DL008F

matrix_integer_minor_prefix_cell_at_coordinates

Every actual decoded minor output is its genuine cofactor cell at any valid quotient/remainder coordinates, by unique coordinates and beta functionality.

Alpha v34 checked-use · first admitted v27 · 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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ q. ∀ j. ∀ u. ∀ v. ∀ k. ∀ r. ∀ s. ∀ a. MatrixMinorPrefix(b,c,S q,0,j,u,v,q,q · q) → Lt(k,q · q) → k = q · r + s → Lt(s,q) → BetaAt(u,v,k,a) → MatrixMinorCell(b,c,S q,0,j,r,s,a)

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

Definition DAG

MatrixMinorCell(b,c,w,r,d,i,j,z) · 2MatrixMinorPrefix(b,c,w,r,d,u,v,q,l) · 1Lt(a,b) · 3BetaAt(b,c,i,x) · 2

Actual proof prerequisites

division_remainder_unique · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c q j u v k r s a. (forall ff_index_mdm_prefix_mdre_minor_prefix_source. (exists ff_gap_mdm_lt_mdre_minor_prefix_source_index_bound. ff_gap_mdm_lt_mdre_minor_prefix_source_index_bound + S (ff_index_mdm_prefix_mdre_minor_prefix_source) = ((q) * (q))) -> exists ff_row_mdm_prefix_mdre_minor_prefix_source ff_column_mdm_prefix_mdre_minor_prefix_source ff_value_mdm_prefix_mdre_minor_prefix_source. (ff_index_mdm_prefix_mdre_minor_prefix_source = (q) * ff_row_mdm_prefix_mdre_minor_prefix_source + ff_column_mdm_prefix_mdre_minor_prefix_source /\ ((exists ff_gap_mdm_lt_mdre_minor_prefix_source_column_bound. ff_gap_mdm_lt_mdre_minor_prefix_source_column_bound + S (ff_column_mdm_prefix_mdre_minor_prefix_source) = (q)) /\ ((exists ff_row_mdm_cell_mdre_minor_prefix_source_cell ff_column_mdm_cell_mdre_minor_prefix_source_cell. (((((exists ff_gap_mdm_lt_mdre_minor_prefix_source_cell_row_before. ff_gap_mdm_lt_mdre_minor_prefix_source_cell_row_before + S (ff_row_mdm_prefix_mdre_minor_prefix_source) = (0)) /\ ff_row_mdm_cell_mdre_minor_prefix_source_cell = ff_row_mdm_prefix_mdre_minor_prefix_source) \/ ((exists ff_gap_mdm_le_mdre_minor_prefix_source_cell_row_after. ff_gap_mdm_le_mdre_minor_prefix_source_cell_row_after + (0) = (ff_row_mdm_prefix_mdre_minor_prefix_source)) /\ ff_row_mdm_cell_mdre_minor_prefix_source_cell = S ff_row_mdm_prefix_mdre_minor_prefix_source))) /\ (((((exists ff_gap_mdm_lt_mdre_minor_prefix_source_cell_column_before. ff_gap_mdm_lt_mdre_minor_prefix_source_cell_column_before + S (ff_column_mdm_prefix_mdre_minor_prefix_source) = (j)) /\ ff_column_mdm_cell_mdre_minor_prefix_source_cell = ff_column_mdm_prefix_mdre_minor_prefix_source) \/ ((exists ff_gap_mdm_le_mdre_minor_prefix_source_cell_column_after. ff_gap_mdm_le_mdre_minor_prefix_source_cell_column_after + (j) = (ff_column_mdm_prefix_mdre_minor_prefix_source)) /\ ff_column_mdm_cell_mdre_minor_prefix_source_cell = S ff_column_mdm_prefix_mdre_minor_prefix_source))) /\ (((exists ff_h_mdm_mdre_minor_prefix_source_cell_source. ff_h_mdm_mdre_minor_prefix_source_cell_source + S (ff_value_mdm_prefix_mdre_minor_prefix_source) = S ((S ((ff_row_mdm_cell_mdre_minor_prefix_source_cell) * (S (q)) + (ff_column_mdm_cell_mdre_minor_prefix_source_cell))) * c)) /\ exists ff_q_mdm_mdre_minor_prefix_source_cell_source. b = ff_q_mdm_mdre_minor_prefix_source_cell_source * S ((S ((ff_row_mdm_cell_mdre_minor_prefix_source_cell) * (S (q)) + (ff_column_mdm_cell_mdre_minor_prefix_source_cell))) * c) + (ff_value_mdm_prefix_mdre_minor_prefix_source)))))) /\ (((exists ff_h_mdm_mdre_minor_prefix_source_target. ff_h_mdm_mdre_minor_prefix_source_target + S (ff_value_mdm_prefix_mdre_minor_prefix_source) = S ((S (ff_index_mdm_prefix_mdre_minor_prefix_source)) * v)) /\ exists ff_q_mdm_mdre_minor_prefix_source_target. u = ff_q_mdm_mdre_minor_prefix_source_target * S ((S (ff_index_mdm_prefix_mdre_minor_prefix_source)) * v) + (ff_value_mdm_prefix_mdre_minor_prefix_source))))))) -> (exists mdr_gap_minor_flat_bound. mdr_gap_minor_flat_bound + S (k) = (q * q)) -> k = q * r + s -> (exists mdr_gap_minor_coordinate_bound. mdr_gap_minor_coordinate_bound + S (s) = (q)) -> (((exists ff_h_mdr_minor_output_value. ff_h_mdr_minor_output_value + S (a) = S ((S (k)) * v)) /\ exists ff_q_mdr_minor_output_value. u = ff_q_mdr_minor_output_value * S ((S (k)) * v) + (a))) -> (exists ff_row_mdm_cell_mdre_minor_cell_result ff_column_mdm_cell_mdre_minor_cell_result. (((((exists ff_gap_mdm_lt_mdre_minor_cell_result_row_before. ff_gap_mdm_lt_mdre_minor_cell_result_row_before + S (r) = (0)) /\ ff_row_mdm_cell_mdre_minor_cell_result = r) \/ ((exists ff_gap_mdm_le_mdre_minor_cell_result_row_after. ff_gap_mdm_le_mdre_minor_cell_result_row_after + (0) = (r)) /\ ff_row_mdm_cell_mdre_minor_cell_result = S r))) /\ (((((exists ff_gap_mdm_lt_mdre_minor_cell_result_column_before. ff_gap_mdm_lt_mdre_minor_cell_result_column_before + S (s) = (j)) /\ ff_column_mdm_cell_mdre_minor_cell_result = s) \/ ((exists ff_gap_mdm_le_mdre_minor_cell_result_column_after. ff_gap_mdm_le_mdre_minor_cell_result_column_after + (j) = (s)) /\ ff_column_mdm_cell_mdre_minor_cell_result = S s))) /\ (((exists ff_h_mdm_mdre_minor_cell_result_source. ff_h_mdm_mdre_minor_cell_result_source + S (a) = S ((S ((ff_row_mdm_cell_mdre_minor_cell_result) * (S (q)) + (ff_column_mdm_cell_mdre_minor_cell_result))) * c)) /\ exists ff_q_mdm_mdre_minor_cell_result_source. b = ff_q_mdm_mdre_minor_cell_result_source * S ((S ((ff_row_mdm_cell_mdre_minor_cell_result) * (S (q)) + (ff_column_mdm_cell_mdre_minor_cell_result))) * c) + (a))))))

Complete tactic proof in conservative notation

All 58 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

58 script commands · 10 reading checkpoints · 3 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 assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro q
  4. L4
    intro j
  5. L5
    intro u
  6. L6
    intro v
  7. L7
    intro k
  8. L8
    intro r
  9. L9
    intro s
  10. L10
    intro a
02Fix variables and assumptionsL11–15

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

  1. L11
    intro hminor
  2. L12
    intro hk
  3. L13
    intro hcoordinate
  4. L14
    intro hs
  5. L15
    intro ha
03Establish hentryL16–19

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

  1. L16
    have hentry : ∃ R. ∃ C. ∃ A. k = q · R + C ∧ (Lt(C,q) ∧ (MatrixMinorCell(b,c,S q,0,j,R,C,A) ∧ BetaAt(u,v,k,A)))Definitions: Lt(C,q)MatrixMinorCell(b,c,S q,0,j,R,C,A)BetaAt(u,v,k,A)Original native command in the exact edition
  2. L17
    specialize hminor (k)
  3. L18
    apply hminor
  4. L19
    exact hk
04Separate the logical casesL20–25

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

  1. L20
    cases hentry
  2. L21
    cases hentry_witness
  3. L22
    cases hentry_witness_witness
  4. L23
    cases hentry_witness_witness_witness
  5. L24
    cases hentry_witness_witness_witness_right
  6. L25
    cases hentry_witness_witness_witness_right_right
05Establish hcoordinatesL26–35

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

  1. L26
    have hcoordinates : r = x /\ s = x1
  2. L27
    specialize division_remainder_unique (q)
  3. L28
    specialize division_remainder_unique (k)
  4. L29
    specialize division_remainder_unique (r)
  5. L30
    specialize division_remainder_unique (s)
  6. L31
    specialize division_remainder_unique (x)
  7. L32
    specialize division_remainder_unique (x1)
  8. L33
    apply division_remainder_unique
  9. L34
    exact hcoordinate
  10. L35
    exact hs
06Use earlier factsL36–37

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

  1. L36
    exact hentry_witness_witness_witness_left
  2. L37
    exact hentry_witness_witness_witness_right_left
07Separate the logical casesL38–38

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

  1. L38
    cases hcoordinates
08Establish hvalueL39–48

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L39
    have hvalue : a = x2
  2. L40
    specialize beta_at_unique (u)
  3. L41
    specialize beta_at_unique (v)
  4. L42
    specialize beta_at_unique (k)
  5. L43
    specialize beta_at_unique (a)
  6. L44
    specialize beta_at_unique (x2)
  7. L45
    apply beta_at_unique
  8. L46
    exact ha
  9. L47
    exact hentry_witness_witness_witness_right_right_right
  10. L48
    rewrite hcoordinates_left
09Calculate and transport equalitiesL49–57

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L49
    rewrite hcoordinates_left
  2. L50
    rewrite hcoordinates_left
  3. L51
    rewrite hcoordinates_left
  4. L52
    rewrite hcoordinates_right
  5. L53
    rewrite hcoordinates_right
  6. L54
    rewrite hcoordinates_right
  7. L55
    rewrite hcoordinates_right
  8. L56
    rewrite hvalue
  9. L57
    rewrite hvalue
10Use earlier factsL58–58

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

  1. L58
    exact hentry_witness_witness_witness_right_right_left

Library-wide reading audit

Original defined command ledger · 58 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro q
  4. 0004intro j
  5. 0005intro u
  6. 0006intro v
  7. 0007intro k
  8. 0008intro r
  9. 0009intro s
  10. 0010intro a
  11. 0011intro hminor
  12. 0012intro hk
  13. 0013intro hcoordinate
  14. 0014intro hs
  15. 0015intro ha
  16. 0016have hentry : ∃ R. ∃ C. ∃ A. k = q · R + C ∧ (Lt(C,q) ∧ (MatrixMinorCell(b,c,S q,0,j,R,C,A) ∧ BetaAt(u,v,k,A)))
  17. 0017specialize hminor (k)
  18. 0018apply hminor
  19. 0019exact hk
  20. 0020cases hentry
  21. 0021cases hentry_witness
  22. 0022cases hentry_witness_witness
  23. 0023cases hentry_witness_witness_witness
  24. 0024cases hentry_witness_witness_witness_right
  25. 0025cases hentry_witness_witness_witness_right_right
  26. 0026have hcoordinates : r = x /\ s = x1
  27. 0027specialize division_remainder_unique (q)
  28. 0028specialize division_remainder_unique (k)
  29. 0029specialize division_remainder_unique (r)
  30. 0030specialize division_remainder_unique (s)
  31. 0031specialize division_remainder_unique (x)
  32. 0032specialize division_remainder_unique (x1)
  33. 0033apply division_remainder_unique
  34. 0034exact hcoordinate
  35. 0035exact hs
  36. 0036exact hentry_witness_witness_witness_left
  37. 0037exact hentry_witness_witness_witness_right_left
  38. 0038cases hcoordinates
  39. 0039have hvalue : a = x2
  40. 0040specialize beta_at_unique (u)
  41. 0041specialize beta_at_unique (v)
  42. 0042specialize beta_at_unique (k)
  43. 0043specialize beta_at_unique (a)
  44. 0044specialize beta_at_unique (x2)
  45. 0045apply beta_at_unique
  46. 0046exact ha
  47. 0047exact hentry_witness_witness_witness_right_right_right
  48. 0048rewrite hcoordinates_left
  49. 0049rewrite hcoordinates_left
  50. 0050rewrite hcoordinates_left
  51. 0051rewrite hcoordinates_left
  52. 0052rewrite hcoordinates_right
  53. 0053rewrite hcoordinates_right
  54. 0054rewrite hcoordinates_right
  55. 0055rewrite hcoordinates_right
  56. 0056rewrite hvalue
  57. 0057rewrite hvalue
  58. 0058exact hentry_witness_witness_witness_right_right_left