CE000E

signed_alternating_cofactor_term_even

At every even cofactor column, the signed alternating product has the unswapped 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) → Even(i) → p = ap · bp + an · bn ∧ n = ap · bn + an · bp

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

Definition DAG

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

Actual proof prerequisites

even_not_odd · 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) -> (p = ap * bp + an * bn /\ n = ap * bn + an * bp)

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 heven
02Separate the logical casesL10–11

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

  1. L10
    cases hterm
  2. L11
    cases hterm_left
03Use earlier factsL12–12

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

  1. L12
    exact hterm_left_right
04Separate the logical casesL13–14

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

  1. L13
    cases hterm_right
  2. L14
    exfalso
05Use earlier factsL15–18

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

  1. L15
    specialize even_not_odd i
  2. L16
    apply even_not_odd
  3. L17
    exact heven
  4. L18
    exact hterm_right_left

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 heven
  10. 0010cases hterm
  11. 0011cases hterm_left
  12. 0012exact hterm_left_right
  13. 0013cases hterm_right
  14. 0014exfalso
  15. 0015specialize even_not_odd i
  16. 0016apply even_not_odd
  17. 0017exact heven
  18. 0018exact hterm_right_left