MN000C

beta_matrix_minor_exists

Deleting any valid row and column from an unrestricted square beta-coded natural matrix constructs its complete exact square cofactor minor.

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. ∀ q. ∀ r. ∀ d. Lt(r,S q) → Lt(d,S q) → ∃ x. ∃ y. MatrixMinorPrefix(b,c,S q,r,d,x,y,q,q · q)

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

Definition DAG

MatrixMinorPrefix(b,c,w,r,d,u,v,q,l) · 1Lt(a,b) · 2

Actual proof prerequisites

beta_matrix_minor_prefix_exists
Original expanded first-order statement 
forall b c q r d. (exists ff_gap_mdm_lt_removed_row. ff_gap_mdm_lt_removed_row + S (r) = (S q)) -> (exists ff_gap_mdm_lt_removed_column. ff_gap_mdm_lt_removed_column + S (d) = (S q)) -> exists u v. (forall ff_index_mdm_prefix_square_minor. (exists ff_gap_mdm_lt_square_minor_index_bound. ff_gap_mdm_lt_square_minor_index_bound + S (ff_index_mdm_prefix_square_minor) = (q * q)) -> exists ff_row_mdm_prefix_square_minor ff_column_mdm_prefix_square_minor ff_value_mdm_prefix_square_minor. (ff_index_mdm_prefix_square_minor = (q) * ff_row_mdm_prefix_square_minor + ff_column_mdm_prefix_square_minor /\ ((exists ff_gap_mdm_lt_square_minor_column_bound. ff_gap_mdm_lt_square_minor_column_bound + S (ff_column_mdm_prefix_square_minor) = (q)) /\ ((exists ff_row_mdm_cell_square_minor_cell ff_column_mdm_cell_square_minor_cell. (((((exists ff_gap_mdm_lt_square_minor_cell_row_before. ff_gap_mdm_lt_square_minor_cell_row_before + S (ff_row_mdm_prefix_square_minor) = (r)) /\ ff_row_mdm_cell_square_minor_cell = ff_row_mdm_prefix_square_minor) \/ ((exists ff_gap_mdm_le_square_minor_cell_row_after. ff_gap_mdm_le_square_minor_cell_row_after + (r) = (ff_row_mdm_prefix_square_minor)) /\ ff_row_mdm_cell_square_minor_cell = S ff_row_mdm_prefix_square_minor))) /\ (((((exists ff_gap_mdm_lt_square_minor_cell_column_before. ff_gap_mdm_lt_square_minor_cell_column_before + S (ff_column_mdm_prefix_square_minor) = (d)) /\ ff_column_mdm_cell_square_minor_cell = ff_column_mdm_prefix_square_minor) \/ ((exists ff_gap_mdm_le_square_minor_cell_column_after. ff_gap_mdm_le_square_minor_cell_column_after + (d) = (ff_column_mdm_prefix_square_minor)) /\ ff_column_mdm_cell_square_minor_cell = S ff_column_mdm_prefix_square_minor))) /\ (((exists ff_h_mdm_square_minor_cell_source. ff_h_mdm_square_minor_cell_source + S (ff_value_mdm_prefix_square_minor) = S ((S ((ff_row_mdm_cell_square_minor_cell) * (S q) + (ff_column_mdm_cell_square_minor_cell))) * c)) /\ exists ff_q_mdm_square_minor_cell_source. b = ff_q_mdm_square_minor_cell_source * S ((S ((ff_row_mdm_cell_square_minor_cell) * (S q) + (ff_column_mdm_cell_square_minor_cell))) * c) + (ff_value_mdm_prefix_square_minor)))))) /\ (((exists ff_h_mdm_square_minor_target. ff_h_mdm_square_minor_target + S (ff_value_mdm_prefix_square_minor) = S ((S (ff_index_mdm_prefix_square_minor)) * v)) /\ exists ff_q_mdm_square_minor_target. u = ff_q_mdm_square_minor_target * S ((S (ff_index_mdm_prefix_square_minor)) * v) + (ff_value_mdm_prefix_square_minor)))))))

Complete unchanged native tactic proof

All 15 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

15 script commands · 2 reading checkpoints · 0 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)
beta matrix minor prefix exists
01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro q
  4. L4
    intro r
  5. L5
    intro d
  6. L6
    intro hrow
  7. L7
    intro hcolumn
02Use earlier factsL8–15

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

  1. L8
    specialize beta_matrix_minor_prefix_exists b
  2. L9
    specialize beta_matrix_minor_prefix_exists c
  3. L10
    specialize beta_matrix_minor_prefix_exists (S q)
  4. L11
    specialize beta_matrix_minor_prefix_exists r
  5. L12
    specialize beta_matrix_minor_prefix_exists d
  6. L13
    specialize beta_matrix_minor_prefix_exists q
  7. L14
    specialize beta_matrix_minor_prefix_exists q
  8. L15
    exact beta_matrix_minor_prefix_exists

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro q
  4. 0004intro r
  5. 0005intro d
  6. 0006intro hrow
  7. 0007intro hcolumn
  8. 0008specialize beta_matrix_minor_prefix_exists b
  9. 0009specialize beta_matrix_minor_prefix_exists c
  10. 0010specialize beta_matrix_minor_prefix_exists (S q)
  11. 0011specialize beta_matrix_minor_prefix_exists r
  12. 0012specialize beta_matrix_minor_prefix_exists d
  13. 0013specialize beta_matrix_minor_prefix_exists q
  14. 0014specialize beta_matrix_minor_prefix_exists q
  15. 0015exact beta_matrix_minor_prefix_exists