GF000C

gaussian_zero_valid

The canonical Gaussian zero belongs to the actual signed-pair carrier.

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

ZPairValid(0)

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

Definition DAG

ZPairValid(z) · 1

Actual proof prerequisites

gaussian_zero_representationgaussian_representation_is_gaussian · checked external prerequisite
Original expanded first-order statement 
exists ge_real_positive_ring_zero_valid ge_real_negative_ring_zero_valid ge_imaginary_positive_ring_zero_valid ge_imaginary_negative_ring_zero_valid. (exists ge_real_code_ring_zero_validdecode ge_imaginary_code_ring_zero_validdecode. (((0) = ((ge_real_code_ring_zero_validdecode) + (ge_imaginary_code_ring_zero_validdecode)) * S ((ge_real_code_ring_zero_validdecode) + (ge_imaginary_code_ring_zero_validdecode)) + ((ge_imaginary_code_ring_zero_validdecode) + (ge_imaginary_code_ring_zero_validdecode))) /\ (((((ge_real_code_ring_zero_validdecode) = 2 * (ge_real_positive_ring_zero_valid) /\ (ge_real_negative_ring_zero_valid) = 0) \/ exists ge_signed_half_ge_ring_zero_validdecode_real. (((ge_real_code_ring_zero_validdecode) = 2 * ge_signed_half_ge_ring_zero_validdecode_real + 1 /\ (ge_real_positive_ring_zero_valid) = 0) /\ (ge_real_negative_ring_zero_valid) = S ge_signed_half_ge_ring_zero_validdecode_real))) /\ ((((ge_imaginary_code_ring_zero_validdecode) = 2 * (ge_imaginary_positive_ring_zero_valid) /\ (ge_imaginary_negative_ring_zero_valid) = 0) \/ exists ge_signed_half_ge_ring_zero_validdecode_imaginary. (((ge_imaginary_code_ring_zero_validdecode) = 2 * ge_signed_half_ge_ring_zero_validdecode_imaginary + 1 /\ (ge_imaginary_positive_ring_zero_valid) = 0) /\ (ge_imaginary_negative_ring_zero_valid) = S ge_signed_half_ge_ring_zero_validdecode_imaginary))))))

Complete tactic proof in conservative notation

All 7 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

7 script commands · 1 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 zero representation
01Use earlier factsL1–7

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

  1. L1
    specialize gaussian_representation_is_gaussian (0)
  2. L2
    specialize gaussian_representation_is_gaussian (0)
  3. L3
    specialize gaussian_representation_is_gaussian (0)
  4. L4
    specialize gaussian_representation_is_gaussian (0)
  5. L5
    specialize gaussian_representation_is_gaussian (0)
  6. L6
    apply gaussian_representation_is_gaussian
  7. L7
    exact gaussian_zero_representation

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001specialize gaussian_representation_is_gaussian (0)
  2. 0002specialize gaussian_representation_is_gaussian (0)
  3. 0003specialize gaussian_representation_is_gaussian (0)
  4. 0004specialize gaussian_representation_is_gaussian (0)
  5. 0005specialize gaussian_representation_is_gaussian (0)
  6. 0006apply gaussian_representation_is_gaussian
  7. 0007exact gaussian_zero_representation