CE0006

signed_cofactor_minor_prefix_empty

Every beta code vacuously describes the genuinely empty signed cofactor-minor prefix.

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,0)

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

add_eq_zero_right · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall pb pc nb nc q u v. (forall ff_index_mce_family_empty. (exists ff_gap_mce_empty_index. ff_gap_mce_empty_index + S (ff_index_mce_family_empty) = (0)) -> exists ff_value_mce_family_empty. ((((exists ff_h_mce_empty_entry. ff_h_mce_empty_entry + S (ff_value_mce_family_empty) = S ((S (ff_index_mce_family_empty)) * v)) /\ exists ff_q_mce_empty_entry. u = ff_q_mce_empty_entry * S ((S (ff_index_mce_family_empty)) * v) + (ff_value_mce_family_empty))) /\ (exists ff_up_mce_record_empty_record ff_us_mce_record_empty_record ff_un_mce_record_empty_record ff_ut_mce_record_empty_record. ((ff_value_mce_family_empty = ((((ff_up_mce_record_empty_record) + (ff_us_mce_record_empty_record)) * S ((ff_up_mce_record_empty_record) + (ff_us_mce_record_empty_record)) + ((ff_us_mce_record_empty_record) + (ff_us_mce_record_empty_record))) + (((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) * S ((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) + ((ff_ut_mce_record_empty_record) + (ff_ut_mce_record_empty_record)))) * S ((((ff_up_mce_record_empty_record) + (ff_us_mce_record_empty_record)) * S ((ff_up_mce_record_empty_record) + (ff_us_mce_record_empty_record)) + ((ff_us_mce_record_empty_record) + (ff_us_mce_record_empty_record))) + (((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) * S ((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) + ((ff_ut_mce_record_empty_record) + (ff_ut_mce_record_empty_record)))) + ((((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) * S ((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) + ((ff_ut_mce_record_empty_record) + (ff_ut_mce_record_empty_record))) + (((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) * S ((ff_un_mce_record_empty_record) + (ff_ut_mce_record_empty_record)) + ((ff_ut_mce_record_empty_record) + (ff_ut_mce_record_empty_record))))) /\ (((forall ff_index_mdm_prefix_mce_empty_record_minor_positive. (exists ff_gap_mdm_lt_mce_empty_record_minor_positive_index_bound. ff_gap_mdm_lt_mce_empty_record_minor_positive_index_bound + S (ff_index_mdm_prefix_mce_empty_record_minor_positive) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_empty_record_minor_positive ff_column_mdm_prefix_mce_empty_record_minor_positive ff_value_mdm_prefix_mce_empty_record_minor_positive. (ff_index_mdm_prefix_mce_empty_record_minor_positive = (q) * ff_row_mdm_prefix_mce_empty_record_minor_positive + ff_column_mdm_prefix_mce_empty_record_minor_positive /\ ((exists ff_gap_mdm_lt_mce_empty_record_minor_positive_column_bound. ff_gap_mdm_lt_mce_empty_record_minor_positive_column_bound + S (ff_column_mdm_prefix_mce_empty_record_minor_positive) = (q)) /\ ((exists ff_row_mdm_cell_mce_empty_record_minor_positive_cell ff_column_mdm_cell_mce_empty_record_minor_positive_cell. (((((exists ff_gap_mdm_lt_mce_empty_record_minor_positive_cell_row_before. ff_gap_mdm_lt_mce_empty_record_minor_positive_cell_row_before + S (ff_row_mdm_prefix_mce_empty_record_minor_positive) = (0)) /\ ff_row_mdm_cell_mce_empty_record_minor_positive_cell = ff_row_mdm_prefix_mce_empty_record_minor_positive) \/ ((exists ff_gap_mdm_le_mce_empty_record_minor_positive_cell_row_after. ff_gap_mdm_le_mce_empty_record_minor_positive_cell_row_after + (0) = (ff_row_mdm_prefix_mce_empty_record_minor_positive)) /\ ff_row_mdm_cell_mce_empty_record_minor_positive_cell = S ff_row_mdm_prefix_mce_empty_record_minor_positive))) /\ (((((exists ff_gap_mdm_lt_mce_empty_record_minor_positive_cell_column_before. ff_gap_mdm_lt_mce_empty_record_minor_positive_cell_column_before + S (ff_column_mdm_prefix_mce_empty_record_minor_positive) = (ff_index_mce_family_empty)) /\ ff_column_mdm_cell_mce_empty_record_minor_positive_cell = ff_column_mdm_prefix_mce_empty_record_minor_positive) \/ ((exists ff_gap_mdm_le_mce_empty_record_minor_positive_cell_column_after. ff_gap_mdm_le_mce_empty_record_minor_positive_cell_column_after + (ff_index_mce_family_empty) = (ff_column_mdm_prefix_mce_empty_record_minor_positive)) /\ ff_column_mdm_cell_mce_empty_record_minor_positive_cell = S ff_column_mdm_prefix_mce_empty_record_minor_positive))) /\ (((exists ff_h_mdm_mce_empty_record_minor_positive_cell_source. ff_h_mdm_mce_empty_record_minor_positive_cell_source + S (ff_value_mdm_prefix_mce_empty_record_minor_positive) = S ((S ((ff_row_mdm_cell_mce_empty_record_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_empty_record_minor_positive_cell))) * pc)) /\ exists ff_q_mdm_mce_empty_record_minor_positive_cell_source. pb = ff_q_mdm_mce_empty_record_minor_positive_cell_source * S ((S ((ff_row_mdm_cell_mce_empty_record_minor_positive_cell) * (S (q)) + (ff_column_mdm_cell_mce_empty_record_minor_positive_cell))) * pc) + (ff_value_mdm_prefix_mce_empty_record_minor_positive)))))) /\ (((exists ff_h_mdm_mce_empty_record_minor_positive_target. ff_h_mdm_mce_empty_record_minor_positive_target + S (ff_value_mdm_prefix_mce_empty_record_minor_positive) = S ((S (ff_index_mdm_prefix_mce_empty_record_minor_positive)) * ff_us_mce_record_empty_record)) /\ exists ff_q_mdm_mce_empty_record_minor_positive_target. ff_up_mce_record_empty_record = ff_q_mdm_mce_empty_record_minor_positive_target * S ((S (ff_index_mdm_prefix_mce_empty_record_minor_positive)) * ff_us_mce_record_empty_record) + (ff_value_mdm_prefix_mce_empty_record_minor_positive))))))) /\ (forall ff_index_mdm_prefix_mce_empty_record_minor_negative. (exists ff_gap_mdm_lt_mce_empty_record_minor_negative_index_bound. ff_gap_mdm_lt_mce_empty_record_minor_negative_index_bound + S (ff_index_mdm_prefix_mce_empty_record_minor_negative) = ((q) * (q))) -> exists ff_row_mdm_prefix_mce_empty_record_minor_negative ff_column_mdm_prefix_mce_empty_record_minor_negative ff_value_mdm_prefix_mce_empty_record_minor_negative. (ff_index_mdm_prefix_mce_empty_record_minor_negative = (q) * ff_row_mdm_prefix_mce_empty_record_minor_negative + ff_column_mdm_prefix_mce_empty_record_minor_negative /\ ((exists ff_gap_mdm_lt_mce_empty_record_minor_negative_column_bound. ff_gap_mdm_lt_mce_empty_record_minor_negative_column_bound + S (ff_column_mdm_prefix_mce_empty_record_minor_negative) = (q)) /\ ((exists ff_row_mdm_cell_mce_empty_record_minor_negative_cell ff_column_mdm_cell_mce_empty_record_minor_negative_cell. (((((exists ff_gap_mdm_lt_mce_empty_record_minor_negative_cell_row_before. ff_gap_mdm_lt_mce_empty_record_minor_negative_cell_row_before + S (ff_row_mdm_prefix_mce_empty_record_minor_negative) = (0)) /\ ff_row_mdm_cell_mce_empty_record_minor_negative_cell = ff_row_mdm_prefix_mce_empty_record_minor_negative) \/ ((exists ff_gap_mdm_le_mce_empty_record_minor_negative_cell_row_after. ff_gap_mdm_le_mce_empty_record_minor_negative_cell_row_after + (0) = (ff_row_mdm_prefix_mce_empty_record_minor_negative)) /\ ff_row_mdm_cell_mce_empty_record_minor_negative_cell = S ff_row_mdm_prefix_mce_empty_record_minor_negative))) /\ (((((exists ff_gap_mdm_lt_mce_empty_record_minor_negative_cell_column_before. ff_gap_mdm_lt_mce_empty_record_minor_negative_cell_column_before + S (ff_column_mdm_prefix_mce_empty_record_minor_negative) = (ff_index_mce_family_empty)) /\ ff_column_mdm_cell_mce_empty_record_minor_negative_cell = ff_column_mdm_prefix_mce_empty_record_minor_negative) \/ ((exists ff_gap_mdm_le_mce_empty_record_minor_negative_cell_column_after. ff_gap_mdm_le_mce_empty_record_minor_negative_cell_column_after + (ff_index_mce_family_empty) = (ff_column_mdm_prefix_mce_empty_record_minor_negative)) /\ ff_column_mdm_cell_mce_empty_record_minor_negative_cell = S ff_column_mdm_prefix_mce_empty_record_minor_negative))) /\ (((exists ff_h_mdm_mce_empty_record_minor_negative_cell_source. ff_h_mdm_mce_empty_record_minor_negative_cell_source + S (ff_value_mdm_prefix_mce_empty_record_minor_negative) = S ((S ((ff_row_mdm_cell_mce_empty_record_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_empty_record_minor_negative_cell))) * nc)) /\ exists ff_q_mdm_mce_empty_record_minor_negative_cell_source. nb = ff_q_mdm_mce_empty_record_minor_negative_cell_source * S ((S ((ff_row_mdm_cell_mce_empty_record_minor_negative_cell) * (S (q)) + (ff_column_mdm_cell_mce_empty_record_minor_negative_cell))) * nc) + (ff_value_mdm_prefix_mce_empty_record_minor_negative)))))) /\ (((exists ff_h_mdm_mce_empty_record_minor_negative_target. ff_h_mdm_mce_empty_record_minor_negative_target + S (ff_value_mdm_prefix_mce_empty_record_minor_negative) = S ((S (ff_index_mdm_prefix_mce_empty_record_minor_negative)) * ff_ut_mce_record_empty_record)) /\ exists ff_q_mdm_mce_empty_record_minor_negative_target. ff_un_mce_record_empty_record = ff_q_mdm_mce_empty_record_minor_negative_target * S ((S (ff_index_mdm_prefix_mce_empty_record_minor_negative)) * ff_ut_mce_record_empty_record) + (ff_value_mdm_prefix_mce_empty_record_minor_negative)))))))))))))

Complete unchanged native tactic proof

All 19 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

19 script commands · 3 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 assumptionsL1–9

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
  6. L6
    intro u
  7. L7
    intro v
  8. L8
    intro i
  9. L9
    intro hi
02Separate the logical casesL10–11

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

  1. L10
    exfalso
  2. L11
    cases hi
03Establish hzeroL12–19

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

  1. L12
    have hzero : S i = 0
  2. L13
    specialize add_eq_zero_right x
  3. L14
    specialize add_eq_zero_right (S i)
  4. L15
    apply add_eq_zero_right
  5. L16
    exact hi_witness
  6. L17
    specialize succ_ne_zero i
  7. L18
    apply succ_ne_zero
  8. L19
    exact hzero

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro pb
  2. 0002intro pc
  3. 0003intro nb
  4. 0004intro nc
  5. 0005intro q
  6. 0006intro u
  7. 0007intro v
  8. 0008intro i
  9. 0009intro hi
  10. 0010exfalso
  11. 0011cases hi
  12. 0012have hzero : S i = 0
  13. 0013specialize add_eq_zero_right x
  14. 0014specialize add_eq_zero_right (S i)
  15. 0015apply add_eq_zero_right
  16. 0016exact hi_witness
  17. 0017specialize succ_ne_zero i
  18. 0018apply succ_ne_zero
  19. 0019exact hzero