GI0044

gaussian_decode_representation

Every normalized Gaussian decoding also represents its two actual integer coordinates.

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. ZPairDecode(z,a,b,c,d) → ZPairRep(z,a,b,c,d)

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

Definition DAG

ZPairDecode(z,ap,an,bp,bn) · 1ZPairRep(z,ap,an,bp,bn) · 1

Actual proof prerequisites

signed_decode_to_balance · checked external prerequisite
Original expanded first-order statement 
forall z a b c d. (exists ge_real_code_decoded_representation ge_imaginary_code_decoded_representation. (((z) = ((ge_real_code_decoded_representation) + (ge_imaginary_code_decoded_representation)) * S ((ge_real_code_decoded_representation) + (ge_imaginary_code_decoded_representation)) + ((ge_imaginary_code_decoded_representation) + (ge_imaginary_code_decoded_representation))) /\ (((((ge_real_code_decoded_representation) = 2 * (a) /\ (b) = 0) \/ exists ge_signed_half_ge_decoded_representation_real. (((ge_real_code_decoded_representation) = 2 * ge_signed_half_ge_decoded_representation_real + 1 /\ (a) = 0) /\ (b) = S ge_signed_half_ge_decoded_representation_real))) /\ ((((ge_imaginary_code_decoded_representation) = 2 * (c) /\ (d) = 0) \/ exists ge_signed_half_ge_decoded_representation_imaginary. (((ge_imaginary_code_decoded_representation) = 2 * ge_signed_half_ge_decoded_representation_imaginary + 1 /\ (c) = 0) /\ (d) = S ge_signed_half_ge_decoded_representation_imaginary)))))) -> (exists ge_representation_real_code_decoded_representation_target ge_representation_imaginary_code_decoded_representation_target. (((z) = ((ge_representation_real_code_decoded_representation_target) + (ge_representation_imaginary_code_decoded_representation_target)) * S ((ge_representation_real_code_decoded_representation_target) + (ge_representation_imaginary_code_decoded_representation_target)) + ((ge_representation_imaginary_code_decoded_representation_target) + (ge_representation_imaginary_code_decoded_representation_target))) /\ ((exists ge_balance_positive_decoded_representation_targetreal ge_balance_negative_decoded_representation_targetreal. (((((ge_representation_real_code_decoded_representation_target) = 2 * (ge_balance_positive_decoded_representation_targetreal) /\ (ge_balance_negative_decoded_representation_targetreal) = 0) \/ exists ge_signed_half_decoded_representation_targetrealdecode. (((ge_representation_real_code_decoded_representation_target) = 2 * ge_signed_half_decoded_representation_targetrealdecode + 1 /\ (ge_balance_positive_decoded_representation_targetreal) = 0) /\ (ge_balance_negative_decoded_representation_targetreal) = S ge_signed_half_decoded_representation_targetrealdecode))) /\ ((a) + ge_balance_negative_decoded_representation_targetreal = (b) + ge_balance_positive_decoded_representation_targetreal))) /\ (exists ge_balance_positive_decoded_representation_targetimaginary ge_balance_negative_decoded_representation_targetimaginary. (((((ge_representation_imaginary_code_decoded_representation_target) = 2 * (ge_balance_positive_decoded_representation_targetimaginary) /\ (ge_balance_negative_decoded_representation_targetimaginary) = 0) \/ exists ge_signed_half_decoded_representation_targetimaginarydecode. (((ge_representation_imaginary_code_decoded_representation_target) = 2 * ge_signed_half_decoded_representation_targetimaginarydecode + 1 /\ (ge_balance_positive_decoded_representation_targetimaginary) = 0) /\ (ge_balance_negative_decoded_representation_targetimaginary) = S ge_signed_half_decoded_representation_targetimaginarydecode))) /\ ((c) + ge_balance_negative_decoded_representation_targetimaginary = (d) + ge_balance_positive_decoded_representation_targetimaginary))))))

Complete tactic proof in conservative notation

All 25 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

25 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 assumptionsL1–6

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 hdecode
02Separate the logical casesL7–10

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

  1. L7
    cases hdecode
  2. L8
    cases hdecode_witness
  3. L9
    cases hdecode_witness_witness
  4. L10
    cases hdecode_witness_witness_right
03Construct an explicit witnessL11–12

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

  1. L11
    exists x
  2. L12
    exists x1
04Separate the logical casesL13–13

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

  1. L13
    split
05Use earlier factsL14–14

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

  1. L14
    exact hdecode_witness_witness_left
06Separate the logical casesL15–15

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

  1. L15
    split
07Use earlier factsL16–25

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

  1. L16
    specialize signed_decode_to_balance x
  2. L17
    specialize signed_decode_to_balance a
  3. L18
    specialize signed_decode_to_balance b
  4. L19
    apply signed_decode_to_balance
  5. L20
    exact hdecode_witness_witness_right_left
  6. L21
    specialize signed_decode_to_balance x1
  7. L22
    specialize signed_decode_to_balance c
  8. L23
    specialize signed_decode_to_balance d
  9. L24
    apply signed_decode_to_balance
  10. L25
    exact hdecode_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro z
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro hdecode
  7. 0007cases hdecode
  8. 0008cases hdecode_witness
  9. 0009cases hdecode_witness_witness
  10. 0010cases hdecode_witness_witness_right
  11. 0011exists x
  12. 0012exists x1
  13. 0013split
  14. 0014exact hdecode_witness_witness_left
  15. 0015split
  16. 0016specialize signed_decode_to_balance x
  17. 0017specialize signed_decode_to_balance a
  18. 0018specialize signed_decode_to_balance b
  19. 0019apply signed_decode_to_balance
  20. 0020exact hdecode_witness_witness_right_left
  21. 0021specialize signed_decode_to_balance x1
  22. 0022specialize signed_decode_to_balance c
  23. 0023specialize signed_decode_to_balance d
  24. 0024apply signed_decode_to_balance
  25. 0025exact hdecode_witness_witness_right_right