MN0006

beta_matrix_minor_cell_functional

The decoded value of a beta-coded cofactor minor is independent of every skipped-coordinate witness.

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

∀ b. ∀ c. ∀ w. ∀ r. ∀ d. ∀ i. ∀ j. ∀ z. ∀ t. MatrixMinorCell(b,c,w,r,d,i,j,z) → MatrixMinorCell(b,c,w,r,d,i,j,t) → z = t

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

Definition DAG

MatrixMinorCell(b,c,w,r,d,i,j,z) · 2

Actual proof prerequisites

matrix_skip_index_functionalbeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c w r d i j z t. (exists ff_row_mdm_cell_cell_result ff_column_mdm_cell_cell_result. (((((exists ff_gap_mdm_lt_cell_result_row_before. ff_gap_mdm_lt_cell_result_row_before + S (i) = (r)) /\ ff_row_mdm_cell_cell_result = i) \/ ((exists ff_gap_mdm_le_cell_result_row_after. ff_gap_mdm_le_cell_result_row_after + (r) = (i)) /\ ff_row_mdm_cell_cell_result = S i))) /\ (((((exists ff_gap_mdm_lt_cell_result_column_before. ff_gap_mdm_lt_cell_result_column_before + S (j) = (d)) /\ ff_column_mdm_cell_cell_result = j) \/ ((exists ff_gap_mdm_le_cell_result_column_after. ff_gap_mdm_le_cell_result_column_after + (d) = (j)) /\ ff_column_mdm_cell_cell_result = S j))) /\ (((exists ff_h_mdm_cell_result_source. ff_h_mdm_cell_result_source + S (z) = S ((S ((ff_row_mdm_cell_cell_result) * (w) + (ff_column_mdm_cell_cell_result))) * c)) /\ exists ff_q_mdm_cell_result_source. b = ff_q_mdm_cell_result_source * S ((S ((ff_row_mdm_cell_cell_result) * (w) + (ff_column_mdm_cell_cell_result))) * c) + (z)))))) -> (exists ff_row_mdm_cell_cell_other ff_column_mdm_cell_cell_other. (((((exists ff_gap_mdm_lt_cell_other_row_before. ff_gap_mdm_lt_cell_other_row_before + S (i) = (r)) /\ ff_row_mdm_cell_cell_other = i) \/ ((exists ff_gap_mdm_le_cell_other_row_after. ff_gap_mdm_le_cell_other_row_after + (r) = (i)) /\ ff_row_mdm_cell_cell_other = S i))) /\ (((((exists ff_gap_mdm_lt_cell_other_column_before. ff_gap_mdm_lt_cell_other_column_before + S (j) = (d)) /\ ff_column_mdm_cell_cell_other = j) \/ ((exists ff_gap_mdm_le_cell_other_column_after. ff_gap_mdm_le_cell_other_column_after + (d) = (j)) /\ ff_column_mdm_cell_cell_other = S j))) /\ (((exists ff_h_mdm_cell_other_source. ff_h_mdm_cell_other_source + S (t) = S ((S ((ff_row_mdm_cell_cell_other) * (w) + (ff_column_mdm_cell_cell_other))) * c)) /\ exists ff_q_mdm_cell_other_source. b = ff_q_mdm_cell_other_source * S ((S ((ff_row_mdm_cell_cell_other) * (w) + (ff_column_mdm_cell_cell_other))) * c) + (t)))))) -> z = t

Complete unchanged native tactic proof

All 47 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

47 script commands · 7 reading checkpoints · 2 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.

Named ingredients (1)
matrix skip index functional
01Fix variables and assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro w
  4. L4
    intro r
  5. L5
    intro d
  6. L6
    intro i
  7. L7
    intro j
  8. L8
    intro z
  9. L9
    intro t
  10. L10
    intro hfirst
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hsecond
03Separate the logical casesL12–19

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

  1. L12
    cases hfirst
  2. L13
    cases hfirst_witness
  3. L14
    cases hfirst_witness_witness
  4. L15
    cases hfirst_witness_witness_right
  5. L16
    cases hsecond
  6. L17
    cases hsecond_witness
  7. L18
    cases hsecond_witness_witness
  8. L19
    cases hsecond_witness_witness_right
04Establish hrowL20–27

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix skip index functional.

  1. L20
    have hrow : x = x2
  2. L21
    specialize matrix_skip_index_functional i
  3. L22
    specialize matrix_skip_index_functional r
  4. L23
    specialize matrix_skip_index_functional x
  5. L24
    specialize matrix_skip_index_functional x2
  6. L25
    apply matrix_skip_index_functional
  7. L26
    exact hfirst_witness_witness_left
  8. L27
    exact hsecond_witness_witness_left
05Establish hcolumnL28–37

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix skip index functional.

  1. L28
    have hcolumn : x1 = x3
  2. L29
    specialize matrix_skip_index_functional j
  3. L30
    specialize matrix_skip_index_functional d
  4. L31
    specialize matrix_skip_index_functional x1
  5. L32
    specialize matrix_skip_index_functional x3
  6. L33
    apply matrix_skip_index_functional
  7. L34
    exact hfirst_witness_witness_right_left
  8. L35
    exact hsecond_witness_witness_right_left
  9. L36
    rewrite hrow at hfirst_witness_witness_right_right
  10. L37
    rewrite hrow at hfirst_witness_witness_right_right
06Calculate and transport equalitiesL38–39

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L38
    rewrite hcolumn at hfirst_witness_witness_right_right
  2. L39
    rewrite hcolumn at hfirst_witness_witness_right_right
07Use earlier factsL40–47

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L40
    specialize beta_at_unique b
  2. L41
    specialize beta_at_unique c
  3. L42
    specialize beta_at_unique (x2 * w + x3)
  4. L43
    specialize beta_at_unique z
  5. L44
    specialize beta_at_unique t
  6. L45
    apply beta_at_unique
  7. L46
    exact hfirst_witness_witness_right_right
  8. L47
    exact hsecond_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 47 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro r
  5. 0005intro d
  6. 0006intro i
  7. 0007intro j
  8. 0008intro z
  9. 0009intro t
  10. 0010intro hfirst
  11. 0011intro hsecond
  12. 0012cases hfirst
  13. 0013cases hfirst_witness
  14. 0014cases hfirst_witness_witness
  15. 0015cases hfirst_witness_witness_right
  16. 0016cases hsecond
  17. 0017cases hsecond_witness
  18. 0018cases hsecond_witness_witness
  19. 0019cases hsecond_witness_witness_right
  20. 0020have hrow : x = x2
  21. 0021specialize matrix_skip_index_functional i
  22. 0022specialize matrix_skip_index_functional r
  23. 0023specialize matrix_skip_index_functional x
  24. 0024specialize matrix_skip_index_functional x2
  25. 0025apply matrix_skip_index_functional
  26. 0026exact hfirst_witness_witness_left
  27. 0027exact hsecond_witness_witness_left
  28. 0028have hcolumn : x1 = x3
  29. 0029specialize matrix_skip_index_functional j
  30. 0030specialize matrix_skip_index_functional d
  31. 0031specialize matrix_skip_index_functional x1
  32. 0032specialize matrix_skip_index_functional x3
  33. 0033apply matrix_skip_index_functional
  34. 0034exact hfirst_witness_witness_right_left
  35. 0035exact hsecond_witness_witness_right_left
  36. 0036rewrite hrow at hfirst_witness_witness_right_right
  37. 0037rewrite hrow at hfirst_witness_witness_right_right
  38. 0038rewrite hcolumn at hfirst_witness_witness_right_right
  39. 0039rewrite hcolumn at hfirst_witness_witness_right_right
  40. 0040specialize beta_at_unique b
  41. 0041specialize beta_at_unique c
  42. 0042specialize beta_at_unique (x2 * w + x3)
  43. 0043specialize beta_at_unique z
  44. 0044specialize beta_at_unique t
  45. 0045apply beta_at_unique
  46. 0046exact hfirst_witness_witness_right_right
  47. 0047exact hsecond_witness_witness_right_right