CE000F

signed_alternating_cofactor_term_odd

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

odd_not_even Stable theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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

Separate complete second-wave branches: Full T13 proof · Alpha v27.