GF0002

gaussian_code_representation_transport

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

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

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 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))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 0 declared prerequisites and contains 10 exact native proof lines.

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

Proof neighborhood

Direct dependencies

none

Direct dependents

GF000B gaussian_zero_representation GF000E gaussian_one_representation

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

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.

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 exact 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