GI001C

gaussian_signed_norm_conjugate

Complex conjugation preserves the actual squared Gaussian norm, for every signed representative.

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

∀ a. ∀ b. ∀ c. ∀ d. ∀ N. GaussianSignedNorm(a,b,c,d,N) → GaussianSignedNorm(a,b,d,c,N)

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

Definition DAG

GaussianSignedNorm(ap,an,bp,bn,N) · 2

Actual proof prerequisites

gaussian_signed_square_negated
Original expanded first-order statement 
forall a b c d N. (exists ge_real_square_conjugate_source ge_imaginary_square_conjugate_source. ((((((a) * (a))) + (((b) * (b)))) = ((ge_real_square_conjugate_source) + (((((a) * (b))) + (((b) * (a))))))) /\ ((((((c) * (c))) + (((d) * (d)))) = ((ge_imaginary_square_conjugate_source) + (((((c) * (d))) + (((d) * (c))))))) /\ ((N) = ge_real_square_conjugate_source + ge_imaginary_square_conjugate_source)))) -> (exists ge_real_square_conjugate_target ge_imaginary_square_conjugate_target. ((((((a) * (a))) + (((b) * (b)))) = ((ge_real_square_conjugate_target) + (((((a) * (b))) + (((b) * (a))))))) /\ ((((((d) * (d))) + (((c) * (c)))) = ((ge_imaginary_square_conjugate_target) + (((((d) * (c))) + (((c) * (d))))))) /\ ((N) = ge_real_square_conjugate_target + ge_imaginary_square_conjugate_target))))

Complete tactic proof in conservative notation

All 21 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

21 script commands · 7 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.

Named ingredients (1)
gaussian signed square negated
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
03Construct an explicit witnessL11–12

Supply the displayed value, then prove that it has the required property.

  1. L11
    exists x
  2. L12
    exists x1
04Separate the logical casesL13–13

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

  1. L13
    split
05Use earlier factsL14–14

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

  1. L14
    exact hnorm_witness_witness_left
06Separate the logical casesL15–15

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

  1. L15
    split
07Use earlier factsL16–21

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

  1. L16
    specialize gaussian_signed_square_negated c
  2. L17
    specialize gaussian_signed_square_negated d
  3. L18
    specialize gaussian_signed_square_negated x1
  4. L19
    apply gaussian_signed_square_negated
  5. L20
    exact hnorm_witness_witness_right_left
  6. L21
    exact hnorm_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 21 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. 0011exists x
  12. 0012exists x1
  13. 0013split
  14. 0014exact hnorm_witness_witness_left
  15. 0015split
  16. 0016specialize gaussian_signed_square_negated c
  17. 0017specialize gaussian_signed_square_negated d
  18. 0018specialize gaussian_signed_square_negated x1
  19. 0019apply gaussian_signed_square_negated
  20. 0020exact hnorm_witness_witness_right_left
  21. 0021exact hnorm_witness_witness_right_right