GI0048

gaussian_decode_zero_iff

The unchanged normalized signed-coordinate decoding detects the zero pair code in both directions.

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) → (z = 0 → a = b ∧ c = d) ∧ (a = b ∧ c = d → z = 0)

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

Definition DAG

ZPairDecode(z,ap,an,bp,bn) · 1

Actual proof prerequisites

gaussian_decode_representationgaussian_representation_zero_iff
Original expanded first-order statement 
forall z a b c d. (exists ge_real_code_decode_zero ge_imaginary_code_decode_zero. (((z) = ((ge_real_code_decode_zero) + (ge_imaginary_code_decode_zero)) * S ((ge_real_code_decode_zero) + (ge_imaginary_code_decode_zero)) + ((ge_imaginary_code_decode_zero) + (ge_imaginary_code_decode_zero))) /\ (((((ge_real_code_decode_zero) = 2 * (a) /\ (b) = 0) \/ exists ge_signed_half_ge_decode_zero_real. (((ge_real_code_decode_zero) = 2 * ge_signed_half_ge_decode_zero_real + 1 /\ (a) = 0) /\ (b) = S ge_signed_half_ge_decode_zero_real))) /\ ((((ge_imaginary_code_decode_zero) = 2 * (c) /\ (d) = 0) \/ exists ge_signed_half_ge_decode_zero_imaginary. (((ge_imaginary_code_decode_zero) = 2 * ge_signed_half_ge_decode_zero_imaginary + 1 /\ (c) = 0) /\ (d) = S ge_signed_half_ge_decode_zero_imaginary)))))) -> ((z = 0 -> (a = b /\ c = d)) /\ ((a = b /\ c = d) -> z = 0))

Complete tactic proof in conservative notation

All 19 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

19 script commands · 3 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 (2)
gaussian representation zero iffgaussian decode representation
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
02Use earlier factsL7–16

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

  1. L7
    specialize gaussian_representation_zero_iff z
  2. L8
    specialize gaussian_representation_zero_iff a
  3. L9
    specialize gaussian_representation_zero_iff b
  4. L10
    specialize gaussian_representation_zero_iff c
  5. L11
    specialize gaussian_representation_zero_iff d
  6. L12
    apply gaussian_representation_zero_iff
  7. L13
    specialize gaussian_decode_representation z
  8. L14
    specialize gaussian_decode_representation a
  9. L15
    specialize gaussian_decode_representation b
  10. L16
    specialize gaussian_decode_representation c
03Use earlier factsL17–19

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

  1. L17
    specialize gaussian_decode_representation d
  2. L18
    apply gaussian_decode_representation
  3. L19
    exact hdecode

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro z
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro hdecode
  7. 0007specialize gaussian_representation_zero_iff z
  8. 0008specialize gaussian_representation_zero_iff a
  9. 0009specialize gaussian_representation_zero_iff b
  10. 0010specialize gaussian_representation_zero_iff c
  11. 0011specialize gaussian_representation_zero_iff d
  12. 0012apply gaussian_representation_zero_iff
  13. 0013specialize gaussian_decode_representation z
  14. 0014specialize gaussian_decode_representation a
  15. 0015specialize gaussian_decode_representation b
  16. 0016specialize gaussian_decode_representation c
  17. 0017specialize gaussian_decode_representation d
  18. 0018apply gaussian_decode_representation
  19. 0019exact hdecode