CE0010

signed_alternating_cofactor_term_exists_unique

Every arbitrary genuinely signed cofactor term has exactly one parity-correct subtraction-free component pair.

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) ∧ (∀ x. ∀ y. SignedAlternatingCofactorTerm(ap,an,bp,bn,i,x,y) → p = x ∧ n = y)

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

Definition DAG

SignedAlternatingCofactorTerm(ap,an,bp,bn,i,p,n) · 2

Actual proof prerequisites

signed_alternating_cofactor_term_existssigned_alternating_cofactor_term_functional
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)))) /\ forall r s. (((exists ff_even_mce_term_other. i = 2 * ff_even_mce_term_other) /\ (r = (ap) * (bp) + (an) * (bn) /\ s = (ap) * (bn) + (an) * (bp))) \/ ((exists ff_odd_mce_term_other. i = 2 * ff_odd_mce_term_other + 1) /\ (r = (ap) * (bn) + (an) * (bp) /\ s = (ap) * (bp) + (an) * (bn)))) -> (p = r /\ n = s))

Complete unchanged native tactic proof

All 31 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

31 script commands · 9 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.

Named ingredients (2)
signed alternating cofactor term existssigned alternating cofactor term functional
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 factsL6–10

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

  1. L6
    specialize signed_alternating_cofactor_term_exists ap
  2. L7
    specialize signed_alternating_cofactor_term_exists an
  3. L8
    specialize signed_alternating_cofactor_term_exists bp
  4. L9
    specialize signed_alternating_cofactor_term_exists bn
  5. L10
    specialize signed_alternating_cofactor_term_exists i
03Separate the logical casesL11–12

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

  1. L11
    cases signed_alternating_cofactor_term_exists
  2. L12
    cases signed_alternating_cofactor_term_exists_witness
04Construct an explicit witnessL13–14

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

  1. L13
    exists x
  2. L14
    exists x1
05Separate the logical casesL15–15

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

  1. L15
    split
06Use earlier factsL16–16

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

  1. L16
    exact signed_alternating_cofactor_term_exists_witness_witness
07Fix variables and assumptionsL17–19

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

  1. L17
    intro r
  2. L18
    intro s
  3. L19
    intro hother
08Use earlier factsL20–29

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

  1. L20
    specialize signed_alternating_cofactor_term_functional ap
  2. L21
    specialize signed_alternating_cofactor_term_functional an
  3. L22
    specialize signed_alternating_cofactor_term_functional bp
  4. L23
    specialize signed_alternating_cofactor_term_functional bn
  5. L24
    specialize signed_alternating_cofactor_term_functional i
  6. L25
    specialize signed_alternating_cofactor_term_functional x
  7. L26
    specialize signed_alternating_cofactor_term_functional x1
  8. L27
    specialize signed_alternating_cofactor_term_functional r
  9. L28
    specialize signed_alternating_cofactor_term_functional s
  10. L29
    apply signed_alternating_cofactor_term_functional
09Use earlier factsL30–31

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

  1. L30
    exact signed_alternating_cofactor_term_exists_witness_witness
  2. L31
    exact hother

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro i
  6. 0006specialize signed_alternating_cofactor_term_exists ap
  7. 0007specialize signed_alternating_cofactor_term_exists an
  8. 0008specialize signed_alternating_cofactor_term_exists bp
  9. 0009specialize signed_alternating_cofactor_term_exists bn
  10. 0010specialize signed_alternating_cofactor_term_exists i
  11. 0011cases signed_alternating_cofactor_term_exists
  12. 0012cases signed_alternating_cofactor_term_exists_witness
  13. 0013exists x
  14. 0014exists x1
  15. 0015split
  16. 0016exact signed_alternating_cofactor_term_exists_witness_witness
  17. 0017intro r
  18. 0018intro s
  19. 0019intro hother
  20. 0020specialize signed_alternating_cofactor_term_functional ap
  21. 0021specialize signed_alternating_cofactor_term_functional an
  22. 0022specialize signed_alternating_cofactor_term_functional bp
  23. 0023specialize signed_alternating_cofactor_term_functional bn
  24. 0024specialize signed_alternating_cofactor_term_functional i
  25. 0025specialize signed_alternating_cofactor_term_functional x
  26. 0026specialize signed_alternating_cofactor_term_functional x1
  27. 0027specialize signed_alternating_cofactor_term_functional r
  28. 0028specialize signed_alternating_cofactor_term_functional s
  29. 0029apply signed_alternating_cofactor_term_functional
  30. 0030exact signed_alternating_cofactor_term_exists_witness_witness
  31. 0031exact hother