GI0052

gaussian_sum_integer_congruence

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

Actual complex addition respects represented integer equality in both coordinates and both inputs.

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 a b c d e f g h i j k l o p q r. (((((a) + (j)) = ((i) + (b))) /\ (((c) + (l)) = ((k) + (d))))) -> (((((e) + (p)) = ((o) + (f))) /\ (((g) + (r)) = ((q) + (h))))) -> (((((((a) + (e))) + (((j) + (p)))) = ((((i) + (o))) + (((b) + (f))))) /\ (((((c) + (g))) + (((l) + (r)))) = ((((k) + (q))) + (((d) + (h)))))))

Constructive proof overview

Generated structural guide

Actual complex addition respects represented integer equality in both coordinates and both inputs.

The unchanged tactic script uses 1 declared prerequisite and contains 43 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

integer_span_pair_add_congruence Alpha theorem; checked-use authorized

Direct dependents

GI0054 gaussian_add_for_representations

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

43 script commands · 6 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 a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
  9. L9
    intro i
  10. L10
    intro j
02Fix variables and assumptionsL11–18

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

  1. L11
    intro k
  2. L12
    intro l
  3. L13
    intro o
  4. L14
    intro p
  5. L15
    intro q
  6. L16
    intro r
  7. L17
    intro hfirst
  8. L18
    intro hsecond
03Separate the logical casesL19–21

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

  1. L19
    cases hfirst
  2. L20
    cases hsecond
  3. L21
    split
04Use earlier factsL22–31

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

  1. L22
    specialize integer_span_pair_add_congruence a
  2. L23
    specialize integer_span_pair_add_congruence b
  3. L24
    specialize integer_span_pair_add_congruence e
  4. L25
    specialize integer_span_pair_add_congruence f
  5. L26
    specialize integer_span_pair_add_congruence i
  6. L27
    specialize integer_span_pair_add_congruence j
  7. L28
    specialize integer_span_pair_add_congruence o
  8. L29
    specialize integer_span_pair_add_congruence p
  9. L30
    apply integer_span_pair_add_congruence
  10. L31
    exact hfirst_left
05Use earlier factsL32–41

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

  1. L32
    exact hsecond_left
  2. L33
    specialize integer_span_pair_add_congruence c
  3. L34
    specialize integer_span_pair_add_congruence d
  4. L35
    specialize integer_span_pair_add_congruence g
  5. L36
    specialize integer_span_pair_add_congruence h
  6. L37
    specialize integer_span_pair_add_congruence k
  7. L38
    specialize integer_span_pair_add_congruence l
  8. L39
    specialize integer_span_pair_add_congruence q
  9. L40
    specialize integer_span_pair_add_congruence r
  10. L41
    apply integer_span_pair_add_congruence
06Use earlier factsL42–43

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

  1. L42
    exact hfirst_right
  2. L43
    exact hsecond_right

Library-wide reading audit

Original exact command ledger · 43 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009intro i
  10. 0010intro j
  11. 0011intro k
  12. 0012intro l
  13. 0013intro o
  14. 0014intro p
  15. 0015intro q
  16. 0016intro r
  17. 0017intro hfirst
  18. 0018intro hsecond
  19. 0019cases hfirst
  20. 0020cases hsecond
  21. 0021split
  22. 0022specialize integer_span_pair_add_congruence a
  23. 0023specialize integer_span_pair_add_congruence b
  24. 0024specialize integer_span_pair_add_congruence e
  25. 0025specialize integer_span_pair_add_congruence f
  26. 0026specialize integer_span_pair_add_congruence i
  27. 0027specialize integer_span_pair_add_congruence j
  28. 0028specialize integer_span_pair_add_congruence o
  29. 0029specialize integer_span_pair_add_congruence p
  30. 0030apply integer_span_pair_add_congruence
  31. 0031exact hfirst_left
  32. 0032exact hsecond_left
  33. 0033specialize integer_span_pair_add_congruence c
  34. 0034specialize integer_span_pair_add_congruence d
  35. 0035specialize integer_span_pair_add_congruence g
  36. 0036specialize integer_span_pair_add_congruence h
  37. 0037specialize integer_span_pair_add_congruence k
  38. 0038specialize integer_span_pair_add_congruence l
  39. 0039specialize integer_span_pair_add_congruence q
  40. 0040specialize integer_span_pair_add_congruence r
  41. 0041apply integer_span_pair_add_congruence
  42. 0042exact hfirst_right
  43. 0043exact hsecond_right