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. ∀ j. SignedCofactorMinorPrefix(pb,pc,nb,nc,q,u,v,S q) → Lt(j,S q) → ∃ x. Beta(u,v,j,x) ∧ SignedMinorRecord(pb,pc,nb,nc,q,j,x)Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
none
Original expanded first-order statement
forall pb pc nb nc q u v j. (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))))))))))))) -> (exists ff_gap_mce_record_column. ff_gap_mce_record_column + S (j) = (S q)) -> exists z. ((((exists ff_h_mce_entry_result. ff_h_mce_entry_result + S (z) = S ((S (j)) * v)) /\ exists ff_q_mce_entry_result. u = ff_q_mce_entry_result * S ((S (j)) * v) + (z))) /\ (exists ff_up_mce_record_family_entry ff_us_mce_record_family_entry ff_un_mce_record_family_entry ff_ut_mce_record_family_entry. ((z = ((((ff_up_mce_record_family_entry) + (ff_us_mce_record_family_entry)) * S ((ff_up_mce_record_family_entry) + (ff_us_mce_record_family_entry)) + ((ff_us_mce_record_family_entry) + (ff_us_mce_record_family_entry))) + (((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) * S ((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) + ((ff_ut_mce_record_family_entry) + (ff_ut_mce_record_family_entry)))) * S ((((ff_up_mce_record_family_entry) + (ff_us_mce_record_family_entry)) * S ((ff_up_mce_record_family_entry) + (ff_us_mce_record_family_entry)) + ((ff_us_mce_record_family_entry) + (ff_us_mce_record_family_entry))) + (((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) * S ((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) + ((ff_ut_mce_record_family_entry) + (ff_ut_mce_record_family_entry)))) + ((((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) * S ((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) + ((ff_ut_mce_record_family_entry) + (ff_ut_mce_record_family_entry))) + (((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) * S ((ff_un_mce_record_family_entry) + (ff_ut_mce_record_family_entry)) + ((ff_ut_mce_record_family_entry) + (ff_ut_mce_record_family_entry))))) /\ (((forall ff_index_mdm_prefix_mce_family_entry_minor_positive. (exists ff_gap_mdm_lt_mce_family_entry_minor_positive_index_bound. ff_gap_mdm_lt_mce_family_entry_minor_positive_index_bound + S (ff_index_mdm_prefix_mce_family_entry_minor_positive) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_family_entry_minor_positive ff_column_mdm_prefix_mce_family_entry_minor_positive ff_value_mdm_prefix_mce_family_entry_minor_positive. (ff_index_mdm_prefix_mce_family_entry_minor_positive = (q) * ff_row_mdm_prefix_mce_family_entry_minor_positive + ff_column_mdm_prefix_mce_family_entry_minor_positive /\ ((exists ff_gap_mdm_lt_mce_family_entry_minor_positive_column_bound. ff_gap_mdm_lt_mce_family_entry_minor_positive_column_bound + S (ff_column_mdm_prefix_mce_family_entry_minor_positive) = (q)) /\ ((exists ff_row_mdm_cell_mce_family_entry_minor_positive_cell ff_column_mdm_cell_mce_family_entry_minor_positive_cell. (((((exists ff_gap_mdm_lt_mce_family_entry_minor_positive_cell_row_before. ff_gap_mdm_lt_mce_family_entry_minor_positive_cell_row_before + S (ff_row_mdm_prefix_mce_family_entry_minor_positive) = (0)) /\ ff_row_mdm_cell_mce_family_entry_minor_positive_cell = ff_row_mdm_prefix_mce_family_entry_minor_positive) \/ ((exists ff_gap_mdm_le_mce_family_entry_minor_positive_cell_row_after. ff_gap_mdm_le_mce_family_entry_minor_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mce_family_entry_minor_positive)) /\ ff_row_mdm_cell_mce_family_entry_minor_positive_cell = S ff_row_mdm_prefix_mce_family_entry_minor_positive))) /\ (((((exists ff_gap_mdm_lt_mce_family_entry_minor_positive_cell_column_before. ff_gap_mdm_lt_mce_family_entry_minor_positive_cell_column_before + S (ff_column_mdm_prefix_mce_family_entry_minor_positive) = (j)) /\ ff_column_mdm_cell_mce_family_entry_minor_positive_cell = ff_column_mdm_prefix_mce_family_entry_minor_positive) \/ ((exists ff_gap_mdm_le_mce_family_entry_minor_positive_cell_column_after. ff_gap_mdm_le_mce_family_entry_minor_positive_cell_column_after + (j) = (ff_column_mdm_prefix_mce_family_entry_minor_positive)) /\ ff_column_mdm_cell_mce_family_entry_minor_positive_cell = S ff_column_mdm_prefix_mce_family_entry_minor_positive))) /\ (((exists ff_h_mdm_mce_family_entry_minor_positive_cell_source. ff_h_mdm_mce_family_entry_minor_positive_cell_source + S (ff_value_mdm_prefix_mce_family_entry_minor_positive) = S ((S ((ff_row_mdm_cell_mce_family_entry_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_family_entry_minor_positive_cell))) * pc)) /\ exists ff_q_mdm_mce_family_entry_minor_positive_cell_source. pb = ff_q_mdm_mce_family_entry_minor_positive_cell_source * S ((S ((ff_row_mdm_cell_mce_family_entry_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_family_entry_minor_positive_cell))) * pc) + (ff_value_mdm_prefix_mce_family_entry_minor_positive)))))) /\ (((exists ff_h_mdm_mce_family_entry_minor_positive_target. ff_h_mdm_mce_family_entry_minor_positive_target + S (ff_value_mdm_prefix_mce_family_entry_minor_positive) = S ((S (ff_index_mdm_prefix_mce_family_entry_minor_positive)) * ff_us_mce_record_family_entry)) /\ exists ff_q_mdm_mce_family_entry_minor_positive_target. ff_up_mce_record_family_entry = ff_q_mdm_mce_family_entry_minor_positive_target * S ((S (ff_index_mdm_prefix_mce_family_entry_minor_positive)) * ff_us_mce_record_family_entry) + (ff_value_mdm_prefix_mce_family_entry_minor_positive))))))) /\ (forall ff_index_mdm_prefix_mce_family_entry_minor_negative. (exists ff_gap_mdm_lt_mce_family_entry_minor_negative_index_bound. ff_gap_mdm_lt_mce_family_entry_minor_negative_index_bound + S (ff_index_mdm_prefix_mce_family_entry_minor_negative) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_family_entry_minor_negative ff_column_mdm_prefix_mce_family_entry_minor_negative ff_value_mdm_prefix_mce_family_entry_minor_negative. (ff_index_mdm_prefix_mce_family_entry_minor_negative = (q) * ff_row_mdm_prefix_mce_family_entry_minor_negative + ff_column_mdm_prefix_mce_family_entry_minor_negative /\ ((exists ff_gap_mdm_lt_mce_family_entry_minor_negative_column_bound. ff_gap_mdm_lt_mce_family_entry_minor_negative_column_bound + S (ff_column_mdm_prefix_mce_family_entry_minor_negative) = (q)) /\ ((exists ff_row_mdm_cell_mce_family_entry_minor_negative_cell ff_column_mdm_cell_mce_family_entry_minor_negative_cell. (((((exists ff_gap_mdm_lt_mce_family_entry_minor_negative_cell_row_before. ff_gap_mdm_lt_mce_family_entry_minor_negative_cell_row_before + S (ff_row_mdm_prefix_mce_family_entry_minor_negative) = (0)) /\ ff_row_mdm_cell_mce_family_entry_minor_negative_cell = ff_row_mdm_prefix_mce_family_entry_minor_negative) \/ ((exists ff_gap_mdm_le_mce_family_entry_minor_negative_cell_row_after. ff_gap_mdm_le_mce_family_entry_minor_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mce_family_entry_minor_negative)) /\ ff_row_mdm_cell_mce_family_entry_minor_negative_cell = S ff_row_mdm_prefix_mce_family_entry_minor_negative))) /\ (((((exists ff_gap_mdm_lt_mce_family_entry_minor_negative_cell_column_before. ff_gap_mdm_lt_mce_family_entry_minor_negative_cell_column_before + S (ff_column_mdm_prefix_mce_family_entry_minor_negative) = (j)) /\ ff_column_mdm_cell_mce_family_entry_minor_negative_cell = ff_column_mdm_prefix_mce_family_entry_minor_negative) \/ ((exists ff_gap_mdm_le_mce_family_entry_minor_negative_cell_column_after. ff_gap_mdm_le_mce_family_entry_minor_negative_cell_column_after + (j) = (ff_column_mdm_prefix_mce_family_entry_minor_negative)) /\ ff_column_mdm_cell_mce_family_entry_minor_negative_cell = S ff_column_mdm_prefix_mce_family_entry_minor_negative))) /\ (((exists ff_h_mdm_mce_family_entry_minor_negative_cell_source. ff_h_mdm_mce_family_entry_minor_negative_cell_source + S (ff_value_mdm_prefix_mce_family_entry_minor_negative) = S ((S ((ff_row_mdm_cell_mce_family_entry_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_family_entry_minor_negative_cell))) * nc)) /\ exists ff_q_mdm_mce_family_entry_minor_negative_cell_source. nb = ff_q_mdm_mce_family_entry_minor_negative_cell_source * S ((S ((ff_row_mdm_cell_mce_family_entry_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_family_entry_minor_negative_cell))) * nc) + (ff_value_mdm_prefix_mce_family_entry_minor_negative)))))) /\ (((exists ff_h_mdm_mce_family_entry_minor_negative_target. ff_h_mdm_mce_family_entry_minor_negative_target + S (ff_value_mdm_prefix_mce_family_entry_minor_negative) = S ((S (ff_index_mdm_prefix_mce_family_entry_minor_negative)) * ff_ut_mce_record_family_entry)) /\ exists ff_q_mdm_mce_family_entry_minor_negative_target. ff_un_mce_record_family_entry = ff_q_mdm_mce_family_entry_minor_negative_target * S ((S (ff_index_mdm_prefix_mce_family_entry_minor_negative)) * ff_ut_mce_record_family_entry) + (ff_value_mdm_prefix_mce_family_entry_minor_negative))))))))))))Complete unchanged native tactic proof
All 13 lines are the exact independently kernel-checked original script.
Read the argument
Proof checkpoints
13 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.