GI004D

gaussian_norm_of_representation

An actual represented pair and its actual squared modulus construct the canonical Gaussian norm graph.

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

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

Definition DAG

GaussianSignedNorm(ap,an,bp,bn,N) · 1ZPairRep(z,ap,an,bp,bn) · 1GNorm(z,n) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall z a b c d N. (exists ge_representation_real_code_norm_intro_rep ge_representation_imaginary_code_norm_intro_rep. (((z) = ((ge_representation_real_code_norm_intro_rep) + (ge_representation_imaginary_code_norm_intro_rep)) * S ((ge_representation_real_code_norm_intro_rep) + (ge_representation_imaginary_code_norm_intro_rep)) + ((ge_representation_imaginary_code_norm_intro_rep) + (ge_representation_imaginary_code_norm_intro_rep))) /\ ((exists ge_balance_positive_norm_intro_repreal ge_balance_negative_norm_intro_repreal. (((((ge_representation_real_code_norm_intro_rep) = 2 * (ge_balance_positive_norm_intro_repreal) /\ (ge_balance_negative_norm_intro_repreal) = 0) \/ exists ge_signed_half_norm_intro_reprealdecode. (((ge_representation_real_code_norm_intro_rep) = 2 * ge_signed_half_norm_intro_reprealdecode + 1 /\ (ge_balance_positive_norm_intro_repreal) = 0) /\ (ge_balance_negative_norm_intro_repreal) = S ge_signed_half_norm_intro_reprealdecode))) /\ ((a) + ge_balance_negative_norm_intro_repreal = (b) + ge_balance_positive_norm_intro_repreal))) /\ (exists ge_balance_positive_norm_intro_repimaginary ge_balance_negative_norm_intro_repimaginary. (((((ge_representation_imaginary_code_norm_intro_rep) = 2 * (ge_balance_positive_norm_intro_repimaginary) /\ (ge_balance_negative_norm_intro_repimaginary) = 0) \/ exists ge_signed_half_norm_intro_repimaginarydecode. (((ge_representation_imaginary_code_norm_intro_rep) = 2 * ge_signed_half_norm_intro_repimaginarydecode + 1 /\ (ge_balance_positive_norm_intro_repimaginary) = 0) /\ (ge_balance_negative_norm_intro_repimaginary) = S ge_signed_half_norm_intro_repimaginarydecode))) /\ ((c) + ge_balance_negative_norm_intro_repimaginary = (d) + ge_balance_positive_norm_intro_repimaginary)))))) -> (exists ge_real_square_norm_intro_raw ge_imaginary_square_norm_intro_raw. ((((((a) * (a))) + (((b) * (b)))) = ((ge_real_square_norm_intro_raw) + (((((a) * (b))) + (((b) * (a))))))) /\ ((((((c) * (c))) + (((d) * (d)))) = ((ge_imaginary_square_norm_intro_raw) + (((((c) * (d))) + (((d) * (c))))))) /\ ((N) = ge_real_square_norm_intro_raw + ge_imaginary_square_norm_intro_raw)))) -> (exists ge_norm_rp_norm_intro_code ge_norm_rn_norm_intro_code ge_norm_ip_norm_intro_code ge_norm_in_norm_intro_code. ((exists ge_representation_real_code_norm_intro_coderepresentation ge_representation_imaginary_code_norm_intro_coderepresentation. (((z) = ((ge_representation_real_code_norm_intro_coderepresentation) + (ge_representation_imaginary_code_norm_intro_coderepresentation)) * S ((ge_representation_real_code_norm_intro_coderepresentation) + (ge_representation_imaginary_code_norm_intro_coderepresentation)) + ((ge_representation_imaginary_code_norm_intro_coderepresentation) + (ge_representation_imaginary_code_norm_intro_coderepresentation))) /\ ((exists ge_balance_positive_norm_intro_coderepresentationreal ge_balance_negative_norm_intro_coderepresentationreal. (((((ge_representation_real_code_norm_intro_coderepresentation) = 2 * (ge_balance_positive_norm_intro_coderepresentationreal) /\ (ge_balance_negative_norm_intro_coderepresentationreal) = 0) \/ exists ge_signed_half_norm_intro_coderepresentationrealdecode. (((ge_representation_real_code_norm_intro_coderepresentation) = 2 * ge_signed_half_norm_intro_coderepresentationrealdecode + 1 /\ (ge_balance_positive_norm_intro_coderepresentationreal) = 0) /\ (ge_balance_negative_norm_intro_coderepresentationreal) = S ge_signed_half_norm_intro_coderepresentationrealdecode))) /\ ((ge_norm_rp_norm_intro_code) + ge_balance_negative_norm_intro_coderepresentationreal = (ge_norm_rn_norm_intro_code) + ge_balance_positive_norm_intro_coderepresentationreal))) /\ (exists ge_balance_positive_norm_intro_coderepresentationimaginary ge_balance_negative_norm_intro_coderepresentationimaginary. (((((ge_representation_imaginary_code_norm_intro_coderepresentation) = 2 * (ge_balance_positive_norm_intro_coderepresentationimaginary) /\ (ge_balance_negative_norm_intro_coderepresentationimaginary) = 0) \/ exists ge_signed_half_norm_intro_coderepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_intro_coderepresentation) = 2 * ge_signed_half_norm_intro_coderepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_intro_coderepresentationimaginary) = 0) /\ (ge_balance_negative_norm_intro_coderepresentationimaginary) = S ge_signed_half_norm_intro_coderepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_intro_code) + ge_balance_negative_norm_intro_coderepresentationimaginary = (ge_norm_in_norm_intro_code) + ge_balance_positive_norm_intro_coderepresentationimaginary)))))) /\ (exists ge_real_square_norm_intro_codesquare ge_imaginary_square_norm_intro_codesquare. ((((((ge_norm_rp_norm_intro_code) * (ge_norm_rp_norm_intro_code))) + (((ge_norm_rn_norm_intro_code) * (ge_norm_rn_norm_intro_code)))) = ((ge_real_square_norm_intro_codesquare) + (((((ge_norm_rp_norm_intro_code) * (ge_norm_rn_norm_intro_code))) + (((ge_norm_rn_norm_intro_code) * (ge_norm_rp_norm_intro_code))))))) /\ ((((((ge_norm_ip_norm_intro_code) * (ge_norm_ip_norm_intro_code))) + (((ge_norm_in_norm_intro_code) * (ge_norm_in_norm_intro_code)))) = ((ge_imaginary_square_norm_intro_codesquare) + (((((ge_norm_ip_norm_intro_code) * (ge_norm_in_norm_intro_code))) + (((ge_norm_in_norm_intro_code) * (ge_norm_ip_norm_intro_code))))))) /\ ((N) = ge_real_square_norm_intro_codesquare + ge_imaginary_square_norm_intro_codesquare))))))

Complete tactic proof in conservative notation

All 15 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

15 script commands · 4 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 assumptionsL1–8

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

  1. L1
    intro z
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro d
  6. L6
    intro N
  7. L7
    intro hrep
  8. L8
    intro hnorm
02Construct an explicit witnessL9–12

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

  1. L9
    exists a
  2. L10
    exists b
  3. L11
    exists c
  4. L12
    exists d
03Separate the logical casesL13–13

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

  1. L13
    split
04Use earlier factsL14–15

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

  1. L14
    exact hrep
  2. L15
    exact hnorm

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro z
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro N
  7. 0007intro hrep
  8. 0008intro hnorm
  9. 0009exists a
  10. 0010exists b
  11. 0011exists c
  12. 0012exists d
  13. 0013split
  14. 0014exact hrep
  15. 0015exact hnorm