CE000F

signed_alternating_cofactor_term_odd

At every odd cofactor column, the signed alternating product swaps its exact positive and negative components.

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) → Odd(i) → p = ap · bn + an · bp ∧ n = ap · bp + an · bn

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

Definition DAG

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

Actual proof prerequisites

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

Complete unchanged native tactic proof

All 18 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

18 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 assumptionsL1–9

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
  6. L6
    intro p
  7. L7
    intro n
  8. L8
    intro hterm
  9. L9
    intro hodd
02Separate the logical casesL10–12

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

  1. L10
    cases hterm
  2. L11
    cases hterm_left
  3. L12
    exfalso
03Use earlier factsL13–16

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

  1. L13
    specialize odd_not_even i
  2. L14
    apply odd_not_even
  3. L15
    exact hodd
  4. L16
    exact hterm_left_left
04Separate the logical casesL17–17

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

  1. L17
    cases hterm_right
05Use earlier factsL18–18

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

  1. L18
    exact hterm_right_right

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro i
  6. 0006intro p
  7. 0007intro n
  8. 0008intro hterm
  9. 0009intro hodd
  10. 0010cases hterm
  11. 0011cases hterm_left
  12. 0012exfalso
  13. 0013specialize odd_not_even i
  14. 0014apply odd_not_even
  15. 0015exact hodd
  16. 0016exact hterm_left_left
  17. 0017cases hterm_right
  18. 0018exact hterm_right_right