DL008D

matrix_integer_minor_cell_at_source

Any genuine cofactor cell is the actual source beta value at the uniquely determined skipped row and column.

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

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

Definition DAG

MatrixSkipIndex(i,r,s) · 2MatrixMinorCell(b,c,w,r,d,i,j,z) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

matrix_skip_index_functional · checked external prerequisite
Original expanded first-order statement 
forall b c q j r s u v a. ((((exists ff_gap_mdm_lt_minor_selected_row_before. ff_gap_mdm_lt_minor_selected_row_before + S (r) = (0)) /\ u = r) \/ ((exists ff_gap_mdm_le_minor_selected_row_after. ff_gap_mdm_le_minor_selected_row_after + (0) = (r)) /\ u = S r))) -> ((((exists ff_gap_mdm_lt_minor_selected_column_before. ff_gap_mdm_lt_minor_selected_column_before + S (s) = (j)) /\ v = s) \/ ((exists ff_gap_mdm_le_minor_selected_column_after. ff_gap_mdm_le_minor_selected_column_after + (j) = (s)) /\ v = S s))) -> (exists ff_row_mdm_cell_mdre_minor_source_cell ff_column_mdm_cell_mdre_minor_source_cell. (((((exists ff_gap_mdm_lt_mdre_minor_source_cell_row_before. ff_gap_mdm_lt_mdre_minor_source_cell_row_before + S (r) = (0)) /\ ff_row_mdm_cell_mdre_minor_source_cell = r) \/ ((exists ff_gap_mdm_le_mdre_minor_source_cell_row_after. ff_gap_mdm_le_mdre_minor_source_cell_row_after + (0) = (r)) /\ ff_row_mdm_cell_mdre_minor_source_cell = S r))) /\ (((((exists ff_gap_mdm_lt_mdre_minor_source_cell_column_before. ff_gap_mdm_lt_mdre_minor_source_cell_column_before + S (s) = (j)) /\ ff_column_mdm_cell_mdre_minor_source_cell = s) \/ ((exists ff_gap_mdm_le_mdre_minor_source_cell_column_after. ff_gap_mdm_le_mdre_minor_source_cell_column_after + (j) = (s)) /\ ff_column_mdm_cell_mdre_minor_source_cell = S s))) /\ (((exists ff_h_mdm_mdre_minor_source_cell_source. ff_h_mdm_mdre_minor_source_cell_source + S (a) = S ((S ((ff_row_mdm_cell_mdre_minor_source_cell) * (S (q)) + (ff_column_mdm_cell_mdre_minor_source_cell))) * c)) /\ exists ff_q_mdm_mdre_minor_source_cell_source. b = ff_q_mdm_mdre_minor_source_cell_source * S ((S ((ff_row_mdm_cell_mdre_minor_source_cell) * (S (q)) + (ff_column_mdm_cell_mdre_minor_source_cell))) * c) + (a)))))) -> (((exists ff_h_mdr_minor_source_value. ff_h_mdr_minor_source_value + S (a) = S ((S (u * (S q) + v)) * c)) /\ exists ff_q_mdr_minor_source_value. b = ff_q_mdr_minor_source_value * S ((S (u * (S q) + v)) * c) + (a)))

Complete tactic proof in conservative notation

All 37 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

37 script commands · 7 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.

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 r
  6. L6
    intro s
  7. L7
    intro u
  8. L8
    intro v
  9. L9
    intro a
  10. L10
    intro hrow
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hcolumn
  2. L12
    intro hcell
03Separate the logical casesL13–16

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

  1. L13
    cases hcell
  2. L14
    cases hcell_witness
  3. L15
    cases hcell_witness_witness
  4. L16
    cases hcell_witness_witness_right
04Establish hroweqL17–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix skip index functional.

  1. L17
    have hroweq : u = x
  2. L18
    specialize matrix_skip_index_functional (r)
  3. L19
    specialize matrix_skip_index_functional (0)
  4. L20
    specialize matrix_skip_index_functional (u)
  5. L21
    specialize matrix_skip_index_functional (x)
  6. L22
    apply matrix_skip_index_functional
  7. L23
    exact hrow
  8. L24
    exact hcell_witness_witness_left
05Establish hcoleqL25–34

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix skip index functional.

  1. L25
    have hcoleq : v = x1
  2. L26
    specialize matrix_skip_index_functional (s)
  3. L27
    specialize matrix_skip_index_functional (j)
  4. L28
    specialize matrix_skip_index_functional (v)
  5. L29
    specialize matrix_skip_index_functional (x1)
  6. L30
    apply matrix_skip_index_functional
  7. L31
    exact hcolumn
  8. L32
    exact hcell_witness_witness_right_left
  9. L33
    rewrite hroweq
  10. L34
    rewrite hroweq
06Calculate and transport equalitiesL35–36

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

  1. L35
    rewrite hcoleq
  2. L36
    rewrite hcoleq
07Use earlier factsL37–37

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

  1. L37
    exact hcell_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 37 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro q
  4. 0004intro j
  5. 0005intro r
  6. 0006intro s
  7. 0007intro u
  8. 0008intro v
  9. 0009intro a
  10. 0010intro hrow
  11. 0011intro hcolumn
  12. 0012intro hcell
  13. 0013cases hcell
  14. 0014cases hcell_witness
  15. 0015cases hcell_witness_witness
  16. 0016cases hcell_witness_witness_right
  17. 0017have hroweq : u = x
  18. 0018specialize matrix_skip_index_functional (r)
  19. 0019specialize matrix_skip_index_functional (0)
  20. 0020specialize matrix_skip_index_functional (u)
  21. 0021specialize matrix_skip_index_functional (x)
  22. 0022apply matrix_skip_index_functional
  23. 0023exact hrow
  24. 0024exact hcell_witness_witness_left
  25. 0025have hcoleq : v = x1
  26. 0026specialize matrix_skip_index_functional (s)
  27. 0027specialize matrix_skip_index_functional (j)
  28. 0028specialize matrix_skip_index_functional (v)
  29. 0029specialize matrix_skip_index_functional (x1)
  30. 0030apply matrix_skip_index_functional
  31. 0031exact hcolumn
  32. 0032exact hcell_witness_witness_right_left
  33. 0033rewrite hroweq
  34. 0034rewrite hroweq
  35. 0035rewrite hcoleq
  36. 0036rewrite hcoleq
  37. 0037exact hcell_witness_witness_right_right