MN0007

beta_matrix_minor_point_exists

Every flat index of a nonempty cofactor-minor row has genuine quotient, remainder and skipped-source witnesses.

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. ∀ k. ¬q = 0 → ∃ x. ∃ y. ∃ z. k = q · x + y ∧ (Lt(y,q) ∧ MatrixMinorCell(b,c,w,r,d,x,y,z))

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

Definition DAG

MatrixMinorCell(b,c,w,r,d,i,j,z) · 1Lt(a,b) · 1

Actual proof prerequisites

division_remainder_exists · checked external prerequisitebeta_matrix_minor_cell_exists
Original expanded first-order statement 
forall b c w r d q k. ~(q = 0) -> (exists ff_row_mdm_point_point_result ff_column_mdm_point_point_result ff_value_mdm_point_point_result. (k = (q) * ff_row_mdm_point_point_result + ff_column_mdm_point_point_result /\ ((exists ff_gap_mdm_lt_point_result_column_bound. ff_gap_mdm_lt_point_result_column_bound + S (ff_column_mdm_point_point_result) = (q)) /\ (exists ff_row_mdm_cell_point_result_cell ff_column_mdm_cell_point_result_cell. (((((exists ff_gap_mdm_lt_point_result_cell_row_before. ff_gap_mdm_lt_point_result_cell_row_before + S (ff_row_mdm_point_point_result) = (r)) /\ ff_row_mdm_cell_point_result_cell = ff_row_mdm_point_point_result) \/ ((exists ff_gap_mdm_le_point_result_cell_row_after. ff_gap_mdm_le_point_result_cell_row_after + (r) = (ff_row_mdm_point_point_result)) /\ ff_row_mdm_cell_point_result_cell = S ff_row_mdm_point_point_result))) /\ (((((exists ff_gap_mdm_lt_point_result_cell_column_before. ff_gap_mdm_lt_point_result_cell_column_before + S (ff_column_mdm_point_point_result) = (d)) /\ ff_column_mdm_cell_point_result_cell = ff_column_mdm_point_point_result) \/ ((exists ff_gap_mdm_le_point_result_cell_column_after. ff_gap_mdm_le_point_result_cell_column_after + (d) = (ff_column_mdm_point_point_result)) /\ ff_column_mdm_cell_point_result_cell = S ff_column_mdm_point_point_result))) /\ (((exists ff_h_mdm_point_result_cell_source. ff_h_mdm_point_result_cell_source + S (ff_value_mdm_point_point_result) = S ((S ((ff_row_mdm_cell_point_result_cell) * (w) + (ff_column_mdm_cell_point_result_cell))) * c)) /\ exists ff_q_mdm_point_result_cell_source. b = ff_q_mdm_point_result_cell_source * S ((S ((ff_row_mdm_cell_point_result_cell) * (w) + (ff_column_mdm_cell_point_result_cell))) * c) + (ff_value_mdm_point_point_result)))))))))

Complete unchanged native tactic proof

All 34 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

34 script commands · 10 reading checkpoints · 2 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 cell exists

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–8

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
  7. L7
    intro k
  8. L8
    intro hq
02Establish hcoordinatesL9–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder exists.

  1. L9
    have hcoordinates : exists i j. k = q * i + j /\ exists gap. gap + S j = q
  2. L10
    specialize division_remainder_exists q
  3. L11
    specialize division_remainder_exists k
  4. L12
    apply division_remainder_exists
  5. L13
    exact hq
03Separate the logical casesL14–16

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

  1. L14
    cases hcoordinates
  2. L15
    cases hcoordinates_witness
  3. L16
    cases hcoordinates_witness_witness
04Establish hcellL17–25

Establish this local claim before using it. It is not an additional assumption.

  1. L17
    have hcell : ∃ z. MatrixMinorCell(b,c,w,r,d,x,x1,z)Definitions: MatrixMinorCellOriginal native command in the exact edition
  2. L18
    specialize beta_matrix_minor_cell_exists b
  3. L19
    specialize beta_matrix_minor_cell_exists c
  4. L20
    specialize beta_matrix_minor_cell_exists w
  5. L21
    specialize beta_matrix_minor_cell_exists r
  6. L22
    specialize beta_matrix_minor_cell_exists d
  7. L23
    specialize beta_matrix_minor_cell_exists x
  8. L24
    specialize beta_matrix_minor_cell_exists x1
  9. L25
    exact beta_matrix_minor_cell_exists
05Separate the logical casesL26–26

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

  1. L26
    cases hcell
06Construct an explicit witnessL27–29

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

  1. L27
    exists x
  2. L28
    exists x1
  3. L29
    exists x2
07Separate the logical casesL30–30

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

  1. L30
    split
08Use earlier factsL31–31

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

  1. L31
    exact hcoordinates_witness_witness_left
09Separate the logical casesL32–32

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

  1. L32
    split
10Use earlier factsL33–34

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

  1. L33
    exact hcoordinates_witness_witness_right
  2. L34
    exact hcell_witness

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro r
  5. 0005intro d
  6. 0006intro q
  7. 0007intro k
  8. 0008intro hq
  9. 0009have hcoordinates : exists i j. k = q * i + j /\ exists gap. gap + S j = q
  10. 0010specialize division_remainder_exists q
  11. 0011specialize division_remainder_exists k
  12. 0012apply division_remainder_exists
  13. 0013exact hq
  14. 0014cases hcoordinates
  15. 0015cases hcoordinates_witness
  16. 0016cases hcoordinates_witness_witness
  17. 0017have hcell : exists z. (exists ff_row_mdm_cell_point_have_cell ff_column_mdm_cell_point_have_cell. (((((exists ff_gap_mdm_lt_point_have_cell_row_before. ff_gap_mdm_lt_point_have_cell_row_before + S (x) = (r)) /\ ff_row_mdm_cell_point_have_cell = x) \/ ((exists ff_gap_mdm_le_point_have_cell_row_after. ff_gap_mdm_le_point_have_cell_row_after + (r) = (x)) /\ ff_row_mdm_cell_point_have_cell = S x))) /\ (((((exists ff_gap_mdm_lt_point_have_cell_column_before. ff_gap_mdm_lt_point_have_cell_column_before + S (x1) = (d)) /\ ff_column_mdm_cell_point_have_cell = x1) \/ ((exists ff_gap_mdm_le_point_have_cell_column_after. ff_gap_mdm_le_point_have_cell_column_after + (d) = (x1)) /\ ff_column_mdm_cell_point_have_cell = S x1))) /\ (((exists ff_h_mdm_point_have_cell_source. ff_h_mdm_point_have_cell_source + S (z) = S ((S ((ff_row_mdm_cell_point_have_cell) * (w) + (ff_column_mdm_cell_point_have_cell))) * c)) /\ exists ff_q_mdm_point_have_cell_source. b = ff_q_mdm_point_have_cell_source * S ((S ((ff_row_mdm_cell_point_have_cell) * (w) + (ff_column_mdm_cell_point_have_cell))) * c) + (z))))))
  18. 0018specialize beta_matrix_minor_cell_exists b
  19. 0019specialize beta_matrix_minor_cell_exists c
  20. 0020specialize beta_matrix_minor_cell_exists w
  21. 0021specialize beta_matrix_minor_cell_exists r
  22. 0022specialize beta_matrix_minor_cell_exists d
  23. 0023specialize beta_matrix_minor_cell_exists x
  24. 0024specialize beta_matrix_minor_cell_exists x1
  25. 0025exact beta_matrix_minor_cell_exists
  26. 0026cases hcell
  27. 0027exists x
  28. 0028exists x1
  29. 0029exists x2
  30. 0030split
  31. 0031exact hcoordinates_witness_witness_left
  32. 0032split
  33. 0033exact hcoordinates_witness_witness_right
  34. 0034exact hcell_witness