GI0046

gaussian_pair_zero_codes

The zero canonical pair code has exactly zero real and imaginary signed codes.

Alpha v34 checked-use · first admitted v28 · 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.

The natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ z. ∀ rc. ∀ ic. z = (rc + ic) · S (rc + ic) + (ic + ic) → z = 0 → rc = 0 ∧ ic = 0

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

Definition DAG

none

Actual proof prerequisites

pair_code_injective · checked external prerequisitemul_zero_left · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall z rc ic. z = ((rc) + (ic)) * S ((rc) + (ic)) + ((ic) + (ic)) -> z = 0 -> rc = 0 /\ ic = 0

Complete tactic proof in conservative notation

All 14 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

14 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 assumptionsL1–5

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

  1. L1
    intro z
  2. L2
    intro rc
  3. L3
    intro ic
  4. L4
    intro hpair
  5. L5
    intro hzero
02Use earlier factsL6–12

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

  1. L6
    specialize pair_code_injective z
  2. L7
    specialize pair_code_injective rc
  3. L8
    specialize pair_code_injective ic
  4. L9
    specialize pair_code_injective 0
  5. L10
    specialize pair_code_injective 0
  6. L11
    apply pair_code_injective
  7. L12
    exact hpair
03Calculate and transport equalitiesL13–14

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

  1. L13
    rewrite hzero
  2. L14
    simp [mul_zero_left, zero_add]

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro z
  2. 0002intro rc
  3. 0003intro ic
  4. 0004intro hpair
  5. 0005intro hzero
  6. 0006specialize pair_code_injective z
  7. 0007specialize pair_code_injective rc
  8. 0008specialize pair_code_injective ic
  9. 0009specialize pair_code_injective 0
  10. 0010specialize pair_code_injective 0
  11. 0011apply pair_code_injective
  12. 0012exact hpair
  13. 0013rewrite hzero
  14. 0014simp [mul_zero_left, zero_add]