GI002F

gaussian_signed_norm_balance

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

An actual Gaussian norm is exactly the difference of its positive-square and negative-cross blocks.

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 a b c d N. (exists ge_real_square_norm_balance ge_imaginary_square_norm_balance. ((((((a) * (a))) + (((b) * (b)))) = ((ge_real_square_norm_balance) + (((((a) * (b))) + (((b) * (a))))))) /\ ((((((c) * (c))) + (((d) * (d)))) = ((ge_imaginary_square_norm_balance) + (((((c) * (d))) + (((d) * (c))))))) /\ ((N) = ge_real_square_norm_balance + ge_imaginary_square_norm_balance)))) -> ((((((a) * (a))) + (((b) * (b))))) + (((((c) * (c))) + (((d) * (d)))))) = ((N) + (((((((a) * (b))) + (((b) * (a))))) + (((((c) * (d))) + (((d) * (c))))))))

Constructive proof overview

Generated structural guide

An actual Gaussian norm is exactly the difference of its positive-square and negative-cross blocks.

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

add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

GI0030 gaussian_conjugate_product_is_norm

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 assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro N
  6. L6
    intro hnorm
02Separate the logical casesL7–10

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

  1. L7
    cases hnorm
  2. L8
    cases hnorm_witness
  3. L9
    cases hnorm_witness_witness
  4. L10
    cases hnorm_witness_witness_right
03Calculate and transport equalitiesL11–14

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

  1. L11
    rewrite hnorm_witness_witness_left
  2. L12
    rewrite hnorm_witness_witness_right_left
  3. L13
    rewrite hnorm_witness_witness_right_right
  4. L14
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro N
  6. 0006intro hnorm
  7. 0007cases hnorm
  8. 0008cases hnorm_witness
  9. 0009cases hnorm_witness_witness
  10. 0010cases hnorm_witness_witness_right
  11. 0011rewrite hnorm_witness_witness_left
  12. 0012rewrite hnorm_witness_witness_right_left
  13. 0013rewrite hnorm_witness_witness_right_right
  14. 0014simp [add_assoc, add_comm, four_square_add_swap_right_tail]