GF000C

gaussian_zero_valid

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 7 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

GF000B gaussian_zero_representation gaussian_representation_is_gaussian Alpha theorem; checked-use authorized

Direct dependents

GF0053 gaussian_divides_decidable GF005A gaussian_mutual_divisibility_associate GF005F gaussian_gcd_bezout_zero_right

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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