GI002E

gaussian_difference_integer_congruence

Actual Gaussian subtraction preserves integer representative equivalence in both operands.

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. ∀ m. ∀ n. ∀ o. ∀ p. a + f = e + b ∧ c + h = g + d → i + n = m + j ∧ k + p = o + l → a + j + (f + m) = e + n + (b + i) ∧ c + l + (h + o) = g + p + (d + k)

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

Definition DAG

none

Actual proof prerequisites

integer_span_pair_add_congruence · checked external prerequisitematrix_integer_pair_negation_balance · checked external prerequisite
Original expanded first-order statement 
forall a b c d e f g h i j k l m n o p. (((((a) + (f)) = ((e) + (b))) /\ (((c) + (h)) = ((g) + (d))))) -> (((((i) + (n)) = ((m) + (j))) /\ (((k) + (p)) = ((o) + (l))))) -> (((((((a) + (j))) + (((f) + (m)))) = ((((e) + (n))) + (((b) + (i))))) /\ (((((c) + (l))) + (((h) + (o)))) = ((((g) + (p))) + (((d) + (k)))))))

Complete tactic proof in conservative notation

All 53 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

53 script commands · 7 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 m
  4. L14
    intro n
  5. L15
    intro o
  6. L16
    intro p
  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 j
  4. L25
    specialize integer_span_pair_add_congruence i
  5. L26
    specialize integer_span_pair_add_congruence e
  6. L27
    specialize integer_span_pair_add_congruence f
  7. L28
    specialize integer_span_pair_add_congruence n
  8. L29
    specialize integer_span_pair_add_congruence m
  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
    specialize matrix_integer_pair_negation_balance i
  2. L33
    specialize matrix_integer_pair_negation_balance j
  3. L34
    specialize matrix_integer_pair_negation_balance m
  4. L35
    specialize matrix_integer_pair_negation_balance n
  5. L36
    apply matrix_integer_pair_negation_balance
  6. L37
    exact hsecond_left
  7. L38
    specialize integer_span_pair_add_congruence c
  8. L39
    specialize integer_span_pair_add_congruence d
  9. L40
    specialize integer_span_pair_add_congruence l
  10. L41
    specialize integer_span_pair_add_congruence k
06Use earlier factsL42–51

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

  1. L42
    specialize integer_span_pair_add_congruence g
  2. L43
    specialize integer_span_pair_add_congruence h
  3. L44
    specialize integer_span_pair_add_congruence p
  4. L45
    specialize integer_span_pair_add_congruence o
  5. L46
    apply integer_span_pair_add_congruence
  6. L47
    exact hfirst_right
  7. L48
    specialize matrix_integer_pair_negation_balance k
  8. L49
    specialize matrix_integer_pair_negation_balance l
  9. L50
    specialize matrix_integer_pair_negation_balance o
  10. L51
    specialize matrix_integer_pair_negation_balance p
07Use earlier factsL52–53

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

  1. L52
    apply matrix_integer_pair_negation_balance
  2. L53
    exact hsecond_right

Library-wide reading audit

Original defined command ledger · 53 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 m
  14. 0014intro n
  15. 0015intro o
  16. 0016intro p
  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 j
  25. 0025specialize integer_span_pair_add_congruence i
  26. 0026specialize integer_span_pair_add_congruence e
  27. 0027specialize integer_span_pair_add_congruence f
  28. 0028specialize integer_span_pair_add_congruence n
  29. 0029specialize integer_span_pair_add_congruence m
  30. 0030apply integer_span_pair_add_congruence
  31. 0031exact hfirst_left
  32. 0032specialize matrix_integer_pair_negation_balance i
  33. 0033specialize matrix_integer_pair_negation_balance j
  34. 0034specialize matrix_integer_pair_negation_balance m
  35. 0035specialize matrix_integer_pair_negation_balance n
  36. 0036apply matrix_integer_pair_negation_balance
  37. 0037exact hsecond_left
  38. 0038specialize integer_span_pair_add_congruence c
  39. 0039specialize integer_span_pair_add_congruence d
  40. 0040specialize integer_span_pair_add_congruence l
  41. 0041specialize integer_span_pair_add_congruence k
  42. 0042specialize integer_span_pair_add_congruence g
  43. 0043specialize integer_span_pair_add_congruence h
  44. 0044specialize integer_span_pair_add_congruence p
  45. 0045specialize integer_span_pair_add_congruence o
  46. 0046apply integer_span_pair_add_congruence
  47. 0047exact hfirst_right
  48. 0048specialize matrix_integer_pair_negation_balance k
  49. 0049specialize matrix_integer_pair_negation_balance l
  50. 0050specialize matrix_integer_pair_negation_balance o
  51. 0051specialize matrix_integer_pair_negation_balance p
  52. 0052apply matrix_integer_pair_negation_balance
  53. 0053exact hsecond_right