GF0003

gaussian_norm_input_valid

An actual norm witness certifies membership in the Gaussian 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

∀ z. ∀ N. GNorm(z,N) → ZPairValid(z)

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

Definition DAG

ZPairValid(z) · 1GNorm(z,n) · 1

Actual proof prerequisites

gaussian_representation_is_gaussian · checked external prerequisite
Original expanded first-order 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)))))))

Complete tactic proof in conservative notation

All 15 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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