GI0050

gaussian_norm_functional

The actual canonical Gaussian squared norm is unique across all arbitrary signed representatives.

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. ∀ N. ∀ M. GNorm(z,N) → GNorm(z,M) → N = M

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

Definition DAG

GNorm(z,n) · 2

Actual proof prerequisites

gaussian_signed_norm_functionalgaussian_norm_for_representation
Original expanded first-order statement 
forall z N M. (exists ge_norm_rp_norm_unique_first ge_norm_rn_norm_unique_first ge_norm_ip_norm_unique_first ge_norm_in_norm_unique_first. ((exists ge_representation_real_code_norm_unique_firstrepresentation ge_representation_imaginary_code_norm_unique_firstrepresentation. (((z) = ((ge_representation_real_code_norm_unique_firstrepresentation) + (ge_representation_imaginary_code_norm_unique_firstrepresentation)) * S ((ge_representation_real_code_norm_unique_firstrepresentation) + (ge_representation_imaginary_code_norm_unique_firstrepresentation)) + ((ge_representation_imaginary_code_norm_unique_firstrepresentation) + (ge_representation_imaginary_code_norm_unique_firstrepresentation))) /\ ((exists ge_balance_positive_norm_unique_firstrepresentationreal ge_balance_negative_norm_unique_firstrepresentationreal. (((((ge_representation_real_code_norm_unique_firstrepresentation) = 2 * (ge_balance_positive_norm_unique_firstrepresentationreal) /\ (ge_balance_negative_norm_unique_firstrepresentationreal) = 0) \/ exists ge_signed_half_norm_unique_firstrepresentationrealdecode. (((ge_representation_real_code_norm_unique_firstrepresentation) = 2 * ge_signed_half_norm_unique_firstrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_unique_firstrepresentationreal) = 0) /\ (ge_balance_negative_norm_unique_firstrepresentationreal) = S ge_signed_half_norm_unique_firstrepresentationrealdecode))) /\ ((ge_norm_rp_norm_unique_first) + ge_balance_negative_norm_unique_firstrepresentationreal = (ge_norm_rn_norm_unique_first) + ge_balance_positive_norm_unique_firstrepresentationreal))) /\ (exists ge_balance_positive_norm_unique_firstrepresentationimaginary ge_balance_negative_norm_unique_firstrepresentationimaginary. (((((ge_representation_imaginary_code_norm_unique_firstrepresentation) = 2 * (ge_balance_positive_norm_unique_firstrepresentationimaginary) /\ (ge_balance_negative_norm_unique_firstrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_unique_firstrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_unique_firstrepresentation) = 2 * ge_signed_half_norm_unique_firstrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_unique_firstrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_unique_firstrepresentationimaginary) = S ge_signed_half_norm_unique_firstrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_unique_first) + ge_balance_negative_norm_unique_firstrepresentationimaginary = (ge_norm_in_norm_unique_first) + ge_balance_positive_norm_unique_firstrepresentationimaginary)))))) /\ (exists ge_real_square_norm_unique_firstsquare ge_imaginary_square_norm_unique_firstsquare. ((((((ge_norm_rp_norm_unique_first) * (ge_norm_rp_norm_unique_first))) + (((ge_norm_rn_norm_unique_first) * (ge_norm_rn_norm_unique_first)))) = ((ge_real_square_norm_unique_firstsquare) + (((((ge_norm_rp_norm_unique_first) * (ge_norm_rn_norm_unique_first))) + (((ge_norm_rn_norm_unique_first) * (ge_norm_rp_norm_unique_first))))))) /\ ((((((ge_norm_ip_norm_unique_first) * (ge_norm_ip_norm_unique_first))) + (((ge_norm_in_norm_unique_first) * (ge_norm_in_norm_unique_first)))) = ((ge_imaginary_square_norm_unique_firstsquare) + (((((ge_norm_ip_norm_unique_first) * (ge_norm_in_norm_unique_first))) + (((ge_norm_in_norm_unique_first) * (ge_norm_ip_norm_unique_first))))))) /\ ((N) = ge_real_square_norm_unique_firstsquare + ge_imaginary_square_norm_unique_firstsquare)))))) -> (exists ge_norm_rp_norm_unique_second ge_norm_rn_norm_unique_second ge_norm_ip_norm_unique_second ge_norm_in_norm_unique_second. ((exists ge_representation_real_code_norm_unique_secondrepresentation ge_representation_imaginary_code_norm_unique_secondrepresentation. (((z) = ((ge_representation_real_code_norm_unique_secondrepresentation) + (ge_representation_imaginary_code_norm_unique_secondrepresentation)) * S ((ge_representation_real_code_norm_unique_secondrepresentation) + (ge_representation_imaginary_code_norm_unique_secondrepresentation)) + ((ge_representation_imaginary_code_norm_unique_secondrepresentation) + (ge_representation_imaginary_code_norm_unique_secondrepresentation))) /\ ((exists ge_balance_positive_norm_unique_secondrepresentationreal ge_balance_negative_norm_unique_secondrepresentationreal. (((((ge_representation_real_code_norm_unique_secondrepresentation) = 2 * (ge_balance_positive_norm_unique_secondrepresentationreal) /\ (ge_balance_negative_norm_unique_secondrepresentationreal) = 0) \/ exists ge_signed_half_norm_unique_secondrepresentationrealdecode. (((ge_representation_real_code_norm_unique_secondrepresentation) = 2 * ge_signed_half_norm_unique_secondrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_unique_secondrepresentationreal) = 0) /\ (ge_balance_negative_norm_unique_secondrepresentationreal) = S ge_signed_half_norm_unique_secondrepresentationrealdecode))) /\ ((ge_norm_rp_norm_unique_second) + ge_balance_negative_norm_unique_secondrepresentationreal = (ge_norm_rn_norm_unique_second) + ge_balance_positive_norm_unique_secondrepresentationreal))) /\ (exists ge_balance_positive_norm_unique_secondrepresentationimaginary ge_balance_negative_norm_unique_secondrepresentationimaginary. (((((ge_representation_imaginary_code_norm_unique_secondrepresentation) = 2 * (ge_balance_positive_norm_unique_secondrepresentationimaginary) /\ (ge_balance_negative_norm_unique_secondrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_unique_secondrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_unique_secondrepresentation) = 2 * ge_signed_half_norm_unique_secondrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_unique_secondrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_unique_secondrepresentationimaginary) = S ge_signed_half_norm_unique_secondrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_unique_second) + ge_balance_negative_norm_unique_secondrepresentationimaginary = (ge_norm_in_norm_unique_second) + ge_balance_positive_norm_unique_secondrepresentationimaginary)))))) /\ (exists ge_real_square_norm_unique_secondsquare ge_imaginary_square_norm_unique_secondsquare. ((((((ge_norm_rp_norm_unique_second) * (ge_norm_rp_norm_unique_second))) + (((ge_norm_rn_norm_unique_second) * (ge_norm_rn_norm_unique_second)))) = ((ge_real_square_norm_unique_secondsquare) + (((((ge_norm_rp_norm_unique_second) * (ge_norm_rn_norm_unique_second))) + (((ge_norm_rn_norm_unique_second) * (ge_norm_rp_norm_unique_second))))))) /\ ((((((ge_norm_ip_norm_unique_second) * (ge_norm_ip_norm_unique_second))) + (((ge_norm_in_norm_unique_second) * (ge_norm_in_norm_unique_second)))) = ((ge_imaginary_square_norm_unique_secondsquare) + (((((ge_norm_ip_norm_unique_second) * (ge_norm_in_norm_unique_second))) + (((ge_norm_in_norm_unique_second) * (ge_norm_ip_norm_unique_second))))))) /\ ((M) = ge_real_square_norm_unique_secondsquare + ge_imaginary_square_norm_unique_secondsquare)))))) -> N = M

Complete tactic proof in conservative notation

All 27 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

27 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.

Named ingredients (2)
gaussian signed norm functionalgaussian norm for representation
01Fix variables and assumptionsL1–5

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

  1. L1
    intro z
  2. L2
    intro N
  3. L3
    intro M
  4. L4
    intro hfirst
  5. L5
    intro hsecond
02Separate the logical casesL6–10

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

  1. L6
    cases hfirst
  2. L7
    cases hfirst_witness
  3. L8
    cases hfirst_witness_witness
  4. L9
    cases hfirst_witness_witness_witness
  5. L10
    cases hfirst_witness_witness_witness_witness
03Use earlier factsL11–20

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

  1. L11
    specialize gaussian_signed_norm_functional x
  2. L12
    specialize gaussian_signed_norm_functional x1
  3. L13
    specialize gaussian_signed_norm_functional x2
  4. L14
    specialize gaussian_signed_norm_functional x3
  5. L15
    specialize gaussian_signed_norm_functional N
  6. L16
    specialize gaussian_signed_norm_functional M
  7. L17
    apply gaussian_signed_norm_functional
  8. L18
    exact hfirst_witness_witness_witness_witness_right
  9. L19
    specialize gaussian_norm_for_representation z
  10. L20
    specialize gaussian_norm_for_representation x
04Use earlier factsL21–27

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

  1. L21
    specialize gaussian_norm_for_representation x1
  2. L22
    specialize gaussian_norm_for_representation x2
  3. L23
    specialize gaussian_norm_for_representation x3
  4. L24
    specialize gaussian_norm_for_representation M
  5. L25
    apply gaussian_norm_for_representation
  6. L26
    exact hfirst_witness_witness_witness_witness_left
  7. L27
    exact hsecond

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro z
  2. 0002intro N
  3. 0003intro M
  4. 0004intro hfirst
  5. 0005intro hsecond
  6. 0006cases hfirst
  7. 0007cases hfirst_witness
  8. 0008cases hfirst_witness_witness
  9. 0009cases hfirst_witness_witness_witness
  10. 0010cases hfirst_witness_witness_witness_witness
  11. 0011specialize gaussian_signed_norm_functional x
  12. 0012specialize gaussian_signed_norm_functional x1
  13. 0013specialize gaussian_signed_norm_functional x2
  14. 0014specialize gaussian_signed_norm_functional x3
  15. 0015specialize gaussian_signed_norm_functional N
  16. 0016specialize gaussian_signed_norm_functional M
  17. 0017apply gaussian_signed_norm_functional
  18. 0018exact hfirst_witness_witness_witness_witness_right
  19. 0019specialize gaussian_norm_for_representation z
  20. 0020specialize gaussian_norm_for_representation x
  21. 0021specialize gaussian_norm_for_representation x1
  22. 0022specialize gaussian_norm_for_representation x2
  23. 0023specialize gaussian_norm_for_representation x3
  24. 0024specialize gaussian_norm_for_representation M
  25. 0025apply gaussian_norm_for_representation
  26. 0026exact hfirst_witness_witness_witness_witness_left
  27. 0027exact hsecond