GF0001

gaussian_valid_has_representation

Every valid canonical Gaussian code has actual signed-coordinate representatives.

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. ZPairValid(z) → ∃ x. ∃ y. ∃ n. ∃ m. ZPairRep(z,x,y,n,m)

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

Definition DAG

ZPairValid(z) · 1ZPairRep(z,ap,an,bp,bn) · 1

Actual proof prerequisites

gaussian_decode_representation · checked external prerequisite
Original expanded first-order statement 
forall z. (exists ge_real_positive_ring_valid ge_real_negative_ring_valid ge_imaginary_positive_ring_valid ge_imaginary_negative_ring_valid. (exists ge_real_code_ring_validdecode ge_imaginary_code_ring_validdecode. (((z) = ((ge_real_code_ring_validdecode) + (ge_imaginary_code_ring_validdecode)) * S ((ge_real_code_ring_validdecode) + (ge_imaginary_code_ring_validdecode)) + ((ge_imaginary_code_ring_validdecode) + (ge_imaginary_code_ring_validdecode))) /\ (((((ge_real_code_ring_validdecode) = 2 * (ge_real_positive_ring_valid) /\ (ge_real_negative_ring_valid) = 0) \/ exists ge_signed_half_ge_ring_validdecode_real. (((ge_real_code_ring_validdecode) = 2 * ge_signed_half_ge_ring_validdecode_real + 1 /\ (ge_real_positive_ring_valid) = 0) /\ (ge_real_negative_ring_valid) = S ge_signed_half_ge_ring_validdecode_real))) /\ ((((ge_imaginary_code_ring_validdecode) = 2 * (ge_imaginary_positive_ring_valid) /\ (ge_imaginary_negative_ring_valid) = 0) \/ exists ge_signed_half_ge_ring_validdecode_imaginary. (((ge_imaginary_code_ring_validdecode) = 2 * ge_signed_half_ge_ring_validdecode_imaginary + 1 /\ (ge_imaginary_positive_ring_valid) = 0) /\ (ge_imaginary_negative_ring_valid) = S ge_signed_half_ge_ring_validdecode_imaginary))))))) -> exists ap an bp bn. (exists ge_representation_real_code_ring_representation ge_representation_imaginary_code_ring_representation. (((z) = ((ge_representation_real_code_ring_representation) + (ge_representation_imaginary_code_ring_representation)) * S ((ge_representation_real_code_ring_representation) + (ge_representation_imaginary_code_ring_representation)) + ((ge_representation_imaginary_code_ring_representation) + (ge_representation_imaginary_code_ring_representation))) /\ ((exists ge_balance_positive_ring_representationreal ge_balance_negative_ring_representationreal. (((((ge_representation_real_code_ring_representation) = 2 * (ge_balance_positive_ring_representationreal) /\ (ge_balance_negative_ring_representationreal) = 0) \/ exists ge_signed_half_ring_representationrealdecode. (((ge_representation_real_code_ring_representation) = 2 * ge_signed_half_ring_representationrealdecode + 1 /\ (ge_balance_positive_ring_representationreal) = 0) /\ (ge_balance_negative_ring_representationreal) = S ge_signed_half_ring_representationrealdecode))) /\ ((ap) + ge_balance_negative_ring_representationreal = (an) + ge_balance_positive_ring_representationreal))) /\ (exists ge_balance_positive_ring_representationimaginary ge_balance_negative_ring_representationimaginary. (((((ge_representation_imaginary_code_ring_representation) = 2 * (ge_balance_positive_ring_representationimaginary) /\ (ge_balance_negative_ring_representationimaginary) = 0) \/ exists ge_signed_half_ring_representationimaginarydecode. (((ge_representation_imaginary_code_ring_representation) = 2 * ge_signed_half_ring_representationimaginarydecode + 1 /\ (ge_balance_positive_ring_representationimaginary) = 0) /\ (ge_balance_negative_ring_representationimaginary) = S ge_signed_half_ring_representationimaginarydecode))) /\ ((bp) + ge_balance_negative_ring_representationimaginary = (bn) + ge_balance_positive_ring_representationimaginary))))))

Complete tactic proof in conservative notation

All 17 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

17 script commands · 4 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 assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro h
02Separate the logical casesL3–6

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

  1. L3
    cases h
  2. L4
    cases h_witness
  3. L5
    cases h_witness_witness
  4. L6
    cases h_witness_witness_witness
03Construct an explicit witnessL7–10

Supply the displayed value, then prove that it has the required property.

  1. L7
    exists (x)
  2. L8
    exists (x1)
  3. L9
    exists (x2)
  4. L10
    exists (x3)
04Use earlier factsL11–17

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

  1. L11
    specialize gaussian_decode_representation (z)
  2. L12
    specialize gaussian_decode_representation (x)
  3. L13
    specialize gaussian_decode_representation (x1)
  4. L14
    specialize gaussian_decode_representation (x2)
  5. L15
    specialize gaussian_decode_representation (x3)
  6. L16
    apply gaussian_decode_representation
  7. L17
    exact h_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro z
  2. 0002intro h
  3. 0003cases h
  4. 0004cases h_witness
  5. 0005cases h_witness_witness
  6. 0006cases h_witness_witness_witness
  7. 0007exists (x)
  8. 0008exists (x1)
  9. 0009exists (x2)
  10. 0010exists (x3)
  11. 0011specialize gaussian_decode_representation (z)
  12. 0012specialize gaussian_decode_representation (x)
  13. 0013specialize gaussian_decode_representation (x1)
  14. 0014specialize gaussian_decode_representation (x2)
  15. 0015specialize gaussian_decode_representation (x3)
  16. 0016apply gaussian_decode_representation
  17. 0017exact h_witness_witness_witness_witness