MN000E

signed_matrix_four_cofactor_expansion_exists

Four arbitrary signed first-row entries and four signed minor determinants have their exact alternating subtraction-free Laplace cofactor expansion.

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

∀ ap. ∀ an. ∀ bp. ∀ bn. ∀ cp. ∀ cn. ∀ dp. ∀ dn. ∀ up. ∀ un. ∀ vp. ∀ vn. ∀ wp. ∀ wn. ∀ xp. ∀ xn. ∃ p. ∃ n. p = ap · up + an · un + (bp · vn + bn · vp) + (cp · wp + cn · wn) + (dp · xn + dn · xp) ∧ n = ap · un + an · up + (bp · vp + bn · vn) + (cp · wn + cn · wp) + (dp · xp + dn · xn)

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

Definition DAG

none

Actual proof prerequisites

none
Original expanded first-order statement 
forall ap an bp bn cp cn dp dn up un vp vn wp wn xp xn. exists p n. (p = ((((((((ap) * (up) + (an) * (un))) + (((bp) * (vn) + (bn) * (vp))))) + (((cp) * (wp) + (cn) * (wn))))) + (((dp) * (xn) + (dn) * (xp)))) /\ n = ((((((((ap) * (un) + (an) * (up))) + (((bp) * (vp) + (bn) * (vn))))) + (((cp) * (wn) + (cn) * (wp))))) + (((dp) * (xp) + (dn) * (xn)))))

Complete unchanged native tactic proof

All 21 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

21 script commands · 5 reading checkpoints · 0 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 ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro cp
  6. L6
    intro cn
  7. L7
    intro dp
  8. L8
    intro dn
  9. L9
    intro up
  10. L10
    intro un
02Fix variables and assumptionsL11–16

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

  1. L11
    intro vp
  2. L12
    intro vn
  3. L13
    intro wp
  4. L14
    intro wn
  5. L15
    intro xp
  6. L16
    intro xn
03Construct an explicit witnessL17–18

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

  1. L17
    exists ((((((((ap) * (up) + (an) * (un))) + (((bp) * (vn) + (bn) * (vp))))) + (((cp) * (wp) + (cn) * (wn))))) + (((dp) * (xn) + (dn) * (xp))))
  2. L18
    exists ((((((((ap) * (un) + (an) * (up))) + (((bp) * (vp) + (bn) * (vn))))) + (((cp) * (wn) + (cn) * (wp))))) + (((dp) * (xp) + (dn) * (xn))))
04Separate the logical casesL19–19

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

  1. L19
    split
05Calculate and transport equalitiesL20–21

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

  1. L20
    refl
  2. L21
    refl

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro cp
  6. 0006intro cn
  7. 0007intro dp
  8. 0008intro dn
  9. 0009intro up
  10. 0010intro un
  11. 0011intro vp
  12. 0012intro vn
  13. 0013intro wp
  14. 0014intro wn
  15. 0015intro xp
  16. 0016intro xn
  17. 0017exists ((((((((ap) * (up) + (an) * (un))) + (((bp) * (vn) + (bn) * (vp))))) + (((cp) * (wp) + (cn) * (wn))))) + (((dp) * (xn) + (dn) * (xp))))
  18. 0018exists ((((((((ap) * (un) + (an) * (up))) + (((bp) * (vp) + (bn) * (vn))))) + (((cp) * (wn) + (cn) * (wp))))) + (((dp) * (xp) + (dn) * (xn))))
  19. 0019split
  20. 0020refl
  21. 0021refl