GF000B

gaussian_zero_representation

The actual canonical Gaussian zero code is 0, with its signed coordinates proved rather than asserted.

Alpha v34 checked-use · first admitted v30 · 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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

ZPairRep(0,0,0,0,0)

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

Definition DAG

ZPairRep(z,ap,an,bp,bn) · 1

Actual proof prerequisites

gaussian_code_representation_transportgaussian_natural_real_representation
Original expanded first-order statement 
exists ge_representation_real_code_ring_zero_representation ge_representation_imaginary_code_ring_zero_representation. (((0) = ((ge_representation_real_code_ring_zero_representation) + (ge_representation_imaginary_code_ring_zero_representation)) * S ((ge_representation_real_code_ring_zero_representation) + (ge_representation_imaginary_code_ring_zero_representation)) + ((ge_representation_imaginary_code_ring_zero_representation) + (ge_representation_imaginary_code_ring_zero_representation))) /\ ((exists ge_balance_positive_ring_zero_representationreal ge_balance_negative_ring_zero_representationreal. (((((ge_representation_real_code_ring_zero_representation) = 2 * (ge_balance_positive_ring_zero_representationreal) /\ (ge_balance_negative_ring_zero_representationreal) = 0) \/ exists ge_signed_half_ring_zero_representationrealdecode. (((ge_representation_real_code_ring_zero_representation) = 2 * ge_signed_half_ring_zero_representationrealdecode + 1 /\ (ge_balance_positive_ring_zero_representationreal) = 0) /\ (ge_balance_negative_ring_zero_representationreal) = S ge_signed_half_ring_zero_representationrealdecode))) /\ ((0) + ge_balance_negative_ring_zero_representationreal = (0) + ge_balance_positive_ring_zero_representationreal))) /\ (exists ge_balance_positive_ring_zero_representationimaginary ge_balance_negative_ring_zero_representationimaginary. (((((ge_representation_imaginary_code_ring_zero_representation) = 2 * (ge_balance_positive_ring_zero_representationimaginary) /\ (ge_balance_negative_ring_zero_representationimaginary) = 0) \/ exists ge_signed_half_ring_zero_representationimaginarydecode. (((ge_representation_imaginary_code_ring_zero_representation) = 2 * ge_signed_half_ring_zero_representationimaginarydecode + 1 /\ (ge_balance_positive_ring_zero_representationimaginary) = 0) /\ (ge_balance_negative_ring_zero_representationimaginary) = S ge_signed_half_ring_zero_representationimaginarydecode))) /\ ((0) + ge_balance_negative_ring_zero_representationimaginary = (0) + ge_balance_positive_ring_zero_representationimaginary)))))

Complete tactic proof in conservative notation

All 10 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

10 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 code representation transportgaussian natural real representation
01Use earlier factsL1–7

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

  1. L1
    specialize gaussian_code_representation_transport (((2*0) + (0)) * S ((2*0) + (0)) + ((0) + (0)))
  2. L2
    specialize gaussian_code_representation_transport (0)
  3. L3
    specialize gaussian_code_representation_transport (0)
  4. L4
    specialize gaussian_code_representation_transport (0)
  5. L5
    specialize gaussian_code_representation_transport (0)
  6. L6
    specialize gaussian_code_representation_transport (0)
  7. L7
    apply gaussian_code_representation_transport
02Calculate and transport equalitiesL8–8

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L8
    norm_num
03Use earlier factsL9–10

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

  1. L9
    specialize gaussian_natural_real_representation (0)
  2. L10
    apply gaussian_natural_real_representation

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001specialize gaussian_code_representation_transport (((2*0) + (0)) * S ((2*0) + (0)) + ((0) + (0)))
  2. 0002specialize gaussian_code_representation_transport (0)
  3. 0003specialize gaussian_code_representation_transport (0)
  4. 0004specialize gaussian_code_representation_transport (0)
  5. 0005specialize gaussian_code_representation_transport (0)
  6. 0006specialize gaussian_code_representation_transport (0)
  7. 0007apply gaussian_code_representation_transport
  8. 0008norm_num
  9. 0009specialize gaussian_natural_real_representation (0)
  10. 0010apply gaussian_natural_real_representation