GI004E

gaussian_norm_for_representation

The canonical norm is the actual squared modulus of every equal signed representative, not merely its selected witness.

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

gaussian_signed_norm_integer_transportgaussian_representation_equal
Original expanded first-order statement 
forall z a b c d N. (exists ge_representation_real_code_norm_fixed_rep ge_representation_imaginary_code_norm_fixed_rep. (((z) = ((ge_representation_real_code_norm_fixed_rep) + (ge_representation_imaginary_code_norm_fixed_rep)) * S ((ge_representation_real_code_norm_fixed_rep) + (ge_representation_imaginary_code_norm_fixed_rep)) + ((ge_representation_imaginary_code_norm_fixed_rep) + (ge_representation_imaginary_code_norm_fixed_rep))) /\ ((exists ge_balance_positive_norm_fixed_repreal ge_balance_negative_norm_fixed_repreal. (((((ge_representation_real_code_norm_fixed_rep) = 2 * (ge_balance_positive_norm_fixed_repreal) /\ (ge_balance_negative_norm_fixed_repreal) = 0) \/ exists ge_signed_half_norm_fixed_reprealdecode. (((ge_representation_real_code_norm_fixed_rep) = 2 * ge_signed_half_norm_fixed_reprealdecode + 1 /\ (ge_balance_positive_norm_fixed_repreal) = 0) /\ (ge_balance_negative_norm_fixed_repreal) = S ge_signed_half_norm_fixed_reprealdecode))) /\ ((a) + ge_balance_negative_norm_fixed_repreal = (b) + ge_balance_positive_norm_fixed_repreal))) /\ (exists ge_balance_positive_norm_fixed_repimaginary ge_balance_negative_norm_fixed_repimaginary. (((((ge_representation_imaginary_code_norm_fixed_rep) = 2 * (ge_balance_positive_norm_fixed_repimaginary) /\ (ge_balance_negative_norm_fixed_repimaginary) = 0) \/ exists ge_signed_half_norm_fixed_repimaginarydecode. (((ge_representation_imaginary_code_norm_fixed_rep) = 2 * ge_signed_half_norm_fixed_repimaginarydecode + 1 /\ (ge_balance_positive_norm_fixed_repimaginary) = 0) /\ (ge_balance_negative_norm_fixed_repimaginary) = S ge_signed_half_norm_fixed_repimaginarydecode))) /\ ((c) + ge_balance_negative_norm_fixed_repimaginary = (d) + ge_balance_positive_norm_fixed_repimaginary)))))) -> (exists ge_norm_rp_norm_fixed_code ge_norm_rn_norm_fixed_code ge_norm_ip_norm_fixed_code ge_norm_in_norm_fixed_code. ((exists ge_representation_real_code_norm_fixed_coderepresentation ge_representation_imaginary_code_norm_fixed_coderepresentation. (((z) = ((ge_representation_real_code_norm_fixed_coderepresentation) + (ge_representation_imaginary_code_norm_fixed_coderepresentation)) * S ((ge_representation_real_code_norm_fixed_coderepresentation) + (ge_representation_imaginary_code_norm_fixed_coderepresentation)) + ((ge_representation_imaginary_code_norm_fixed_coderepresentation) + (ge_representation_imaginary_code_norm_fixed_coderepresentation))) /\ ((exists ge_balance_positive_norm_fixed_coderepresentationreal ge_balance_negative_norm_fixed_coderepresentationreal. (((((ge_representation_real_code_norm_fixed_coderepresentation) = 2 * (ge_balance_positive_norm_fixed_coderepresentationreal) /\ (ge_balance_negative_norm_fixed_coderepresentationreal) = 0) \/ exists ge_signed_half_norm_fixed_coderepresentationrealdecode. (((ge_representation_real_code_norm_fixed_coderepresentation) = 2 * ge_signed_half_norm_fixed_coderepresentationrealdecode + 1 /\ (ge_balance_positive_norm_fixed_coderepresentationreal) = 0) /\ (ge_balance_negative_norm_fixed_coderepresentationreal) = S ge_signed_half_norm_fixed_coderepresentationrealdecode))) /\ ((ge_norm_rp_norm_fixed_code) + ge_balance_negative_norm_fixed_coderepresentationreal = (ge_norm_rn_norm_fixed_code) + ge_balance_positive_norm_fixed_coderepresentationreal))) /\ (exists ge_balance_positive_norm_fixed_coderepresentationimaginary ge_balance_negative_norm_fixed_coderepresentationimaginary. (((((ge_representation_imaginary_code_norm_fixed_coderepresentation) = 2 * (ge_balance_positive_norm_fixed_coderepresentationimaginary) /\ (ge_balance_negative_norm_fixed_coderepresentationimaginary) = 0) \/ exists ge_signed_half_norm_fixed_coderepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_fixed_coderepresentation) = 2 * ge_signed_half_norm_fixed_coderepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_fixed_coderepresentationimaginary) = 0) /\ (ge_balance_negative_norm_fixed_coderepresentationimaginary) = S ge_signed_half_norm_fixed_coderepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_fixed_code) + ge_balance_negative_norm_fixed_coderepresentationimaginary = (ge_norm_in_norm_fixed_code) + ge_balance_positive_norm_fixed_coderepresentationimaginary)))))) /\ (exists ge_real_square_norm_fixed_codesquare ge_imaginary_square_norm_fixed_codesquare. ((((((ge_norm_rp_norm_fixed_code) * (ge_norm_rp_norm_fixed_code))) + (((ge_norm_rn_norm_fixed_code) * (ge_norm_rn_norm_fixed_code)))) = ((ge_real_square_norm_fixed_codesquare) + (((((ge_norm_rp_norm_fixed_code) * (ge_norm_rn_norm_fixed_code))) + (((ge_norm_rn_norm_fixed_code) * (ge_norm_rp_norm_fixed_code))))))) /\ ((((((ge_norm_ip_norm_fixed_code) * (ge_norm_ip_norm_fixed_code))) + (((ge_norm_in_norm_fixed_code) * (ge_norm_in_norm_fixed_code)))) = ((ge_imaginary_square_norm_fixed_codesquare) + (((((ge_norm_ip_norm_fixed_code) * (ge_norm_in_norm_fixed_code))) + (((ge_norm_in_norm_fixed_code) * (ge_norm_ip_norm_fixed_code))))))) /\ ((N) = ge_real_square_norm_fixed_codesquare + ge_imaginary_square_norm_fixed_codesquare)))))) -> (exists ge_real_square_norm_fixed_raw ge_imaginary_square_norm_fixed_raw. ((((((a) * (a))) + (((b) * (b)))) = ((ge_real_square_norm_fixed_raw) + (((((a) * (b))) + (((b) * (a))))))) /\ ((((((c) * (c))) + (((d) * (d)))) = ((ge_imaginary_square_norm_fixed_raw) + (((((c) * (d))) + (((d) * (c))))))) /\ ((N) = ge_real_square_norm_fixed_raw + ge_imaginary_square_norm_fixed_raw))))

Complete tactic proof in conservative notation

All 36 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

36 script commands · 5 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 (2)
gaussian signed norm integer transportgaussian representation equal
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
02Separate the logical casesL9–13

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

  1. L9
    cases hnorm
  2. L10
    cases hnorm_witness
  3. L11
    cases hnorm_witness_witness
  4. L12
    cases hnorm_witness_witness_witness
  5. L13
    cases hnorm_witness_witness_witness_witness
03Use earlier factsL14–23

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

  1. L14
    specialize gaussian_signed_norm_integer_transport x
  2. L15
    specialize gaussian_signed_norm_integer_transport x1
  3. L16
    specialize gaussian_signed_norm_integer_transport x2
  4. L17
    specialize gaussian_signed_norm_integer_transport x3
  5. L18
    specialize gaussian_signed_norm_integer_transport a
  6. L19
    specialize gaussian_signed_norm_integer_transport b
  7. L20
    specialize gaussian_signed_norm_integer_transport c
  8. L21
    specialize gaussian_signed_norm_integer_transport d
  9. L22
    specialize gaussian_signed_norm_integer_transport N
  10. L23
    apply gaussian_signed_norm_integer_transport
04Use earlier factsL24–33

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

  1. L24
    specialize gaussian_representation_equal z
  2. L25
    specialize gaussian_representation_equal x
  3. L26
    specialize gaussian_representation_equal x1
  4. L27
    specialize gaussian_representation_equal x2
  5. L28
    specialize gaussian_representation_equal x3
  6. L29
    specialize gaussian_representation_equal a
  7. L30
    specialize gaussian_representation_equal b
  8. L31
    specialize gaussian_representation_equal c
  9. L32
    specialize gaussian_representation_equal d
  10. L33
    apply gaussian_representation_equal
05Use earlier factsL34–36

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

  1. L34
    exact hnorm_witness_witness_witness_witness_left
  2. L35
    exact hrep
  3. L36
    exact hnorm_witness_witness_witness_witness_right

Library-wide reading audit

Original defined command ledger · 36 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. 0009cases hnorm
  10. 0010cases hnorm_witness
  11. 0011cases hnorm_witness_witness
  12. 0012cases hnorm_witness_witness_witness
  13. 0013cases hnorm_witness_witness_witness_witness
  14. 0014specialize gaussian_signed_norm_integer_transport x
  15. 0015specialize gaussian_signed_norm_integer_transport x1
  16. 0016specialize gaussian_signed_norm_integer_transport x2
  17. 0017specialize gaussian_signed_norm_integer_transport x3
  18. 0018specialize gaussian_signed_norm_integer_transport a
  19. 0019specialize gaussian_signed_norm_integer_transport b
  20. 0020specialize gaussian_signed_norm_integer_transport c
  21. 0021specialize gaussian_signed_norm_integer_transport d
  22. 0022specialize gaussian_signed_norm_integer_transport N
  23. 0023apply gaussian_signed_norm_integer_transport
  24. 0024specialize gaussian_representation_equal z
  25. 0025specialize gaussian_representation_equal x
  26. 0026specialize gaussian_representation_equal x1
  27. 0027specialize gaussian_representation_equal x2
  28. 0028specialize gaussian_representation_equal x3
  29. 0029specialize gaussian_representation_equal a
  30. 0030specialize gaussian_representation_equal b
  31. 0031specialize gaussian_representation_equal c
  32. 0032specialize gaussian_representation_equal d
  33. 0033apply gaussian_representation_equal
  34. 0034exact hnorm_witness_witness_witness_witness_left
  35. 0035exact hrep
  36. 0036exact hnorm_witness_witness_witness_witness_right