MN000A

beta_matrix_minor_prefix_empty_exists

The zero-dimensional cofactor minor has an unconditional constructive empty beta code.

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. ∀ q. ∃ u. ∃ v. MatrixMinorPrefix(b,c,w,r,d,u,v,q,0)

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

Definition DAG

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

Actual proof prerequisites

add_eq_zero_right · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall b c w r d q. exists u v. (forall ff_index_mdm_prefix_prefix_empty. (exists ff_gap_mdm_lt_prefix_empty_index_bound. ff_gap_mdm_lt_prefix_empty_index_bound + S (ff_index_mdm_prefix_prefix_empty) = (0)) -> exists ff_row_mdm_prefix_prefix_empty ff_column_mdm_prefix_prefix_empty ff_value_mdm_prefix_prefix_empty. (ff_index_mdm_prefix_prefix_empty = (q) * ff_row_mdm_prefix_prefix_empty + ff_column_mdm_prefix_prefix_empty /\ ((exists ff_gap_mdm_lt_prefix_empty_column_bound. ff_gap_mdm_lt_prefix_empty_column_bound + S (ff_column_mdm_prefix_prefix_empty) = (q)) /\ ((exists ff_row_mdm_cell_prefix_empty_cell ff_column_mdm_cell_prefix_empty_cell. (((((exists ff_gap_mdm_lt_prefix_empty_cell_row_before. ff_gap_mdm_lt_prefix_empty_cell_row_before + S (ff_row_mdm_prefix_prefix_empty) = (r)) /\ ff_row_mdm_cell_prefix_empty_cell = ff_row_mdm_prefix_prefix_empty) \/ ((exists ff_gap_mdm_le_prefix_empty_cell_row_after. ff_gap_mdm_le_prefix_empty_cell_row_after + (r) = (ff_row_mdm_prefix_prefix_empty)) /\ ff_row_mdm_cell_prefix_empty_cell = S ff_row_mdm_prefix_prefix_empty))) /\ (((((exists ff_gap_mdm_lt_prefix_empty_cell_column_before. ff_gap_mdm_lt_prefix_empty_cell_column_before + S (ff_column_mdm_prefix_prefix_empty) = (d)) /\ ff_column_mdm_cell_prefix_empty_cell = ff_column_mdm_prefix_prefix_empty) \/ ((exists ff_gap_mdm_le_prefix_empty_cell_column_after. ff_gap_mdm_le_prefix_empty_cell_column_after + (d) = (ff_column_mdm_prefix_prefix_empty)) /\ ff_column_mdm_cell_prefix_empty_cell = S ff_column_mdm_prefix_prefix_empty))) /\ (((exists ff_h_mdm_prefix_empty_cell_source. ff_h_mdm_prefix_empty_cell_source + S (ff_value_mdm_prefix_prefix_empty) = S ((S ((ff_row_mdm_cell_prefix_empty_cell) * (w) + (ff_column_mdm_cell_prefix_empty_cell))) * c)) /\ exists ff_q_mdm_prefix_empty_cell_source. b = ff_q_mdm_prefix_empty_cell_source * S ((S ((ff_row_mdm_cell_prefix_empty_cell) * (w) + (ff_column_mdm_cell_prefix_empty_cell))) * c) + (ff_value_mdm_prefix_prefix_empty)))))) /\ (((exists ff_h_mdm_prefix_empty_target. ff_h_mdm_prefix_empty_target + S (ff_value_mdm_prefix_prefix_empty) = S ((S (ff_index_mdm_prefix_prefix_empty)) * v)) /\ exists ff_q_mdm_prefix_empty_target. u = ff_q_mdm_prefix_empty_target * S ((S (ff_index_mdm_prefix_prefix_empty)) * v) + (ff_value_mdm_prefix_prefix_empty)))))))

Complete unchanged native tactic proof

All 20 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

20 script commands · 5 reading checkpoints · 1 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 assumptionsL1–6

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 q
02Construct an explicit witnessL7–8

Supply the displayed value, then prove that it has the required property.

  1. L7
    exists 0
  2. L8
    exists 0
03Fix variables and assumptionsL9–10

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

  1. L9
    intro k
  2. L10
    intro hk
04Separate the logical casesL11–12

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

  1. L11
    exfalso
  2. L12
    cases hk
05Establish hzeroL13–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add eq zero right.

  1. L13
    have hzero : S k = 0
  2. L14
    specialize add_eq_zero_right x
  3. L15
    specialize add_eq_zero_right (S k)
  4. L16
    apply add_eq_zero_right
  5. L17
    exact hk_witness
  6. L18
    specialize succ_ne_zero k
  7. L19
    apply succ_ne_zero
  8. L20
    exact hzero

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro r
  5. 0005intro d
  6. 0006intro q
  7. 0007exists 0
  8. 0008exists 0
  9. 0009intro k
  10. 0010intro hk
  11. 0011exfalso
  12. 0012cases hk
  13. 0013have hzero : S k = 0
  14. 0014specialize add_eq_zero_right x
  15. 0015specialize add_eq_zero_right (S k)
  16. 0016apply add_eq_zero_right
  17. 0017exact hk_witness
  18. 0018specialize succ_ne_zero k
  19. 0019apply succ_ne_zero
  20. 0020exact hzero