GF0018

gaussian_code_zero_implies_norm_zero

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

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

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

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

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

Proof neighborhood

Direct dependencies

gaussian_norm_functional Alpha theorem; checked-use authorized GF000D gaussian_zero_norm

Direct dependents

GF001D gaussian_unit_nonzero GF0082 gaussian_nonunit_divisor_strict_quotient

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

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.

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 exact 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