GF0002

gaussian_code_representation_transport

Proved equality of canonical natural codes preserves the actual signed representation.

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. ∀ w. ∀ ap. ∀ an. ∀ bp. ∀ bn. z = w → ZPairRep(z,ap,an,bp,bn) → ZPairRep(w,ap,an,bp,bn)

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

Definition DAG

ZPairRep(z,ap,an,bp,bn) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall z w ap an bp bn. z=w -> (exists ge_representation_real_code_ring_old_code ge_representation_imaginary_code_ring_old_code. (((z) = ((ge_representation_real_code_ring_old_code) + (ge_representation_imaginary_code_ring_old_code)) * S ((ge_representation_real_code_ring_old_code) + (ge_representation_imaginary_code_ring_old_code)) + ((ge_representation_imaginary_code_ring_old_code) + (ge_representation_imaginary_code_ring_old_code))) /\ ((exists ge_balance_positive_ring_old_codereal ge_balance_negative_ring_old_codereal. (((((ge_representation_real_code_ring_old_code) = 2 * (ge_balance_positive_ring_old_codereal) /\ (ge_balance_negative_ring_old_codereal) = 0) \/ exists ge_signed_half_ring_old_coderealdecode. (((ge_representation_real_code_ring_old_code) = 2 * ge_signed_half_ring_old_coderealdecode + 1 /\ (ge_balance_positive_ring_old_codereal) = 0) /\ (ge_balance_negative_ring_old_codereal) = S ge_signed_half_ring_old_coderealdecode))) /\ ((ap) + ge_balance_negative_ring_old_codereal = (an) + ge_balance_positive_ring_old_codereal))) /\ (exists ge_balance_positive_ring_old_codeimaginary ge_balance_negative_ring_old_codeimaginary. (((((ge_representation_imaginary_code_ring_old_code) = 2 * (ge_balance_positive_ring_old_codeimaginary) /\ (ge_balance_negative_ring_old_codeimaginary) = 0) \/ exists ge_signed_half_ring_old_codeimaginarydecode. (((ge_representation_imaginary_code_ring_old_code) = 2 * ge_signed_half_ring_old_codeimaginarydecode + 1 /\ (ge_balance_positive_ring_old_codeimaginary) = 0) /\ (ge_balance_negative_ring_old_codeimaginary) = S ge_signed_half_ring_old_codeimaginarydecode))) /\ ((bp) + ge_balance_negative_ring_old_codeimaginary = (bn) + ge_balance_positive_ring_old_codeimaginary)))))) -> (exists ge_representation_real_code_ring_new_code ge_representation_imaginary_code_ring_new_code. (((w) = ((ge_representation_real_code_ring_new_code) + (ge_representation_imaginary_code_ring_new_code)) * S ((ge_representation_real_code_ring_new_code) + (ge_representation_imaginary_code_ring_new_code)) + ((ge_representation_imaginary_code_ring_new_code) + (ge_representation_imaginary_code_ring_new_code))) /\ ((exists ge_balance_positive_ring_new_codereal ge_balance_negative_ring_new_codereal. (((((ge_representation_real_code_ring_new_code) = 2 * (ge_balance_positive_ring_new_codereal) /\ (ge_balance_negative_ring_new_codereal) = 0) \/ exists ge_signed_half_ring_new_coderealdecode. (((ge_representation_real_code_ring_new_code) = 2 * ge_signed_half_ring_new_coderealdecode + 1 /\ (ge_balance_positive_ring_new_codereal) = 0) /\ (ge_balance_negative_ring_new_codereal) = S ge_signed_half_ring_new_coderealdecode))) /\ ((ap) + ge_balance_negative_ring_new_codereal = (an) + ge_balance_positive_ring_new_codereal))) /\ (exists ge_balance_positive_ring_new_codeimaginary ge_balance_negative_ring_new_codeimaginary. (((((ge_representation_imaginary_code_ring_new_code) = 2 * (ge_balance_positive_ring_new_codeimaginary) /\ (ge_balance_negative_ring_new_codeimaginary) = 0) \/ exists ge_signed_half_ring_new_codeimaginarydecode. (((ge_representation_imaginary_code_ring_new_code) = 2 * ge_signed_half_ring_new_codeimaginarydecode + 1 /\ (ge_balance_positive_ring_new_codeimaginary) = 0) /\ (ge_balance_negative_ring_new_codeimaginary) = S ge_signed_half_ring_new_codeimaginarydecode))) /\ ((bp) + ge_balance_negative_ring_new_codeimaginary = (bn) + ge_balance_positive_ring_new_codeimaginary))))))

Complete tactic proof in conservative notation

All 10 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

10 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 assumptionsL1–8

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

  1. L1
    intro z
  2. L2
    intro w
  3. L3
    intro ap
  4. L4
    intro an
  5. L5
    intro bp
  6. L6
    intro bn
  7. L7
    intro heq
  8. L8
    intro h
02Calculate and transport equalitiesL9–9

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

  1. L9
    rewrite heq at h
03Use earlier factsL10–10

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

  1. L10
    exact h

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro z
  2. 0002intro w
  3. 0003intro ap
  4. 0004intro an
  5. 0005intro bp
  6. 0006intro bn
  7. 0007intro heq
  8. 0008intro h
  9. 0009rewrite heq at h
  10. 0010exact h