GI0052

gaussian_sum_integer_congruence

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

Alpha v34 checked-use · first admitted v28 · 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.

The natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ 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)

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

Definition DAG

none

Actual proof prerequisites

integer_span_pair_add_congruence · checked external prerequisite
Original expanded first-order 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)))))))

Complete tactic proof in conservative notation

All 43 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

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 assumptionsL1–10

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 defined 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