GI0046

gaussian_pair_zero_codes

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

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

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 rc ic. z = ((rc) + (ic)) * S ((rc) + (ic)) + ((ic) + (ic)) -> z = 0 -> rc = 0 /\ ic = 0

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

pair_code_injective Alpha theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

GI0047 gaussian_representation_zero_iff

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

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.

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 exact 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]