MN0003

matrix_skip_index_avoids_removed

A skipped matrix coordinate never equals the row or column that was actually deleted.

Alpha v34 checked-use · first admitted v24 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Historical partial components only: this chapter proves arbitrary signed cofactor minors and exact signed determinants through dimension four. T13 is now closed by the separate Alpha-v27 integer-linear-algebra branch: arbitrary determinant data, rank, and integer column spans, without a claim of lattice index or normal forms. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ i. ∀ r. ∀ s. MatrixSkipIndex(i,r,s) → ¬s = r

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

MatrixSkipIndex(i,r,s) · 1

Actual proof prerequisites

lt_irrefl_expanded · checked external prerequisitesucc_le_succ · checked external prerequisite
Original expanded first-order statement 
forall i r s. ((((exists ff_gap_mdm_lt_result_before. ff_gap_mdm_lt_result_before + S (i) = (r)) /\ s = i) \/ ((exists ff_gap_mdm_le_result_after. ff_gap_mdm_le_result_after + (r) = (i)) /\ s = S i))) -> ~(s = r)

Complete unchanged native tactic proof

All 31 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

31 script commands · 6 reading checkpoints · 3 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro i
  2. L2
    intro r
  3. L3
    intro s
  4. L4
    intro hskip
  5. L5
    intro hequal
02Separate the logical casesL6–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    cases hskip
  2. L7
    cases hskip_left
03Establish heqiL8–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply lt irrefl expanded.

  1. L8
    have heqi : i = r
  2. L9
    trans s
  3. L10
    symm
  4. L11
    exact hskip_left_right
  5. L12
    exact hequal
  6. L13
    rewrite heqi at hskip_left_left
  7. L14
    specialize lt_irrefl_expanded r
  8. L15
    apply lt_irrefl_expanded
  9. L16
    exact hskip_left_left
04Separate the logical casesL17–17

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L17
    cases hskip_right
05Establish hboundL18–22

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply succ le succ.

  1. L18
    have hbound : exists gap. gap + S r = S i
  2. L19
    specialize succ_le_succ r
  3. L20
    specialize succ_le_succ i
  4. L21
    apply succ_le_succ
  5. L22
    exact hskip_right_left
06Establish heqiL23–31

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply lt irrefl expanded.

  1. L23
    have heqi : S i = r
  2. L24
    trans s
  3. L25
    symm
  4. L26
    exact hskip_right_right
  5. L27
    exact hequal
  6. L28
    rewrite heqi at hbound
  7. L29
    specialize lt_irrefl_expanded r
  8. L30
    apply lt_irrefl_expanded
  9. L31
    exact hbound

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro i
  2. 0002intro r
  3. 0003intro s
  4. 0004intro hskip
  5. 0005intro hequal
  6. 0006cases hskip
  7. 0007cases hskip_left
  8. 0008have heqi : i = r
  9. 0009trans s
  10. 0010symm
  11. 0011exact hskip_left_right
  12. 0012exact hequal
  13. 0013rewrite heqi at hskip_left_left
  14. 0014specialize lt_irrefl_expanded r
  15. 0015apply lt_irrefl_expanded
  16. 0016exact hskip_left_left
  17. 0017cases hskip_right
  18. 0018have hbound : exists gap. gap + S r = S i
  19. 0019specialize succ_le_succ r
  20. 0020specialize succ_le_succ i
  21. 0021apply succ_le_succ
  22. 0022exact hskip_right_left
  23. 0023have heqi : S i = r
  24. 0024trans s
  25. 0025symm
  26. 0026exact hskip_right_right
  27. 0027exact hequal
  28. 0028rewrite heqi at hbound
  29. 0029specialize lt_irrefl_expanded r
  30. 0030apply lt_irrefl_expanded
  31. 0031exact hbound