GF0018

gaussian_code_zero_implies_norm_zero

The actual norm of the zero canonical code is zero, not a positive auxiliary value.

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

∀ z. ∀ N. GNorm(z,N) → z = 0 → N = 0

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

Definition DAG

GNorm(z,n) · 1

Actual proof prerequisites

gaussian_norm_functional · checked external prerequisitegaussian_zero_norm
Original expanded first-order statement 
forall z N. (exists ge_norm_rp_code_zero_norm ge_norm_rn_code_zero_norm ge_norm_ip_code_zero_norm ge_norm_in_code_zero_norm. ((exists ge_representation_real_code_code_zero_normrepresentation ge_representation_imaginary_code_code_zero_normrepresentation. (((z) = ((ge_representation_real_code_code_zero_normrepresentation) + (ge_representation_imaginary_code_code_zero_normrepresentation)) * S ((ge_representation_real_code_code_zero_normrepresentation) + (ge_representation_imaginary_code_code_zero_normrepresentation)) + ((ge_representation_imaginary_code_code_zero_normrepresentation) + (ge_representation_imaginary_code_code_zero_normrepresentation))) /\ ((exists ge_balance_positive_code_zero_normrepresentationreal ge_balance_negative_code_zero_normrepresentationreal. (((((ge_representation_real_code_code_zero_normrepresentation) = 2 * (ge_balance_positive_code_zero_normrepresentationreal) /\ (ge_balance_negative_code_zero_normrepresentationreal) = 0) \/ exists ge_signed_half_code_zero_normrepresentationrealdecode. (((ge_representation_real_code_code_zero_normrepresentation) = 2 * ge_signed_half_code_zero_normrepresentationrealdecode + 1 /\ (ge_balance_positive_code_zero_normrepresentationreal) = 0) /\ (ge_balance_negative_code_zero_normrepresentationreal) = S ge_signed_half_code_zero_normrepresentationrealdecode))) /\ ((ge_norm_rp_code_zero_norm) + ge_balance_negative_code_zero_normrepresentationreal = (ge_norm_rn_code_zero_norm) + ge_balance_positive_code_zero_normrepresentationreal))) /\ (exists ge_balance_positive_code_zero_normrepresentationimaginary ge_balance_negative_code_zero_normrepresentationimaginary. (((((ge_representation_imaginary_code_code_zero_normrepresentation) = 2 * (ge_balance_positive_code_zero_normrepresentationimaginary) /\ (ge_balance_negative_code_zero_normrepresentationimaginary) = 0) \/ exists ge_signed_half_code_zero_normrepresentationimaginarydecode. (((ge_representation_imaginary_code_code_zero_normrepresentation) = 2 * ge_signed_half_code_zero_normrepresentationimaginarydecode + 1 /\ (ge_balance_positive_code_zero_normrepresentationimaginary) = 0) /\ (ge_balance_negative_code_zero_normrepresentationimaginary) = S ge_signed_half_code_zero_normrepresentationimaginarydecode))) /\ ((ge_norm_ip_code_zero_norm) + ge_balance_negative_code_zero_normrepresentationimaginary = (ge_norm_in_code_zero_norm) + ge_balance_positive_code_zero_normrepresentationimaginary)))))) /\ (exists ge_real_square_code_zero_normsquare ge_imaginary_square_code_zero_normsquare. ((((((ge_norm_rp_code_zero_norm) * (ge_norm_rp_code_zero_norm))) + (((ge_norm_rn_code_zero_norm) * (ge_norm_rn_code_zero_norm)))) = ((ge_real_square_code_zero_normsquare) + (((((ge_norm_rp_code_zero_norm) * (ge_norm_rn_code_zero_norm))) + (((ge_norm_rn_code_zero_norm) * (ge_norm_rp_code_zero_norm))))))) /\ ((((((ge_norm_ip_code_zero_norm) * (ge_norm_ip_code_zero_norm))) + (((ge_norm_in_code_zero_norm) * (ge_norm_in_code_zero_norm)))) = ((ge_imaginary_square_code_zero_normsquare) + (((((ge_norm_ip_code_zero_norm) * (ge_norm_in_code_zero_norm))) + (((ge_norm_in_code_zero_norm) * (ge_norm_ip_code_zero_norm))))))) /\ ((N) = ge_real_square_code_zero_normsquare + ge_imaginary_square_code_zero_normsquare)))))) -> z=0 -> N=0

Complete tactic proof in conservative notation

All 11 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

11 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 (1)
gaussian zero norm
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro z
  2. L2
    intro N
  3. L3
    intro h
  4. L4
    intro hz
02Calculate and transport equalitiesL5–5

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

  1. L5
    rewrite hz at h
03Use earlier factsL6–11

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

  1. L6
    specialize gaussian_norm_functional (0)
  2. L7
    specialize gaussian_norm_functional (N)
  3. L8
    specialize gaussian_norm_functional (0)
  4. L9
    apply gaussian_norm_functional
  5. L10
    exact h
  6. L11
    exact gaussian_zero_norm

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro z
  2. 0002intro N
  3. 0003intro h
  4. 0004intro hz
  5. 0005rewrite hz at h
  6. 0006specialize gaussian_norm_functional (0)
  7. 0007specialize gaussian_norm_functional (N)
  8. 0008specialize gaussian_norm_functional (0)
  9. 0009apply gaussian_norm_functional
  10. 0010exact h
  11. 0011exact gaussian_zero_norm