MN000F

signed_matrix_four_cofactor_expansion_functional

Both natural components of a four-term signed Laplace cofactor expansion are unique.

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

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 p n q m. (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))))) -> (q = ((((((((ap) * (up) + (an) * (un))) + (((bp) * (vn) + (bn) * (vp))))) + (((cp) * (wp) + (cn) * (wn))))) + (((dp) * (xn) + (dn) * (xp)))) /\ m = ((((((((ap) * (un) + (an) * (up))) + (((bp) * (vp) + (bn) * (vn))))) + (((cp) * (wn) + (cn) * (wp))))) + (((dp) * (xp) + (dn) * (xn))))) -> (p = q /\ n = m)

Complete unchanged native tactic proof

All 33 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

33 script commands · 12 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 assumptionsL11–20

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
  7. L17
    intro p
  8. L18
    intro n
  9. L19
    intro q
  10. L20
    intro m
03Fix variables and assumptionsL21–22

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

  1. L21
    intro hfirst
  2. L22
    intro hsecond
04Separate the logical casesL23–25

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

  1. L23
    cases hfirst
  2. L24
    cases hsecond
  3. L25
    split
05Calculate and transport equalitiesL26–26

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

  1. L26
    trans ((((((((ap) * (up) + (an) * (un))) + (((bp) * (vn) + (bn) * (vp))))) + (((cp) * (wp) + (cn) * (wn))))) + (((dp) * (xn) + (dn) * (xp))))
06Use earlier factsL27–27

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

  1. L27
    exact hfirst_left
07Calculate and transport equalitiesL28–28

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

  1. L28
    symm
08Use earlier factsL29–29

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

  1. L29
    exact hsecond_left
09Calculate and transport equalitiesL30–30

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

  1. L30
    trans ((((((((ap) * (un) + (an) * (up))) + (((bp) * (vp) + (bn) * (vn))))) + (((cp) * (wn) + (cn) * (wp))))) + (((dp) * (xp) + (dn) * (xn))))
10Use earlier factsL31–31

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

  1. L31
    exact hfirst_right
11Calculate and transport equalitiesL32–32

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

  1. L32
    symm
12Use earlier factsL33–33

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

  1. L33
    exact hsecond_right

Library-wide reading audit

Original defined command ledger · 33 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. 0017intro p
  18. 0018intro n
  19. 0019intro q
  20. 0020intro m
  21. 0021intro hfirst
  22. 0022intro hsecond
  23. 0023cases hfirst
  24. 0024cases hsecond
  25. 0025split
  26. 0026trans ((((((((ap) * (up) + (an) * (un))) + (((bp) * (vn) + (bn) * (vp))))) + (((cp) * (wp) + (cn) * (wn))))) + (((dp) * (xn) + (dn) * (xp))))
  27. 0027exact hfirst_left
  28. 0028symm
  29. 0029exact hsecond_left
  30. 0030trans ((((((((ap) * (un) + (an) * (up))) + (((bp) * (vp) + (bn) * (vn))))) + (((cp) * (wn) + (cn) * (wp))))) + (((dp) * (xp) + (dn) * (xn))))
  31. 0031exact hfirst_right
  32. 0032symm
  33. 0033exact hsecond_right