CE001B

signed_matrix_first_row_components_exists

Every arbitrary-dimensional signed square matrix has two complete beta-coded first-row natural-component streams.

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. ∃ ab. ∃ ac. ∃ db. ∃ dc. MatrixAffineSlice(pb,pc,0,1,ab,ac,S q) ∧ MatrixAffineSlice(nb,nc,0,1,db,dc,S q)

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

Definition DAG

MatrixAffineSlice(b,c,s,d,u,v,l) · 2

Actual proof prerequisites

beta_affine_matrix_slice_exists · checked external prerequisite
Original expanded first-order statement 
forall pb pc nb nc q. exists ab ac db dc. ((forall ff_index_mcp_mce_first_row_positive ff_source_mcp_mce_first_row_positive ff_target_mcp_mce_first_row_positive. (exists mcp_gap_mce_first_row_positive_bound. mcp_gap_mce_first_row_positive_bound + S (ff_index_mcp_mce_first_row_positive) = (S q)) -> (((exists fs_h_mcp_mce_first_row_positive_source. fs_h_mcp_mce_first_row_positive_source + S (ff_source_mcp_mce_first_row_positive) = S ((S ((0) + (1) * ff_index_mcp_mce_first_row_positive)) * pc)) /\ exists fs_q_mcp_mce_first_row_positive_source. pb = fs_q_mcp_mce_first_row_positive_source * S ((S ((0) + (1) * ff_index_mcp_mce_first_row_positive)) * pc) + (ff_source_mcp_mce_first_row_positive))) -> (((exists fs_h_mcp_mce_first_row_positive_target. fs_h_mcp_mce_first_row_positive_target + S (ff_target_mcp_mce_first_row_positive) = S ((S (ff_index_mcp_mce_first_row_positive)) * ac)) /\ exists fs_q_mcp_mce_first_row_positive_target. ab = fs_q_mcp_mce_first_row_positive_target * S ((S (ff_index_mcp_mce_first_row_positive)) * ac) + (ff_target_mcp_mce_first_row_positive))) -> ff_target_mcp_mce_first_row_positive = ff_source_mcp_mce_first_row_positive) /\ (forall ff_index_mcp_mce_first_row_negative ff_source_mcp_mce_first_row_negative ff_target_mcp_mce_first_row_negative. (exists mcp_gap_mce_first_row_negative_bound. mcp_gap_mce_first_row_negative_bound + S (ff_index_mcp_mce_first_row_negative) = (S q)) -> (((exists fs_h_mcp_mce_first_row_negative_source. fs_h_mcp_mce_first_row_negative_source + S (ff_source_mcp_mce_first_row_negative) = S ((S ((0) + (1) * ff_index_mcp_mce_first_row_negative)) * nc)) /\ exists fs_q_mcp_mce_first_row_negative_source. nb = fs_q_mcp_mce_first_row_negative_source * S ((S ((0) + (1) * ff_index_mcp_mce_first_row_negative)) * nc) + (ff_source_mcp_mce_first_row_negative))) -> (((exists fs_h_mcp_mce_first_row_negative_target. fs_h_mcp_mce_first_row_negative_target + S (ff_target_mcp_mce_first_row_negative) = S ((S (ff_index_mcp_mce_first_row_negative)) * dc)) /\ exists fs_q_mcp_mce_first_row_negative_target. db = fs_q_mcp_mce_first_row_negative_target * S ((S (ff_index_mcp_mce_first_row_negative)) * dc) + (ff_target_mcp_mce_first_row_negative))) -> ff_target_mcp_mce_first_row_negative = ff_source_mcp_mce_first_row_negative))

Complete unchanged native tactic proof

All 30 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

30 script commands · 8 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.

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 assumptionsL1–5

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
02Establish hpositiveL6–12

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

  1. L6
    have hpositive : ∃ ab. ∃ ac. MatrixAffineSlice(pb,pc,0,1,ab,ac,S q)Definitions: MatrixAffineSliceOriginal native command in the exact edition
  2. L7
    specialize beta_affine_matrix_slice_exists pb
  3. L8
    specialize beta_affine_matrix_slice_exists pc
  4. L9
    specialize beta_affine_matrix_slice_exists 0
  5. L10
    specialize beta_affine_matrix_slice_exists 1
  6. L11
    specialize beta_affine_matrix_slice_exists (S q)
  7. L12
    exact beta_affine_matrix_slice_exists
03Separate the logical casesL13–14

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

  1. L13
    cases hpositive
  2. L14
    cases hpositive_witness
04Establish hnegativeL15–21

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

  1. L15
    have hnegative : ∃ db. ∃ dc. MatrixAffineSlice(nb,nc,0,1,db,dc,S q)Definitions: MatrixAffineSliceOriginal native command in the exact edition
  2. L16
    specialize beta_affine_matrix_slice_exists nb
  3. L17
    specialize beta_affine_matrix_slice_exists nc
  4. L18
    specialize beta_affine_matrix_slice_exists 0
  5. L19
    specialize beta_affine_matrix_slice_exists 1
  6. L20
    specialize beta_affine_matrix_slice_exists (S q)
  7. L21
    exact beta_affine_matrix_slice_exists
05Separate the logical casesL22–23

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

  1. L22
    cases hnegative
  2. L23
    cases hnegative_witness
06Construct an explicit witnessL24–27

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

  1. L24
    exists x
  2. L25
    exists x1
  3. L26
    exists x2
  4. L27
    exists x3
07Separate the logical casesL28–28

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

  1. L28
    split
08Use earlier factsL29–30

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

  1. L29
    exact hpositive_witness_witness
  2. L30
    exact hnegative_witness_witness

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro pb
  2. 0002intro pc
  3. 0003intro nb
  4. 0004intro nc
  5. 0005intro q
  6. 0006have hpositive : exists ab ac. (forall ff_index_mcp_mce_have_row_positive ff_source_mcp_mce_have_row_positive ff_target_mcp_mce_have_row_positive. (exists mcp_gap_mce_have_row_positive_bound. mcp_gap_mce_have_row_positive_bound + S (ff_index_mcp_mce_have_row_positive) = (S q)) -> (((exists fs_h_mcp_mce_have_row_positive_source. fs_h_mcp_mce_have_row_positive_source + S (ff_source_mcp_mce_have_row_positive) = S ((S ((0) + (1) * ff_index_mcp_mce_have_row_positive)) * pc)) /\ exists fs_q_mcp_mce_have_row_positive_source. pb = fs_q_mcp_mce_have_row_positive_source * S ((S ((0) + (1) * ff_index_mcp_mce_have_row_positive)) * pc) + (ff_source_mcp_mce_have_row_positive))) -> (((exists fs_h_mcp_mce_have_row_positive_target. fs_h_mcp_mce_have_row_positive_target + S (ff_target_mcp_mce_have_row_positive) = S ((S (ff_index_mcp_mce_have_row_positive)) * ac)) /\ exists fs_q_mcp_mce_have_row_positive_target. ab = fs_q_mcp_mce_have_row_positive_target * S ((S (ff_index_mcp_mce_have_row_positive)) * ac) + (ff_target_mcp_mce_have_row_positive))) -> ff_target_mcp_mce_have_row_positive = ff_source_mcp_mce_have_row_positive)
  7. 0007specialize beta_affine_matrix_slice_exists pb
  8. 0008specialize beta_affine_matrix_slice_exists pc
  9. 0009specialize beta_affine_matrix_slice_exists 0
  10. 0010specialize beta_affine_matrix_slice_exists 1
  11. 0011specialize beta_affine_matrix_slice_exists (S q)
  12. 0012exact beta_affine_matrix_slice_exists
  13. 0013cases hpositive
  14. 0014cases hpositive_witness
  15. 0015have hnegative : exists db dc. (forall ff_index_mcp_mce_have_row_negative ff_source_mcp_mce_have_row_negative ff_target_mcp_mce_have_row_negative. (exists mcp_gap_mce_have_row_negative_bound. mcp_gap_mce_have_row_negative_bound + S (ff_index_mcp_mce_have_row_negative) = (S q)) -> (((exists fs_h_mcp_mce_have_row_negative_source. fs_h_mcp_mce_have_row_negative_source + S (ff_source_mcp_mce_have_row_negative) = S ((S ((0) + (1) * ff_index_mcp_mce_have_row_negative)) * nc)) /\ exists fs_q_mcp_mce_have_row_negative_source. nb = fs_q_mcp_mce_have_row_negative_source * S ((S ((0) + (1) * ff_index_mcp_mce_have_row_negative)) * nc) + (ff_source_mcp_mce_have_row_negative))) -> (((exists fs_h_mcp_mce_have_row_negative_target. fs_h_mcp_mce_have_row_negative_target + S (ff_target_mcp_mce_have_row_negative) = S ((S (ff_index_mcp_mce_have_row_negative)) * dc)) /\ exists fs_q_mcp_mce_have_row_negative_target. db = fs_q_mcp_mce_have_row_negative_target * S ((S (ff_index_mcp_mce_have_row_negative)) * dc) + (ff_target_mcp_mce_have_row_negative))) -> ff_target_mcp_mce_have_row_negative = ff_source_mcp_mce_have_row_negative)
  16. 0016specialize beta_affine_matrix_slice_exists nb
  17. 0017specialize beta_affine_matrix_slice_exists nc
  18. 0018specialize beta_affine_matrix_slice_exists 0
  19. 0019specialize beta_affine_matrix_slice_exists 1
  20. 0020specialize beta_affine_matrix_slice_exists (S q)
  21. 0021exact beta_affine_matrix_slice_exists
  22. 0022cases hnegative
  23. 0023cases hnegative_witness
  24. 0024exists x
  25. 0025exists x1
  26. 0026exists x2
  27. 0027exists x3
  28. 0028split
  29. 0029exact hpositive_witness_witness
  30. 0030exact hnegative_witness_witness