GI0041

gaussian_representation_integer_transport

A canonical Gaussian coordinate code is invariant under arbitrary equal signed representatives.

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

∀ z. ∀ a. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. a + f = e + b ∧ c + h = g + d → ZPairRep(z,a,b,c,d) → ZPairRep(z,e,f,g,h)

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

Definition DAG

ZPairRep(z,ap,an,bp,bn) · 2

Actual proof prerequisites

gaussian_signed_balance_integer_transport
Original expanded first-order statement 
forall z a b c d e f g h. (((((a) + (f)) = ((e) + (b))) /\ (((c) + (h)) = ((g) + (d))))) -> (exists ge_representation_real_code_representation_transport_source ge_representation_imaginary_code_representation_transport_source. (((z) = ((ge_representation_real_code_representation_transport_source) + (ge_representation_imaginary_code_representation_transport_source)) * S ((ge_representation_real_code_representation_transport_source) + (ge_representation_imaginary_code_representation_transport_source)) + ((ge_representation_imaginary_code_representation_transport_source) + (ge_representation_imaginary_code_representation_transport_source))) /\ ((exists ge_balance_positive_representation_transport_sourcereal ge_balance_negative_representation_transport_sourcereal. (((((ge_representation_real_code_representation_transport_source) = 2 * (ge_balance_positive_representation_transport_sourcereal) /\ (ge_balance_negative_representation_transport_sourcereal) = 0) \/ exists ge_signed_half_representation_transport_sourcerealdecode. (((ge_representation_real_code_representation_transport_source) = 2 * ge_signed_half_representation_transport_sourcerealdecode + 1 /\ (ge_balance_positive_representation_transport_sourcereal) = 0) /\ (ge_balance_negative_representation_transport_sourcereal) = S ge_signed_half_representation_transport_sourcerealdecode))) /\ ((a) + ge_balance_negative_representation_transport_sourcereal = (b) + ge_balance_positive_representation_transport_sourcereal))) /\ (exists ge_balance_positive_representation_transport_sourceimaginary ge_balance_negative_representation_transport_sourceimaginary. (((((ge_representation_imaginary_code_representation_transport_source) = 2 * (ge_balance_positive_representation_transport_sourceimaginary) /\ (ge_balance_negative_representation_transport_sourceimaginary) = 0) \/ exists ge_signed_half_representation_transport_sourceimaginarydecode. (((ge_representation_imaginary_code_representation_transport_source) = 2 * ge_signed_half_representation_transport_sourceimaginarydecode + 1 /\ (ge_balance_positive_representation_transport_sourceimaginary) = 0) /\ (ge_balance_negative_representation_transport_sourceimaginary) = S ge_signed_half_representation_transport_sourceimaginarydecode))) /\ ((c) + ge_balance_negative_representation_transport_sourceimaginary = (d) + ge_balance_positive_representation_transport_sourceimaginary)))))) -> (exists ge_representation_real_code_representation_transport_target ge_representation_imaginary_code_representation_transport_target. (((z) = ((ge_representation_real_code_representation_transport_target) + (ge_representation_imaginary_code_representation_transport_target)) * S ((ge_representation_real_code_representation_transport_target) + (ge_representation_imaginary_code_representation_transport_target)) + ((ge_representation_imaginary_code_representation_transport_target) + (ge_representation_imaginary_code_representation_transport_target))) /\ ((exists ge_balance_positive_representation_transport_targetreal ge_balance_negative_representation_transport_targetreal. (((((ge_representation_real_code_representation_transport_target) = 2 * (ge_balance_positive_representation_transport_targetreal) /\ (ge_balance_negative_representation_transport_targetreal) = 0) \/ exists ge_signed_half_representation_transport_targetrealdecode. (((ge_representation_real_code_representation_transport_target) = 2 * ge_signed_half_representation_transport_targetrealdecode + 1 /\ (ge_balance_positive_representation_transport_targetreal) = 0) /\ (ge_balance_negative_representation_transport_targetreal) = S ge_signed_half_representation_transport_targetrealdecode))) /\ ((e) + ge_balance_negative_representation_transport_targetreal = (f) + ge_balance_positive_representation_transport_targetreal))) /\ (exists ge_balance_positive_representation_transport_targetimaginary ge_balance_negative_representation_transport_targetimaginary. (((((ge_representation_imaginary_code_representation_transport_target) = 2 * (ge_balance_positive_representation_transport_targetimaginary) /\ (ge_balance_negative_representation_transport_targetimaginary) = 0) \/ exists ge_signed_half_representation_transport_targetimaginarydecode. (((ge_representation_imaginary_code_representation_transport_target) = 2 * ge_signed_half_representation_transport_targetimaginarydecode + 1 /\ (ge_balance_positive_representation_transport_targetimaginary) = 0) /\ (ge_balance_negative_representation_transport_targetimaginary) = S ge_signed_half_representation_transport_targetimaginarydecode))) /\ ((g) + ge_balance_negative_representation_transport_targetimaginary = (h) + ge_balance_positive_representation_transport_targetimaginary))))))

Complete tactic proof in conservative notation

All 37 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

37 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 (1)
gaussian signed balance integer transport
01Fix variables and assumptionsL1–10

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

  1. L1
    intro z
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro d
  6. L6
    intro e
  7. L7
    intro f
  8. L8
    intro g
  9. L9
    intro h
  10. L10
    intro hequal
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hrepresentation
03Separate the logical casesL12–16

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

  1. L12
    cases hequal
  2. L13
    cases hrepresentation
  3. L14
    cases hrepresentation_witness
  4. L15
    cases hrepresentation_witness_witness
  5. L16
    cases hrepresentation_witness_witness_right
04Construct an explicit witnessL17–18

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

  1. L17
    exists x
  2. L18
    exists x1
05Separate the logical casesL19–19

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

  1. L19
    split
06Use earlier factsL20–20

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

  1. L20
    exact hrepresentation_witness_witness_left
07Separate the logical casesL21–21

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

  1. L21
    split
08Use earlier factsL22–31

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

  1. L22
    specialize gaussian_signed_balance_integer_transport x
  2. L23
    specialize gaussian_signed_balance_integer_transport a
  3. L24
    specialize gaussian_signed_balance_integer_transport b
  4. L25
    specialize gaussian_signed_balance_integer_transport e
  5. L26
    specialize gaussian_signed_balance_integer_transport f
  6. L27
    apply gaussian_signed_balance_integer_transport
  7. L28
    exact hequal_left
  8. L29
    exact hrepresentation_witness_witness_right_left
  9. L30
    specialize gaussian_signed_balance_integer_transport x1
  10. L31
    specialize gaussian_signed_balance_integer_transport c
09Use earlier factsL32–37

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

  1. L32
    specialize gaussian_signed_balance_integer_transport d
  2. L33
    specialize gaussian_signed_balance_integer_transport g
  3. L34
    specialize gaussian_signed_balance_integer_transport h
  4. L35
    apply gaussian_signed_balance_integer_transport
  5. L36
    exact hequal_right
  6. L37
    exact hrepresentation_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 37 lines
  1. 0001intro z
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro e
  7. 0007intro f
  8. 0008intro g
  9. 0009intro h
  10. 0010intro hequal
  11. 0011intro hrepresentation
  12. 0012cases hequal
  13. 0013cases hrepresentation
  14. 0014cases hrepresentation_witness
  15. 0015cases hrepresentation_witness_witness
  16. 0016cases hrepresentation_witness_witness_right
  17. 0017exists x
  18. 0018exists x1
  19. 0019split
  20. 0020exact hrepresentation_witness_witness_left
  21. 0021split
  22. 0022specialize gaussian_signed_balance_integer_transport x
  23. 0023specialize gaussian_signed_balance_integer_transport a
  24. 0024specialize gaussian_signed_balance_integer_transport b
  25. 0025specialize gaussian_signed_balance_integer_transport e
  26. 0026specialize gaussian_signed_balance_integer_transport f
  27. 0027apply gaussian_signed_balance_integer_transport
  28. 0028exact hequal_left
  29. 0029exact hrepresentation_witness_witness_right_left
  30. 0030specialize gaussian_signed_balance_integer_transport x1
  31. 0031specialize gaussian_signed_balance_integer_transport c
  32. 0032specialize gaussian_signed_balance_integer_transport d
  33. 0033specialize gaussian_signed_balance_integer_transport g
  34. 0034specialize gaussian_signed_balance_integer_transport h
  35. 0035apply gaussian_signed_balance_integer_transport
  36. 0036exact hequal_right
  37. 0037exact hrepresentation_witness_witness_right_right