CE000C

signed_alternating_cofactor_term_exists

Every genuinely signed row/cofactor pair has its exact parity-correct alternating product.

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

∀ ap. ∀ an. ∀ bp. ∀ bn. ∀ i. ∃ p. ∃ n. SignedAlternatingCofactorTerm(ap,an,bp,bn,i,p,n)

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

Definition DAG

SignedAlternatingCofactorTerm(ap,an,bp,bn,i,p,n) · 1

Actual proof prerequisites

parity_cases · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn i. exists p n. (((exists ff_even_mce_term_result. i = 2 * ff_even_mce_term_result) /\ (p = (ap) * (bp) + (an) * (bn) /\ n = (ap) * (bn) + (an) * (bp))) \/ ((exists ff_odd_mce_term_result. i = 2 * ff_odd_mce_term_result + 1) /\ (p = (ap) * (bn) + (an) * (bp) /\ n = (ap) * (bp) + (an) * (bn))))

Complete unchanged native tactic proof

All 26 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

26 script commands · 15 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 assumptionsL1–5

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 i
02Use earlier factsL6–6

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

  1. L6
    specialize parity_cases i
03Separate the logical casesL7–8

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

  1. L7
    cases parity_cases
  2. L8
    cases parity_cases_witness
04Construct an explicit witnessL9–10

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

  1. L9
    exists ap * bp + an * bn
  2. L10
    exists ap * bn + an * bp
05Separate the logical casesL11–12

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

  1. L11
    left
  2. L12
    split
06Construct an explicit witnessL13–13

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

  1. L13
    exists x
07Use earlier factsL14–14

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

  1. L14
    exact parity_cases_witness_left
08Separate the logical casesL15–15

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

  1. L15
    split
09Calculate and transport equalitiesL16–17

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

  1. L16
    refl
  2. L17
    refl
10Construct an explicit witnessL18–19

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

  1. L18
    exists ap * bn + an * bp
  2. L19
    exists ap * bp + an * bn
11Separate the logical casesL20–21

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

  1. L20
    right
  2. L21
    split
12Construct an explicit witnessL22–22

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

  1. L22
    exists x
13Use earlier factsL23–23

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

  1. L23
    exact parity_cases_witness_right
14Separate the logical casesL24–24

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

  1. L24
    split
15Calculate and transport equalitiesL25–26

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

  1. L25
    refl
  2. L26
    refl

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro i
  6. 0006specialize parity_cases i
  7. 0007cases parity_cases
  8. 0008cases parity_cases_witness
  9. 0009exists ap * bp + an * bn
  10. 0010exists ap * bn + an * bp
  11. 0011left
  12. 0012split
  13. 0013exists x
  14. 0014exact parity_cases_witness_left
  15. 0015split
  16. 0016refl
  17. 0017refl
  18. 0018exists ap * bn + an * bp
  19. 0019exists ap * bp + an * bn
  20. 0020right
  21. 0021split
  22. 0022exists x
  23. 0023exact parity_cases_witness_right
  24. 0024split
  25. 0025refl
  26. 0026refl