-790e71275301381d.parquet",
271
+ "sha256": "790e71275301381d5696df9c1fae571ee74af2539788ceee2174b7755b104421",
272
+ "bytes": 41187,
273
+ "rows": 1,
274
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
275
+ },
276
+ {
277
+ "path": "data/internal_test/mathlib-001259-0000-a31c26af1c7d679a.parquet",
278
+ "sha256": "a31c26af1c7d679a199cc590e9ced5d1d2b48f19b71ef55da3930ce8b7435fbd",
279
+ "bytes": 131176,
280
+ "rows": 12,
281
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
282
+ }
283
+ ],
284
+ "proof_ids": [
285
+ "mathlib:comp_dualTensorHom",
286
+ "mathlib:cross_anticomm",
287
+ "mathlib:cross_anticomm'",
288
+ "mathlib:cross_dot_cross",
289
+ "mathlib:cross_self",
290
+ "mathlib:dot_cross_self",
291
+ "mathlib:dot_self_cross",
292
+ "mathlib:dualTensorHomEquivOfBasis_symm_cancel_left",
293
+ "mathlib:dualTensorHomEquivOfBasis_symm_cancel_right",
294
+ "mathlib:dualTensorHom_prodMap_zero",
295
+ "mathlib:homTensorHomEquiv_apply",
296
+ "mathlib:homTensorHomEquiv_toLinearMap",
297
+ "mathlib:lTensorHomEquivHomLTensor_apply",
298
+ "mathlib:leibniz_cross",
299
+ "mathlib:map_dualTensorHom",
300
+ "mathlib:rTensorHomEquivHomRTensor_apply",
301
+ "mathlib:toMatrix_dualTensorHom",
302
+ "mathlib:transpose_dualTensorHom",
303
+ "mathlib:triple_product_eq_det",
304
+ "mathlib:triple_product_permutation",
305
+ "mathlib:zero_prodMap_dualTensorHom"
306
+ ],
307
+ "proof_keys": [
308
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"comp_dualTensorHom\"]",
309
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"dualTensorHomEquivOfBasis_symm_cancel_left\"]",
310
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"dualTensorHomEquivOfBasis_symm_cancel_right\"]",
311
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"dualTensorHom_prodMap_zero\"]",
312
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"homTensorHomEquiv_apply\"]",
313
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"homTensorHomEquiv_toLinearMap\"]",
314
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"lTensorHomEquivHomLTensor_apply\"]",
315
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"map_dualTensorHom\"]",
316
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"rTensorHomEquivHomRTensor_apply\"]",
317
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"toMatrix_dualTensorHom\"]",
318
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"transpose_dualTensorHom\"]",
319
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Contraction.lean\",null,null,\"zero_prodMap_dualTensorHom\"]",
320
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"cross_anticomm\"]",
321
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"cross_anticomm'\"]",
322
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"cross_dot_cross\"]",
323
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"cross_self\"]",
324
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"dot_cross_self\"]",
325
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"dot_self_cross\"]",
326
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"leibniz_cross\"]",
327
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"triple_product_eq_det\"]",
328
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/CrossProduct.lean\",null,null,\"triple_product_permutation\"]"
329
+ ]
330
+ }
manifests/batches/mathlib/001260.json ADDED
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manifests/batches/mathlib/001261.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001262.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001263.json ADDED
@@ -0,0 +1,599 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001263",
4
+ "batch_fingerprint": "886c2d4a38f35813f615bcd430a9c153d85f3ec8654668a33a8fb2cb1411fcaf",
5
+ "evidence": {
6
+ "batch_fingerprint": "886c2d4a38f35813f615bcd430a9c153d85f3ec8654668a33a8fb2cb1411fcaf",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "FiniteDimensional.rank_eq_ofNat_iff_finrank_eq_ofNat",
12
+ "id": "FiniteDimensional.rank_eq_ofNat_iff_finrank_eq_ofNat",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 70,
18
+ 31
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
21
+ "full_name": "FiniteDimensional.rank_eq_ofNat_iff_finrank_eq_ofNat",
22
+ "start": [
23
+ 67,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Finrank.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "FiniteDimensional.rank_eq_one_iff_finrank_eq_one",
34
+ "id": "FiniteDimensional.rank_eq_one_iff_finrank_eq_one",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 65,
40
+ 29
41
+ ],
42
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
43
+ "full_name": "FiniteDimensional.rank_eq_one_iff_finrank_eq_one",
44
+ "start": [
45
+ 64,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Finrank.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "LinearEquiv.finrank_map_eq",
56
+ "id": "LinearEquiv.finrank_map_eq",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
60
+ "end": [
61
+ 121,
62
+ 37
63
+ ],
64
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
65
+ "full_name": "LinearEquiv.finrank_map_eq",
66
+ "start": [
67
+ 119,
68
+ 1
69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Finrank.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "Submodule.finrank_map_subtype_eq",
78
+ "id": "Submodule.finrank_map_subtype_eq",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
82
+ "end": [
83
+ 134,
84
+ 50
85
+ ],
86
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
87
+ "full_name": "Submodule.finrank_map_subtype_eq",
88
+ "start": [
89
+ 132,
90
+ 1
91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Finrank.lean"
96
+ },
97
+ {
98
+ "error": "theorem has no selected tactic transitions",
99
+ "full_name": "FiniteDimensional.nonempty_linearEquiv_iff_finrank_eq",
100
+ "id": "FiniteDimensional.nonempty_linearEquiv_iff_finrank_eq",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 167,
106
+ 75
107
+ ],
108
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
109
+ "full_name": "FiniteDimensional.nonempty_linearEquiv_iff_finrank_eq",
110
+ "start": [
111
+ 165,
112
+ 1
113
+ ],
114
+ "traced_tactics": [],
115
+ "url": "https://github.com/leanprover-community/mathlib4"
116
+ },
117
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
118
+ },
119
+ {
120
+ "error": "theorem has no selected tactic transitions",
121
+ "full_name": "LinearEquiv.nonempty_equiv_iff_lift_rank_eq",
122
+ "id": "LinearEquiv.nonempty_equiv_iff_lift_rank_eq",
123
+ "reason": "import_failure",
124
+ "selection": {
125
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
126
+ "end": [
127
+ 148,
128
+ 91
129
+ ],
130
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
131
+ "full_name": "LinearEquiv.nonempty_equiv_iff_lift_rank_eq",
132
+ "start": [
133
+ 146,
134
+ 1
135
+ ],
136
+ "traced_tactics": [],
137
+ "url": "https://github.com/leanprover-community/mathlib4"
138
+ },
139
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
140
+ },
141
+ {
142
+ "error": "theorem has no selected tactic transitions",
143
+ "full_name": "LinearEquiv.nonempty_equiv_iff_rank_eq",
144
+ "id": "LinearEquiv.nonempty_equiv_iff_rank_eq",
145
+ "reason": "import_failure",
146
+ "selection": {
147
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
148
+ "end": [
149
+ 154,
150
+ 81
151
+ ],
152
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
153
+ "full_name": "LinearEquiv.nonempty_equiv_iff_rank_eq",
154
+ "start": [
155
+ 152,
156
+ 1
157
+ ],
158
+ "traced_tactics": [],
159
+ "url": "https://github.com/leanprover-community/mathlib4"
160
+ },
161
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
162
+ },
163
+ {
164
+ "error": "theorem has no selected tactic transitions",
165
+ "full_name": "Module.Free.rank_eq_card_chooseBasisIndex",
166
+ "id": "Module.Free.rank_eq_card_chooseBasisIndex",
167
+ "reason": "import_failure",
168
+ "selection": {
169
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
170
+ "end": [
171
+ 84,
172
+ 38
173
+ ],
174
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
175
+ "full_name": "Module.Free.rank_eq_card_chooseBasisIndex",
176
+ "start": [
177
+ 83,
178
+ 1
179
+ ],
180
+ "traced_tactics": [],
181
+ "url": "https://github.com/leanprover-community/mathlib4"
182
+ },
183
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
184
+ },
185
+ {
186
+ "error": "no internal trace for tactic 'simp'",
187
+ "full_name": "Module.Free.rank_eq_mk_of_infinite_lt",
188
+ "id": "Module.Free.rank_eq_mk_of_infinite_lt",
189
+ "reason": "import_failure",
190
+ "selection": {
191
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
192
+ "end": [
193
+ 102,
194
+ 42
195
+ ],
196
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
197
+ "full_name": "Module.Free.rank_eq_mk_of_infinite_lt",
198
+ "start": [
199
+ 93,
200
+ 1
201
+ ],
202
+ "traced_tactics": [
203
+ {
204
+ "annotated_tactic": [
205
+ "\"have : <a>Infinite<\\/a> M := infinite_iff.mpr <| lift_le.mp <| <a>le_trans<\\/a> (by simp) h_lt.le\"",
206
+ "[{\"full_name\":\"Infinite\",\"def_path\":\"Mathlib\\/Data\\/Finite\\/Defs.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,15]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]}]"
207
+ ],
208
+ "state_after": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\n\u22a2 Module.rank R M = #M",
209
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\n\u22a2 Module.rank R M = #M",
210
+ "tactic": "have : Infinite M := infinite_iff.mpr <| lift_le.mp <| le_trans (by simp) h_lt.le"
211
+ },
212
+ {
213
+ "annotated_tactic": [
214
+ "\"have h : <a>lift<\\/a> #M = <a>lift<\\/a> #(<a>ChooseBasisIndex<\\/a> R M \\u2192\\u2080 R) := lift_mk_eq'.mpr \\u27e8(<a>chooseBasis<\\/a> R M).<a>repr<\\/a>\\u27e9\"",
215
+ "[{\"full_name\":\"Cardinal.lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,9]},{\"full_name\":\"Cardinal.lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,9]},{\"full_name\":\"Module.Free.ChooseBasisIndex\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Basic.lean\",\"def_pos\":[75,5],\"def_end_pos\":[75,21]},{\"full_name\":\"Module.Free.chooseBasis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Basic.lean\",\"def_pos\":[85,19],\"def_end_pos\":[85,30]},{\"full_name\":\"Basis.repr\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[93,5],\"def_end_pos\":[93,9]}]"
216
+ ],
217
+ "state_after": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\nh : lift.{max u v, v} #M = lift.{v, max u v} #(ChooseBasisIndex R M \u2192\u2080 R)\n\u22a2 Module.rank R M = #M",
218
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\n\u22a2 Module.rank R M = #M",
219
+ "tactic": "have h : lift #M = lift #(ChooseBasisIndex R M \u2192\u2080 R) := lift_mk_eq'.mpr \u27e8(chooseBasis R M).repr\u27e9"
220
+ },
221
+ {
222
+ "annotated_tactic": [
223
+ "\"simp only [<a>mk_finsupp_lift_of_infinite'<\\/a>, <a>lift_id'<\\/a>, \\u2190 <a>rank_eq_card_chooseBasisIndex<\\/a>, <a>lift_max<\\/a>,\\n <a>lift_lift<\\/a>] at h\"",
224
+ "[{\"full_name\":\"Cardinal.mk_finsupp_lift_of_infinite'\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Ordinal.lean\",\"def_pos\":[1227,9],\"def_end_pos\":[1227,37]},{\"full_name\":\"Cardinal.lift_id'\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[218,9],\"def_end_pos\":[218,17]},{\"full_name\":\"Module.Free.rank_eq_card_chooseBasisIndex\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Free.lean\",\"def_pos\":[83,9],\"def_end_pos\":[83,38]},{\"full_name\":\"Cardinal.lift_max\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1192,9],\"def_end_pos\":[1192,17]},{\"full_name\":\"Cardinal.lift_lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,18]}]"
225
+ ],
226
+ "state_after": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\nh : lift.{max u v, v} #M = max (lift.{max u v, v} (Module.rank R M)) (lift.{v, u} #R)\n\u22a2 Module.rank R M = #M",
227
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\nh : lift.{max u v, v} #M = lift.{v, max u v} #(ChooseBasisIndex R M \u2192\u2080 R)\n\u22a2 Module.rank R M = #M",
228
+ "tactic": "simp only [mk_finsupp_lift_of_infinite', lift_id', \u2190 rank_eq_card_chooseBasisIndex, lift_max,\n lift_lift] at h"
229
+ },
230
+ {
231
+ "annotated_tactic": [
232
+ "\"refine lift_inj.mp ((max_eq_iff.mp h.symm).<a>resolve_right<\\/a> <| <a>not_and_of_not_left<\\/a> _ ?_).<a>left<\\/a>\"",
233
+ "[{\"full_name\":\"Or.resolve_right\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[556,9],\"def_end_pos\":[556,25]},{\"full_name\":\"not_and_of_not_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[48,9],\"def_end_pos\":[48,28]},{\"full_name\":\"And.left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[521,3],\"def_end_pos\":[521,7]}]"
234
+ ],
235
+ "state_after": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\nh : lift.{max u v, v} #M = max (lift.{max u v, v} (Module.rank R M)) (lift.{v, u} #R)\n\u22a2 \u00aclift.{v, u} #R = lift.{max u v, v} #M",
236
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\nh : lift.{max u v, v} #M = max (lift.{max u v, v} (Module.rank R M)) (lift.{v, u} #R)\n\u22a2 Module.rank R M = #M",
237
+ "tactic": "refine lift_inj.mp ((max_eq_iff.mp h.symm).resolve_right <| not_and_of_not_left _ ?_).left"
238
+ },
239
+ {
240
+ "annotated_tactic": [
241
+ "\"exact (<a>lift_umax<\\/a>.{v, u}.<a>symm<\\/a> \\u25b8 h_lt).<a>ne<\\/a>\"",
242
+ "[{\"full_name\":\"Cardinal.lift_umax\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[202,9],\"def_end_pos\":[202,18]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]}]"
243
+ ],
244
+ "state_after": "no goals",
245
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\nthis : Infinite M\nh : lift.{max u v, v} #M = max (lift.{max u v, v} (Module.rank R M)) (lift.{v, u} #R)\n\u22a2 \u00aclift.{v, u} #R = lift.{max u v, v} #M",
246
+ "tactic": "exact (lift_umax.{v, u}.symm \u25b8 h_lt).ne"
247
+ },
248
+ {
249
+ "annotated_tactic": [
250
+ "\"simp\"",
251
+ "[]"
252
+ ],
253
+ "state_after": "no goals",
254
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : StrongRankCondition R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Free R M\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Free R M'\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Free R M\u2081\ninst\u271d : Infinite R\nh_lt : lift.{v, u} #R < lift.{u, v} #M\n\u22a2 lift.{u, v} \u2135\u2080 \u2264 lift.{v, u} #R",
255
+ "tactic": "simp"
256
+ }
257
+ ],
258
+ "url": "https://github.com/leanprover-community/mathlib4"
259
+ },
260
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
261
+ },
262
+ {
263
+ "error": "no internal trace for tactic \"rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']\"",
264
+ "full_name": "nonempty_linearEquiv_of_lift_rank_eq",
265
+ "id": "nonempty_linearEquiv_of_lift_rank_eq",
266
+ "reason": "import_failure",
267
+ "selection": {
268
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
269
+ "end": [
270
+ 118,
271
+ 65
272
+ ],
273
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
274
+ "full_name": "nonempty_linearEquiv_of_lift_rank_eq",
275
+ "start": [
276
+ 111,
277
+ 1
278
+ ],
279
+ "traced_tactics": [
280
+ {
281
+ "annotated_tactic": [
282
+ "\"obtain \\u27e8\\u27e8\\u03b1, B\\u27e9\\u27e9 := <a>Module.Free.exists_basis<\\/a> (R := R) (M := M)\"",
283
+ "[{\"full_name\":\"Module.Free.exists_basis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Basic.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,15]}]"
284
+ ],
285
+ "state_after": "case intro.mk\nR : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
286
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
287
+ "tactic": "obtain \u27e8\u27e8\u03b1, B\u27e9\u27e9 := Module.Free.exists_basis (R := R) (M := M)"
288
+ },
289
+ {
290
+ "annotated_tactic": [
291
+ "\"obtain \\u27e8\\u27e8\\u03b2, B'\\u27e9\\u27e9 := <a>Module.Free.exists_basis<\\/a> (R := R) (M := M')\"",
292
+ "[{\"full_name\":\"Module.Free.exists_basis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Basic.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,15]}]"
293
+ ],
294
+ "state_after": "case intro.mk.intro.mk\nR : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u03b2 : Type v'\nB' : Basis \u03b2 R M'\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
295
+ "state_before": "case intro.mk\nR : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
296
+ "tactic": "obtain \u27e8\u27e8\u03b2, B'\u27e9\u27e9 := Module.Free.exists_basis (R := R) (M := M')"
297
+ },
298
+ {
299
+ "annotated_tactic": [
300
+ "\"have : <a>Cardinal.lift<\\/a>.{v', v} #\\u03b1 = <a>Cardinal.lift<\\/a>.{v, v'} #\\u03b2 := by\\n rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']\"",
301
+ "[{\"full_name\":\"Cardinal.lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,9]},{\"full_name\":\"Cardinal.lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,9]}]"
302
+ ],
303
+ "state_after": "case intro.mk.intro.mk\nR : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u03b2 : Type v'\nB' : Basis \u03b2 R M'\nthis : lift.{v', v} #\u03b1 = lift.{v, v'} #\u03b2\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
304
+ "state_before": "case intro.mk.intro.mk\nR : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u03b2 : Type v'\nB' : Basis \u03b2 R M'\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
305
+ "tactic": "have : Cardinal.lift.{v', v} #\u03b1 = Cardinal.lift.{v, v'} #\u03b2 := by\n rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']"
306
+ },
307
+ {
308
+ "annotated_tactic": [
309
+ "\"exact (<a>Cardinal.lift_mk_eq<\\/a>.{v, v', 0}.1 this).<a>map<\\/a> (B.equiv B')\"",
310
+ "[{\"full_name\":\"Cardinal.lift_mk_eq\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[324,9],\"def_end_pos\":[324,19]},{\"full_name\":\"Nonempty.map\",\"def_path\":\"Mathlib\\/Logic\\/Nonempty.lean\",\"def_pos\":[105,9],\"def_end_pos\":[105,21]}]"
311
+ ],
312
+ "state_after": "no goals",
313
+ "state_before": "case intro.mk.intro.mk\nR : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u03b2 : Type v'\nB' : Basis \u03b2 R M'\nthis : lift.{v', v} #\u03b1 = lift.{v, v'} #\u03b2\n\u22a2 Nonempty (M \u2243\u2097[R] M')",
314
+ "tactic": "exact (Cardinal.lift_mk_eq.{v, v', 0}.1 this).map (B.equiv B')"
315
+ },
316
+ {
317
+ "annotated_tactic": [
318
+ "\"rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']\"",
319
+ "[]"
320
+ ],
321
+ "state_after": "no goals",
322
+ "state_before": "R : Type u\nM M\u2081 : Type v\nM' : Type v'\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : StrongRankCondition R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module.Free R M'\ninst\u271d\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module.Free R M\u2081\ncnd : lift.{v', v} (Module.rank R M) = lift.{v, v'} (Module.rank R M')\n\u03b1 : Type v\nB : Basis \u03b1 R M\n\u03b2 : Type v'\nB' : Basis \u03b2 R M'\n\u22a2 lift.{v', v} #\u03b1 = lift.{v, v'} #\u03b2",
323
+ "tactic": "rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']"
324
+ }
325
+ ],
326
+ "url": "https://github.com/leanprover-community/mathlib4"
327
+ },
328
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
329
+ },
330
+ {
331
+ "error": "theorem has no selected tactic transitions",
332
+ "full_name": "nonempty_linearEquiv_of_rank_eq",
333
+ "id": "nonempty_linearEquiv_of_rank_eq",
334
+ "reason": "import_failure",
335
+ "selection": {
336
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
337
+ "end": [
338
+ 124,
339
+ 59
340
+ ],
341
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Free.lean",
342
+ "full_name": "nonempty_linearEquiv_of_rank_eq",
343
+ "start": [
344
+ 122,
345
+ 1
346
+ ],
347
+ "traced_tactics": [],
348
+ "url": "https://github.com/leanprover-community/mathlib4"
349
+ },
350
+ "source_file": "Mathlib/LinearAlgebra/Dimension/Free.lean"
351
+ }
352
+ ],
353
+ "input_proofs": 31,
354
+ "input_units": 2,
355
+ "proofs": 20,
356
+ "published_proofs": 17,
357
+ "published_transitions": 28,
358
+ "replay_failures": [],
359
+ "transitions": 39,
360
+ "validation_failures": [
361
+ {
362
+ "errors": [
363
+ "internal transition boundary mismatch",
364
+ "internal transition boundary mismatch"
365
+ ],
366
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
367
+ "proof_id": "mathlib:FiniteDimensional.finrank_le_of_rank_le",
368
+ "provenance": {
369
+ "selection": {
370
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
371
+ "end": [
372
+ 75,
373
+ 26
374
+ ],
375
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
376
+ "full_name": "FiniteDimensional.finrank_le_of_rank_le",
377
+ "start": [
378
+ 72,
379
+ 1
380
+ ],
381
+ "traced_tactics": [
382
+ {
383
+ "annotated_tactic": [
384
+ "\"rwa [\\u2190 <a>Cardinal.toNat_le_iff_le_of_lt_aleph0<\\/a>, <a>toNat_natCast<\\/a>] at h\"",
385
+ "[{\"full_name\":\"Cardinal.toNat_le_iff_le_of_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,37]},{\"full_name\":\"Cardinal.toNat_natCast\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[38,28],\"def_end_pos\":[38,41]}]"
386
+ ],
387
+ "state_after": "case hc\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M \u2264 \u2191n\n\u22a2 Module.rank R M < \u2135\u2080\n\ncase hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M \u2264 \u2191n\n\u22a2 \u2191n < \u2135\u2080",
388
+ "state_before": "R : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M \u2264 \u2191n\n\u22a2 finrank R M \u2264 n",
389
+ "tactic": "rwa [\u2190 Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h"
390
+ },
391
+ {
392
+ "annotated_tactic": [
393
+ "\"exact h.trans_lt (<a>nat_lt_aleph0<\\/a> n)\"",
394
+ "[{\"full_name\":\"Cardinal.nat_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1526,9],\"def_end_pos\":[1526,22]}]"
395
+ ],
396
+ "state_after": "no goals",
397
+ "state_before": "case hc\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M \u2264 \u2191n\n\u22a2 Module.rank R M < \u2135\u2080",
398
+ "tactic": "exact h.trans_lt (nat_lt_aleph0 n)"
399
+ },
400
+ {
401
+ "annotated_tactic": [
402
+ "\"exact <a>nat_lt_aleph0<\\/a> n\"",
403
+ "[{\"full_name\":\"Cardinal.nat_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1526,9],\"def_end_pos\":[1526,22]}]"
404
+ ],
405
+ "state_after": "no goals",
406
+ "state_before": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M \u2264 \u2191n\n\u22a2 \u2191n < \u2135\u2080",
407
+ "tactic": "exact nat_lt_aleph0 n"
408
+ }
409
+ ],
410
+ "url": "https://github.com/leanprover-community/mathlib4"
411
+ }
412
+ },
413
+ "theorem_name": "FiniteDimensional.finrank_le_of_rank_le"
414
+ },
415
+ {
416
+ "errors": [
417
+ "internal transition boundary mismatch",
418
+ "internal transition boundary mismatch"
419
+ ],
420
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
421
+ "proof_id": "mathlib:FiniteDimensional.finrank_lt_of_rank_lt",
422
+ "provenance": {
423
+ "selection": {
424
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
425
+ "end": [
426
+ 81,
427
+ 26
428
+ ],
429
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
430
+ "full_name": "FiniteDimensional.finrank_lt_of_rank_lt",
431
+ "start": [
432
+ 78,
433
+ 1
434
+ ],
435
+ "traced_tactics": [
436
+ {
437
+ "annotated_tactic": [
438
+ "\"rwa [\\u2190 <a>Cardinal.toNat_lt_iff_lt_of_lt_aleph0<\\/a>, <a>toNat_natCast<\\/a>] at h\"",
439
+ "[{\"full_name\":\"Cardinal.toNat_lt_iff_lt_of_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,37]},{\"full_name\":\"Cardinal.toNat_natCast\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[38,28],\"def_end_pos\":[38,41]}]"
440
+ ],
441
+ "state_after": "case hc\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M < \u2191n\n\u22a2 Module.rank R M < \u2135\u2080\n\ncase hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M < \u2191n\n\u22a2 \u2191n < \u2135\u2080",
442
+ "state_before": "R : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M < \u2191n\n\u22a2 finrank R M < n",
443
+ "tactic": "rwa [\u2190 Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_natCast] at h"
444
+ },
445
+ {
446
+ "annotated_tactic": [
447
+ "\"exact h.trans (<a>nat_lt_aleph0<\\/a> n)\"",
448
+ "[{\"full_name\":\"Cardinal.nat_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1526,9],\"def_end_pos\":[1526,22]}]"
449
+ ],
450
+ "state_after": "no goals",
451
+ "state_before": "case hc\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M < \u2191n\n\u22a2 Module.rank R M < \u2135\u2080",
452
+ "tactic": "exact h.trans (nat_lt_aleph0 n)"
453
+ },
454
+ {
455
+ "annotated_tactic": [
456
+ "\"exact <a>nat_lt_aleph0<\\/a> n\"",
457
+ "[{\"full_name\":\"Cardinal.nat_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1526,9],\"def_end_pos\":[1526,22]}]"
458
+ ],
459
+ "state_after": "no goals",
460
+ "state_before": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : Module.rank R M < \u2191n\n\u22a2 \u2191n < \u2135\u2080",
461
+ "tactic": "exact nat_lt_aleph0 n"
462
+ }
463
+ ],
464
+ "url": "https://github.com/leanprover-community/mathlib4"
465
+ }
466
+ },
467
+ "theorem_name": "FiniteDimensional.finrank_lt_of_rank_lt"
468
+ },
469
+ {
470
+ "errors": [
471
+ "internal transition boundary mismatch",
472
+ "internal transition boundary mismatch"
473
+ ],
474
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
475
+ "proof_id": "mathlib:FiniteDimensional.lt_rank_of_lt_finrank",
476
+ "provenance": {
477
+ "selection": {
478
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
479
+ "end": [
480
+ 89,
481
+ 20
482
+ ],
483
+ "file_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean",
484
+ "full_name": "FiniteDimensional.lt_rank_of_lt_finrank",
485
+ "start": [
486
+ 84,
487
+ 1
488
+ ],
489
+ "traced_tactics": [
490
+ {
491
+ "annotated_tactic": [
492
+ "\"rwa [\\u2190 <a>Cardinal.toNat_lt_iff_lt_of_lt_aleph0<\\/a>, <a>toNat_natCast<\\/a>]\"",
493
+ "[{\"full_name\":\"Cardinal.toNat_lt_iff_lt_of_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,37]},{\"full_name\":\"Cardinal.toNat_natCast\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[38,28],\"def_end_pos\":[38,41]}]"
494
+ ],
495
+ "state_after": "case hc\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : n < finrank R M\n\u22a2 \u2191n < \u2135\u2080\n\ncase hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : n < finrank R M\n\u22a2 Module.rank R M < \u2135\u2080",
496
+ "state_before": "R : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : n < finrank R M\n\u22a2 \u2191n < Module.rank R M",
497
+ "tactic": "rwa [\u2190 Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_natCast]"
498
+ },
499
+ {
500
+ "annotated_tactic": [
501
+ "\"exact <a>nat_lt_aleph0<\\/a> n\"",
502
+ "[{\"full_name\":\"Cardinal.nat_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1526,9],\"def_end_pos\":[1526,22]}]"
503
+ ],
504
+ "state_after": "no goals",
505
+ "state_before": "case hc\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : n < finrank R M\n\u22a2 \u2191n < \u2135\u2080",
506
+ "tactic": "exact nat_lt_aleph0 n"
507
+ },
508
+ {
509
+ "annotated_tactic": [
510
+ "\"contrapose! h\"",
511
+ "[]"
512
+ ],
513
+ "state_after": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank R M\n\u22a2 finrank R M \u2264 n",
514
+ "state_before": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : n < finrank R M\n\u22a2 Module.rank R M < \u2135\u2080",
515
+ "tactic": "contrapose! h"
516
+ },
517
+ {
518
+ "annotated_tactic": [
519
+ "\"rw [<a>finrank<\\/a>, <a>Cardinal.toNat_apply_of_aleph0_le<\\/a> h]\"",
520
+ "[{\"full_name\":\"FiniteDimensional.finrank\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Finrank.lean\",\"def_pos\":[54,19],\"def_end_pos\":[54,26]},{\"full_name\":\"Cardinal.toNat_apply_of_aleph0_le\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[57,9],\"def_end_pos\":[57,33]}]"
521
+ ],
522
+ "state_after": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank R M\n\u22a2 0 \u2264 n",
523
+ "state_before": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank R M\n\u22a2 finrank R M \u2264 n",
524
+ "tactic": "rw [finrank, Cardinal.toNat_apply_of_aleph0_le h]"
525
+ },
526
+ {
527
+ "annotated_tactic": [
528
+ "\"exact n.zero_le\"",
529
+ "[]"
530
+ ],
531
+ "state_after": "no goals",
532
+ "state_before": "case hd\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank R M\n\u22a2 0 \u2264 n",
533
+ "tactic": "exact n.zero_le"
534
+ }
535
+ ],
536
+ "url": "https://github.com/leanprover-community/mathlib4"
537
+ }
538
+ },
539
+ "theorem_name": "FiniteDimensional.lt_rank_of_lt_finrank"
540
+ }
541
+ ],
542
+ "validation_rejected_proofs": 3,
543
+ "zero_transition_proofs": 0
544
+ },
545
+ "shards": [
546
+ {
547
+ "path": "data/train/mathlib-001263-0000-7cd2925c9ae14269.parquet",
548
+ "sha256": "7cd2925c9ae14269e568a41a859f230b2eec3aa28b173e92c0575e53fab8c4d0",
549
+ "bytes": 91625,
550
+ "rows": 22,
551
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
552
+ },
553
+ {
554
+ "path": "data/internal_test/mathlib-001263-0000-0229231aefdeec7d.parquet",
555
+ "sha256": "0229231aefdeec7d649970493e0ab18d57970e1927b66ab07ab85ddcaedb1781",
556
+ "bytes": 67286,
557
+ "rows": 6,
558
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
559
+ }
560
+ ],
561
+ "proof_ids": [
562
+ "mathlib:FiniteDimensional.basisUnique_repr_eq_zero_iff",
563
+ "mathlib:FiniteDimensional.finrank_eq_card_chooseBasisIndex",
564
+ "mathlib:FiniteDimensional.finrank_eq_of_rank_eq",
565
+ "mathlib:FiniteDimensional.finrank_le_finrank_of_rank_le_rank",
566
+ "mathlib:FiniteDimensional.finrank_mul_finrank",
567
+ "mathlib:FiniteDimensional.finrank_of_not_finite",
568
+ "mathlib:FiniteDimensional.nonempty_linearEquiv_of_finrank_eq",
569
+ "mathlib:FiniteDimensional.one_lt_rank_of_one_lt_finrank",
570
+ "mathlib:LinearEquiv.finrank_eq",
571
+ "mathlib:LinearMap.finrank_range_of_inj",
572
+ "mathlib:Module.finite_iff_of_rank_eq_nsmul",
573
+ "mathlib:Module.finite_of_finrank_eq_succ",
574
+ "mathlib:Module.finite_of_finrank_pos",
575
+ "mathlib:Module.rank_lt_alpeh0_iff",
576
+ "mathlib:finrank_top",
577
+ "mathlib:lift_rank_mul_lift_rank",
578
+ "mathlib:rank_mul_rank"
579
+ ],
580
+ "proof_keys": [
581
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Finrank.lean\",null,null,\"FiniteDimensional.finrank_eq_of_rank_eq\"]",
582
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Finrank.lean\",null,null,\"FiniteDimensional.finrank_le_finrank_of_rank_le_rank\"]",
583
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Finrank.lean\",null,null,\"FiniteDimensional.one_lt_rank_of_one_lt_finrank\"]",
584
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Finrank.lean\",null,null,\"LinearEquiv.finrank_eq\"]",
585
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Finrank.lean\",null,null,\"LinearMap.finrank_range_of_inj\"]",
586
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Finrank.lean\",null,null,\"finrank_top\"]",
587
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"FiniteDimensional.basisUnique_repr_eq_zero_iff\"]",
588
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"FiniteDimensional.finrank_eq_card_chooseBasisIndex\"]",
589
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"FiniteDimensional.finrank_mul_finrank\"]",
590
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"FiniteDimensional.finrank_of_not_finite\"]",
591
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"FiniteDimensional.nonempty_linearEquiv_of_finrank_eq\"]",
592
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"Module.finite_iff_of_rank_eq_nsmul\"]",
593
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"Module.finite_of_finrank_eq_succ\"]",
594
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"Module.finite_of_finrank_pos\"]",
595
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"Module.rank_lt_alpeh0_iff\"]",
596
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"lift_rank_mul_lift_rank\"]",
597
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Dimension/Free.lean\",null,null,\"rank_mul_rank\"]"
598
+ ]
599
+ }
manifests/batches/mathlib/001264.json ADDED
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manifests/batches/mathlib/001265.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001266.json ADDED
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manifests/batches/mathlib/001267.json ADDED
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manifests/batches/mathlib/001268.json ADDED
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manifests/batches/mathlib/001269.json ADDED
@@ -0,0 +1,605 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001269",
4
+ "batch_fingerprint": "28f700b16d3b3272170ef51296a017475618d747c732eb22f781bd285d5b7fa0",
5
+ "evidence": {
6
+ "batch_fingerprint": "28f700b16d3b3272170ef51296a017475618d747c732eb22f781bd285d5b7fa0",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "Module.End.hasEigenvalue_iff_isRoot",
12
+ "id": "Module.End.hasEigenvalue_iff_isRoot",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 95,
18
+ 53
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean",
21
+ "full_name": "Module.End.hasEigenvalue_iff_isRoot",
22
+ "start": [
23
+ 94,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean"
30
+ },
31
+ {
32
+ "error": "no internal trace for tactic \"cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu\"",
33
+ "full_name": "Module.End.hasEigenvalue_of_isRoot",
34
+ "id": "Module.End.hasEigenvalue_of_isRoot",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 91,
40
+ 8
41
+ ],
42
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean",
43
+ "full_name": "Module.End.hasEigenvalue_of_isRoot",
44
+ "start": [
45
+ 75,
46
+ 1
47
+ ],
48
+ "traced_tactics": [
49
+ {
50
+ "annotated_tactic": [
51
+ "\"cases' <a>dvd_iff_isRoot<\\/a>.2 h with p hp\"",
52
+ "[{\"full_name\":\"Polynomial.dvd_iff_isRoot\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Div.lean\",\"def_pos\":[629,9],\"def_end_pos\":[629,23]}]"
53
+ ],
54
+ "state_after": "case intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\n\u22a2 f.HasEigenvalue \u03bc",
55
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\n\u22a2 f.HasEigenvalue \u03bc",
56
+ "tactic": "cases' dvd_iff_isRoot.2 h with p hp"
57
+ },
58
+ {
59
+ "annotated_tactic": [
60
+ "\"rw [<a>HasEigenvalue<\\/a>, <a>eigenspace<\\/a>]\"",
61
+ "[{\"full_name\":\"Module.End.HasEigenvalue\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[80,5],\"def_end_pos\":[80,18]},{\"full_name\":\"Module.End.eigenspace\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[65,5],\"def_end_pos\":[65,15]}]"
62
+ ],
63
+ "state_after": "case intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\n\u22a2 LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) \u2260 \u22a5",
64
+ "state_before": "case intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\n\u22a2 f.HasEigenvalue \u03bc",
65
+ "tactic": "rw [HasEigenvalue, eigenspace]"
66
+ },
67
+ {
68
+ "annotated_tactic": [
69
+ "\"intro con\"",
70
+ "[]"
71
+ ],
72
+ "state_after": "case intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\n\u22a2 False",
73
+ "state_before": "case intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\n\u22a2 LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) \u2260 \u22a5",
74
+ "tactic": "intro con"
75
+ },
76
+ {
77
+ "annotated_tactic": [
78
+ "\"cases' (<a>LinearMap.isUnit_iff_ker_eq_bot<\\/a> _).2 con with u hu\"",
79
+ "[{\"full_name\":\"LinearMap.isUnit_iff_ker_eq_bot\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FiniteDimensional.lean\",\"def_pos\":[781,9],\"def_end_pos\":[781,30]}]"
80
+ ],
81
+ "state_after": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\n\u22a2 False",
82
+ "state_before": "case intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\n\u22a2 False",
83
+ "tactic": "cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu"
84
+ },
85
+ {
86
+ "annotated_tactic": [
87
+ "\"have p_ne_0 : p \\u2260 0 := by\\n intro con\\n apply <a>minpoly.ne_zero<\\/a> (<a>Algebra.IsIntegral.isIntegral<\\/a> (R := K) f)\\n rw [hp, con, <a>mul_zero<\\/a>]\"",
88
+ "[{\"full_name\":\"minpoly.ne_zero\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,16]},{\"full_name\":\"Algebra.IsIntegral.isIntegral\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[69,3],\"def_end_pos\":[69,13]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
89
+ ],
90
+ "state_after": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\n\u22a2 False",
91
+ "state_before": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\n\u22a2 False",
92
+ "tactic": "have p_ne_0 : p \u2260 0 := by\n intro con\n apply minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := K) f)\n rw [hp, con, mul_zero]"
93
+ },
94
+ {
95
+ "annotated_tactic": [
96
+ "\"have : (<a>aeval<\\/a> f) p = 0 := by\\n have h_aeval := <a>minpoly.aeval<\\/a> K f\\n revert h_aeval\\n simp [hp, \\u2190 hu]\"",
97
+ "[{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]},{\"full_name\":\"minpoly.aeval\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,14]}]"
98
+ ],
99
+ "state_after": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\n\u22a2 False",
100
+ "state_before": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\n\u22a2 False",
101
+ "tactic": "have : (aeval f) p = 0 := by\n have h_aeval := minpoly.aeval K f\n revert h_aeval\n simp [hp, \u2190 hu]"
102
+ },
103
+ {
104
+ "annotated_tactic": [
105
+ "\"have h_deg := <a>minpoly.degree_le_of_ne_zero<\\/a> K f p_ne_0 this\"",
106
+ "[{\"full_name\":\"minpoly.degree_le_of_ne_zero\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Field.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,29]}]"
107
+ ],
108
+ "state_after": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\nh_deg : (minpoly K f).degree \u2264 p.degree\n\u22a2 False",
109
+ "state_before": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\n\u22a2 False",
110
+ "tactic": "have h_deg := minpoly.degree_le_of_ne_zero K f p_ne_0 this"
111
+ },
112
+ {
113
+ "annotated_tactic": [
114
+ "\"rw [hp, <a>degree_mul<\\/a>, <a>degree_X_sub_C<\\/a>, <a>Polynomial.degree_eq_natDegree<\\/a> p_ne_0] at h_deg\"",
115
+ "[{\"full_name\":\"Polynomial.degree_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1651,9],\"def_end_pos\":[1651,19]},{\"full_name\":\"Polynomial.degree_X_sub_C\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1603,9],\"def_end_pos\":[1603,23]},{\"full_name\":\"Polynomial.degree_eq_natDegree\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[132,9],\"def_end_pos\":[132,28]}]"
116
+ ],
117
+ "state_after": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\nh_deg : 1 + \u2191p.natDegree \u2264 \u2191p.natDegree\n\u22a2 False",
118
+ "state_before": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\nh_deg : (minpoly K f).degree \u2264 p.degree\n\u22a2 False",
119
+ "tactic": "rw [hp, degree_mul, degree_X_sub_C, Polynomial.degree_eq_natDegree p_ne_0] at h_deg"
120
+ },
121
+ {
122
+ "annotated_tactic": [
123
+ "\"norm_cast at h_deg\"",
124
+ "[]"
125
+ ],
126
+ "state_after": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\nh_deg : 1 + p.natDegree \u2264 p.natDegree\n\u22a2 False",
127
+ "state_before": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\nh_deg : 1 + \u2191p.natDegree \u2264 \u2191p.natDegree\n\u22a2 False",
128
+ "tactic": "norm_cast at h_deg"
129
+ },
130
+ {
131
+ "annotated_tactic": [
132
+ "\"omega\"",
133
+ "[]"
134
+ ],
135
+ "state_after": "no goals",
136
+ "state_before": "case intro.intro\nK : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nthis : (aeval f) p = 0\nh_deg : 1 + p.natDegree \u2264 p.natDegree\n\u22a2 False",
137
+ "tactic": "omega"
138
+ },
139
+ {
140
+ "annotated_tactic": [
141
+ "\"intro con\"",
142
+ "[]"
143
+ ],
144
+ "state_after": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon\u271d : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\ncon : p = 0\n\u22a2 False",
145
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\n\u22a2 p \u2260 0",
146
+ "tactic": "intro con"
147
+ },
148
+ {
149
+ "annotated_tactic": [
150
+ "\"apply <a>minpoly.ne_zero<\\/a> (<a>Algebra.IsIntegral.isIntegral<\\/a> (R := K) f)\"",
151
+ "[{\"full_name\":\"minpoly.ne_zero\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,16]},{\"full_name\":\"Algebra.IsIntegral.isIntegral\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[69,3],\"def_end_pos\":[69,13]}]"
152
+ ],
153
+ "state_after": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon\u271d : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\ncon : p = 0\n\u22a2 minpoly K f = 0",
154
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon\u271d : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\ncon : p = 0\n\u22a2 False",
155
+ "tactic": "apply minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := K) f)"
156
+ },
157
+ {
158
+ "annotated_tactic": [
159
+ "\"rw [hp, con, <a>mul_zero<\\/a>]\"",
160
+ "[{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
161
+ ],
162
+ "state_after": "no goals",
163
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon\u271d : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\ncon : p = 0\n\u22a2 minpoly K f = 0",
164
+ "tactic": "rw [hp, con, mul_zero]"
165
+ },
166
+ {
167
+ "annotated_tactic": [
168
+ "\"have h_aeval := <a>minpoly.aeval<\\/a> K f\"",
169
+ "[{\"full_name\":\"minpoly.aeval\",\"def_path\":\"Mathlib\\/FieldTheory\\/Minpoly\\/Basic.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,14]}]"
170
+ ],
171
+ "state_after": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nh_aeval : (aeval f) (minpoly K f) = 0\n\u22a2 (aeval f) p = 0",
172
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\n\u22a2 (aeval f) p = 0",
173
+ "tactic": "have h_aeval := minpoly.aeval K f"
174
+ },
175
+ {
176
+ "annotated_tactic": [
177
+ "\"revert h_aeval\"",
178
+ "[]"
179
+ ],
180
+ "state_after": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\n\u22a2 (aeval f) (minpoly K f) = 0 \u2192 (aeval f) p = 0",
181
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\nh_aeval : (aeval f) (minpoly K f) = 0\n\u22a2 (aeval f) p = 0",
182
+ "tactic": "revert h_aeval"
183
+ },
184
+ {
185
+ "annotated_tactic": [
186
+ "\"simp [hp, \\u2190 hu]\"",
187
+ "[]"
188
+ ],
189
+ "state_after": "no goals",
190
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u03bc : K\nh : (minpoly K f).IsRoot \u03bc\np : K[X]\nhp : minpoly K f = (X - C \u03bc) * p\ncon : LinearMap.ker (f - (algebraMap K (End K V)) \u03bc) = \u22a5\nu : (V \u2192\u2097[K] V)\u02e3\nhu : \u2191u = f - (algebraMap K (End K V)) \u03bc\np_ne_0 : p \u2260 0\n\u22a2 (aeval f) (minpoly K f) = 0 \u2192 (aeval f) p = 0",
191
+ "tactic": "simp [hp, \u2190 hu]"
192
+ }
193
+ ],
194
+ "url": "https://github.com/leanprover-community/mathlib4"
195
+ },
196
+ "source_file": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean"
197
+ },
198
+ {
199
+ "error": "no internal trace for tactic 'simp'",
200
+ "full_name": "Module.End.IsSemisimple.genEigenspace_eq_eigenspace",
201
+ "id": "Module.End.IsSemisimple.genEigenspace_eq_eigenspace",
202
+ "reason": "import_failure",
203
+ "selection": {
204
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
205
+ "end": [
206
+ 53,
207
+ 101
208
+ ],
209
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean",
210
+ "full_name": "Module.End.IsSemisimple.genEigenspace_eq_eigenspace",
211
+ "start": [
212
+ 49,
213
+ 1
214
+ ],
215
+ "traced_tactics": [
216
+ {
217
+ "annotated_tactic": [
218
+ "\"refine <a>le_antisymm<\\/a> (fun m hm \\u21a6 mem_eigenspace_iff.mpr ?_) (<a>eigenspace_le_genEigenspace<\\/a> hk)\"",
219
+ "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"Module.End.eigenspace_le_genEigenspace\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[259,9],\"def_end_pos\":[259,36]}]"
220
+ ],
221
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\nhf : f.IsSemisimple\n\u03bc : R\nk : \u2115\nhk : 0 < k\nm : M\nhm : m \u2208 (f.genEigenspace \u03bc) k\n\u22a2 f m = \u03bc \u2022 m",
222
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\nhf : f.IsSemisimple\n\u03bc : R\nk : \u2115\nhk : 0 < k\n\u22a2 (f.genEigenspace \u03bc) k = f.eigenspace \u03bc",
223
+ "tactic": "refine le_antisymm (fun m hm \u21a6 mem_eigenspace_iff.mpr ?_) (eigenspace_le_genEigenspace hk)"
224
+ },
225
+ {
226
+ "annotated_tactic": [
227
+ "\"exact <a>apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub<\\/a> hm <a>rfl<\\/a> hf (by simp)\"",
228
+ "[{\"full_name\":\"Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Semisimple.lean\",\"def_pos\":[28,7],\"def_end_pos\":[28,79]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
229
+ ],
230
+ "state_after": "no goals",
231
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\nhf : f.IsSemisimple\n\u03bc : R\nk : \u2115\nhk : 0 < k\nm : M\nhm : m \u2208 (f.genEigenspace \u03bc) k\n\u22a2 f m = \u03bc \u2022 m",
232
+ "tactic": "exact apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub hm rfl hf (by simp)"
233
+ },
234
+ {
235
+ "annotated_tactic": [
236
+ "\"simp\"",
237
+ "[]"
238
+ ],
239
+ "state_after": "no goals",
240
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\nhf : f.IsSemisimple\n\u03bc : R\nk : \u2115\nhk : 0 < k\nm : M\nhm : m \u2208 (f.genEigenspace \u03bc) k\n\u22a2 IsNilpotent (f - f)",
241
+ "tactic": "simp"
242
+ }
243
+ ],
244
+ "url": "https://github.com/leanprover-community/mathlib4"
245
+ },
246
+ "source_file": "Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean"
247
+ },
248
+ {
249
+ "error": "no internal trace for tactic 'replace this : g.restrict h\u2081 - algebraMap R (End R p) \u03bc = 0 :=\\n eq_zero_of_isNilpotent_isSemisimple this (by simpa using hss.restrict)'",
250
+ "full_name": "Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub",
251
+ "id": "Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub",
252
+ "reason": "import_failure",
253
+ "selection": {
254
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
255
+ "end": [
256
+ 47,
257
+ 90
258
+ ],
259
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean",
260
+ "full_name": "Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub",
261
+ "start": [
262
+ 28,
263
+ 1
264
+ ],
265
+ "traced_tactics": [
266
+ {
267
+ "annotated_tactic": [
268
+ "\"set p := f.genEigenspace \\u03bc k\"",
269
+ "[]"
270
+ ],
271
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\n\u22a2 g m = \u03bc \u2022 m",
272
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhm : m \u2208 (f.genEigenspace \u03bc) k\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\n\u22a2 g m = \u03bc \u2022 m",
273
+ "tactic": "set p := f.genEigenspace \u03bc k"
274
+ },
275
+ {
276
+ "annotated_tactic": [
277
+ "\"have h\\u2081 : <a>MapsTo<\\/a> g p p := <a>mapsTo_genEigenspace_of_comm<\\/a> hfg \\u03bc k\"",
278
+ "[{\"full_name\":\"Set.MapsTo\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[267,5],\"def_end_pos\":[267,11]},{\"full_name\":\"Module.End.mapsTo_genEigenspace_of_comm\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[306,7],\"def_end_pos\":[306,35]}]"
279
+ ],
280
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
281
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\n\u22a2 g m = \u03bc \u2022 m",
282
+ "tactic": "have h\u2081 : MapsTo g p p := mapsTo_genEigenspace_of_comm hfg \u03bc k"
283
+ },
284
+ {
285
+ "annotated_tactic": [
286
+ "\"have h\\u2082 : <a>MapsTo<\\/a> (g - <a>algebraMap<\\/a> R (<a>End<\\/a> R M) \\u03bc) p p :=\\n <a>mapsTo_genEigenspace_of_comm<\\/a> (hfg.sub_right <| <a>Algebra.commute_algebraMap_right<\\/a> \\u03bc f) \\u03bc k\"",
287
+ "[{\"full_name\":\"Set.MapsTo\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[267,5],\"def_end_pos\":[267,11]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Module.End\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[29,8],\"def_end_pos\":[29,18]},{\"full_name\":\"Module.End.mapsTo_genEigenspace_of_comm\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[306,7],\"def_end_pos\":[306,35]},{\"full_name\":\"Algebra.commute_algebraMap_right\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[348,7],\"def_end_pos\":[348,31]}]"
288
+ ],
289
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
290
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
291
+ "tactic": "have h\u2082 : MapsTo (g - algebraMap R (End R M) \u03bc) p p :=\n mapsTo_genEigenspace_of_comm (hfg.sub_right <| Algebra.commute_algebraMap_right \u03bc f) \u03bc k"
292
+ },
293
+ {
294
+ "annotated_tactic": [
295
+ "\"have h\\u2083 : <a>MapsTo<\\/a> (f - g) p p :=\\n <a>mapsTo_genEigenspace_of_comm<\\/a> (<a>Commute.sub_right<\\/a> <a>rfl<\\/a> hfg) \\u03bc k\"",
296
+ "[{\"full_name\":\"Set.MapsTo\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[267,5],\"def_end_pos\":[267,11]},{\"full_name\":\"Module.End.mapsTo_genEigenspace_of_comm\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[306,7],\"def_end_pos\":[306,35]},{\"full_name\":\"Commute.sub_right\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Commute.lean\",\"def_pos\":[135,9],\"def_end_pos\":[135,18]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
297
+ ],
298
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
299
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
300
+ "tactic": "have h\u2083 : MapsTo (f - g) p p :=\n mapsTo_genEigenspace_of_comm (Commute.sub_right rfl hfg) \u03bc k"
301
+ },
302
+ {
303
+ "annotated_tactic": [
304
+ "\"have h\\u2084 : <a>MapsTo<\\/a> (f - <a>algebraMap<\\/a> R (<a>End<\\/a> R M) \\u03bc) p p :=\\n <a>mapsTo_genEigenspace_of_comm<\\/a> (<a>Algebra.mul_sub_algebraMap_commutes<\\/a> f \\u03bc) \\u03bc k\"",
305
+ "[{\"full_name\":\"Set.MapsTo\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[267,5],\"def_end_pos\":[267,11]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Module.End\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[29,8],\"def_end_pos\":[29,18]},{\"full_name\":\"Module.End.mapsTo_genEigenspace_of_comm\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[306,7],\"def_end_pos\":[306,35]},{\"full_name\":\"Algebra.mul_sub_algebraMap_commutes\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,36]}]"
306
+ ],
307
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
308
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
309
+ "tactic": "have h\u2084 : MapsTo (f - algebraMap R (End R M) \u03bc) p p :=\n mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f \u03bc) \u03bc k"
310
+ },
311
+ {
312
+ "annotated_tactic": [
313
+ "\"replace hfg : <a>Commute<\\/a> (f - <a>algebraMap<\\/a> R (<a>End<\\/a> R M) \\u03bc) (f - g) :=\\n (<a>Commute.sub_right<\\/a> <a>rfl<\\/a> hfg).<a>sub_left<\\/a> <| <a>Algebra.commute_algebraMap_left<\\/a> \\u03bc (f - g)\"",
314
+ "[{\"full_name\":\"Commute\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[37,5],\"def_end_pos\":[37,12]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Module.End\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[29,8],\"def_end_pos\":[29,18]},{\"full_name\":\"Commute.sub_right\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Commute.lean\",\"def_pos\":[135,9],\"def_end_pos\":[135,18]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"Commute.sub_left\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Commute.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,17]},{\"full_name\":\"Algebra.commute_algebraMap_left\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[345,7],\"def_end_pos\":[345,30]}]"
315
+ ],
316
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\n\u22a2 g m = \u03bc \u2022 m",
317
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhfg : Commute f g\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\n\u22a2 g m = \u03bc \u2022 m",
318
+ "tactic": "replace hfg : Commute (f - algebraMap R (End R M) \u03bc) (f - g) :=\n (Commute.sub_right rfl hfg).sub_left <| Algebra.commute_algebraMap_left \u03bc (f - g)"
319
+ },
320
+ {
321
+ "annotated_tactic": [
322
+ "\"suffices <a>IsNilpotent<\\/a> ((g - <a>algebraMap<\\/a> R (<a>End<\\/a> R M) \\u03bc).<a>restrict<\\/a> h\\u2082) by\\n replace this : g.restrict h\\u2081 - <a>algebraMap<\\/a> R (<a>End<\\/a> R p) \\u03bc = 0 :=\\n <a>eq_zero_of_isNilpotent_isSemisimple<\\/a> this (by simpa using hss.restrict)\\n simpa [<a>LinearMap.restrict_apply<\\/a>, <a>sub_eq_zero<\\/a>] using <a>LinearMap.congr_fun<\\/a> this \\u27e8m, hm\\u27e9\"",
323
+ "[{\"full_name\":\"IsNilpotent\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Defs.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,16]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Module.End\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[29,8],\"def_end_pos\":[29,18]},{\"full_name\":\"LinearMap.restrict\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[172,5],\"def_end_pos\":[172,13]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Module.End\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[29,8],\"def_end_pos\":[29,18]},{\"full_name\":\"Module.End.eq_zero_of_isNilpotent_isSemisimple\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Semisimple.lean\",\"def_pos\":[79,7],\"def_end_pos\":[79,42]},{\"full_name\":\"LinearMap.restrict_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[183,9],\"def_end_pos\":[183,23]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"LinearMap.congr_fun\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[344,19],\"def_end_pos\":[344,28]}]"
324
+ ],
325
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\n\u22a2 IsNilpotent (LinearMap.restrict (g - (algebraMap R (End R M)) \u03bc) h\u2082)",
326
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\n\u22a2 g m = \u03bc \u2022 m",
327
+ "tactic": "suffices IsNilpotent ((g - algebraMap R (End R M) \u03bc).restrict h\u2082) by\n replace this : g.restrict h\u2081 - algebraMap R (End R p) \u03bc = 0 :=\n eq_zero_of_isNilpotent_isSemisimple this (by simpa using hss.restrict)\n simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this \u27e8m, hm\u27e9"
328
+ },
329
+ {
330
+ "annotated_tactic": [
331
+ "\"simpa [<a>LinearMap.restrict_sub<\\/a> h\\u2084 h\\u2083] using (<a>LinearMap.restrict_commute<\\/a> hfg h\\u2084 h\\u2083).<a>isNilpotent_sub<\\/a>\\n (f.isNilpotent_restrict_sub_algebraMap \\u03bc k) (<a>Module.End.isNilpotent.restrict<\\/a> h\\u2083 hnil)\"",
332
+ "[{\"full_name\":\"LinearMap.restrict_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[188,7],\"def_end_pos\":[188,19]},{\"full_name\":\"LinearMap.restrict_commute\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[204,7],\"def_end_pos\":[204,23]},{\"full_name\":\"Commute.isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[174,9],\"def_end_pos\":[174,24]},{\"full_name\":\"Module.End.isNilpotent.restrict\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Lemmas.lean\",\"def_pos\":[114,7],\"def_end_pos\":[114,27]}]"
333
+ ],
334
+ "state_after": "no goals",
335
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\n\u22a2 IsNilpotent (LinearMap.restrict (g - (algebraMap R (End R M)) \u03bc) h\u2082)",
336
+ "tactic": "simpa [LinearMap.restrict_sub h\u2084 h\u2083] using (LinearMap.restrict_commute hfg h\u2084 h\u2083).isNilpotent_sub\n (f.isNilpotent_restrict_sub_algebraMap \u03bc k) (Module.End.isNilpotent.restrict h\u2083 hnil)"
337
+ },
338
+ {
339
+ "annotated_tactic": [
340
+ "\"replace this : g.restrict h\\u2081 - <a>algebraMap<\\/a> R (<a>End<\\/a> R p) \\u03bc = 0 :=\\n <a>eq_zero_of_isNilpotent_isSemisimple<\\/a> this (by simpa using hss.restrict)\"",
341
+ "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Module.End\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[29,8],\"def_end_pos\":[29,18]},{\"full_name\":\"Module.End.eq_zero_of_isNilpotent_isSemisimple\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Semisimple.lean\",\"def_pos\":[79,7],\"def_end_pos\":[79,42]}]"
342
+ ],
343
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\nthis : LinearMap.restrict g h\u2081 - (algebraMap R (End R \u21a5p)) \u03bc = 0\n\u22a2 g m = \u03bc \u2022 m",
344
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\nthis : IsNilpotent (LinearMap.restrict (g - (algebraMap R (End R M)) \u03bc) h\u2082)\n\u22a2 g m = \u03bc \u2022 m",
345
+ "tactic": "replace this : g.restrict h\u2081 - algebraMap R (End R p) \u03bc = 0 :=\n eq_zero_of_isNilpotent_isSemisimple this (by simpa using hss.restrict)"
346
+ },
347
+ {
348
+ "annotated_tactic": [
349
+ "\"simpa [<a>LinearMap.restrict_apply<\\/a>, <a>sub_eq_zero<\\/a>] using <a>LinearMap.congr_fun<\\/a> this \\u27e8m, hm\\u27e9\"",
350
+ "[{\"full_name\":\"LinearMap.restrict_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[183,9],\"def_end_pos\":[183,23]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"LinearMap.congr_fun\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[344,19],\"def_end_pos\":[344,28]}]"
351
+ ],
352
+ "state_after": "no goals",
353
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\nthis : LinearMap.restrict g h\u2081 - (algebraMap R (End R \u21a5p)) \u03bc = 0\n\u22a2 g m = \u03bc \u2022 m",
354
+ "tactic": "simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this \u27e8m, hm\u27e9"
355
+ },
356
+ {
357
+ "annotated_tactic": [
358
+ "\"simpa using hss.restrict\"",
359
+ "[]"
360
+ ],
361
+ "state_after": "no goals",
362
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf g : End R M\n\u03bc : R\nk : \u2115\nm : M\nhss : g.IsSemisimple\nhnil : IsNilpotent (f - g)\np : Submodule R M := (f.genEigenspace \u03bc) k\nhm : m \u2208 p\nh\u2081 : MapsTo \u21d1g \u2191p \u2191p\nh\u2082 : MapsTo \u21d1(g - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nh\u2083 : MapsTo \u21d1(f - g) \u2191p \u2191p\nh\u2084 : MapsTo \u21d1(f - (algebraMap R (End R M)) \u03bc) \u2191p \u2191p\nhfg : Commute (f - (algebraMap R (End R M)) \u03bc) (f - g)\nthis : IsNilpotent (LinearMap.restrict (g - (algebraMap R (End R M)) \u03bc) h\u2082)\n\u22a2 IsSemisimple (LinearMap.restrict g h\u2081 - (algebraMap R (End R \u21a5p)) \u03bc)",
363
+ "tactic": "simpa using hss.restrict"
364
+ }
365
+ ],
366
+ "url": "https://github.com/leanprover-community/mathlib4"
367
+ },
368
+ "source_file": "Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean"
369
+ }
370
+ ],
371
+ "input_proofs": 12,
372
+ "input_units": 2,
373
+ "proofs": 8,
374
+ "published_proofs": 6,
375
+ "published_transitions": 18,
376
+ "replay_failures": [],
377
+ "transitions": 33,
378
+ "validation_failures": [
379
+ {
380
+ "errors": [
381
+ "internal transition boundary mismatch",
382
+ "internal transition boundary mismatch",
383
+ "internal transition boundary mismatch"
384
+ ],
385
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean",
386
+ "proof_id": "mathlib:Module.End.aeval_apply_of_hasEigenvector",
387
+ "provenance": {
388
+ "selection": {
389
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
390
+ "end": [
391
+ 62,
392
+ 56
393
+ ],
394
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean",
395
+ "full_name": "Module.End.aeval_apply_of_hasEigenvector",
396
+ "start": [
397
+ 54,
398
+ 1
399
+ ],
400
+ "traced_tactics": [
401
+ {
402
+ "annotated_tactic": [
403
+ "\"refine p.induction_on ?_ ?_ ?_\"",
404
+ "[]"
405
+ ],
406
+ "state_after": "case refine_1\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 \u2200 (a : K), ((aeval f) (C a)) x = eval \u03bc (C a) \u2022 x\n\ncase refine_2\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 \u2200 (p q : K[X]),\n ((aeval f) p) x = eval \u03bc p \u2022 x \u2192 ((aeval f) q) x = eval \u03bc q \u2022 x \u2192 ((aeval f) (p + q)) x = eval \u03bc (p + q) \u2022 x\n\ncase refine_3\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 \u2200 (n : \u2115) (a : K),\n ((aeval f) (C a * X ^ n)) x = eval \u03bc (C a * X ^ n) \u2022 x \u2192\n ((aeval f) (C a * X ^ (n + 1))) x = eval \u03bc (C a * X ^ (n + 1)) \u2022 x",
407
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 ((aeval f) p) x = eval \u03bc p \u2022 x",
408
+ "tactic": "refine p.induction_on ?_ ?_ ?_"
409
+ },
410
+ {
411
+ "annotated_tactic": [
412
+ "\"intro a\"",
413
+ "[]"
414
+ ],
415
+ "state_after": "case refine_1\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\na : K\n\u22a2 ((aeval f) (C a)) x = eval \u03bc (C a) \u2022 x",
416
+ "state_before": "case refine_1\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 \u2200 (a : K), ((aeval f) (C a)) x = eval \u03bc (C a) \u2022 x",
417
+ "tactic": "intro a"
418
+ },
419
+ {
420
+ "annotated_tactic": [
421
+ "\"simp [<a>Module.algebraMap_end_apply<\\/a>]\"",
422
+ "[{\"full_name\":\"Module.algebraMap_end_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,29]}]"
423
+ ],
424
+ "state_after": "no goals",
425
+ "state_before": "case refine_1\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\na : K\n\u22a2 ((aeval f) (C a)) x = eval \u03bc (C a) \u2022 x",
426
+ "tactic": "simp [Module.algebraMap_end_apply]"
427
+ },
428
+ {
429
+ "annotated_tactic": [
430
+ "\"intro p q hp hq\"",
431
+ "[]"
432
+ ],
433
+ "state_after": "case refine_2\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np\u271d : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\np q : K[X]\nhp : ((aeval f) p) x = eval \u03bc p \u2022 x\nhq : ((aeval f) q) x = eval \u03bc q \u2022 x\n\u22a2 ((aeval f) (p + q)) x = eval \u03bc (p + q) \u2022 x",
434
+ "state_before": "case refine_2\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 \u2200 (p q : K[X]),\n ((aeval f) p) x = eval \u03bc p \u2022 x \u2192 ((aeval f) q) x = eval \u03bc q \u2022 x \u2192 ((aeval f) (p + q)) x = eval \u03bc (p + q) \u2022 x",
435
+ "tactic": "intro p q hp hq"
436
+ },
437
+ {
438
+ "annotated_tactic": [
439
+ "\"simp [hp, hq, <a>add_smul<\\/a>]\"",
440
+ "[{\"full_name\":\"add_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,17]}]"
441
+ ],
442
+ "state_after": "no goals",
443
+ "state_before": "case refine_2\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np\u271d : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\np q : K[X]\nhp : ((aeval f) p) x = eval \u03bc p \u2022 x\nhq : ((aeval f) q) x = eval \u03bc q \u2022 x\n\u22a2 ((aeval f) (p + q)) x = eval \u03bc (p + q) \u2022 x",
444
+ "tactic": "simp [hp, hq, add_smul]"
445
+ },
446
+ {
447
+ "annotated_tactic": [
448
+ "\"intro n a hna\"",
449
+ "[]"
450
+ ],
451
+ "state_after": "case refine_3\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\nn : \u2115\na : K\nhna : ((aeval f) (C a * X ^ n)) x = eval \u03bc (C a * X ^ n) \u2022 x\n\u22a2 ((aeval f) (C a * X ^ (n + 1))) x = eval \u03bc (C a * X ^ (n + 1)) \u2022 x",
452
+ "state_before": "case refine_3\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\n\u22a2 \u2200 (n : \u2115) (a : K),\n ((aeval f) (C a * X ^ n)) x = eval \u03bc (C a * X ^ n) \u2022 x \u2192\n ((aeval f) (C a * X ^ (n + 1))) x = eval \u03bc (C a * X ^ (n + 1)) \u2022 x",
453
+ "tactic": "intro n a hna"
454
+ },
455
+ {
456
+ "annotated_tactic": [
457
+ "\"rw [<a>mul_comm<\\/a>, <a>pow_succ'<\\/a>, <a>mul_assoc<\\/a>, <a>AlgHom.map_mul<\\/a>, <a>LinearMap.mul_apply<\\/a>, <a>mul_comm<\\/a>, hna]\"",
458
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"pow_succ'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[667,34],\"def_end_pos\":[667,43]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"LinearMap.mul_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/End.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
459
+ ],
460
+ "state_after": "case refine_3\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\nn : \u2115\na : K\nhna : ((aeval f) (C a * X ^ n)) x = eval \u03bc (C a * X ^ n) \u2022 x\n\u22a2 ((aeval f) X) (eval \u03bc (C a * X ^ n) \u2022 x) = eval \u03bc (X * (C a * X ^ n)) \u2022 x",
461
+ "state_before": "case refine_3\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\nn : \u2115\na : K\nhna : ((aeval f) (C a * X ^ n)) x = eval \u03bc (C a * X ^ n) \u2022 x\n\u22a2 ((aeval f) (C a * X ^ (n + 1))) x = eval \u03bc (C a * X ^ (n + 1)) \u2022 x",
462
+ "tactic": "rw [mul_comm, pow_succ', mul_assoc, AlgHom.map_mul, LinearMap.mul_apply, mul_comm, hna]"
463
+ },
464
+ {
465
+ "annotated_tactic": [
466
+ "\"simp only [<a>mem_eigenspace_iff<\\/a>.1 h.1, <a>smul_smul<\\/a>, <a>aeval_X<\\/a>, <a>eval_mul<\\/a>, <a>eval_C<\\/a>, <a>eval_pow<\\/a>, <a>eval_X<\\/a>,\\n <a>LinearMap.map_smul\\u209b\\u2097<\\/a>, <a>RingHom.id_apply<\\/a>, <a>mul_comm<\\/a>]\"",
467
+ "[{\"full_name\":\"Module.End.mem_eigenspace_iff\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[105,9],\"def_end_pos\":[105,27]},{\"full_name\":\"smul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[446,7],\"def_end_pos\":[446,16]},{\"full_name\":\"Polynomial.aeval_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[209,9],\"def_end_pos\":[209,16]},{\"full_name\":\"Polynomial.eval_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Eval.lean\",\"def_pos\":[1106,9],\"def_end_pos\":[1106,17]},{\"full_name\":\"Polynomial.eval_C\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Eval.lean\",\"def_pos\":[359,9],\"def_end_pos\":[359,15]},{\"full_name\":\"Polynomial.eval_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Eval.lean\",\"def_pos\":[1125,9],\"def_end_pos\":[1125,17]},{\"full_name\":\"Polynomial.eval_X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Eval.lean\",\"def_pos\":[377,9],\"def_end_pos\":[377,15]},{\"full_name\":\"LinearMap.map_smul\\u209b\\u2097\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[372,19],\"def_end_pos\":[372,29]},{\"full_name\":\"RingHom.id_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[638,9],\"def_end_pos\":[638,17]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
468
+ ],
469
+ "state_after": "no goals",
470
+ "state_before": "case refine_3\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\np : K[X]\n\u03bc : K\nx : V\nh : f.HasEigenvector \u03bc x\nn : \u2115\na : K\nhna : ((aeval f) (C a * X ^ n)) x = eval \u03bc (C a * X ^ n) \u2022 x\n\u22a2 ((aeval f) X) (eval \u03bc (C a * X ^ n) \u2022 x) = eval \u03bc (X * (C a * X ^ n)) \u2022 x",
471
+ "tactic": "simp only [mem_eigenspace_iff.1 h.1, smul_smul, aeval_X, eval_mul, eval_C, eval_pow, eval_X,\n LinearMap.map_smul\u209b\u2097, RingHom.id_apply, mul_comm]"
472
+ }
473
+ ],
474
+ "url": "https://github.com/leanprover-community/mathlib4"
475
+ }
476
+ },
477
+ "theorem_name": "Module.End.aeval_apply_of_hasEigenvector"
478
+ },
479
+ {
480
+ "errors": [
481
+ "internal transition boundary mismatch",
482
+ "internal transition boundary mismatch"
483
+ ],
484
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean",
485
+ "proof_id": "mathlib:Module.End.eigenspace_aeval_polynomial_degree_1",
486
+ "provenance": {
487
+ "selection": {
488
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
489
+ "end": [
490
+ 43,
491
+ 76
492
+ ],
493
+ "file_path": "Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean",
494
+ "full_name": "Module.End.eigenspace_aeval_polynomial_degree_1",
495
+ "start": [
496
+ 32,
497
+ 1
498
+ ],
499
+ "traced_tactics": [
500
+ {
501
+ "annotated_tactic": [
502
+ "\"rw [<a>eigenspace_div<\\/a>]\"",
503
+ "[{\"full_name\":\"Module.End.eigenspace_div\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Eigenspace\\/Basic.lean\",\"def_pos\":[155,9],\"def_end_pos\":[155,23]}]"
504
+ ],
505
+ "state_after": "case hb\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 q.leadingCoeff \u2260 0",
506
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 f.eigenspace (-q.coeff 0 / q.leadingCoeff) =\n LinearMap.ker (q.leadingCoeff \u2022 f - (algebraMap K (End K V)) (-q.coeff 0))",
507
+ "tactic": "rw [eigenspace_div]"
508
+ },
509
+ {
510
+ "annotated_tactic": [
511
+ "\"intro h\"",
512
+ "[]"
513
+ ],
514
+ "state_after": "case hb\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\nh : q.leadingCoeff = 0\n\u22a2 False",
515
+ "state_before": "case hb\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 q.leadingCoeff \u2260 0",
516
+ "tactic": "intro h"
517
+ },
518
+ {
519
+ "annotated_tactic": [
520
+ "\"rw [<a>leadingCoeff_eq_zero_iff_deg_eq_bot<\\/a>.1 h] at hq\"",
521
+ "[{\"full_name\":\"Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[698,9],\"def_end_pos\":[698,44]}]"
522
+ ],
523
+ "state_after": "case hb\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : \u22a5 = 1\nh : q.leadingCoeff = 0\n\u22a2 False",
524
+ "state_before": "case hb\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\nh : q.leadingCoeff = 0\n\u22a2 False",
525
+ "tactic": "rw [leadingCoeff_eq_zero_iff_deg_eq_bot.1 h] at hq"
526
+ },
527
+ {
528
+ "annotated_tactic": [
529
+ "\"cases hq\"",
530
+ "[]"
531
+ ],
532
+ "state_after": "no goals",
533
+ "state_before": "case hb\nK : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : \u22a5 = 1\nh : q.leadingCoeff = 0\n\u22a2 False",
534
+ "tactic": "cases hq"
535
+ },
536
+ {
537
+ "annotated_tactic": [
538
+ "\"rw [<a>C_mul'<\\/a>, <a>aeval_def<\\/a>]\"",
539
+ "[{\"full_name\":\"Polynomial.C_mul'\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Coeff.lean\",\"def_pos\":[178,9],\"def_end_pos\":[178,15]},{\"full_name\":\"Polynomial.aeval_def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,18]}]"
540
+ ],
541
+ "state_after": "K : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 LinearMap.ker (q.leadingCoeff \u2022 f - (algebraMap K (End K V)) (-q.coeff 0)) =\n LinearMap.ker (eval\u2082 (algebraMap K (End K V)) f (q.leadingCoeff \u2022 X + C (q.coeff 0)))",
542
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 LinearMap.ker (q.leadingCoeff \u2022 f - (algebraMap K (End K V)) (-q.coeff 0)) =\n LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0)))",
543
+ "tactic": "rw [C_mul', aeval_def]"
544
+ },
545
+ {
546
+ "annotated_tactic": [
547
+ "\"simp [<a>algebraMap<\\/a>, Algebra.toRingHom]\"",
548
+ "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]}]"
549
+ ],
550
+ "state_after": "no goals",
551
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 LinearMap.ker (q.leadingCoeff \u2022 f - (algebraMap K (End K V)) (-q.coeff 0)) =\n LinearMap.ker (eval\u2082 (algebraMap K (End K V)) f (q.leadingCoeff \u2022 X + C (q.coeff 0)))",
552
+ "tactic": "simp [algebraMap, Algebra.toRingHom]"
553
+ },
554
+ {
555
+ "annotated_tactic": [
556
+ "\"rwa [\\u2190 <a>eq_X_add_C_of_degree_eq_one<\\/a>]\"",
557
+ "[{\"full_name\":\"Polynomial.eq_X_add_C_of_degree_eq_one\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[457,9],\"def_end_pos\":[457,36]}]"
558
+ ],
559
+ "state_after": "no goals",
560
+ "state_before": "K : Type v\nV : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nq : K[X]\nhq : q.degree = 1\n\u22a2 LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0))) = LinearMap.ker ((aeval f) q)",
561
+ "tactic": "rwa [\u2190 eq_X_add_C_of_degree_eq_one]"
562
+ }
563
+ ],
564
+ "url": "https://github.com/leanprover-community/mathlib4"
565
+ }
566
+ },
567
+ "theorem_name": "Module.End.eigenspace_aeval_polynomial_degree_1"
568
+ }
569
+ ],
570
+ "validation_rejected_proofs": 2,
571
+ "zero_transition_proofs": 0
572
+ },
573
+ "shards": [
574
+ {
575
+ "path": "data/train/mathlib-001269-0000-12e48659a0e6dd40.parquet",
576
+ "sha256": "12e48659a0e6dd402afeab8b848998ec7ee6b0882d0088e10422ac3117cd6568",
577
+ "bytes": 63131,
578
+ "rows": 11,
579
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
580
+ },
581
+ {
582
+ "path": "data/internal_test/mathlib-001269-0000-8c6c9702c4ea82b8.parquet",
583
+ "sha256": "8c6c9702c4ea82b84a32632952bfe2e66349cd9f6f39510a2a2b02472ee2dbfc",
584
+ "bytes": 65707,
585
+ "rows": 7,
586
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
587
+ }
588
+ ],
589
+ "proof_ids": [
590
+ "mathlib:Matrix.finite_spectrum",
591
+ "mathlib:Module.End.IsSemisimple.maxGenEigenspace_eq_eigenspace",
592
+ "mathlib:Module.End.finite_hasEigenvalue",
593
+ "mathlib:Module.End.finite_spectrum",
594
+ "mathlib:Module.End.isRoot_of_hasEigenvalue",
595
+ "mathlib:Module.End.ker_aeval_ring_hom'_unit_polynomial"
596
+ ],
597
+ "proof_keys": [
598
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean\",null,null,\"Matrix.finite_spectrum\"]",
599
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean\",null,null,\"Module.End.finite_hasEigenvalue\"]",
600
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean\",null,null,\"Module.End.finite_spectrum\"]",
601
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean\",null,null,\"Module.End.isRoot_of_hasEigenvalue\"]",
602
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean\",null,null,\"Module.End.ker_aeval_ring_hom'_unit_polynomial\"]",
603
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean\",null,null,\"Module.End.IsSemisimple.maxGenEigenspace_eq_eigenspace\"]"
604
+ ]
605
+ }
manifests/batches/mathlib/001270.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001271.json ADDED
@@ -0,0 +1,1107 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
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+ ],
537
+ "state_after": "case mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 \u22c0[R]^i M\n\u22a2 (lift\u03b9 R M) \u2191\u27e8x, hx\u27e9 = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i) \u27e8x, hx\u27e9",
538
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : \u21a5(\u22c0[R]^i M)\n\u22a2 (lift\u03b9 R M) \u2191x = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i) x",
539
+ "tactic": "cases' x with x hx"
540
+ },
541
+ {
542
+ "annotated_tactic": [
543
+ "\"dsimp only [<a>Subtype.coe_mk<\\/a>, <a>DirectSum.lof_eq_of<\\/a>]\"",
544
+ "[{\"full_name\":\"Subtype.coe_mk\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,15]},{\"full_name\":\"DirectSum.lof_eq_of\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Module.lean\",\"def_pos\":[71,9],\"def_end_pos\":[71,18]}]"
545
+ ],
546
+ "state_after": "case mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 \u22c0[R]^i M\n\u22a2 (lift\u03b9 R M) x = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i) \u27e8x, hx\u27e9",
547
+ "state_before": "case mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 \u22c0[R]^i M\n\u22a2 (lift\u03b9 R M) \u2191\u27e8x, hx\u27e9 = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i) \u27e8x, hx\u27e9",
548
+ "tactic": "dsimp only [Subtype.coe_mk, DirectSum.lof_eq_of]"
549
+ },
550
+ {
551
+ "annotated_tactic": [
552
+ "\"induction hx using <a>Submodule.pow_induction_on_left'<\\/a> with\\n | algebraMap => simp_rw [<a>AlgHom.commutes<\\/a>, <a>DirectSum.algebraMap_apply<\\/a>]; rfl\\n -- FIXME: specialized `map_add` to avoid a (whole-declaration) timeout\\n | <a>add<\\/a> _ _ _ _ _ ihx ihy => simp_rw [<a>AlgHom.map_add<\\/a>, ihx, ihy, \\u2190 <a>AddMonoidHom.map_add<\\/a>]; rfl\\n | mem_mul _ hm _ _ _ ih =>\\n obtain \\u27e8_, rfl\\u27e9 := hm\\n simp_rw [<a>AlgHom.map_mul<\\/a>, ih, <a>GradedAlgebra.lift\\u03b9<\\/a>, <a>lift_\\u03b9_apply<\\/a>, <a>GradedAlgebra.\\u03b9_apply<\\/a> R M,\\n <a>DirectSum.of_mul_of<\\/a>]\\n exact <a>DirectSum.of_eq_of_gradedMonoid_eq<\\/a> (<a>Sigma.subtype_ext<\\/a> (<a>add_comm<\\/a> _ _) <a>rfl<\\/a>)\"",
553
+ "[{\"full_name\":\"Submodule.pow_induction_on_left'\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Operations.lean\",\"def_pos\":[471,19],\"def_end_pos\":[471,41]},{\"full_name\":\"AlgHom.commutes\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,17]},{\"full_name\":\"DirectSum.algebraMap_apply\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Algebra.lean\",\"def_pos\":[96,9],\"def_end_pos\":[96,25]},{\"full_name\":\"Submodule.add\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[248,10],\"def_end_pos\":[248,13]},{\"full_name\":\"AlgHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[238,19],\"def_end_pos\":[238,26]},{\"full_name\":\"AddMonoidHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[770,3],\"def_end_pos\":[770,14]},{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"ExteriorAlgebra.GradedAlgebra.lift\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Grading.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,24]},{\"full_name\":\"ExteriorAlgebra.lift_\\u03b9_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[119,9],\"def_end_pos\":[119,21]},{\"full_name\":\"ExteriorAlgebra.GradedAlgebra.\\u03b9_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Grading.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,30]},{\"full_name\":\"DirectSum.of_mul_of\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Ring.lean\",\"def_pos\":[216,9],\"def_end_pos\":[216,18]},{\"full_name\":\"DirectSum.of_eq_of_gradedMonoid_eq\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Ring.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,33]},{\"full_name\":\"Sigma.subtype_ext\",\"def_path\":\"Mathlib\\/Data\\/Sigma\\/Basic.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,20]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
554
+ ],
555
+ "state_after": "no goals",
556
+ "state_before": "case mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 \u22c0[R]^i M\n\u22a2 (lift\u03b9 R M) x = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i) \u27e8x, hx\u27e9",
557
+ "tactic": "induction hx using Submodule.pow_induction_on_left' with\n| algebraMap => simp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl\n| add _ _ _ _ _ ihx ihy => simp_rw [AlgHom.map_add, ihx, ihy, \u2190 AddMonoidHom.map_add]; rfl\n| mem_mul _ hm _ _ _ ih =>\n obtain \u27e8_, rfl\u27e9 := hm\n simp_rw [AlgHom.map_mul, ih, GradedAlgebra.lift\u03b9, lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M,\n DirectSum.of_mul_of]\n exact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)"
558
+ },
559
+ {
560
+ "annotated_tactic": [
561
+ "\"simp_rw [<a>AlgHom.commutes<\\/a>, <a>DirectSum.algebraMap_apply<\\/a>]\"",
562
+ "[{\"full_name\":\"AlgHom.commutes\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,17]},{\"full_name\":\"DirectSum.algebraMap_apply\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Algebra.lean\",\"def_pos\":[96,9],\"def_end_pos\":[96,25]}]"
563
+ ],
564
+ "state_after": "case mk.algebraMap\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nr\u271d : R\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) 0) (DirectSum.GAlgebra.toFun r\u271d) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) 0) \u27e8(algebraMap R (ExteriorAlgebra R M)) r\u271d, \u22ef\u27e9",
565
+ "state_before": "case mk.algebraMap\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nr\u271d : R\n\u22a2 (lift\u03b9 R M) ((algebraMap R (ExteriorAlgebra R M)) r\u271d) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) 0) \u27e8(algebraMap R (ExteriorAlgebra R M)) r\u271d, \u22ef\u27e9",
566
+ "tactic": "simp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]"
567
+ },
568
+ {
569
+ "annotated_tactic": [
570
+ "\"rfl\"",
571
+ "[]"
572
+ ],
573
+ "state_after": "no goals",
574
+ "state_before": "case mk.algebraMap\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nr\u271d : R\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) 0) (DirectSum.GAlgebra.toFun r\u271d) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) 0) \u27e8(algebraMap R (ExteriorAlgebra R M)) r\u271d, \u22ef\u27e9",
575
+ "tactic": "rfl"
576
+ },
577
+ {
578
+ "annotated_tactic": [
579
+ "\"simp_rw [<a>AlgHom.map_add<\\/a>, ihx, ihy, \\u2190 <a>AddMonoidHom.map_add<\\/a>]\"",
580
+ "[{\"full_name\":\"AlgHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[238,19],\"def_end_pos\":[238,26]},{\"full_name\":\"AddMonoidHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[770,3],\"def_end_pos\":[770,14]}]"
581
+ ],
582
+ "state_after": "case mk.add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx x\u271d y\u271d : ExteriorAlgebra R M\ni\u271d : \u2115\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nhy\u271d : y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nihx : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nihy : (lift\u03b9 R M) y\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8y\u271d, hy\u271d\u27e9\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) (\u27e8x\u271d, hx\u271d\u27e9 + \u27e8y\u271d, hy\u271d\u27e9) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d + y\u271d, \u22ef\u27e9",
583
+ "state_before": "case mk.add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx x\u271d y\u271d : ExteriorAlgebra R M\ni\u271d : \u2115\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nhy\u271d : y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nihx : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nihy : (lift\u03b9 R M) y\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8y\u271d, hy\u271d\u27e9\n\u22a2 (lift\u03b9 R M) (x\u271d + y\u271d) = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d + y\u271d, \u22ef\u27e9",
584
+ "tactic": "simp_rw [AlgHom.map_add, ihx, ihy, \u2190 AddMonoidHom.map_add]"
585
+ },
586
+ {
587
+ "annotated_tactic": [
588
+ "\"rfl\"",
589
+ "[]"
590
+ ],
591
+ "state_after": "no goals",
592
+ "state_before": "case mk.add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx x\u271d y\u271d : ExteriorAlgebra R M\ni\u271d : \u2115\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nhy\u271d : y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nihx : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nihy : (lift\u03b9 R M) y\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8y\u271d, hy\u271d\u27e9\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) (\u27e8x\u271d, hx\u271d\u27e9 + \u27e8y\u271d, hy\u271d\u27e9) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d + y\u271d, \u22ef\u27e9",
593
+ "tactic": "rfl"
594
+ },
595
+ {
596
+ "annotated_tactic": [
597
+ "\"obtain \\u27e8_, rfl\\u27e9 := hm\"",
598
+ "[]"
599
+ ],
600
+ "state_after": "case mk.mem_mul.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nw\u271d : M\n\u22a2 (lift\u03b9 R M) ((\u03b9 R) w\u271d * x\u271d) = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d.succ) \u27e8(\u03b9 R) w\u271d * x\u271d, \u22ef\u27e9",
601
+ "state_before": "case mk.mem_mul\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx m\u271d : ExteriorAlgebra R M\nhm : m\u271d \u2208 LinearMap.range (\u03b9 R)\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\n\u22a2 (lift\u03b9 R M) (m\u271d * x\u271d) = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d.succ) \u27e8m\u271d * x\u271d, \u22ef\u27e9",
602
+ "tactic": "obtain \u27e8_, rfl\u27e9 := hm"
603
+ },
604
+ {
605
+ "annotated_tactic": [
606
+ "\"simp_rw [<a>AlgHom.map_mul<\\/a>, ih, <a>GradedAlgebra.lift\\u03b9<\\/a>, <a>lift_\\u03b9_apply<\\/a>, <a>GradedAlgebra.\\u03b9_apply<\\/a> R M,\\n <a>DirectSum.of_mul_of<\\/a>]\"",
607
+ "[{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"ExteriorAlgebra.GradedAlgebra.lift\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Grading.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,24]},{\"full_name\":\"ExteriorAlgebra.lift_\\u03b9_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[119,9],\"def_end_pos\":[119,21]},{\"full_name\":\"ExteriorAlgebra.GradedAlgebra.\\u03b9_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Grading.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,30]},{\"full_name\":\"DirectSum.of_mul_of\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Ring.lean\",\"def_pos\":[216,9],\"def_end_pos\":[216,18]}]"
608
+ ],
609
+ "state_after": "case mk.mem_mul.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nw\u271d : M\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) (1 + i\u271d)) (GradedMonoid.GMul.mul \u27e8(\u03b9 R) w\u271d, \u22ef\u27e9 \u27e8x\u271d, hx\u271d\u27e9) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d.succ) \u27e8(\u03b9 R) w\u271d * x\u271d, \u22ef\u27e9",
610
+ "state_before": "case mk.mem_mul.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nw\u271d : M\n\u22a2 (lift\u03b9 R M) ((\u03b9 R) w\u271d * x\u271d) = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d.succ) \u27e8(\u03b9 R) w\u271d * x\u271d, \u22ef\u27e9",
611
+ "tactic": "simp_rw [AlgHom.map_mul, ih, GradedAlgebra.lift\u03b9, lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M,\n DirectSum.of_mul_of]"
612
+ },
613
+ {
614
+ "annotated_tactic": [
615
+ "\"exact <a>DirectSum.of_eq_of_gradedMonoid_eq<\\/a> (<a>Sigma.subtype_ext<\\/a> (<a>add_comm<\\/a> _ _) <a>rfl<\\/a>)\"",
616
+ "[{\"full_name\":\"DirectSum.of_eq_of_gradedMonoid_eq\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Ring.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,33]},{\"full_name\":\"Sigma.subtype_ext\",\"def_path\":\"Mathlib\\/Data\\/Sigma\\/Basic.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,20]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
617
+ ],
618
+ "state_after": "no goals",
619
+ "state_before": "case mk.mem_mul.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih : (lift\u03b9 R M) x\u271d = (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d) \u27e8x\u271d, hx\u271d\u27e9\nw\u271d : M\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) (1 + i\u271d)) (GradedMonoid.GMul.mul \u27e8(\u03b9 R) w\u271d, \u22ef\u27e9 \u27e8x\u271d, hx\u271d\u27e9) =\n (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) i\u271d.succ) \u27e8(\u03b9 R) w\u271d * x\u271d, \u22ef\u27e9",
620
+ "tactic": "exact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)"
621
+ }
622
+ ],
623
+ "url": "https://github.com/leanprover-community/mathlib4"
624
+ },
625
+ "source_file": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean"
626
+ },
627
+ {
628
+ "full_name": "ExteriorAlgebra.GradedAlgebra.\u03b9_apply",
629
+ "id": "ExteriorAlgebra.GradedAlgebra.\u03b9_apply",
630
+ "reason": "missing_trace",
631
+ "selection": {
632
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
633
+ "end": [
634
+ 40,
635
+ 6
636
+ ],
637
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean",
638
+ "full_name": "ExteriorAlgebra.GradedAlgebra.\u03b9_apply",
639
+ "start": [
640
+ 36,
641
+ 1
642
+ ],
643
+ "traced_tactics": [
644
+ {
645
+ "annotated_tactic": [
646
+ "\"simpa only [<a>pow_one<\\/a>] using <a>LinearMap.mem_range_self<\\/a> _ m\"",
647
+ "[{\"full_name\":\"pow_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[663,7],\"def_end_pos\":[663,14]},{\"full_name\":\"LinearMap.mem_range_self\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,23]}]"
648
+ ],
649
+ "state_after": "no goals",
650
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 (\u03b9 R) m \u2208 \u22c0[R]^1 M",
651
+ "tactic": "simpa only [pow_one] using LinearMap.mem_range_self _ m"
652
+ }
653
+ ],
654
+ "url": "https://github.com/leanprover-community/mathlib4"
655
+ },
656
+ "source_file": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean"
657
+ },
658
+ {
659
+ "full_name": "ExteriorAlgebra.GradedAlgebra.\u03b9_sq_zero",
660
+ "id": "ExteriorAlgebra.GradedAlgebra.\u03b9_sq_zero",
661
+ "reason": "missing_trace",
662
+ "selection": {
663
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
664
+ "end": [
665
+ 54,
666
+ 81
667
+ ],
668
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean",
669
+ "full_name": "ExteriorAlgebra.GradedAlgebra.\u03b9_sq_zero",
670
+ "start": [
671
+ 52,
672
+ 1
673
+ ],
674
+ "traced_tactics": [
675
+ {
676
+ "annotated_tactic": [
677
+ "\"rw [<a>GradedAlgebra.\\u03b9_apply<\\/a>, <a>DirectSum.of_mul_of<\\/a>]\"",
678
+ "[{\"full_name\":\"ExteriorAlgebra.GradedAlgebra.\\u03b9_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Grading.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,30]},{\"full_name\":\"DirectSum.of_mul_of\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Ring.lean\",\"def_pos\":[216,9],\"def_end_pos\":[216,18]}]"
679
+ ],
680
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) (1 + 1)) (GradedMonoid.GMul.mul \u27e8(\u03b9 R) m, \u22ef\u27e9 \u27e8(\u03b9 R) m, \u22ef\u27e9) = 0",
681
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 (GradedAlgebra.\u03b9 R M) m * (GradedAlgebra.\u03b9 R M) m = 0",
682
+ "tactic": "rw [GradedAlgebra.\u03b9_apply, DirectSum.of_mul_of]"
683
+ },
684
+ {
685
+ "annotated_tactic": [
686
+ "\"exact DFinsupp.single_eq_zero.mpr (<a>Subtype.ext<\\/a> <| <a>ExteriorAlgebra.\\u03b9_sq_zero<\\/a> _)\"",
687
+ "[{\"full_name\":\"Subtype.ext\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[74,19],\"def_end_pos\":[74,22]},{\"full_name\":\"ExteriorAlgebra.\\u03b9_sq_zero\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,18]}]"
688
+ ],
689
+ "state_after": "no goals",
690
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 (DirectSum.of (fun i => \u21a5(\u22c0[R]^i M)) (1 + 1)) (GradedMonoid.GMul.mul \u27e8(\u03b9 R) m, \u22ef\u27e9 \u27e8(\u03b9 R) m, \u22ef\u27e9) = 0",
691
+ "tactic": "exact DFinsupp.single_eq_zero.mpr (Subtype.ext <| ExteriorAlgebra.\u03b9_sq_zero _)"
692
+ }
693
+ ],
694
+ "url": "https://github.com/leanprover-community/mathlib4"
695
+ },
696
+ "source_file": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean"
697
+ },
698
+ {
699
+ "full_name": "ExteriorAlgebra.\u03b9Multi_span",
700
+ "id": "ExteriorAlgebra.\u03b9Multi_span",
701
+ "reason": "missing_trace",
702
+ "selection": {
703
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
704
+ "end": [
705
+ 112,
706
+ 24
707
+ ],
708
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean",
709
+ "full_name": "ExteriorAlgebra.\u03b9Multi_span",
710
+ "start": [
711
+ 97,
712
+ 1
713
+ ],
714
+ "traced_tactics": [
715
+ {
716
+ "annotated_tactic": [
717
+ "\"rw [<a>Submodule.eq_top_iff'<\\/a>]\"",
718
+ "[{\"full_name\":\"Submodule.eq_top_iff'\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Lattice.lean\",\"def_pos\":[168,9],\"def_end_pos\":[168,20]}]"
719
+ ],
720
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 \u2200 (x : ExteriorAlgebra R M), x \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
721
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd) = \u22a4",
722
+ "tactic": "rw [Submodule.eq_top_iff']"
723
+ },
724
+ {
725
+ "annotated_tactic": [
726
+ "\"intro x\"",
727
+ "[]"
728
+ ],
729
+ "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx : ExteriorAlgebra R M\n\u22a2 x \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
730
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 \u2200 (x : ExteriorAlgebra R M), x \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
731
+ "tactic": "intro x"
732
+ },
733
+ {
734
+ "annotated_tactic": [
735
+ "\"induction x using <a>DirectSum.Decomposition.inductionOn<\\/a> fun i => \\u22c0[R]^i M with\\n | h_zero => exact <a>Submodule.zero_mem<\\/a> _\\n | h_add _ _ hm hm' => exact <a>Submodule.add_mem<\\/a> _ hm hm'\\n | h_homogeneous hm =>\\n let \\u27e8m, hm\\u27e9 := hm\\n apply <a>Set.mem_of_mem_of_subset<\\/a> hm\\n rw [\\u2190 <a>\\u03b9Multi_span_fixedDegree<\\/a>]\\n refine <a>Submodule.span_mono<\\/a> fun _ hx \\u21a6 ?_\\n obtain \\u27e8y, rfl\\u27e9 := hx\\n exact \\u27e8\\u27e8_, y\\u27e9, <a>rfl<\\/a>\\u27e9\"",
736
+ "[{\"full_name\":\"DirectSum.Decomposition.inductionOn\",\"def_path\":\"Mathlib\\/Algebra\\/DirectSum\\/Decomposition.lean\",\"def_pos\":[99,19],\"def_end_pos\":[99,44]},{\"full_name\":\"Submodule.zero_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[216,19],\"def_end_pos\":[216,27]},{\"full_name\":\"Submodule.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[220,19],\"def_end_pos\":[220,26]},{\"full_name\":\"Set.mem_of_mem_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,29]},{\"full_name\":\"ExteriorAlgebra.\\u03b9Multi_span_fixedDegree\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[363,7],\"def_end_pos\":[363,30]},{\"full_name\":\"Submodule.span_mono\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,18]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
737
+ ],
738
+ "state_after": "no goals",
739
+ "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx : ExteriorAlgebra R M\n\u22a2 x \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
740
+ "tactic": "induction x using DirectSum.Decomposition.inductionOn fun i => \u22c0[R]^i M with\n| h_zero => exact Submodule.zero_mem _\n| h_add _ _ hm hm' => exact Submodule.add_mem _ hm hm'\n| h_homogeneous hm =>\n let \u27e8m, hm\u27e9 := hm\n apply Set.mem_of_mem_of_subset hm\n rw [\u2190 \u03b9Multi_span_fixedDegree]\n refine Submodule.span_mono fun _ hx \u21a6 ?_\n obtain \u27e8y, rfl\u27e9 := hx\n exact \u27e8\u27e8_, y\u27e9, rfl\u27e9"
741
+ },
742
+ {
743
+ "annotated_tactic": [
744
+ "\"exact <a>Submodule.zero_mem<\\/a> _\"",
745
+ "[{\"full_name\":\"Submodule.zero_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[216,19],\"def_end_pos\":[216,27]}]"
746
+ ],
747
+ "state_after": "no goals",
748
+ "state_before": "case h_zero\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 0 \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
749
+ "tactic": "exact Submodule.zero_mem _"
750
+ },
751
+ {
752
+ "annotated_tactic": [
753
+ "\"exact <a>Submodule.add_mem<\\/a> _ hm hm'\"",
754
+ "[{\"full_name\":\"Submodule.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[220,19],\"def_end_pos\":[220,26]}]"
755
+ ],
756
+ "state_after": "no goals",
757
+ "state_before": "case h_add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm\u271d m'\u271d : ExteriorAlgebra R M\nhm : m\u271d \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)\nhm' : m'\u271d \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)\n\u22a2 m\u271d + m'\u271d \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
758
+ "tactic": "exact Submodule.add_mem _ hm hm'"
759
+ },
760
+ {
761
+ "annotated_tactic": [
762
+ "\"let \\u27e8m, hm\\u27e9 := hm\"",
763
+ "[]"
764
+ ],
765
+ "state_after": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\n\u22a2 \u2191\u27e8m, hm\u27e9 \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
766
+ "state_before": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm : \u21a5(\u22c0[R]^i\u271d M)\n\u22a2 \u2191hm \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
767
+ "tactic": "let \u27e8m, hm\u27e9 := hm"
768
+ },
769
+ {
770
+ "annotated_tactic": [
771
+ "\"apply <a>Set.mem_of_mem_of_subset<\\/a> hm\"",
772
+ "[{\"full_name\":\"Set.mem_of_mem_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,29]}]"
773
+ ],
774
+ "state_after": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\n\u22a2 \u2191(\u22c0[R]^i\u271d M) \u2286 \u2191(Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd))",
775
+ "state_before": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\n\u22a2 \u2191\u27e8m, hm\u27e9 \u2208 Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd)",
776
+ "tactic": "apply Set.mem_of_mem_of_subset hm"
777
+ },
778
+ {
779
+ "annotated_tactic": [
780
+ "\"rw [\\u2190 <a>\\u03b9Multi_span_fixedDegree<\\/a>]\"",
781
+ "[{\"full_name\":\"ExteriorAlgebra.\\u03b9Multi_span_fixedDegree\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[363,7],\"def_end_pos\":[363,30]}]"
782
+ ],
783
+ "state_after": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\n\u22a2 \u2191(Submodule.span R (Set.range \u21d1(\u03b9Multi R i\u271d))) \u2286 \u2191(Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd))",
784
+ "state_before": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\n\u22a2 \u2191(\u22c0[R]^i\u271d M) \u2286 \u2191(Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd))",
785
+ "tactic": "rw [\u2190 \u03b9Multi_span_fixedDegree]"
786
+ },
787
+ {
788
+ "annotated_tactic": [
789
+ "\"refine <a>Submodule.span_mono<\\/a> fun _ hx \\u21a6 ?_\"",
790
+ "[{\"full_name\":\"Submodule.span_mono\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,18]}]"
791
+ ],
792
+ "state_after": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\nx\u271d : ExteriorAlgebra R M\nhx : x\u271d \u2208 Set.range \u21d1(\u03b9Multi R i\u271d)\n\u22a2 x\u271d \u2208 Set.range fun x => (\u03b9Multi R x.fst) x.snd",
793
+ "state_before": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\n\u22a2 \u2191(Submodule.span R (Set.range \u21d1(\u03b9Multi R i\u271d))) \u2286 \u2191(Submodule.span R (Set.range fun x => (\u03b9Multi R x.fst) x.snd))",
794
+ "tactic": "refine Submodule.span_mono fun _ hx \u21a6 ?_"
795
+ },
796
+ {
797
+ "annotated_tactic": [
798
+ "\"obtain \\u27e8y, rfl\\u27e9 := hx\"",
799
+ "[]"
800
+ ],
801
+ "state_after": "case h_homogeneous.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\ny : Fin i\u271d \u2192 M\n\u22a2 (\u03b9Multi R i\u271d) y \u2208 Set.range fun x => (\u03b9Multi R x.fst) x.snd",
802
+ "state_before": "case h_homogeneous\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\nx\u271d : ExteriorAlgebra R M\nhx : x\u271d \u2208 Set.range \u21d1(\u03b9Multi R i\u271d)\n\u22a2 x\u271d \u2208 Set.range fun x => (\u03b9Multi R x.fst) x.snd",
803
+ "tactic": "obtain \u27e8y, rfl\u27e9 := hx"
804
+ },
805
+ {
806
+ "annotated_tactic": [
807
+ "\"exact \\u27e8\\u27e8_, y\\u27e9, <a>rfl<\\/a>\\u27e9\"",
808
+ "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
809
+ ],
810
+ "state_after": "no goals",
811
+ "state_before": "case h_homogeneous.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni\u271d : \u2115\nhm\u271d : \u21a5(\u22c0[R]^i\u271d M)\nm : ExteriorAlgebra R M\nhm : m \u2208 \u22c0[R]^i\u271d M\ny : Fin i\u271d \u2192 M\n\u22a2 (\u03b9Multi R i\u271d) y \u2208 Set.range fun x => (\u03b9Multi R x.fst) x.snd",
812
+ "tactic": "exact \u27e8\u27e8_, y\u27e9, rfl\u27e9"
813
+ }
814
+ ],
815
+ "url": "https://github.com/leanprover-community/mathlib4"
816
+ },
817
+ "source_file": "Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean"
818
+ }
819
+ ],
820
+ "input_proofs": 44,
821
+ "input_units": 2,
822
+ "proofs": 17,
823
+ "published_proofs": 15,
824
+ "published_transitions": 46,
825
+ "replay_failures": [],
826
+ "transitions": 64,
827
+ "validation_failures": [
828
+ {
829
+ "errors": [
830
+ "internal transition boundary mismatch",
831
+ "internal transition boundary mismatch"
832
+ ],
833
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean",
834
+ "proof_id": "mathlib:ExteriorAlgebra.\u03b9_eq_algebraMap_iff",
835
+ "provenance": {
836
+ "selection": {
837
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
838
+ "end": [
839
+ 250,
840
+ 46
841
+ ],
842
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean",
843
+ "full_name": "ExteriorAlgebra.\u03b9_eq_algebraMap_iff",
844
+ "start": [
845
+ 241,
846
+ 1
847
+ ],
848
+ "traced_tactics": [
849
+ {
850
+ "annotated_tactic": [
851
+ "\"refine \\u27e8fun h => ?_, ?_\\u27e9\"",
852
+ "[]"
853
+ ],
854
+ "state_after": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\n\u22a2 x = 0 \u2227 r = 0\n\ncase refine_2\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\n\u22a2 x = 0 \u2227 r = 0 \u2192 (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r",
855
+ "state_before": "R : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\n\u22a2 (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r \u2194 x = 0 \u2227 r = 0",
856
+ "tactic": "refine \u27e8fun h => ?_, ?_\u27e9"
857
+ },
858
+ {
859
+ "annotated_tactic": [
860
+ "\"letI : <a>Module<\\/a> R\\u1d50\\u1d52\\u1d56 M := <a>Module.compHom<\\/a> _ ((<a>RingHom.id<\\/a> R).<a>fromOpposite<\\/a> <a>mul_comm<\\/a>)\"",
861
+ "[{\"full_name\":\"Module\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[57,7],\"def_end_pos\":[57,13]},{\"full_name\":\"Module.compHom\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[158,8],\"def_end_pos\":[158,22]},{\"full_name\":\"RingHom.id\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[626,5],\"def_end_pos\":[626,7]},{\"full_name\":\"RingHom.fromOpposite\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Opposite.lean\",\"def_pos\":[253,5],\"def_end_pos\":[253,25]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
862
+ ],
863
+ "state_after": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\n\u22a2 x = 0 \u2227 r = 0",
864
+ "state_before": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\n\u22a2 x = 0 \u2227 r = 0",
865
+ "tactic": "letI : Module R\u1d50\u1d52\u1d56 M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)"
866
+ },
867
+ {
868
+ "annotated_tactic": [
869
+ "\"haveI : <a>IsCentralScalar<\\/a> R M := \\u27e8fun r m => <a>rfl<\\/a>\\u27e9\"",
870
+ "[{\"full_name\":\"IsCentralScalar\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[277,7],\"def_end_pos\":[277,22]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
871
+ ],
872
+ "state_after": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis : IsCentralScalar R M\n\u22a2 x = 0 \u2227 r = 0",
873
+ "state_before": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\n\u22a2 x = 0 \u2227 r = 0",
874
+ "tactic": "haveI : IsCentralScalar R M := \u27e8fun r m => rfl\u27e9"
875
+ },
876
+ {
877
+ "annotated_tactic": [
878
+ "\"have hf0 : <a>toTrivSqZeroExt<\\/a> (<a>\\u03b9<\\/a> R x) = (0, x) := <a>toTrivSqZeroExt_\\u03b9<\\/a> _\"",
879
+ "[{\"full_name\":\"ExteriorAlgebra.toTrivSqZeroExt\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[199,5],\"def_end_pos\":[199,20]},{\"full_name\":\"ExteriorAlgebra.\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[67,8],\"def_end_pos\":[67,9]},{\"full_name\":\"ExteriorAlgebra.toTrivSqZeroExt_\\u03b9\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[205,9],\"def_end_pos\":[205,26]}]"
880
+ ],
881
+ "state_after": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis : IsCentralScalar R M\nhf0 : toTrivSqZeroExt ((\u03b9 R) x) = (0, x)\n\u22a2 x = 0 \u2227 r = 0",
882
+ "state_before": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis : IsCentralScalar R M\n\u22a2 x = 0 \u2227 r = 0",
883
+ "tactic": "have hf0 : toTrivSqZeroExt (\u03b9 R x) = (0, x) := toTrivSqZeroExt_\u03b9 _"
884
+ },
885
+ {
886
+ "annotated_tactic": [
887
+ "\"rw [h, <a>AlgHom.commutes<\\/a>] at hf0\"",
888
+ "[{\"full_name\":\"AlgHom.commutes\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,17]}]"
889
+ ],
890
+ "state_after": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis : IsCentralScalar R M\nhf0 : (algebraMap R (TrivSqZeroExt R M)) r = (0, x)\n\u22a2 x = 0 \u2227 r = 0",
891
+ "state_before": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis : IsCentralScalar R M\nhf0 : toTrivSqZeroExt ((\u03b9 R) x) = (0, x)\n\u22a2 x = 0 \u2227 r = 0",
892
+ "tactic": "rw [h, AlgHom.commutes] at hf0"
893
+ },
894
+ {
895
+ "annotated_tactic": [
896
+ "\"have : r = 0 \\u2227 0 = x := <a>Prod.ext_iff<\\/a>.1 hf0\"",
897
+ "[{\"full_name\":\"Prod.ext_iff\",\"def_path\":\"Mathlib\\/Data\\/Prod\\/Basic.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,16]}]"
898
+ ],
899
+ "state_after": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d\u00b9 : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis\u271d : IsCentralScalar R M\nhf0 : (algebraMap R (TrivSqZeroExt R M)) r = (0, x)\nthis : r = 0 \u2227 0 = x\n\u22a2 x = 0 \u2227 r = 0",
900
+ "state_before": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis : IsCentralScalar R M\nhf0 : (algebraMap R (TrivSqZeroExt R M)) r = (0, x)\n\u22a2 x = 0 \u2227 r = 0",
901
+ "tactic": "have : r = 0 \u2227 0 = x := Prod.ext_iff.1 hf0"
902
+ },
903
+ {
904
+ "annotated_tactic": [
905
+ "\"exact this.symm.imp_left <a>Eq.symm<\\/a>\"",
906
+ "[{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
907
+ ],
908
+ "state_after": "no goals",
909
+ "state_before": "case refine_1\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nh : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\nthis\u271d\u00b9 : Module R\u1d50\u1d52\u1d56 M := Module.compHom M ((RingHom.id R).fromOpposite \u22ef)\nthis\u271d : IsCentralScalar R M\nhf0 : (algebraMap R (TrivSqZeroExt R M)) r = (0, x)\nthis : r = 0 \u2227 0 = x\n\u22a2 x = 0 \u2227 r = 0",
910
+ "tactic": "exact this.symm.imp_left Eq.symm"
911
+ },
912
+ {
913
+ "annotated_tactic": [
914
+ "\"rintro \\u27e8rfl, rfl\\u27e9\"",
915
+ "[]"
916
+ ],
917
+ "state_after": "case refine_2.intro\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 (\u03b9 R) 0 = (algebraMap R (ExteriorAlgebra R M)) 0",
918
+ "state_before": "case refine_2\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\n\u22a2 x = 0 \u2227 r = 0 \u2192 (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r",
919
+ "tactic": "rintro \u27e8rfl, rfl\u27e9"
920
+ },
921
+ {
922
+ "annotated_tactic": [
923
+ "\"rw [<a>LinearMap.map_zero<\\/a>, <a>RingHom.map_zero<\\/a>]\"",
924
+ "[{\"full_name\":\"LinearMap.map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[363,19],\"def_end_pos\":[363,27]},{\"full_name\":\"RingHom.map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[546,19],\"def_end_pos\":[546,27]}]"
925
+ ],
926
+ "state_after": "no goals",
927
+ "state_before": "case refine_2.intro\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 (\u03b9 R) 0 = (algebraMap R (ExteriorAlgebra R M)) 0",
928
+ "tactic": "rw [LinearMap.map_zero, RingHom.map_zero]"
929
+ }
930
+ ],
931
+ "url": "https://github.com/leanprover-community/mathlib4"
932
+ }
933
+ },
934
+ "theorem_name": "ExteriorAlgebra.\u03b9_eq_algebraMap_iff"
935
+ },
936
+ {
937
+ "errors": [
938
+ "internal transition boundary mismatch",
939
+ "internal transition boundary mismatch",
940
+ "internal transition boundary mismatch",
941
+ "internal transition boundary mismatch"
942
+ ],
943
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean",
944
+ "proof_id": "mathlib:ExteriorAlgebra.\u03b9_mul_prod_list",
945
+ "provenance": {
946
+ "selection": {
947
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
948
+ "end": [
949
+ 286,
950
+ 47
951
+ ],
952
+ "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean",
953
+ "full_name": "ExteriorAlgebra.\u03b9_mul_prod_list",
954
+ "start": [
955
+ 274,
956
+ 1
957
+ ],
958
+ "traced_tactics": [
959
+ {
960
+ "annotated_tactic": [
961
+ "\"induction' n with n hn\"",
962
+ "[]"
963
+ ],
964
+ "state_after": "case zero\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : Fin 0 \u2192 M\ni : Fin 0\n\u22a2 (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\n\ncase succ\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\n\u22a2 (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0",
965
+ "state_before": "R : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin n \u2192 M\ni : Fin n\n\u22a2 (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0",
966
+ "tactic": "induction' n with n hn"
967
+ },
968
+ {
969
+ "annotated_tactic": [
970
+ "\"exact i.elim0\"",
971
+ "[]"
972
+ ],
973
+ "state_after": "no goals",
974
+ "state_before": "case zero\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : Fin 0 \u2192 M\ni : Fin 0\n\u22a2 (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0",
975
+ "tactic": "exact i.elim0"
976
+ },
977
+ {
978
+ "annotated_tactic": [
979
+ "\"rw [<a>List.ofFn_succ<\\/a>, <a>List.prod_cons<\\/a>, \\u2190 <a>mul_assoc<\\/a>]\"",
980
+ "[{\"full_name\":\"List.ofFn_succ\",\"def_path\":\"Mathlib\\/Data\\/List\\/OfFn.lean\",\"def_pos\":[129,9],\"def_end_pos\":[129,18]},{\"full_name\":\"List.prod_cons\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/List.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,18]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
981
+ ],
982
+ "state_after": "case succ\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
983
+ "state_before": "case succ\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\n\u22a2 (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0",
984
+ "tactic": "rw [List.ofFn_succ, List.prod_cons, \u2190 mul_assoc]"
985
+ },
986
+ {
987
+ "annotated_tactic": [
988
+ "\"by_cases h : i = 0\"",
989
+ "[]"
990
+ ],
991
+ "state_after": "case pos\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : i = 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0\n\ncase neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
992
+ "state_before": "case succ\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
993
+ "tactic": "by_cases h : i = 0"
994
+ },
995
+ {
996
+ "annotated_tactic": [
997
+ "\"rw [h, <a>\\u03b9_sq_zero<\\/a>, <a>zero_mul<\\/a>]\"",
998
+ "[{\"full_name\":\"ExteriorAlgebra.\\u03b9_sq_zero\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,18]},{\"full_name\":\"MulZeroClass.zero_mul\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,11]}]"
999
+ ],
1000
+ "state_after": "no goals",
1001
+ "state_before": "case pos\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nhn : \u2200 (f : Fin n \u2192 M) (i : Fin n), (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i)).prod = 0\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : i = 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
1002
+ "tactic": "rw [h, \u03b9_sq_zero, zero_mul]"
1003
+ },
1004
+ {
1005
+ "annotated_tactic": [
1006
+ "\"simp only at hn\"",
1007
+ "[]"
1008
+ ],
1009
+ "state_after": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn : (\u03b9 R) (f 0) * ((\u03b9 R) (f (i.pred h).succ) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod) = (\u03b9 R) (f 0) * 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
1010
+ "state_before": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn :\n (fun x => (\u03b9 R) (f 0) * x) ((\u03b9 R) (f (i.pred h).succ) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod) =\n (fun x => (\u03b9 R) (f 0) * x) 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
1011
+ "tactic": "simp only at hn"
1012
+ },
1013
+ {
1014
+ "annotated_tactic": [
1015
+ "\"rw [<a>Fin.succ_pred<\\/a>, \\u2190 <a>mul_assoc<\\/a>, <a>mul_zero<\\/a>] at hn\"",
1016
+ "[{\"full_name\":\"Fin.succ_pred\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Lemmas.lean\",\"def_pos\":[521,17],\"def_end_pos\":[521,26]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
1017
+ ],
1018
+ "state_after": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn : (\u03b9 R) (f 0) * (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
1019
+ "state_before": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn : (\u03b9 R) (f 0) * ((\u03b9 R) (f (i.pred h).succ) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod) = (\u03b9 R) (f 0) * 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
1020
+ "tactic": "rw [Fin.succ_pred, \u2190 mul_assoc, mul_zero] at hn"
1021
+ },
1022
+ {
1023
+ "annotated_tactic": [
1024
+ "\"refine (<a>eq_zero_iff_eq_zero_of_add_eq_zero<\\/a> ?_).<a>mp<\\/a> hn\"",
1025
+ "[{\"full_name\":\"eq_zero_iff_eq_zero_of_add_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[159,3],\"def_end_pos\":[159,14]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
1026
+ ],
1027
+ "state_after": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn : (\u03b9 R) (f 0) * (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0\n\u22a2 (\u03b9 R) (f 0) * (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod +\n (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod =\n 0",
1028
+ "state_before": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn : (\u03b9 R) (f 0) * (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0\n\u22a2 (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0",
1029
+ "tactic": "refine (eq_zero_iff_eq_zero_of_add_eq_zero ?_).mp hn"
1030
+ },
1031
+ {
1032
+ "annotated_tactic": [
1033
+ "\"rw [\\u2190 <a>add_mul<\\/a>, <a>\\u03b9_add_mul_swap<\\/a>, <a>zero_mul<\\/a>]\"",
1034
+ "[{\"full_name\":\"add_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[91,7],\"def_end_pos\":[91,14]},{\"full_name\":\"ExteriorAlgebra.\\u03b9_add_mul_swap\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/ExteriorAlgebra\\/Basic.lean\",\"def_pos\":[270,9],\"def_end_pos\":[270,23]},{\"full_name\":\"MulZeroClass.zero_mul\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,11]}]"
1035
+ ],
1036
+ "state_after": "no goals",
1037
+ "state_before": "case neg\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nn : \u2115\nf : Fin (n + 1) \u2192 M\ni : Fin (n + 1)\nh : \u00aci = 0\nhn : (\u03b9 R) (f 0) * (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod = 0\n\u22a2 (\u03b9 R) (f 0) * (\u03b9 R) (f i) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod +\n (\u03b9 R) (f i) * (\u03b9 R) (f 0) * (List.ofFn fun i => (\u03b9 R) (f i.succ)).prod =\n 0",
1038
+ "tactic": "rw [\u2190 add_mul, \u03b9_add_mul_swap, zero_mul]"
1039
+ }
1040
+ ],
1041
+ "url": "https://github.com/leanprover-community/mathlib4"
1042
+ }
1043
+ },
1044
+ "theorem_name": "ExteriorAlgebra.\u03b9_mul_prod_list"
1045
+ }
1046
+ ],
1047
+ "validation_rejected_proofs": 2,
1048
+ "zero_transition_proofs": 0
1049
+ },
1050
+ "shards": [
1051
+ {
1052
+ "path": "data/train/mathlib-001271-0000-e1175859b49e6cfb.parquet",
1053
+ "sha256": "e1175859b49e6cfbaf56851cb0186666484f87b546d3ac7d897f69f869e10ee6",
1054
+ "bytes": 177461,
1055
+ "rows": 27,
1056
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1057
+ },
1058
+ {
1059
+ "path": "data/dev/mathlib-001271-0000-d0a7db333a3cd041.parquet",
1060
+ "sha256": "d0a7db333a3cd0416472b169d88c4b4a8ff252fd540e41e4c83edc1c72a05a12",
1061
+ "bytes": 69691,
1062
+ "rows": 2,
1063
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1064
+ },
1065
+ {
1066
+ "path": "data/internal_test/mathlib-001271-0000-8785f22bb938a06f.parquet",
1067
+ "sha256": "8785f22bb938a06f852a07f44806175cf71449ca8d0b3227cae56decd8337ac8",
1068
+ "bytes": 169974,
1069
+ "rows": 17,
1070
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
1071
+ }
1072
+ ],
1073
+ "proof_ids": [
1074
+ "mathlib:ExteriorAlgebra.algebraMap_leftInverse",
1075
+ "mathlib:ExteriorAlgebra.comp_\u03b9_sq_zero",
1076
+ "mathlib:ExteriorAlgebra.leftInverse_map_iff",
1077
+ "mathlib:ExteriorAlgebra.map_apply_\u03b9Multi",
1078
+ "mathlib:ExteriorAlgebra.map_comp_\u03b9Multi",
1079
+ "mathlib:ExteriorAlgebra.map_surjective_iff",
1080
+ "mathlib:ExteriorAlgebra.toTrivSqZeroExt_comp_map",
1081
+ "mathlib:ExteriorAlgebra.\u03b9Inv_comp_map",
1082
+ "mathlib:ExteriorAlgebra.\u03b9Multi_range",
1083
+ "mathlib:ExteriorAlgebra.\u03b9Multi_span_fixedDegree",
1084
+ "mathlib:ExteriorAlgebra.\u03b9Multi_succ_apply",
1085
+ "mathlib:ExteriorAlgebra.\u03b9Multi_succ_curryLeft",
1086
+ "mathlib:ExteriorAlgebra.\u03b9_eq_zero_iff",
1087
+ "mathlib:ExteriorAlgebra.\u03b9_leftInverse",
1088
+ "mathlib:TensorAlgebra.toExterior_\u03b9"
1089
+ ],
1090
+ "proof_keys": [
1091
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.algebraMap_leftInverse\"]",
1092
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.comp_\u03b9_sq_zero\"]",
1093
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.leftInverse_map_iff\"]",
1094
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.map_apply_\u03b9Multi\"]",
1095
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.map_comp_\u03b9Multi\"]",
1096
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.map_surjective_iff\"]",
1097
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.toTrivSqZeroExt_comp_map\"]",
1098
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9Inv_comp_map\"]",
1099
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9Multi_range\"]",
1100
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9Multi_span_fixedDegree\"]",
1101
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9Multi_succ_apply\"]",
1102
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9Multi_succ_curryLeft\"]",
1103
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9_eq_zero_iff\"]",
1104
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"ExteriorAlgebra.\u03b9_leftInverse\"]",
1105
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean\",null,null,\"TensorAlgebra.toExterior_\u03b9\"]"
1106
+ ]
1107
+ }
manifests/batches/mathlib/001272.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001273.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/001274.json ADDED
@@ -0,0 +1,382 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001274",
4
+ "batch_fingerprint": "76244fb7eaec2e39b43c4463e919d807b04960f17facff0524b4a124943e6000",
5
+ "evidence": {
6
+ "batch_fingerprint": "76244fb7eaec2e39b43c4463e919d807b04960f17facff0524b4a124943e6000",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "full_name": "Basis.equivFun_symm_stdBasis",
11
+ "id": "Basis.equivFun_symm_stdBasis",
12
+ "reason": "missing_trace",
13
+ "selection": {
14
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
15
+ "end": [
16
+ 170,
17
+ 7
18
+ ],
19
+ "file_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean",
20
+ "full_name": "Basis.equivFun_symm_stdBasis",
21
+ "start": [
22
+ 167,
23
+ 1
24
+ ],
25
+ "traced_tactics": [
26
+ {
27
+ "annotated_tactic": [
28
+ "\"cases <a>nonempty_fintype<\\/a> n\"",
29
+ "[{\"full_name\":\"nonempty_fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[443,9],\"def_end_pos\":[443,25]}]"
30
+ ],
31
+ "state_after": "case intro\nR : Type u_1\nM : Type u_2\nn : Type u_3\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Finite n\nb : Basis n R M\ni : n\nval\u271d : Fintype n\n\u22a2 b.equivFun.symm ((stdBasis R (fun x => R) i) 1) = b i",
32
+ "state_before": "R : Type u_1\nM : Type u_2\nn : Type u_3\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Finite n\nb : Basis n R M\ni : n\n\u22a2 b.equivFun.symm ((stdBasis R (fun x => R) i) 1) = b i",
33
+ "tactic": "cases nonempty_fintype n"
34
+ },
35
+ {
36
+ "annotated_tactic": [
37
+ "\"simp\"",
38
+ "[]"
39
+ ],
40
+ "state_after": "no goals",
41
+ "state_before": "case intro\nR : Type u_1\nM : Type u_2\nn : Type u_3\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Finite n\nb : Basis n R M\ni : n\nval\u271d : Fintype n\n\u22a2 b.equivFun.symm ((stdBasis R (fun x => R) i) 1) = b i",
42
+ "tactic": "simp"
43
+ }
44
+ ],
45
+ "url": "https://github.com/leanprover-community/mathlib4"
46
+ },
47
+ "source_file": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean"
48
+ },
49
+ {
50
+ "full_name": "Finset.sum_single_ite",
51
+ "id": "Finset.sum_single_ite",
52
+ "reason": "missing_trace",
53
+ "selection": {
54
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
55
+ "end": [
56
+ 164,
57
+ 32
58
+ ],
59
+ "file_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean",
60
+ "full_name": "Finset.sum_single_ite",
61
+ "start": [
62
+ 161,
63
+ 1
64
+ ],
65
+ "traced_tactics": [
66
+ {
67
+ "annotated_tactic": [
68
+ "\"simp only [<a>apply_ite<\\/a> (<a>Finsupp.single<\\/a> _), <a>Finsupp.single_zero<\\/a>, <a>Finset.sum_ite_eq<\\/a>,\\n <a>if_pos<\\/a> (<a>Finset.mem_univ<\\/a> _)]\"",
69
+ "[{\"full_name\":\"apply_ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,18]},{\"full_name\":\"Finsupp.single\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[270,5],\"def_end_pos\":[270,11]},{\"full_name\":\"Finsupp.single_zero\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,20]},{\"full_name\":\"Finset.sum_ite_eq\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1319,3],\"def_end_pos\":[1319,14]},{\"full_name\":\"if_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[932,9],\"def_end_pos\":[932,15]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]}]"
70
+ ],
71
+ "state_after": "no goals",
72
+ "state_before": "R : Type u_1\nM : Type u_2\nn : Type u_3\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Fintype n\na : R\ni : n\n\u22a2 \u2211 x : n, Finsupp.single x (if i = x then a else 0) = Finsupp.single i a",
73
+ "tactic": "simp only [apply_ite (Finsupp.single _), Finsupp.single_zero, Finset.sum_ite_eq,\n if_pos (Finset.mem_univ _)]"
74
+ }
75
+ ],
76
+ "url": "https://github.com/leanprover-community/mathlib4"
77
+ },
78
+ "source_file": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean"
79
+ },
80
+ {
81
+ "full_name": "Finsupp.basis_repr",
82
+ "id": "Finsupp.basis_repr",
83
+ "reason": "missing_trace",
84
+ "selection": {
85
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
86
+ "end": [
87
+ 102,
88
+ 6
89
+ ],
90
+ "file_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean",
91
+ "full_name": "Finsupp.basis_repr",
92
+ "start": [
93
+ 100,
94
+ 1
95
+ ],
96
+ "traced_tactics": [],
97
+ "url": "https://github.com/leanprover-community/mathlib4"
98
+ },
99
+ "source_file": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean"
100
+ },
101
+ {
102
+ "full_name": "Finsupp.coe_basis",
103
+ "id": "Finsupp.coe_basis",
104
+ "reason": "missing_trace",
105
+ "selection": {
106
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
107
+ "end": [
108
+ 119,
109
+ 20
110
+ ],
111
+ "file_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean",
112
+ "full_name": "Finsupp.coe_basis",
113
+ "start": [
114
+ 106,
115
+ 1
116
+ ],
117
+ "traced_tactics": [
118
+ {
119
+ "annotated_tactic": [
120
+ "\"ext \\u27e8j, y\\u27e9\"",
121
+ "[]"
122
+ ],
123
+ "state_after": "case h.mk\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
124
+ "state_before": "R : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\n\u22a2 (Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd)) = single \u27e8i, x\u27e9 1",
125
+ "tactic": "ext \u27e8j, y\u27e9"
126
+ },
127
+ {
128
+ "annotated_tactic": [
129
+ "\"by_cases h : i = j\"",
130
+ "[]"
131
+ ],
132
+ "state_after": "case pos\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\nh : i = j\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9\n\ncase neg\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\nh : \u00aci = j\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
133
+ "state_before": "case h.mk\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
134
+ "tactic": "by_cases h : i = j"
135
+ },
136
+ {
137
+ "annotated_tactic": [
138
+ "\"cases h\"",
139
+ "[]"
140
+ ],
141
+ "state_after": "case pos.refl\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx y : \u03c6 i\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8i, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8i, y\u27e9",
142
+ "state_before": "case pos\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\nh : i = j\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
143
+ "tactic": "cases h"
144
+ },
145
+ {
146
+ "annotated_tactic": [
147
+ "\"simp only [<a>basis_repr<\\/a>, <a>single_eq_same<\\/a>, <a>Basis.repr_self<\\/a>,\\n <a>Finsupp.single_apply_left<\\/a> <a>sigma_mk_injective<\\/a>]\"",
148
+ "[{\"full_name\":\"Finsupp.basis_repr\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FinsuppVectorSpace.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,19]},{\"full_name\":\"Finsupp.single_eq_same\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[304,9],\"def_end_pos\":[304,23]},{\"full_name\":\"Basis.repr_self\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,18]},{\"full_name\":\"Finsupp.single_apply_left\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[293,9],\"def_end_pos\":[293,26]},{\"full_name\":\"sigma_mk_injective\",\"def_path\":\"Mathlib\\/Data\\/Sigma\\/Basic.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,34]}]"
149
+ ],
150
+ "state_after": "no goals",
151
+ "state_before": "case pos.refl\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx y : \u03c6 i\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8i, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8i, y\u27e9",
152
+ "tactic": "simp only [basis_repr, single_eq_same, Basis.repr_self,\n Finsupp.single_apply_left sigma_mk_injective]"
153
+ },
154
+ {
155
+ "annotated_tactic": [
156
+ "\"have : <a>Sigma.mk<\\/a> i x \\u2260 <a>Sigma.mk<\\/a> j y := fun h' => h <| <a>congrArg<\\/a> (fun s => s.fst) h'\"",
157
+ "[{\"full_name\":\"Sigma.mk\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[173,3],\"def_end_pos\":[173,5]},{\"full_name\":\"Sigma.mk\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[173,3],\"def_end_pos\":[173,5]},{\"full_name\":\"congrArg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[362,9],\"def_end_pos\":[362,17]}]"
158
+ ],
159
+ "state_after": "case neg\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\nh : \u00aci = j\nthis : \u27e8i, x\u27e9 \u2260 \u27e8j, y\u27e9\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
160
+ "state_before": "case neg\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\nh : \u00aci = j\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
161
+ "tactic": "have : Sigma.mk i x \u2260 Sigma.mk j y := fun h' => h <| congrArg (fun s => s.fst) h'"
162
+ },
163
+ {
164
+ "annotated_tactic": [
165
+ "\"simp only [<a>basis_repr<\\/a>, <a>single_apply<\\/a>, h, this, <a>false_and_iff<\\/a>, <a>if_false<\\/a>, <a>LinearEquiv.map_zero<\\/a>,\\n <a>zero_apply<\\/a>]\"",
166
+ "[{\"full_name\":\"Finsupp.basis_repr\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FinsuppVectorSpace.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,19]},{\"full_name\":\"Finsupp.single_apply\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[287,9],\"def_end_pos\":[287,21]},{\"full_name\":\"false_and_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,22]},{\"full_name\":\"if_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[26,17],\"def_end_pos\":[26,25]},{\"full_name\":\"LinearEquiv.map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[496,19],\"def_end_pos\":[496,27]},{\"full_name\":\"Finsupp.zero_apply\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[157,9],\"def_end_pos\":[157,19]}]"
167
+ ],
168
+ "state_after": "no goals",
169
+ "state_before": "case neg\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nb : (i : \u03b9) \u2192 Basis (\u03c6 i) R M\nx\u271d : (i : \u03b9) \u00d7 \u03c6 i\ni : \u03b9\nx : \u03c6 i\nj : \u03b9\ny : \u03c6 j\nh : \u00aci = j\nthis : \u27e8i, x\u27e9 \u2260 \u27e8j, y\u27e9\n\u22a2 ((Finsupp.basis b).repr (single \u27e8i, x\u27e9.fst ((b \u27e8i, x\u27e9.fst) \u27e8i, x\u27e9.snd))) \u27e8j, y\u27e9 = (single \u27e8i, x\u27e9 1) \u27e8j, y\u27e9",
170
+ "tactic": "simp only [basis_repr, single_apply, h, this, false_and_iff, if_false, LinearEquiv.map_zero,\nzero_apply]"
171
+ }
172
+ ],
173
+ "url": "https://github.com/leanprover-community/mathlib4"
174
+ },
175
+ "source_file": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean"
176
+ },
177
+ {
178
+ "full_name": "Finsupp.coe_basisSingleOne",
179
+ "id": "Finsupp.coe_basisSingleOne",
180
+ "reason": "missing_trace",
181
+ "selection": {
182
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
183
+ "end": [
184
+ 130,
185
+ 45
186
+ ],
187
+ "file_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean",
188
+ "full_name": "Finsupp.coe_basisSingleOne",
189
+ "start": [
190
+ 129,
191
+ 1
192
+ ],
193
+ "traced_tactics": [],
194
+ "url": "https://github.com/leanprover-community/mathlib4"
195
+ },
196
+ "source_file": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean"
197
+ },
198
+ {
199
+ "full_name": "Finsupp.linearIndependent_single",
200
+ "id": "Finsupp.linearIndependent_single",
201
+ "reason": "missing_trace",
202
+ "selection": {
203
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
204
+ "end": [
205
+ 51,
206
+ 50
207
+ ],
208
+ "file_path": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean",
209
+ "full_name": "Finsupp.linearIndependent_single",
210
+ "start": [
211
+ 34,
212
+ 1
213
+ ],
214
+ "traced_tactics": [
215
+ {
216
+ "annotated_tactic": [
217
+ "\"apply @<a>linearIndependent_iUnion_finite<\\/a> R _ _ _ _ \\u03b9 \\u03c6 fun i x => <a>single<\\/a> i (f i x)\"",
218
+ "[{\"full_name\":\"linearIndependent_iUnion_finite\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/LinearIndependent.lean\",\"def_pos\":[833,9],\"def_end_pos\":[833,40]},{\"full_name\":\"Finsupp.single\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[270,5],\"def_end_pos\":[270,11]}]"
219
+ ],
220
+ "state_after": "case hindep\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\n\u22a2 \u2200 (j : \u03b9), LinearIndependent R fun x => single j (f j x)\n\ncase hd\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\n\u22a2 \u2200 (i : \u03b9) (t : Set \u03b9),\n t.Finite \u2192\n i \u2209 t \u2192\n Disjoint (span R (Set.range fun x => single i (f i x))) (\u2a06 i \u2208 t, span R (Set.range fun x => single i (f i x)))",
221
+ "state_before": "R : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\n\u22a2 LinearIndependent R fun ix => single ix.fst (f ix.fst ix.snd)",
222
+ "tactic": "apply @linearIndependent_iUnion_finite R _ _ _ _ \u03b9 \u03c6 fun i x => single i (f i x)"
223
+ },
224
+ {
225
+ "annotated_tactic": [
226
+ "\"intro i\"",
227
+ "[]"
228
+ ],
229
+ "state_after": "case hindep\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\n\u22a2 LinearIndependent R fun x => single i (f i x)",
230
+ "state_before": "case hindep\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\n\u22a2 \u2200 (j : \u03b9), LinearIndependent R fun x => single j (f j x)",
231
+ "tactic": "intro i"
232
+ },
233
+ {
234
+ "annotated_tactic": [
235
+ "\"have h_disjoint : <a>Disjoint<\\/a> (<a>span<\\/a> R (<a>range<\\/a> (f i))) (<a>ker<\\/a> (<a>lsingle<\\/a> i)) := by\\n rw [<a>ker_lsingle<\\/a>]\\n exact <a>disjoint_bot_right<\\/a>\"",
236
+ "[{\"full_name\":\"Disjoint\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,13]},{\"full_name\":\"Submodule.span\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[51,5],\"def_end_pos\":[51,9]},{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"LinearMap.ker\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Ker.lean\",\"def_pos\":[60,5],\"def_end_pos\":[60,8]},{\"full_name\":\"Finsupp.lsingle\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Finsupp.lean\",\"def_pos\":[147,5],\"def_end_pos\":[147,12]},{\"full_name\":\"Finsupp.ker_lsingle\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Finsupp.lean\",\"def_pos\":[241,9],\"def_end_pos\":[241,20]},{\"full_name\":\"disjoint_bot_right\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[68,9],\"def_end_pos\":[68,27]}]"
237
+ ],
238
+ "state_after": "case hindep\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nh_disjoint : Disjoint (span R (Set.range (f i))) (ker (lsingle i))\n\u22a2 LinearIndependent R fun x => single i (f i x)",
239
+ "state_before": "case hindep\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\n\u22a2 LinearIndependent R fun x => single i (f i x)",
240
+ "tactic": "have h_disjoint : Disjoint (span R (range (f i))) (ker (lsingle i)) := by\n rw [ker_lsingle]\n exact disjoint_bot_right"
241
+ },
242
+ {
243
+ "annotated_tactic": [
244
+ "\"apply (hf i).<a>map<\\/a> h_disjoint\"",
245
+ "[{\"full_name\":\"LinearIndependent.map\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/LinearIndependent.lean\",\"def_pos\":[259,9],\"def_end_pos\":[259,30]}]"
246
+ ],
247
+ "state_after": "no goals",
248
+ "state_before": "case hindep\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nh_disjoint : Disjoint (span R (Set.range (f i))) (ker (lsingle i))\n\u22a2 LinearIndependent R fun x => single i (f i x)",
249
+ "tactic": "apply (hf i).map h_disjoint"
250
+ },
251
+ {
252
+ "annotated_tactic": [
253
+ "\"rw [<a>ker_lsingle<\\/a>]\"",
254
+ "[{\"full_name\":\"Finsupp.ker_lsingle\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Finsupp.lean\",\"def_pos\":[241,9],\"def_end_pos\":[241,20]}]"
255
+ ],
256
+ "state_after": "R : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\n\u22a2 Disjoint (span R (Set.range (f i))) \u22a5",
257
+ "state_before": "R : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\n\u22a2 Disjoint (span R (Set.range (f i))) (ker (lsingle i))",
258
+ "tactic": "rw [ker_lsingle]"
259
+ },
260
+ {
261
+ "annotated_tactic": [
262
+ "\"exact <a>disjoint_bot_right<\\/a>\"",
263
+ "[{\"full_name\":\"disjoint_bot_right\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[68,9],\"def_end_pos\":[68,27]}]"
264
+ ],
265
+ "state_after": "no goals",
266
+ "state_before": "R : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\n\u22a2 Disjoint (span R (Set.range (f i))) \u22a5",
267
+ "tactic": "exact disjoint_bot_right"
268
+ },
269
+ {
270
+ "annotated_tactic": [
271
+ "\"intro i t _ hit\"",
272
+ "[]"
273
+ ],
274
+ "state_after": "case hd\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 Disjoint (span R (Set.range fun x => single i (f i x))) (\u2a06 i \u2208 t, span R (Set.range fun x => single i (f i x)))",
275
+ "state_before": "case hd\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\n\u22a2 \u2200 (i : \u03b9) (t : Set \u03b9),\n t.Finite \u2192\n i \u2209 t \u2192\n Disjoint (span R (Set.range fun x => single i (f i x))) (\u2a06 i \u2208 t, span R (Set.range fun x => single i (f i x)))",
276
+ "tactic": "intro i t _ hit"
277
+ },
278
+ {
279
+ "annotated_tactic": [
280
+ "\"refine (<a>disjoint_lsingle_lsingle<\\/a> {i} t (<a>disjoint_singleton_left<\\/a>.2 hit)).<a>mono<\\/a> ?_ ?_\"",
281
+ "[{\"full_name\":\"Finsupp.disjoint_lsingle_lsingle\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Finsupp.lean\",\"def_pos\":[266,9],\"def_end_pos\":[266,33]},{\"full_name\":\"Set.disjoint_singleton_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1569,7],\"def_end_pos\":[1569,30]},{\"full_name\":\"Disjoint.mono\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[71,9],\"def_end_pos\":[71,22]}]"
282
+ ],
283
+ "state_after": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 span R (Set.range fun x => single i (f i x)) \u2264 \u2a06 a \u2208 {i}, LinearMap.range (lsingle a)\n\ncase hd.refine_2\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 \u2a06 i \u2208 t, span R (Set.range fun x => single i (f i x)) \u2264 \u2a06 a \u2208 t, LinearMap.range (lsingle a)",
284
+ "state_before": "case hd\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 Disjoint (span R (Set.range fun x => single i (f i x))) (\u2a06 i \u2208 t, span R (Set.range fun x => single i (f i x)))",
285
+ "tactic": "refine (disjoint_lsingle_lsingle {i} t (disjoint_singleton_left.2 hit)).mono ?_ ?_"
286
+ },
287
+ {
288
+ "annotated_tactic": [
289
+ "\"rw [<a>span_le<\\/a>]\"",
290
+ "[{\"full_name\":\"Submodule.span_le\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,16]}]"
291
+ ],
292
+ "state_after": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 \u2191(\u2a06 a \u2208 {i}, LinearMap.range (lsingle a))",
293
+ "state_before": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 span R (Set.range fun x => single i (f i x)) \u2264 \u2a06 a \u2208 {i}, LinearMap.range (lsingle a)",
294
+ "tactic": "rw [span_le]"
295
+ },
296
+ {
297
+ "annotated_tactic": [
298
+ "\"simp only [<a>iSup_singleton<\\/a>]\"",
299
+ "[{\"full_name\":\"iSup_singleton\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[1426,9],\"def_end_pos\":[1426,23]}]"
300
+ ],
301
+ "state_after": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 \u2191(LinearMap.range (lsingle i))",
302
+ "state_before": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 \u2191(\u2a06 a \u2208 {i}, LinearMap.range (lsingle a))",
303
+ "tactic": "simp only [iSup_singleton]"
304
+ },
305
+ {
306
+ "annotated_tactic": [
307
+ "\"rw [<a>range_coe<\\/a>]\"",
308
+ "[{\"full_name\":\"LinearMap.range_coe\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,18]}]"
309
+ ],
310
+ "state_after": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 Set.range \u21d1(lsingle i)",
311
+ "state_before": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 \u2191(LinearMap.range (lsingle i))",
312
+ "tactic": "rw [range_coe]"
313
+ },
314
+ {
315
+ "annotated_tactic": [
316
+ "\"apply <a>range_comp_subset_range<\\/a> _ (<a>lsingle<\\/a> i)\"",
317
+ "[{\"full_name\":\"Set.range_comp_subset_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[748,9],\"def_end_pos\":[748,32]},{\"full_name\":\"Finsupp.lsingle\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Finsupp.lean\",\"def_pos\":[147,5],\"def_end_pos\":[147,12]}]"
318
+ ],
319
+ "state_after": "no goals",
320
+ "state_before": "case hd.refine_1\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 Set.range \u21d1(lsingle i)",
321
+ "tactic": "apply range_comp_subset_range _ (lsingle i)"
322
+ },
323
+ {
324
+ "annotated_tactic": [
325
+ "\"refine <a>iSup\\u2082_mono<\\/a> fun i hi => ?_\"",
326
+ "[{\"full_name\":\"iSup\\u2082_mono\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[794,9],\"def_end_pos\":[794,19]}]"
327
+ ],
328
+ "state_after": "case hd.refine_2\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni\u271d : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i\u271d \u2209 t\ni : \u03b9\nhi : i \u2208 t\n\u22a2 span R (Set.range fun x => single i (f i x)) \u2264 LinearMap.range (lsingle i)",
329
+ "state_before": "case hd.refine_2\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i \u2209 t\n\u22a2 \u2a06 i \u2208 t, span R (Set.range fun x => single i (f i x)) \u2264 \u2a06 a \u2208 t, LinearMap.range (lsingle a)",
330
+ "tactic": "refine iSup\u2082_mono fun i hi => ?_"
331
+ },
332
+ {
333
+ "annotated_tactic": [
334
+ "\"rw [<a>span_le<\\/a>, <a>range_coe<\\/a>]\"",
335
+ "[{\"full_name\":\"Submodule.span_le\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,16]},{\"full_name\":\"LinearMap.range_coe\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,18]}]"
336
+ ],
337
+ "state_after": "case hd.refine_2\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni\u271d : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i\u271d \u2209 t\ni : \u03b9\nhi : i \u2208 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 Set.range \u21d1(lsingle i)",
338
+ "state_before": "case hd.refine_2\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni\u271d : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i\u271d \u2209 t\ni : \u03b9\nhi : i \u2208 t\n\u22a2 span R (Set.range fun x => single i (f i x)) \u2264 LinearMap.range (lsingle i)",
339
+ "tactic": "rw [span_le, range_coe]"
340
+ },
341
+ {
342
+ "annotated_tactic": [
343
+ "\"apply <a>range_comp_subset_range<\\/a> _ (<a>lsingle<\\/a> i)\"",
344
+ "[{\"full_name\":\"Set.range_comp_subset_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[748,9],\"def_end_pos\":[748,32]},{\"full_name\":\"Finsupp.lsingle\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Finsupp.lean\",\"def_pos\":[147,5],\"def_end_pos\":[147,12]}]"
345
+ ],
346
+ "state_after": "no goals",
347
+ "state_before": "case hd.refine_2\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c6 : \u03b9 \u2192 Type u_4\nf : (\u03b9 : \u03b9) \u2192 \u03c6 \u03b9 \u2192 M\nhf : \u2200 (i : \u03b9), LinearIndependent R (f i)\ni\u271d : \u03b9\nt : Set \u03b9\na\u271d : t.Finite\nhit : i\u271d \u2209 t\ni : \u03b9\nhi : i \u2208 t\n\u22a2 (Set.range fun x => single i (f i x)) \u2286 Set.range \u21d1(lsingle i)",
348
+ "tactic": "apply range_comp_subset_range _ (lsingle i)"
349
+ }
350
+ ],
351
+ "url": "https://github.com/leanprover-community/mathlib4"
352
+ },
353
+ "source_file": "Mathlib/LinearAlgebra/FinsuppVectorSpace.lean"
354
+ }
355
+ ],
356
+ "input_proofs": 7,
357
+ "input_units": 2,
358
+ "proofs": 1,
359
+ "published_proofs": 1,
360
+ "published_transitions": 1,
361
+ "replay_failures": [],
362
+ "transitions": 1,
363
+ "validation_failures": [],
364
+ "validation_rejected_proofs": 0,
365
+ "zero_transition_proofs": 0
366
+ },
367
+ "shards": [
368
+ {
369
+ "path": "data/train/mathlib-001274-0000-c19078baa7f702c3.parquet",
370
+ "sha256": "c19078baa7f702c3e53a5582f546d0926058ee70d6471943f87ea4ab8d1f586a",
371
+ "bytes": 34086,
372
+ "rows": 1,
373
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
374
+ }
375
+ ],
376
+ "proof_ids": [
377
+ "mathlib:FreeAlgebra.rank_eq"
378
+ ],
379
+ "proof_keys": [
380
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeAlgebra.lean\",null,null,\"FreeAlgebra.rank_eq\"]"
381
+ ]
382
+ }
manifests/batches/mathlib/001275.json ADDED
@@ -0,0 +1,168 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "mathlib/001275",
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+ "batch_fingerprint": "29e952615fe22b4a27c5849aa367b07215537a5393d09efa0c13e3a7f3374970",
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+ "evidence": {
6
+ "batch_fingerprint": "29e952615fe22b4a27c5849aa367b07215537a5393d09efa0c13e3a7f3374970",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "Module.Free.infinite",
12
+ "id": "Module.Free.infinite",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 116,
18
+ 84
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean",
21
+ "full_name": "Module.Free.infinite",
22
+ "start": [
23
+ 115,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Basic.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "Module.Free.of_basis",
34
+ "id": "Module.Free.of_basis",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 61,
40
+ 58
41
+ ],
42
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean",
43
+ "full_name": "Module.Free.of_basis",
44
+ "start": [
45
+ 60,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Basic.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "Module.Free.of_equiv",
56
+ "id": "Module.Free.of_equiv",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
60
+ "end": [
61
+ 121,
62
+ 38
63
+ ],
64
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean",
65
+ "full_name": "Module.Free.of_equiv",
66
+ "start": [
67
+ 120,
68
+ 1
69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Basic.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "Module.Free.of_equiv'",
78
+ "id": "Module.Free.of_equiv'",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
82
+ "end": [
83
+ 128,
84
+ 13
85
+ ],
86
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean",
87
+ "full_name": "Module.Free.of_equiv'",
88
+ "start": [
89
+ 126,
90
+ 1
91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Basic.lean"
96
+ },
97
+ {
98
+ "error": "theorem has no selected tactic transitions",
99
+ "full_name": "Module.free_def",
100
+ "id": "Module.free_def",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 50,
106
+ 85
107
+ ],
108
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean",
109
+ "full_name": "Module.free_def",
110
+ "start": [
111
+ 45,
112
+ 1
113
+ ],
114
+ "traced_tactics": [],
115
+ "url": "https://github.com/leanprover-community/mathlib4"
116
+ },
117
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Basic.lean"
118
+ },
119
+ {
120
+ "error": "theorem has no selected tactic transitions",
121
+ "full_name": "Module.free_iff_set",
122
+ "id": "Module.free_iff_set",
123
+ "reason": "import_failure",
124
+ "selection": {
125
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
126
+ "end": [
127
+ 55,
128
+ 47
129
+ ],
130
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean",
131
+ "full_name": "Module.free_iff_set",
132
+ "start": [
133
+ 53,
134
+ 1
135
+ ],
136
+ "traced_tactics": [],
137
+ "url": "https://github.com/leanprover-community/mathlib4"
138
+ },
139
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Basic.lean"
140
+ }
141
+ ],
142
+ "input_proofs": 7,
143
+ "input_units": 2,
144
+ "proofs": 1,
145
+ "published_proofs": 1,
146
+ "published_transitions": 3,
147
+ "replay_failures": [],
148
+ "transitions": 3,
149
+ "validation_failures": [],
150
+ "validation_rejected_proofs": 0,
151
+ "zero_transition_proofs": 0
152
+ },
153
+ "shards": [
154
+ {
155
+ "path": "data/train/mathlib-001275-0000-57e3b15f98981938.parquet",
156
+ "sha256": "57e3b15f989819389aa8a13964af1c8c8d6e3eb48d534b84c1c74452b3c7c0d7",
157
+ "bytes": 61899,
158
+ "rows": 3,
159
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
160
+ }
161
+ ],
162
+ "proof_ids": [
163
+ "mathlib:LinearMap.det_zero''"
164
+ ],
165
+ "proof_keys": [
166
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Determinant.lean\",null,null,\"LinearMap.det_zero''\"]"
167
+ ]
168
+ }
manifests/batches/mathlib/001276.json ADDED
@@ -0,0 +1,239 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001276",
4
+ "batch_fingerprint": "4ec9ae6aeb8cfcdc9994b71790ef1cca860387ec27710fd8b8c8fec6a6f50cb4",
5
+ "evidence": {
6
+ "batch_fingerprint": "4ec9ae6aeb8cfcdc9994b71790ef1cca860387ec27710fd8b8c8fec6a6f50cb4",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "no internal trace for tactic 'refine \u27e8\u27e8Finset.univ.image b, ?_\u27e9\u27e9'",
11
+ "full_name": "Module.Finite.of_basis",
12
+ "id": "Module.Finite.of_basis",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 58,
18
+ 81
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean",
21
+ "full_name": "Module.Finite.of_basis",
22
+ "start": [
23
+ 53,
24
+ 1
25
+ ],
26
+ "traced_tactics": [
27
+ {
28
+ "annotated_tactic": [
29
+ "\"cases <a>nonempty_fintype<\\/a> \\u03b9\"",
30
+ "[{\"full_name\":\"nonempty_fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[443,9],\"def_end_pos\":[443,25]}]"
31
+ ],
32
+ "state_after": "case intro\nR\u271d : Type u\nM\u271d : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : CommRing R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Free R\u271d M\u271d\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R\u271d N\ninst\u271d\u2074 : Free R\u271d N\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 Finite R M",
33
+ "state_before": "R\u271d : Type u\nM\u271d : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : CommRing R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Free R\u271d M\u271d\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R\u271d N\ninst\u271d\u2074 : Free R\u271d N\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\n\u22a2 Finite R M",
34
+ "tactic": "cases nonempty_fintype \u03b9"
35
+ },
36
+ {
37
+ "annotated_tactic": [
38
+ "\"classical\\n refine \\u27e8\\u27e8Finset.univ.image b, ?_\\u27e9\\u27e9\\n simp only [<a>Set.image_univ<\\/a>, <a>Finset.coe_univ<\\/a>, <a>Finset.coe_image<\\/a>, <a>Basis.span_eq<\\/a>]\"",
39
+ "[{\"full_name\":\"Set.image_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[701,9],\"def_end_pos\":[701,19]},{\"full_name\":\"Finset.coe_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,17]},{\"full_name\":\"Finset.coe_image\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[419,9],\"def_end_pos\":[419,18]},{\"full_name\":\"Basis.span_eq\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[571,19],\"def_end_pos\":[571,26]}]"
40
+ ],
41
+ "state_after": "no goals",
42
+ "state_before": "case intro\nR\u271d : Type u\nM\u271d : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : CommRing R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Free R\u271d M\u271d\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R\u271d N\ninst\u271d\u2074 : Free R\u271d N\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 Finite R M",
43
+ "tactic": "classical\n refine \u27e8\u27e8Finset.univ.image b, ?_\u27e9\u27e9\n simp only [Set.image_univ, Finset.coe_univ, Finset.coe_image, Basis.span_eq]"
44
+ },
45
+ {
46
+ "annotated_tactic": [
47
+ "\"refine \\u27e8\\u27e8Finset.univ.image b, ?_\\u27e9\\u27e9\"",
48
+ "[]"
49
+ ],
50
+ "state_after": "case intro\nR\u271d : Type u\nM\u271d : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : CommRing R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Free R\u271d M\u271d\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R\u271d N\ninst\u271d\u2074 : Free R\u271d N\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 Submodule.span R \u2191(Finset.image (\u21d1b) Finset.univ) = \u22a4",
51
+ "state_before": "case intro\nR\u271d : Type u\nM\u271d : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : CommRing R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Free R\u271d M\u271d\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R\u271d N\ninst\u271d\u2074 : Free R\u271d N\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 Finite R M",
52
+ "tactic": "refine \u27e8\u27e8Finset.univ.image b, ?_\u27e9\u27e9"
53
+ },
54
+ {
55
+ "annotated_tactic": [
56
+ "\"simp only [<a>Set.image_univ<\\/a>, <a>Finset.coe_univ<\\/a>, <a>Finset.coe_image<\\/a>, <a>Basis.span_eq<\\/a>]\"",
57
+ "[{\"full_name\":\"Set.image_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[701,9],\"def_end_pos\":[701,19]},{\"full_name\":\"Finset.coe_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,17]},{\"full_name\":\"Finset.coe_image\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[419,9],\"def_end_pos\":[419,18]},{\"full_name\":\"Basis.span_eq\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[571,19],\"def_end_pos\":[571,26]}]"
58
+ ],
59
+ "state_after": "no goals",
60
+ "state_before": "case intro\nR\u271d : Type u\nM\u271d : Type v\nN : Type w\ninst\u271d\u00b9\u2070 : CommRing R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Free R\u271d M\u271d\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R\u271d N\ninst\u271d\u2074 : Free R\u271d N\nR : Type u_1\nM : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 Submodule.span R \u2191(Finset.image (\u21d1b) Finset.univ) = \u22a4",
61
+ "tactic": "simp only [Set.image_univ, Finset.coe_univ, Finset.coe_image, Basis.span_eq]"
62
+ }
63
+ ],
64
+ "url": "https://github.com/leanprover-community/mathlib4"
65
+ },
66
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean"
67
+ }
68
+ ],
69
+ "input_proofs": 8,
70
+ "input_units": 2,
71
+ "proofs": 7,
72
+ "published_proofs": 5,
73
+ "published_transitions": 9,
74
+ "replay_failures": [],
75
+ "transitions": 18,
76
+ "validation_failures": [
77
+ {
78
+ "errors": [
79
+ "internal transition boundary mismatch",
80
+ "internal transition boundary mismatch"
81
+ ],
82
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean",
83
+ "proof_id": "mathlib:card_algHom_le_finrank",
84
+ "provenance": {
85
+ "selection": {
86
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
87
+ "end": [
88
+ 94,
89
+ 77
90
+ ],
91
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean",
92
+ "full_name": "card_algHom_le_finrank",
93
+ "start": [
94
+ 91,
95
+ 1
96
+ ],
97
+ "traced_tactics": [
98
+ {
99
+ "annotated_tactic": [
100
+ "\"convert <a>toNat_le_toNat<\\/a> (<a>cardinal_mk_algHom_le_rank<\\/a> K M L) ?_\"",
101
+ "[{\"full_name\":\"Cardinal.toNat_le_toNat\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,23]},{\"full_name\":\"cardinal_mk_algHom_le_rank\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Finite\\/Matrix.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,35]}]"
102
+ ],
103
+ "state_after": "case h.e'_4\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 finrank K M = toNat (lift.{v, u_2} (Module.rank K M))\n\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 lift.{v, u_2} (Module.rank K M) < \u2135\u2080",
104
+ "state_before": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 Nat.card (M \u2192\u2090[K] L) \u2264 finrank K M",
105
+ "tactic": "convert toNat_le_toNat (cardinal_mk_algHom_le_rank K M L) ?_"
106
+ },
107
+ {
108
+ "annotated_tactic": [
109
+ "\"rw [<a>toNat_lift<\\/a>, <a>finrank<\\/a>]\"",
110
+ "[{\"full_name\":\"Cardinal.toNat_lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/ToNat.lean\",\"def_pos\":[164,9],\"def_end_pos\":[164,19]},{\"full_name\":\"FiniteDimensional.finrank\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Finrank.lean\",\"def_pos\":[54,19],\"def_end_pos\":[54,26]}]"
111
+ ],
112
+ "state_after": "no goals",
113
+ "state_before": "case h.e'_4\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 finrank K M = toNat (lift.{v, u_2} (Module.rank K M))",
114
+ "tactic": "rw [toNat_lift, finrank]"
115
+ },
116
+ {
117
+ "annotated_tactic": [
118
+ "\"rw [<a>lift_lt_aleph0<\\/a>]\"",
119
+ "[{\"full_name\":\"Cardinal.lift_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1294,9],\"def_end_pos\":[1294,23]}]"
120
+ ],
121
+ "state_after": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 Module.rank K M < \u2135\u2080",
122
+ "state_before": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 lift.{v, u_2} (Module.rank K M) < \u2135\u2080",
123
+ "tactic": "rw [lift_lt_aleph0]"
124
+ },
125
+ {
126
+ "annotated_tactic": [
127
+ "\"have := <a>Module.nontrivial<\\/a> K L\"",
128
+ "[{\"full_name\":\"Module.nontrivial\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[309,19],\"def_end_pos\":[309,36]}]"
129
+ ],
130
+ "state_after": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\nthis : Nontrivial K\n\u22a2 Module.rank K M < \u2135\u2080",
131
+ "state_before": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 Module.rank K M < \u2135\u2080",
132
+ "tactic": "have := Module.nontrivial K L"
133
+ },
134
+ {
135
+ "annotated_tactic": [
136
+ "\"apply <a>rank_lt_aleph0<\\/a>\"",
137
+ "[{\"full_name\":\"rank_lt_aleph0\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/StrongRankCondition.lean\",\"def_pos\":[465,9],\"def_end_pos\":[465,23]}]"
138
+ ],
139
+ "state_after": "no goals",
140
+ "state_before": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\nthis : Nontrivial K\n\u22a2 Module.rank K M < \u2135\u2080",
141
+ "tactic": "apply rank_lt_aleph0"
142
+ }
143
+ ],
144
+ "url": "https://github.com/leanprover-community/mathlib4"
145
+ }
146
+ },
147
+ "theorem_name": "card_algHom_le_finrank"
148
+ },
149
+ {
150
+ "errors": [
151
+ "internal transition boundary mismatch",
152
+ "internal transition boundary mismatch"
153
+ ],
154
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean",
155
+ "proof_id": "mathlib:cardinal_mk_algHom_le_rank",
156
+ "provenance": {
157
+ "selection": {
158
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
159
+ "end": [
160
+ 89,
161
+ 56
162
+ ],
163
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean",
164
+ "full_name": "cardinal_mk_algHom_le_rank",
165
+ "start": [
166
+ 85,
167
+ 1
168
+ ],
169
+ "traced_tactics": [
170
+ {
171
+ "annotated_tactic": [
172
+ "\"convert (<a>linearIndependent_algHom_toLinearMap<\\/a> K M L).<a>cardinal_lift_le_rank<\\/a>\"",
173
+ "[{\"full_name\":\"linearIndependent_algHom_toLinearMap\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/LinearIndependent.lean\",\"def_pos\":[1215,7],\"def_end_pos\":[1215,43]},{\"full_name\":\"LinearIndependent.cardinal_lift_le_rank\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Basic.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,30]}]"
174
+ ],
175
+ "state_after": "case h.e'_3\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 #(M \u2192\u2090[K] L) = lift.{max u_2 v, max u_2 v} #(M \u2192\u2090[K] L)\n\ncase h.e'_4\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 lift.{v, u_2} (Module.rank K M) = lift.{max u_2 v, max u_2 v} (Module.rank L (M \u2192\u2097[K] L))",
176
+ "state_before": "R : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 #(M \u2192\u2090[K] L) \u2264 lift.{v, u_2} (Module.rank K M)",
177
+ "tactic": "convert (linearIndependent_algHom_toLinearMap K M L).cardinal_lift_le_rank"
178
+ },
179
+ {
180
+ "annotated_tactic": [
181
+ "\"rw [<a>lift_id<\\/a>]\"",
182
+ "[{\"full_name\":\"Cardinal.lift_id\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,16]}]"
183
+ ],
184
+ "state_after": "no goals",
185
+ "state_before": "case h.e'_3\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 #(M \u2192\u2090[K] L) = lift.{max u_2 v, max u_2 v} #(M \u2192\u2090[K] L)",
186
+ "tactic": "rw [lift_id]"
187
+ },
188
+ {
189
+ "annotated_tactic": [
190
+ "\"have := <a>Module.nontrivial<\\/a> K L\"",
191
+ "[{\"full_name\":\"Module.nontrivial\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[309,19],\"def_end_pos\":[309,36]}]"
192
+ ],
193
+ "state_after": "case h.e'_4\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\nthis : Nontrivial K\n\u22a2 lift.{v, u_2} (Module.rank K M) = lift.{max u_2 v, max u_2 v} (Module.rank L (M \u2192\u2097[K] L))",
194
+ "state_before": "case h.e'_4\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\n\u22a2 lift.{v, u_2} (Module.rank K M) = lift.{max u_2 v, max u_2 v} (Module.rank L (M \u2192\u2097[K] L))",
195
+ "tactic": "have := Module.nontrivial K L"
196
+ },
197
+ {
198
+ "annotated_tactic": [
199
+ "\"rw [<a>lift_id<\\/a>, <a>FiniteDimensional.rank_linearMap_self<\\/a>]\"",
200
+ "[{\"full_name\":\"Cardinal.lift_id\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,16]},{\"full_name\":\"FiniteDimensional.rank_linearMap_self\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/Finite\\/Matrix.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,46]}]"
201
+ ],
202
+ "state_after": "no goals",
203
+ "state_before": "case h.e'_4\nR : Type u\nS : Type u'\nM\u271d : Type v\nN : Type w\nK : Type u_1\nM : Type u_2\nL : Type v\ninst\u271d\u2077 : CommRing K\ninst\u271d\u2076 : Ring M\ninst\u271d\u2075 : Algebra K M\ninst\u271d\u2074 : Module.Free K M\ninst\u271d\u00b3 : Module.Finite K M\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : Algebra K L\nthis : Nontrivial K\n\u22a2 lift.{v, u_2} (Module.rank K M) = lift.{max u_2 v, max u_2 v} (Module.rank L (M \u2192\u2097[K] L))",
204
+ "tactic": "rw [lift_id, FiniteDimensional.rank_linearMap_self]"
205
+ }
206
+ ],
207
+ "url": "https://github.com/leanprover-community/mathlib4"
208
+ }
209
+ },
210
+ "theorem_name": "cardinal_mk_algHom_le_rank"
211
+ }
212
+ ],
213
+ "validation_rejected_proofs": 2,
214
+ "zero_transition_proofs": 0
215
+ },
216
+ "shards": [
217
+ {
218
+ "path": "data/train/mathlib-001276-0000-7df6ab6fe36fb30c.parquet",
219
+ "sha256": "7df6ab6fe36fb30c6aebc57d6306d1532a7239284085ae8e48e1af6e9eb44b29",
220
+ "bytes": 85463,
221
+ "rows": 9,
222
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
223
+ }
224
+ ],
225
+ "proof_ids": [
226
+ "mathlib:FiniteDimensional.finrank_linearMap",
227
+ "mathlib:FiniteDimensional.finrank_linearMap_self",
228
+ "mathlib:FiniteDimensional.rank_linearMap",
229
+ "mathlib:FiniteDimensional.rank_linearMap_self",
230
+ "mathlib:Matrix.rank_vecMulVec"
231
+ ],
232
+ "proof_keys": [
233
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean\",null,null,\"FiniteDimensional.finrank_linearMap\"]",
234
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean\",null,null,\"FiniteDimensional.finrank_linearMap_self\"]",
235
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean\",null,null,\"FiniteDimensional.rank_linearMap\"]",
236
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean\",null,null,\"FiniteDimensional.rank_linearMap_self\"]",
237
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean\",null,null,\"Matrix.rank_vecMulVec\"]"
238
+ ]
239
+ }
manifests/batches/mathlib/001277.json ADDED
@@ -0,0 +1,185 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/001277",
4
+ "batch_fingerprint": "ca228d3d8847714f5b65fb4570cdfe88eaddf1770c39cb20e52dc84e45e3aef6",
5
+ "evidence": {
6
+ "batch_fingerprint": "ca228d3d8847714f5b65fb4570cdfe88eaddf1770c39cb20e52dc84e45e3aef6",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "no internal trace for tactic \"classical\\nrw [\u2190 Matrix.det_diagonal, \u2190 LinearMap.det_toLin b']\\nlet e :=\\n (b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans\\n ((LinearEquiv.coord S S f hf).restrictScalars R)\\nrefine (LinearMap.associated_det_of_eq_comp e _ _ ?_).symm\\ndsimp only [e, LinearEquiv.trans_apply]\\nsimp_rw [\u2190 LinearEquiv.coe_toLinearMap, \u2190 LinearMap.comp_apply, \u2190 LinearMap.ext_iff]\\nrefine b'.ext fun i => ?_\\nsimp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,\\n Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul,\\n zero_smul, Finset.sum_ite_eq', mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]\\nchange _ = f * _\\nerw [mul_comm, \u2190 smul_eq_mul, LinearEquiv.restrictScalars_apply, LinearEquiv.coord_apply_smul,\\n Ideal.selfBasis_def]\\nrfl\"",
11
+ "full_name": "associated_norm_prod_smith",
12
+ "id": "associated_norm_prod_smith",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 50,
18
+ 6
19
+ ],
20
+ "file_path": "Mathlib/LinearAlgebra/FreeModule/Norm.lean",
21
+ "full_name": "associated_norm_prod_smith",
22
+ "start": [
23
+ 30,
24
+ 1
25
+ ],
26
+ "traced_tactics": [
27
+ {
28
+ "annotated_tactic": [
29
+ "\"have hI := span_singleton_eq_bot.not.2 hf\"",
30
+ "[]"
31
+ ],
32
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\n\u22a2 Associated ((Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) \u22ef i)",
33
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\n\u22a2 Associated ((Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) \u22ef i)",
34
+ "tactic": "have hI := span_singleton_eq_bot.not.2 hf"
35
+ },
36
+ {
37
+ "annotated_tactic": [
38
+ "\"let b' := <a>ringBasis<\\/a> b (<a>span<\\/a> {f}) hI\"",
39
+ "[{\"full_name\":\"Ideal.ringBasis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/PID.lean\",\"def_pos\":[624,19],\"def_end_pos\":[624,34]},{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]}]"
40
+ ],
41
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated ((Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) \u22ef i)",
42
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\n\u22a2 Associated ((Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) \u22ef i)",
43
+ "tactic": "let b' := ringBasis b (span {f}) hI"
44
+ },
45
+ {
46
+ "annotated_tactic": [
47
+ "\"classical\\n rw [\\u2190 <a>Matrix.det_diagonal<\\/a>, \\u2190 <a>LinearMap.det_toLin<\\/a> b']\\n let e :=\\n (b'.equiv ((<a>span<\\/a> {f}).<a>selfBasis<\\/a> b hI) <| <a>Equiv.refl<\\/a> _).<a>trans<\\/a>\\n ((<a>LinearEquiv.coord<\\/a> S S f hf).<a>restrictScalars<\\/a> R)\\n refine (<a>LinearMap.associated_det_of_eq_comp<\\/a> e _ _ ?_).<a>symm<\\/a>\\n dsimp only [e, <a>LinearEquiv.trans_apply<\\/a>]\\n simp_rw [\\u2190 <a>LinearEquiv.coe_toLinearMap<\\/a>, \\u2190 <a>LinearMap.comp_apply<\\/a>, \\u2190 <a>LinearMap.ext_iff<\\/a>]\\n refine b'.ext fun i => ?_\\n simp_rw [<a>LinearMap.comp_apply<\\/a>, <a>LinearEquiv.coe_toLinearMap<\\/a>, <a>Matrix.toLin_apply<\\/a>, <a>Basis.repr_self<\\/a>,\\n <a>Finsupp.single_eq_pi_single<\\/a>, <a>Matrix.diagonal_mulVec_single<\\/a>, <a>Pi.single_apply<\\/a>, <a>ite_smul<\\/a>,\\n <a>zero_smul<\\/a>, <a>Finset.sum_ite_eq'<\\/a>, <a>mul_one<\\/a>, <a>if_pos<\\/a> (<a>Finset.mem_univ<\\/a> _), b'.equiv_apply]\\n change _ = f * _\\n -- This used to be `rw`, but we need `erw` after leanprover\\/lean4#2644\\n erw [<a>mul_comm<\\/a>, \\u2190 <a>smul_eq_mul<\\/a>, <a>LinearEquiv.restrictScalars_apply<\\/a>, <a>LinearEquiv.coord_apply_smul<\\/a>,\\n <a>Ideal.selfBasis_def<\\/a>]\\n rfl\"",
48
+ "[{\"full_name\":\"Matrix.det_diagonal\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,21]},{\"full_name\":\"LinearMap.det_toLin\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Determinant.lean\",\"def_pos\":[219,9],\"def_end_pos\":[219,18]},{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]},{\"full_name\":\"Ideal.selfBasis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/PID.lean\",\"def_pos\":[636,19],\"def_end_pos\":[636,34]},{\"full_name\":\"Equiv.refl\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[155,23],\"def_end_pos\":[155,27]},{\"full_name\":\"LinearEquiv.trans\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[337,5],\"def_end_pos\":[337,10]},{\"full_name\":\"LinearEquiv.coord\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[1182,8],\"def_end_pos\":[1182,13]},{\"full_name\":\"LinearEquiv.restrictScalars\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[615,5],\"def_end_pos\":[615,20]},{\"full_name\":\"LinearMap.associated_det_of_eq_comp\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Determinant.lean\",\"def_pos\":[481,9],\"def_end_pos\":[481,44]},{\"full_name\":\"Associated.symm\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[419,19],\"def_end_pos\":[419,23]},{\"full_name\":\"LinearEquiv.trans_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[366,9],\"def_end_pos\":[366,20]},{\"full_name\":\"LinearEquiv.coe_toLinearMap\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,24]},{\"full_name\":\"LinearMap.comp_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,19]},{\"full_name\":\"LinearMap.ext_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[348,9],\"def_end_pos\":[348,16]},{\"full_name\":\"LinearMap.comp_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,19]},{\"full_name\":\"LinearEquiv.coe_toLinearMap\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,24]},{\"full_name\":\"Matrix.toLin_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/ToLin.lean\",\"def_pos\":[673,9],\"def_end_pos\":[673,27]},{\"full_name\":\"Basis.repr_self\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,18]},{\"full_name\":\"Finsupp.single_eq_pi_single\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,28]},{\"full_name\":\"Matrix.diagonal_mulVec_single\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1832,9],\"def_end_pos\":[1832,31]},{\"full_name\":\"Pi.single_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[385,3],\"def_end_pos\":[385,14]},{\"full_name\":\"ite_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[45,30],\"def_end_pos\":[45,38]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]},{\"full_name\":\"Finset.sum_ite_eq'\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"if_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[932,9],\"def_end_pos\":[932,15]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"LinearEquiv.restrictScalars_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[614,3],\"def_end_pos\":[614,8]},{\"full_name\":\"LinearEquiv.coord_apply_smul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[1190,9],\"def_end_pos\":[1190,25]},{\"full_name\":\"Ideal.selfBasis_def\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/PID.lean\",\"def_pos\":[658,9],\"def_end_pos\":[658,28]}]"
49
+ ],
50
+ "state_after": "no goals",
51
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated ((Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) \u22ef i)",
52
+ "tactic": "classical\nrw [\u2190 Matrix.det_diagonal, \u2190 LinearMap.det_toLin b']\nlet e :=\n (b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans\n ((LinearEquiv.coord S S f hf).restrictScalars R)\nrefine (LinearMap.associated_det_of_eq_comp e _ _ ?_).symm\ndsimp only [e, LinearEquiv.trans_apply]\nsimp_rw [\u2190 LinearEquiv.coe_toLinearMap, \u2190 LinearMap.comp_apply, \u2190 LinearMap.ext_iff]\nrefine b'.ext fun i => ?_\nsimp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,\n Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul,\n zero_smul, Finset.sum_ite_eq', mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]\nchange _ = f * _\nerw [mul_comm, \u2190 smul_eq_mul, LinearEquiv.restrictScalars_apply, LinearEquiv.coord_apply_smul,\n Ideal.selfBasis_def]\nrfl"
53
+ },
54
+ {
55
+ "annotated_tactic": [
56
+ "\"rw [\\u2190 <a>Matrix.det_diagonal<\\/a>, \\u2190 <a>LinearMap.det_toLin<\\/a> b']\"",
57
+ "[{\"full_name\":\"Matrix.det_diagonal\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,21]},{\"full_name\":\"LinearMap.det_toLin\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Determinant.lean\",\"def_pos\":[219,9],\"def_end_pos\":[219,18]}]"
58
+ ],
59
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated ((Algebra.norm R) f) (LinearMap.det ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))))",
60
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated ((Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) \u22ef i)",
61
+ "tactic": "rw [\u2190 Matrix.det_diagonal, \u2190 LinearMap.det_toLin b']"
62
+ },
63
+ {
64
+ "annotated_tactic": [
65
+ "\"let e :=\\n (b'.equiv ((<a>span<\\/a> {f}).<a>selfBasis<\\/a> b hI) <| <a>Equiv.refl<\\/a> _).<a>trans<\\/a>\\n ((<a>LinearEquiv.coord<\\/a> S S f hf).<a>restrictScalars<\\/a> R)\"",
66
+ "[{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]},{\"full_name\":\"Ideal.selfBasis\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/PID.lean\",\"def_pos\":[636,19],\"def_end_pos\":[636,34]},{\"full_name\":\"Equiv.refl\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[155,23],\"def_end_pos\":[155,27]},{\"full_name\":\"LinearEquiv.trans\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[337,5],\"def_end_pos\":[337,10]},{\"full_name\":\"LinearEquiv.coord\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[1182,8],\"def_end_pos\":[1182,13]},{\"full_name\":\"LinearEquiv.restrictScalars\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[615,5],\"def_end_pos\":[615,20]}]"
67
+ ],
68
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 Associated ((Algebra.norm R) f) (LinearMap.det ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))))",
69
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated ((Algebra.norm R) f) (LinearMap.det ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))))",
70
+ "tactic": "let e :=\n (b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans\n ((LinearEquiv.coord S S f hf).restrictScalars R)"
71
+ },
72
+ {
73
+ "annotated_tactic": [
74
+ "\"refine (<a>LinearMap.associated_det_of_eq_comp<\\/a> e _ _ ?_).<a>symm<\\/a>\"",
75
+ "[{\"full_name\":\"LinearMap.associated_det_of_eq_comp\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Determinant.lean\",\"def_pos\":[481,9],\"def_end_pos\":[481,44]},{\"full_name\":\"Associated.symm\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[419,19],\"def_end_pos\":[419,23]}]"
76
+ ],
77
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 \u2200 (x : S),\n ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))) x = (\u2191(Algebra.lmul R S).toRingHom f) (e x)",
78
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 Associated ((Algebra.norm R) f) (LinearMap.det ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))))",
79
+ "tactic": "refine (LinearMap.associated_det_of_eq_comp e _ _ ?_).symm"
80
+ },
81
+ {
82
+ "annotated_tactic": [
83
+ "\"dsimp only [e, <a>LinearEquiv.trans_apply<\\/a>]\"",
84
+ "[{\"full_name\":\"LinearEquiv.trans_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[366,9],\"def_end_pos\":[366,20]}]"
85
+ ],
86
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 \u2200 (x : S),\n ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))) x =\n (\u2191(Algebra.lmul R S).toRingHom f)\n ((LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n ((b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9)) x))",
87
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 \u2200 (x : S),\n ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))) x = (\u2191(Algebra.lmul R S).toRingHom f) (e x)",
88
+ "tactic": "dsimp only [e, LinearEquiv.trans_apply]"
89
+ },
90
+ {
91
+ "annotated_tactic": [
92
+ "\"simp_rw [\\u2190 <a>LinearEquiv.coe_toLinearMap<\\/a>, \\u2190 <a>LinearMap.comp_apply<\\/a>, \\u2190 <a>LinearMap.ext_iff<\\/a>]\"",
93
+ "[{\"full_name\":\"LinearEquiv.coe_toLinearMap\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,24]},{\"full_name\":\"LinearMap.comp_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,19]},{\"full_name\":\"LinearMap.ext_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[348,9],\"def_end_pos\":[348,16]}]"
94
+ ],
95
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 \u2191(Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef)) =\n \u2191(Algebra.lmul R S).toRingHom f \u2218\u2097\n \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) \u2218\u2097\n \u2191(b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))",
96
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 \u2200 (x : S),\n ((Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))) x =\n (\u2191(Algebra.lmul R S).toRingHom f)\n ((LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n ((b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9)) x))",
97
+ "tactic": "simp_rw [\u2190 LinearEquiv.coe_toLinearMap, \u2190 LinearMap.comp_apply, \u2190 LinearMap.ext_iff]"
98
+ },
99
+ {
100
+ "annotated_tactic": [
101
+ "\"refine b'.ext fun i => ?_\"",
102
+ "[]"
103
+ ],
104
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 (\u2191(Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))) (b' i) =\n (\u2191(Algebra.lmul R S).toRingHom f \u2218\u2097\n \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) \u2218\u2097\n \u2191(b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9)))\n (b' i)",
105
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\n\u22a2 \u2191(Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef)) =\n \u2191(Algebra.lmul R S).toRingHom f \u2218\u2097\n \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) \u2218\u2097\n \u2191(b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))",
106
+ "tactic": "refine b'.ext fun i => ?_"
107
+ },
108
+ {
109
+ "annotated_tactic": [
110
+ "\"simp_rw [<a>LinearMap.comp_apply<\\/a>, <a>LinearEquiv.coe_toLinearMap<\\/a>, <a>Matrix.toLin_apply<\\/a>, <a>Basis.repr_self<\\/a>,\\n <a>Finsupp.single_eq_pi_single<\\/a>, <a>Matrix.diagonal_mulVec_single<\\/a>, <a>Pi.single_apply<\\/a>, <a>ite_smul<\\/a>,\\n <a>zero_smul<\\/a>, <a>Finset.sum_ite_eq'<\\/a>, <a>mul_one<\\/a>, <a>if_pos<\\/a> (<a>Finset.mem_univ<\\/a> _), b'.equiv_apply]\"",
111
+ "[{\"full_name\":\"LinearMap.comp_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/LinearMap\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,19]},{\"full_name\":\"LinearEquiv.coe_toLinearMap\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,24]},{\"full_name\":\"Matrix.toLin_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/ToLin.lean\",\"def_pos\":[673,9],\"def_end_pos\":[673,27]},{\"full_name\":\"Basis.repr_self\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,18]},{\"full_name\":\"Finsupp.single_eq_pi_single\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,28]},{\"full_name\":\"Matrix.diagonal_mulVec_single\",\"def_path\":\"Mathlib\\/Data\\/Matrix\\/Basic.lean\",\"def_pos\":[1832,9],\"def_end_pos\":[1832,31]},{\"full_name\":\"Pi.single_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[385,3],\"def_end_pos\":[385,14]},{\"full_name\":\"ite_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[45,30],\"def_end_pos\":[45,38]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]},{\"full_name\":\"Finset.sum_ite_eq'\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"if_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[932,9],\"def_end_pos\":[932,15]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]}]"
112
+ ],
113
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) \u22ef i \u2022 b' i =\n (\u2191(Algebra.lmul R S).toRingHom f)\n ((LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) ((selfBasis b (span {f}) hI) ((Equiv.refl \u03b9) i)))",
114
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 (\u2191(Matrix.toLin b' b') (Matrix.diagonal (smithCoeffs b (span {f}) \u22ef))) (b' i) =\n (\u2191(Algebra.lmul R S).toRingHom f \u2218\u2097\n \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) \u2218\u2097\n \u2191(b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9)))\n (b' i)",
115
+ "tactic": "simp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,\n Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul,\n zero_smul, Finset.sum_ite_eq', mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]"
116
+ },
117
+ {
118
+ "annotated_tactic": [
119
+ "\"change _ = f * _\"",
120
+ "[]"
121
+ ],
122
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) \u22ef i \u2022 b' i =\n f * (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) ((selfBasis b (span {f}) hI) ((Equiv.refl \u03b9) i))",
123
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) \u22ef i \u2022 b' i =\n (\u2191(Algebra.lmul R S).toRingHom f)\n ((LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) ((selfBasis b (span {f}) hI) ((Equiv.refl \u03b9) i)))",
124
+ "tactic": "change _ = f * _"
125
+ },
126
+ {
127
+ "annotated_tactic": [
128
+ "\"erw [<a>mul_comm<\\/a>, \\u2190 <a>smul_eq_mul<\\/a>, <a>LinearEquiv.restrictScalars_apply<\\/a>, <a>LinearEquiv.coord_apply_smul<\\/a>,\\n <a>Ideal.selfBasis_def<\\/a>]\"",
129
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"LinearEquiv.restrictScalars_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Equiv.lean\",\"def_pos\":[614,3],\"def_end_pos\":[614,8]},{\"full_name\":\"LinearEquiv.coord_apply_smul\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[1190,9],\"def_end_pos\":[1190,25]},{\"full_name\":\"Ideal.selfBasis_def\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FreeModule\\/PID.lean\",\"def_pos\":[658,9],\"def_end_pos\":[658,28]}]"
130
+ ],
131
+ "state_after": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) \u22ef i \u2022 b' i =\n smithCoeffs b (span {f}) hI ((Equiv.refl \u03b9) i) \u2022 (ringBasis b (span {f}) hI) ((Equiv.refl \u03b9) i)",
132
+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) \u22ef i \u2022 b' i =\n f * (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) ((selfBasis b (span {f}) hI) ((Equiv.refl \u03b9) i))",
133
+ "tactic": "erw [mul_comm, \u2190 smul_eq_mul, LinearEquiv.restrictScalars_apply, LinearEquiv.coord_apply_smul,\n Ideal.selfBasis_def]"
134
+ },
135
+ {
136
+ "annotated_tactic": [
137
+ "\"rfl\"",
138
+ "[]"
139
+ ],
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+ "state_after": "no goals",
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+ "state_before": "R : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n b'.equiv (selfBasis b (span {f}) hI) (Equiv.refl \u03b9) \u226a\u226b\u2097 LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) \u22ef i \u2022 b' i =\n smithCoeffs b (span {f}) hI ((Equiv.refl \u03b9) i) \u2022 (ringBasis b (span {f}) hI) ((Equiv.refl \u03b9) i)",
142
+ "tactic": "rfl"
143
+ }
144
+ ],
145
+ "url": "https://github.com/leanprover-community/mathlib4"
146
+ },
147
+ "source_file": "Mathlib/LinearAlgebra/FreeModule/Norm.lean"
148
+ }
149
+ ],
150
+ "input_proofs": 3,
151
+ "input_units": 2,
152
+ "proofs": 2,
153
+ "published_proofs": 2,
154
+ "published_transitions": 7,
155
+ "replay_failures": [],
156
+ "transitions": 7,
157
+ "validation_failures": [],
158
+ "validation_rejected_proofs": 0,
159
+ "zero_transition_proofs": 0
160
+ },
161
+ "shards": [
162
+ {
163
+ "path": "data/train/mathlib-001277-0000-191ee671e78fdbf2.parquet",
164
+ "sha256": "191ee671e78fdbf230b835ceaaea4bbe6f0e9e39fc1a2e89701f9a27cd463027",
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+ "bytes": 68366,
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+ "rows": 6,
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+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
168
+ },
169
+ {
170
+ "path": "data/internal_test/mathlib-001277-0000-4739e0cfac822842.parquet",
171
+ "sha256": "4739e0cfac822842a266e1fbb629cdec517bef021899fc4fc78cccd8470b7527",
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+ "bytes": 51175,
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+ "rows": 1,
174
+ "revision": "d9462abbb0a50dc3ff4347c425365b9169cc7ce6"
175
+ }
176
+ ],
177
+ "proof_ids": [
178
+ "mathlib:Ideal.finrank_quotient_eq_sum",
179
+ "mathlib:finrank_quotient_span_eq_natDegree_norm"
180
+ ],
181
+ "proof_keys": [
182
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean\",null,null,\"Ideal.finrank_quotient_eq_sum\"]",
183
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/LinearAlgebra/FreeModule/Norm.lean\",null,null,\"finrank_quotient_span_eq_natDegree_norm\"]"
184
+ ]
185
+ }
manifests/batches/mathlib/001280.json ADDED
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