GF0003

gaussian_norm_input_valid

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

An actual norm witness certifies membership in the Gaussian 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

forall z N. (exists ge_norm_rp_ring_norm ge_norm_rn_ring_norm ge_norm_ip_ring_norm ge_norm_in_ring_norm. ((exists ge_representation_real_code_ring_normrepresentation ge_representation_imaginary_code_ring_normrepresentation. (((z) = ((ge_representation_real_code_ring_normrepresentation) + (ge_representation_imaginary_code_ring_normrepresentation)) * S ((ge_representation_real_code_ring_normrepresentation) + (ge_representation_imaginary_code_ring_normrepresentation)) + ((ge_representation_imaginary_code_ring_normrepresentation) + (ge_representation_imaginary_code_ring_normrepresentation))) /\ ((exists ge_balance_positive_ring_normrepresentationreal ge_balance_negative_ring_normrepresentationreal. (((((ge_representation_real_code_ring_normrepresentation) = 2 * (ge_balance_positive_ring_normrepresentationreal) /\ (ge_balance_negative_ring_normrepresentationreal) = 0) \/ exists ge_signed_half_ring_normrepresentationrealdecode. (((ge_representation_real_code_ring_normrepresentation) = 2 * ge_signed_half_ring_normrepresentationrealdecode + 1 /\ (ge_balance_positive_ring_normrepresentationreal) = 0) /\ (ge_balance_negative_ring_normrepresentationreal) = S ge_signed_half_ring_normrepresentationrealdecode))) /\ ((ge_norm_rp_ring_norm) + ge_balance_negative_ring_normrepresentationreal = (ge_norm_rn_ring_norm) + ge_balance_positive_ring_normrepresentationreal))) /\ (exists ge_balance_positive_ring_normrepresentationimaginary ge_balance_negative_ring_normrepresentationimaginary. (((((ge_representation_imaginary_code_ring_normrepresentation) = 2 * (ge_balance_positive_ring_normrepresentationimaginary) /\ (ge_balance_negative_ring_normrepresentationimaginary) = 0) \/ exists ge_signed_half_ring_normrepresentationimaginarydecode. (((ge_representation_imaginary_code_ring_normrepresentation) = 2 * ge_signed_half_ring_normrepresentationimaginarydecode + 1 /\ (ge_balance_positive_ring_normrepresentationimaginary) = 0) /\ (ge_balance_negative_ring_normrepresentationimaginary) = S ge_signed_half_ring_normrepresentationimaginarydecode))) /\ ((ge_norm_ip_ring_norm) + ge_balance_negative_ring_normrepresentationimaginary = (ge_norm_in_ring_norm) + ge_balance_positive_ring_normrepresentationimaginary)))))) /\ (exists ge_real_square_ring_normsquare ge_imaginary_square_ring_normsquare. ((((((ge_norm_rp_ring_norm) * (ge_norm_rp_ring_norm))) + (((ge_norm_rn_ring_norm) * (ge_norm_rn_ring_norm)))) = ((ge_real_square_ring_normsquare) + (((((ge_norm_rp_ring_norm) * (ge_norm_rn_ring_norm))) + (((ge_norm_rn_ring_norm) * (ge_norm_rp_ring_norm))))))) /\ ((((((ge_norm_ip_ring_norm) * (ge_norm_ip_ring_norm))) + (((ge_norm_in_ring_norm) * (ge_norm_in_ring_norm)))) = ((ge_imaginary_square_ring_normsquare) + (((((ge_norm_ip_ring_norm) * (ge_norm_in_ring_norm))) + (((ge_norm_in_ring_norm) * (ge_norm_ip_ring_norm))))))) /\ ((N) = ge_real_square_ring_normsquare + ge_imaginary_square_ring_normsquare)))))) -> (exists ge_real_positive_ring_norm_domain ge_real_negative_ring_norm_domain ge_imaginary_positive_ring_norm_domain ge_imaginary_negative_ring_norm_domain. (exists ge_real_code_ring_norm_domaindecode ge_imaginary_code_ring_norm_domaindecode. (((z) = ((ge_real_code_ring_norm_domaindecode) + (ge_imaginary_code_ring_norm_domaindecode)) * S ((ge_real_code_ring_norm_domaindecode) + (ge_imaginary_code_ring_norm_domaindecode)) + ((ge_imaginary_code_ring_norm_domaindecode) + (ge_imaginary_code_ring_norm_domaindecode))) /\ (((((ge_real_code_ring_norm_domaindecode) = 2 * (ge_real_positive_ring_norm_domain) /\ (ge_real_negative_ring_norm_domain) = 0) \/ exists ge_signed_half_ge_ring_norm_domaindecode_real. (((ge_real_code_ring_norm_domaindecode) = 2 * ge_signed_half_ge_ring_norm_domaindecode_real + 1 /\ (ge_real_positive_ring_norm_domain) = 0) /\ (ge_real_negative_ring_norm_domain) = S ge_signed_half_ge_ring_norm_domaindecode_real))) /\ ((((ge_imaginary_code_ring_norm_domaindecode) = 2 * (ge_imaginary_positive_ring_norm_domain) /\ (ge_imaginary_negative_ring_norm_domain) = 0) \/ exists ge_signed_half_ge_ring_norm_domaindecode_imaginary. (((ge_imaginary_code_ring_norm_domaindecode) = 2 * ge_signed_half_ge_ring_norm_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_ring_norm_domain) = 0) /\ (ge_imaginary_negative_ring_norm_domain) = S ge_signed_half_ge_ring_norm_domaindecode_imaginary)))))))

Constructive proof overview

Generated structural guide

An actual norm witness certifies membership in the Gaussian carrier.

The unchanged tactic script uses 1 declared prerequisite and contains 15 exact native proof lines.

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

Proof neighborhood

Direct dependencies

gaussian_representation_is_gaussian Alpha theorem; checked-use authorized

Direct dependents

GF0052 gaussian_division_divisible_remainder_zero GF0070 gaussian_norm_bounded_coordinates GF0078 gaussian_factor_search_complete GF007E gaussian_irreducible_or_strict_nonunit_factorization GF0080 gaussian_irreducible_divisor_bounded_norm GF0083 gaussian_irreducible_factor_reduction GF0094 gaussian_irreducible_factorization_bounded_norm

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

15 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro z
  2. L2
    intro N
  3. L3
    intro h
02Separate the logical casesL4–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases h
  2. L5
    cases h_witness
  3. L6
    cases h_witness_witness
  4. L7
    cases h_witness_witness_witness
  5. L8
    cases h_witness_witness_witness_witness
03Use earlier factsL9–15

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

  1. L9
    specialize gaussian_representation_is_gaussian (z)
  2. L10
    specialize gaussian_representation_is_gaussian (x)
  3. L11
    specialize gaussian_representation_is_gaussian (x1)
  4. L12
    specialize gaussian_representation_is_gaussian (x2)
  5. L13
    specialize gaussian_representation_is_gaussian (x3)
  6. L14
    apply gaussian_representation_is_gaussian
  7. L15
    exact h_witness_witness_witness_witness_left

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro z
  2. 0002intro N
  3. 0003intro h
  4. 0004cases h
  5. 0005cases h_witness
  6. 0006cases h_witness_witness
  7. 0007cases h_witness_witness_witness
  8. 0008cases h_witness_witness_witness_witness
  9. 0009specialize gaussian_representation_is_gaussian (z)
  10. 0010specialize gaussian_representation_is_gaussian (x)
  11. 0011specialize gaussian_representation_is_gaussian (x1)
  12. 0012specialize gaussian_representation_is_gaussian (x2)
  13. 0013specialize gaussian_representation_is_gaussian (x3)
  14. 0014apply gaussian_representation_is_gaussian
  15. 0015exact h_witness_witness_witness_witness_left