CE000D

signed_alternating_cofactor_term_functional

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

Both components of the signed alternating cofactor product are uniquely determined.

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 r s. (((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 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)

Constructive proof overview

Generated structural guide

Both components of the signed alternating cofactor product are uniquely determined.

The unchanged tactic script uses 2 declared prerequisites and contains 52 exact native proof lines.

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

Proof neighborhood

Direct dependencies

even_not_odd Stable theorem; checked-use authorized odd_not_even Stable theorem; checked-use authorized

Direct dependents

CE0010 signed_alternating_cofactor_term_exists_unique CE0016 signed_alternating_product_prefix_pointwise_functional

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

52 script commands · 24 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 assumptionsL1–10

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 r
  9. L9
    intro s
  10. L10
    intro hfirst
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hsecond
03Separate the logical casesL12–18

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

  1. L12
    cases hfirst
  2. L13
    cases hfirst_left
  3. L14
    cases hfirst_left_right
  4. L15
    cases hsecond
  5. L16
    cases hsecond_left
  6. L17
    cases hsecond_left_right
  7. L18
    split
04Calculate and transport equalitiesL19–19

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

  1. L19
    trans ap * bp + an * bn
05Use earlier factsL20–20

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

  1. L20
    exact hfirst_left_right_left
06Calculate and transport equalitiesL21–21

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

  1. L21
    symm
07Use earlier factsL22–22

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

  1. L22
    exact hsecond_left_right_left
08Calculate and transport equalitiesL23–23

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

  1. L23
    trans ap * bn + an * bp
09Use earlier factsL24–24

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

  1. L24
    exact hfirst_left_right_right
10Calculate and transport equalitiesL25–25

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

  1. L25
    symm
11Use earlier factsL26–26

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

  1. L26
    exact hsecond_left_right_right
12Separate the logical casesL27–28

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

  1. L27
    cases hsecond_right
  2. L28
    exfalso
13Use earlier factsL29–32

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

  1. L29
    specialize even_not_odd i
  2. L30
    apply even_not_odd
  3. L31
    exact hfirst_left_left
  4. L32
    exact hsecond_right_left
14Separate the logical casesL33–37

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

  1. L33
    cases hfirst_right
  2. L34
    cases hfirst_right_right
  3. L35
    cases hsecond
  4. L36
    cases hsecond_left
  5. L37
    exfalso
15Use earlier factsL38–41

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

  1. L38
    specialize odd_not_even i
  2. L39
    apply odd_not_even
  3. L40
    exact hfirst_right_left
  4. L41
    exact hsecond_left_left
16Separate the logical casesL42–44

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

  1. L42
    cases hsecond_right
  2. L43
    cases hsecond_right_right
  3. L44
    split
17Calculate and transport equalitiesL45–45

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

  1. L45
    trans ap * bn + an * bp
18Use earlier factsL46–46

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

  1. L46
    exact hfirst_right_right_left
19Calculate and transport equalitiesL47–47

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

  1. L47
    symm
20Use earlier factsL48–48

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

  1. L48
    exact hsecond_right_right_left
21Calculate and transport equalitiesL49–49

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

  1. L49
    trans ap * bp + an * bn
22Use earlier factsL50–50

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

  1. L50
    exact hfirst_right_right_right
23Calculate and transport equalitiesL51–51

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

  1. L51
    symm
24Use earlier factsL52–52

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

  1. L52
    exact hsecond_right_right_right

Library-wide reading audit

Original exact command ledger · 52 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro i
  6. 0006intro p
  7. 0007intro n
  8. 0008intro r
  9. 0009intro s
  10. 0010intro hfirst
  11. 0011intro hsecond
  12. 0012cases hfirst
  13. 0013cases hfirst_left
  14. 0014cases hfirst_left_right
  15. 0015cases hsecond
  16. 0016cases hsecond_left
  17. 0017cases hsecond_left_right
  18. 0018split
  19. 0019trans ap * bp + an * bn
  20. 0020exact hfirst_left_right_left
  21. 0021symm
  22. 0022exact hsecond_left_right_left
  23. 0023trans ap * bn + an * bp
  24. 0024exact hfirst_left_right_right
  25. 0025symm
  26. 0026exact hsecond_left_right_right
  27. 0027cases hsecond_right
  28. 0028exfalso
  29. 0029specialize even_not_odd i
  30. 0030apply even_not_odd
  31. 0031exact hfirst_left_left
  32. 0032exact hsecond_right_left
  33. 0033cases hfirst_right
  34. 0034cases hfirst_right_right
  35. 0035cases hsecond
  36. 0036cases hsecond_left
  37. 0037exfalso
  38. 0038specialize odd_not_even i
  39. 0039apply odd_not_even
  40. 0040exact hfirst_right_left
  41. 0041exact hsecond_left_left
  42. 0042cases hsecond_right
  43. 0043cases hsecond_right_right
  44. 0044split
  45. 0045trans ap * bn + an * bp
  46. 0046exact hfirst_right_right_left
  47. 0047symm
  48. 0048exact hsecond_right_right_left
  49. 0049trans ap * bp + an * bn
  50. 0050exact hfirst_right_right_right
  51. 0051symm
  52. 0052exact hsecond_right_right_right

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