CE0009

signed_cofactor_minor_family_exists

Every arbitrary-dimensional signed square matrix has one complete beta-coded family containing ALL exact first-row signed cofactor minors.

Alpha v34 checked-use · first admitted v25 · 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 genuine signed first-row minors and unique alternating folds, with supplied cofactor values. T13 is now closed in the separate Alpha-v27 integer-linear-algebra branch with actual arbitrary determinant data, rank, and integer column spans; lattice index and normal forms are not claimed. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ pb. ∀ pc. ∀ nb. ∀ nc. ∀ q. ∃ u. ∃ v. SignedCofactorMinorPrefix(pb,pc,nb,nc,q,u,v,S q)

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

Definition DAG

SignedCofactorMinorPrefix(pb,pc,nb,nc,q,b,c,l) · 1

Actual proof prerequisites

le_refl · checked external prerequisitesigned_cofactor_minor_prefix_exists_bounded
Original expanded first-order statement 
forall pb pc nb nc q. exists u v. (forall ff_index_mce_family_full. (exists ff_gap_mce_full_index. ff_gap_mce_full_index + S (ff_index_mce_family_full) = (S q)) -> exists ff_value_mce_family_full. ((((exists ff_h_mce_full_entry. ff_h_mce_full_entry + S (ff_value_mce_family_full) = S ((S (ff_index_mce_family_full)) * v)) /\ exists ff_q_mce_full_entry. u = ff_q_mce_full_entry * S ((S (ff_index_mce_family_full)) * v) + (ff_value_mce_family_full))) /\ (exists ff_up_mce_record_full_record ff_us_mce_record_full_record ff_un_mce_record_full_record ff_ut_mce_record_full_record. ((ff_value_mce_family_full = ((((ff_up_mce_record_full_record) + (ff_us_mce_record_full_record)) * S ((ff_up_mce_record_full_record) + (ff_us_mce_record_full_record)) + ((ff_us_mce_record_full_record) + (ff_us_mce_record_full_record))) + (((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) * S ((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) + ((ff_ut_mce_record_full_record) + (ff_ut_mce_record_full_record)))) * S ((((ff_up_mce_record_full_record) + (ff_us_mce_record_full_record)) * S ((ff_up_mce_record_full_record) + (ff_us_mce_record_full_record)) + ((ff_us_mce_record_full_record) + (ff_us_mce_record_full_record))) + (((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) * S ((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) + ((ff_ut_mce_record_full_record) + (ff_ut_mce_record_full_record)))) + ((((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) * S ((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) + ((ff_ut_mce_record_full_record) + (ff_ut_mce_record_full_record))) + (((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) * S ((ff_un_mce_record_full_record) + (ff_ut_mce_record_full_record)) + ((ff_ut_mce_record_full_record) + (ff_ut_mce_record_full_record))))) /\ (((forall ff_index_mdm_prefix_mce_full_record_minor_positive. (exists ff_gap_mdm_lt_mce_full_record_minor_positive_index_bound. ff_gap_mdm_lt_mce_full_record_minor_positive_index_bound + S (ff_index_mdm_prefix_mce_full_record_minor_positive) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_full_record_minor_positive ff_column_mdm_prefix_mce_full_record_minor_positive ff_value_mdm_prefix_mce_full_record_minor_positive. (ff_index_mdm_prefix_mce_full_record_minor_positive = (q) * ff_row_mdm_prefix_mce_full_record_minor_positive + ff_column_mdm_prefix_mce_full_record_minor_positive /\ ((exists ff_gap_mdm_lt_mce_full_record_minor_positive_column_bound. ff_gap_mdm_lt_mce_full_record_minor_positive_column_bound + S (ff_column_mdm_prefix_mce_full_record_minor_positive) = (q)) /\ ((exists ff_row_mdm_cell_mce_full_record_minor_positive_cell ff_column_mdm_cell_mce_full_record_minor_positive_cell. (((((exists ff_gap_mdm_lt_mce_full_record_minor_positive_cell_row_before. ff_gap_mdm_lt_mce_full_record_minor_positive_cell_row_before + S (ff_row_mdm_prefix_mce_full_record_minor_positive) = (0)) /\ ff_row_mdm_cell_mce_full_record_minor_positive_cell = ff_row_mdm_prefix_mce_full_record_minor_positive) \/ ((exists ff_gap_mdm_le_mce_full_record_minor_positive_cell_row_after. ff_gap_mdm_le_mce_full_record_minor_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mce_full_record_minor_positive)) /\ ff_row_mdm_cell_mce_full_record_minor_positive_cell = S ff_row_mdm_prefix_mce_full_record_minor_positive))) /\ (((((exists ff_gap_mdm_lt_mce_full_record_minor_positive_cell_column_before. ff_gap_mdm_lt_mce_full_record_minor_positive_cell_column_before + S (ff_column_mdm_prefix_mce_full_record_minor_positive) = (ff_index_mce_family_full)) /\ ff_column_mdm_cell_mce_full_record_minor_positive_cell = ff_column_mdm_prefix_mce_full_record_minor_positive) \/ ((exists ff_gap_mdm_le_mce_full_record_minor_positive_cell_column_after. ff_gap_mdm_le_mce_full_record_minor_positive_cell_column_after + (ff_index_mce_family_full) = (ff_column_mdm_prefix_mce_full_record_minor_positive)) /\ ff_column_mdm_cell_mce_full_record_minor_positive_cell = S ff_column_mdm_prefix_mce_full_record_minor_positive))) /\ (((exists ff_h_mdm_mce_full_record_minor_positive_cell_source. ff_h_mdm_mce_full_record_minor_positive_cell_source + S (ff_value_mdm_prefix_mce_full_record_minor_positive) = S ((S ((ff_row_mdm_cell_mce_full_record_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_full_record_minor_positive_cell))) * pc)) /\ exists ff_q_mdm_mce_full_record_minor_positive_cell_source. pb = ff_q_mdm_mce_full_record_minor_positive_cell_source * S ((S ((ff_row_mdm_cell_mce_full_record_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_full_record_minor_positive_cell))) * pc) + (ff_value_mdm_prefix_mce_full_record_minor_positive)))))) /\ (((exists ff_h_mdm_mce_full_record_minor_positive_target. ff_h_mdm_mce_full_record_minor_positive_target + S (ff_value_mdm_prefix_mce_full_record_minor_positive) = S ((S (ff_index_mdm_prefix_mce_full_record_minor_positive)) * ff_us_mce_record_full_record)) /\ exists ff_q_mdm_mce_full_record_minor_positive_target. ff_up_mce_record_full_record = ff_q_mdm_mce_full_record_minor_positive_target * S ((S (ff_index_mdm_prefix_mce_full_record_minor_positive)) * ff_us_mce_record_full_record) + (ff_value_mdm_prefix_mce_full_record_minor_positive))))))) /\ (forall ff_index_mdm_prefix_mce_full_record_minor_negative. (exists ff_gap_mdm_lt_mce_full_record_minor_negative_index_bound. ff_gap_mdm_lt_mce_full_record_minor_negative_index_bound + S (ff_index_mdm_prefix_mce_full_record_minor_negative) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_full_record_minor_negative ff_column_mdm_prefix_mce_full_record_minor_negative ff_value_mdm_prefix_mce_full_record_minor_negative. (ff_index_mdm_prefix_mce_full_record_minor_negative = (q) * ff_row_mdm_prefix_mce_full_record_minor_negative + ff_column_mdm_prefix_mce_full_record_minor_negative /\ ((exists ff_gap_mdm_lt_mce_full_record_minor_negative_column_bound. ff_gap_mdm_lt_mce_full_record_minor_negative_column_bound + S (ff_column_mdm_prefix_mce_full_record_minor_negative) = (q)) /\ ((exists ff_row_mdm_cell_mce_full_record_minor_negative_cell ff_column_mdm_cell_mce_full_record_minor_negative_cell. (((((exists ff_gap_mdm_lt_mce_full_record_minor_negative_cell_row_before. ff_gap_mdm_lt_mce_full_record_minor_negative_cell_row_before + S (ff_row_mdm_prefix_mce_full_record_minor_negative) = (0)) /\ ff_row_mdm_cell_mce_full_record_minor_negative_cell = ff_row_mdm_prefix_mce_full_record_minor_negative) \/ ((exists ff_gap_mdm_le_mce_full_record_minor_negative_cell_row_after. ff_gap_mdm_le_mce_full_record_minor_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mce_full_record_minor_negative)) /\ ff_row_mdm_cell_mce_full_record_minor_negative_cell = S ff_row_mdm_prefix_mce_full_record_minor_negative))) /\ (((((exists ff_gap_mdm_lt_mce_full_record_minor_negative_cell_column_before. ff_gap_mdm_lt_mce_full_record_minor_negative_cell_column_before + S (ff_column_mdm_prefix_mce_full_record_minor_negative) = (ff_index_mce_family_full)) /\ ff_column_mdm_cell_mce_full_record_minor_negative_cell = ff_column_mdm_prefix_mce_full_record_minor_negative) \/ ((exists ff_gap_mdm_le_mce_full_record_minor_negative_cell_column_after. ff_gap_mdm_le_mce_full_record_minor_negative_cell_column_after + (ff_index_mce_family_full) = (ff_column_mdm_prefix_mce_full_record_minor_negative)) /\ ff_column_mdm_cell_mce_full_record_minor_negative_cell = S ff_column_mdm_prefix_mce_full_record_minor_negative))) /\ (((exists ff_h_mdm_mce_full_record_minor_negative_cell_source. ff_h_mdm_mce_full_record_minor_negative_cell_source + S (ff_value_mdm_prefix_mce_full_record_minor_negative) = S ((S ((ff_row_mdm_cell_mce_full_record_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_full_record_minor_negative_cell))) * nc)) /\ exists ff_q_mdm_mce_full_record_minor_negative_cell_source. nb = ff_q_mdm_mce_full_record_minor_negative_cell_source * S ((S ((ff_row_mdm_cell_mce_full_record_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_full_record_minor_negative_cell))) * nc) + (ff_value_mdm_prefix_mce_full_record_minor_negative)))))) /\ (((exists ff_h_mdm_mce_full_record_minor_negative_target. ff_h_mdm_mce_full_record_minor_negative_target + S (ff_value_mdm_prefix_mce_full_record_minor_negative) = S ((S (ff_index_mdm_prefix_mce_full_record_minor_negative)) * ff_ut_mce_record_full_record)) /\ exists ff_q_mdm_mce_full_record_minor_negative_target. ff_un_mce_record_full_record = ff_q_mdm_mce_full_record_minor_negative_target * S ((S (ff_index_mdm_prefix_mce_full_record_minor_negative)) * ff_ut_mce_record_full_record) + (ff_value_mdm_prefix_mce_full_record_minor_negative)))))))))))))

Complete unchanged native tactic proof

All 14 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

14 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)
signed cofactor minor prefix exists bounded
01Fix variables and assumptionsL1–5

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

  1. L1
    intro pb
  2. L2
    intro pc
  3. L3
    intro nb
  4. L4
    intro nc
  5. L5
    intro q
02Use earlier factsL6–14

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

  1. L6
    specialize signed_cofactor_minor_prefix_exists_bounded pb
  2. L7
    specialize signed_cofactor_minor_prefix_exists_bounded pc
  3. L8
    specialize signed_cofactor_minor_prefix_exists_bounded nb
  4. L9
    specialize signed_cofactor_minor_prefix_exists_bounded nc
  5. L10
    specialize signed_cofactor_minor_prefix_exists_bounded q
  6. L11
    specialize signed_cofactor_minor_prefix_exists_bounded (S q)
  7. L12
    apply signed_cofactor_minor_prefix_exists_bounded
  8. L13
    specialize le_refl (S q)
  9. L14
    exact le_refl

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro pb
  2. 0002intro pc
  3. 0003intro nb
  4. 0004intro nc
  5. 0005intro q
  6. 0006specialize signed_cofactor_minor_prefix_exists_bounded pb
  7. 0007specialize signed_cofactor_minor_prefix_exists_bounded pc
  8. 0008specialize signed_cofactor_minor_prefix_exists_bounded nb
  9. 0009specialize signed_cofactor_minor_prefix_exists_bounded nc
  10. 0010specialize signed_cofactor_minor_prefix_exists_bounded q
  11. 0011specialize signed_cofactor_minor_prefix_exists_bounded (S q)
  12. 0012apply signed_cofactor_minor_prefix_exists_bounded
  13. 0013specialize le_refl (S q)
  14. 0014exact le_refl