GI0051

gaussian_norm_exists_unique

Every canonical Gaussian integer has one genuinely constructed and uniquely determined natural squared norm.

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. ZPairValid(z) → ∃ x. GNorm(z,x) ∧ (∀ y. GNorm(z,y) → y = x)

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

Definition DAG

ZPairValid(z) · 1GNorm(z,n) · 3

Actual proof prerequisites

gaussian_norm_existsgaussian_norm_functional
Original expanded first-order statement 
forall z. (exists ge_real_positive_norm_total_unique_input ge_real_negative_norm_total_unique_input ge_imaginary_positive_norm_total_unique_input ge_imaginary_negative_norm_total_unique_input. (exists ge_real_code_norm_total_unique_inputdecode ge_imaginary_code_norm_total_unique_inputdecode. (((z) = ((ge_real_code_norm_total_unique_inputdecode) + (ge_imaginary_code_norm_total_unique_inputdecode)) * S ((ge_real_code_norm_total_unique_inputdecode) + (ge_imaginary_code_norm_total_unique_inputdecode)) + ((ge_imaginary_code_norm_total_unique_inputdecode) + (ge_imaginary_code_norm_total_unique_inputdecode))) /\ (((((ge_real_code_norm_total_unique_inputdecode) = 2 * (ge_real_positive_norm_total_unique_input) /\ (ge_real_negative_norm_total_unique_input) = 0) \/ exists ge_signed_half_ge_norm_total_unique_inputdecode_real. (((ge_real_code_norm_total_unique_inputdecode) = 2 * ge_signed_half_ge_norm_total_unique_inputdecode_real + 1 /\ (ge_real_positive_norm_total_unique_input) = 0) /\ (ge_real_negative_norm_total_unique_input) = S ge_signed_half_ge_norm_total_unique_inputdecode_real))) /\ ((((ge_imaginary_code_norm_total_unique_inputdecode) = 2 * (ge_imaginary_positive_norm_total_unique_input) /\ (ge_imaginary_negative_norm_total_unique_input) = 0) \/ exists ge_signed_half_ge_norm_total_unique_inputdecode_imaginary. (((ge_imaginary_code_norm_total_unique_inputdecode) = 2 * ge_signed_half_ge_norm_total_unique_inputdecode_imaginary + 1 /\ (ge_imaginary_positive_norm_total_unique_input) = 0) /\ (ge_imaginary_negative_norm_total_unique_input) = S ge_signed_half_ge_norm_total_unique_inputdecode_imaginary))))))) -> exists N. (((exists ge_norm_rp_norm_total_unique_witness ge_norm_rn_norm_total_unique_witness ge_norm_ip_norm_total_unique_witness ge_norm_in_norm_total_unique_witness. ((exists ge_representation_real_code_norm_total_unique_witnessrepresentation ge_representation_imaginary_code_norm_total_unique_witnessrepresentation. (((z) = ((ge_representation_real_code_norm_total_unique_witnessrepresentation) + (ge_representation_imaginary_code_norm_total_unique_witnessrepresentation)) * S ((ge_representation_real_code_norm_total_unique_witnessrepresentation) + (ge_representation_imaginary_code_norm_total_unique_witnessrepresentation)) + ((ge_representation_imaginary_code_norm_total_unique_witnessrepresentation) + (ge_representation_imaginary_code_norm_total_unique_witnessrepresentation))) /\ ((exists ge_balance_positive_norm_total_unique_witnessrepresentationreal ge_balance_negative_norm_total_unique_witnessrepresentationreal. (((((ge_representation_real_code_norm_total_unique_witnessrepresentation) = 2 * (ge_balance_positive_norm_total_unique_witnessrepresentationreal) /\ (ge_balance_negative_norm_total_unique_witnessrepresentationreal) = 0) \/ exists ge_signed_half_norm_total_unique_witnessrepresentationrealdecode. (((ge_representation_real_code_norm_total_unique_witnessrepresentation) = 2 * ge_signed_half_norm_total_unique_witnessrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_total_unique_witnessrepresentationreal) = 0) /\ (ge_balance_negative_norm_total_unique_witnessrepresentationreal) = S ge_signed_half_norm_total_unique_witnessrepresentationrealdecode))) /\ ((ge_norm_rp_norm_total_unique_witness) + ge_balance_negative_norm_total_unique_witnessrepresentationreal = (ge_norm_rn_norm_total_unique_witness) + ge_balance_positive_norm_total_unique_witnessrepresentationreal))) /\ (exists ge_balance_positive_norm_total_unique_witnessrepresentationimaginary ge_balance_negative_norm_total_unique_witnessrepresentationimaginary. (((((ge_representation_imaginary_code_norm_total_unique_witnessrepresentation) = 2 * (ge_balance_positive_norm_total_unique_witnessrepresentationimaginary) /\ (ge_balance_negative_norm_total_unique_witnessrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_total_unique_witnessrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_total_unique_witnessrepresentation) = 2 * ge_signed_half_norm_total_unique_witnessrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_total_unique_witnessrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_total_unique_witnessrepresentationimaginary) = S ge_signed_half_norm_total_unique_witnessrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_total_unique_witness) + ge_balance_negative_norm_total_unique_witnessrepresentationimaginary = (ge_norm_in_norm_total_unique_witness) + ge_balance_positive_norm_total_unique_witnessrepresentationimaginary)))))) /\ (exists ge_real_square_norm_total_unique_witnesssquare ge_imaginary_square_norm_total_unique_witnesssquare. ((((((ge_norm_rp_norm_total_unique_witness) * (ge_norm_rp_norm_total_unique_witness))) + (((ge_norm_rn_norm_total_unique_witness) * (ge_norm_rn_norm_total_unique_witness)))) = ((ge_real_square_norm_total_unique_witnesssquare) + (((((ge_norm_rp_norm_total_unique_witness) * (ge_norm_rn_norm_total_unique_witness))) + (((ge_norm_rn_norm_total_unique_witness) * (ge_norm_rp_norm_total_unique_witness))))))) /\ ((((((ge_norm_ip_norm_total_unique_witness) * (ge_norm_ip_norm_total_unique_witness))) + (((ge_norm_in_norm_total_unique_witness) * (ge_norm_in_norm_total_unique_witness)))) = ((ge_imaginary_square_norm_total_unique_witnesssquare) + (((((ge_norm_ip_norm_total_unique_witness) * (ge_norm_in_norm_total_unique_witness))) + (((ge_norm_in_norm_total_unique_witness) * (ge_norm_ip_norm_total_unique_witness))))))) /\ ((N) = ge_real_square_norm_total_unique_witnesssquare + ge_imaginary_square_norm_total_unique_witnesssquare)))))) /\ (forall M. (exists ge_norm_rp_norm_total_unique_compare ge_norm_rn_norm_total_unique_compare ge_norm_ip_norm_total_unique_compare ge_norm_in_norm_total_unique_compare. ((exists ge_representation_real_code_norm_total_unique_comparerepresentation ge_representation_imaginary_code_norm_total_unique_comparerepresentation. (((z) = ((ge_representation_real_code_norm_total_unique_comparerepresentation) + (ge_representation_imaginary_code_norm_total_unique_comparerepresentation)) * S ((ge_representation_real_code_norm_total_unique_comparerepresentation) + (ge_representation_imaginary_code_norm_total_unique_comparerepresentation)) + ((ge_representation_imaginary_code_norm_total_unique_comparerepresentation) + (ge_representation_imaginary_code_norm_total_unique_comparerepresentation))) /\ ((exists ge_balance_positive_norm_total_unique_comparerepresentationreal ge_balance_negative_norm_total_unique_comparerepresentationreal. (((((ge_representation_real_code_norm_total_unique_comparerepresentation) = 2 * (ge_balance_positive_norm_total_unique_comparerepresentationreal) /\ (ge_balance_negative_norm_total_unique_comparerepresentationreal) = 0) \/ exists ge_signed_half_norm_total_unique_comparerepresentationrealdecode. (((ge_representation_real_code_norm_total_unique_comparerepresentation) = 2 * ge_signed_half_norm_total_unique_comparerepresentationrealdecode + 1 /\ (ge_balance_positive_norm_total_unique_comparerepresentationreal) = 0) /\ (ge_balance_negative_norm_total_unique_comparerepresentationreal) = S ge_signed_half_norm_total_unique_comparerepresentationrealdecode))) /\ ((ge_norm_rp_norm_total_unique_compare) + ge_balance_negative_norm_total_unique_comparerepresentationreal = (ge_norm_rn_norm_total_unique_compare) + ge_balance_positive_norm_total_unique_comparerepresentationreal))) /\ (exists ge_balance_positive_norm_total_unique_comparerepresentationimaginary ge_balance_negative_norm_total_unique_comparerepresentationimaginary. (((((ge_representation_imaginary_code_norm_total_unique_comparerepresentation) = 2 * (ge_balance_positive_norm_total_unique_comparerepresentationimaginary) /\ (ge_balance_negative_norm_total_unique_comparerepresentationimaginary) = 0) \/ exists ge_signed_half_norm_total_unique_comparerepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_total_unique_comparerepresentation) = 2 * ge_signed_half_norm_total_unique_comparerepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_total_unique_comparerepresentationimaginary) = 0) /\ (ge_balance_negative_norm_total_unique_comparerepresentationimaginary) = S ge_signed_half_norm_total_unique_comparerepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_total_unique_compare) + ge_balance_negative_norm_total_unique_comparerepresentationimaginary = (ge_norm_in_norm_total_unique_compare) + ge_balance_positive_norm_total_unique_comparerepresentationimaginary)))))) /\ (exists ge_real_square_norm_total_unique_comparesquare ge_imaginary_square_norm_total_unique_comparesquare. ((((((ge_norm_rp_norm_total_unique_compare) * (ge_norm_rp_norm_total_unique_compare))) + (((ge_norm_rn_norm_total_unique_compare) * (ge_norm_rn_norm_total_unique_compare)))) = ((ge_real_square_norm_total_unique_comparesquare) + (((((ge_norm_rp_norm_total_unique_compare) * (ge_norm_rn_norm_total_unique_compare))) + (((ge_norm_rn_norm_total_unique_compare) * (ge_norm_rp_norm_total_unique_compare))))))) /\ ((((((ge_norm_ip_norm_total_unique_compare) * (ge_norm_ip_norm_total_unique_compare))) + (((ge_norm_in_norm_total_unique_compare) * (ge_norm_in_norm_total_unique_compare)))) = ((ge_imaginary_square_norm_total_unique_comparesquare) + (((((ge_norm_ip_norm_total_unique_compare) * (ge_norm_in_norm_total_unique_compare))) + (((ge_norm_in_norm_total_unique_compare) * (ge_norm_ip_norm_total_unique_compare))))))) /\ ((M) = ge_real_square_norm_total_unique_comparesquare + ge_imaginary_square_norm_total_unique_comparesquare)))))) -> M = N)))

Complete tactic proof in conservative notation

All 18 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

18 script commands · 8 reading checkpoints · 1 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 norm existsgaussian norm functional
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro hvalid
02Establish hnormL3–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gaussian norm exists.

  1. L3
    have hnorm : ∃ N. GNorm(z,N)Definitions: GNorm(z,N)Original native command in the exact edition
  2. L4
    specialize gaussian_norm_exists z
  3. L5
    apply gaussian_norm_exists
  4. L6
    exact hvalid
03Separate the logical casesL7–7

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

  1. L7
    cases hnorm
04Construct an explicit witnessL8–8

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

  1. L8
    exists x
05Separate the logical casesL9–9

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

  1. L9
    split
06Use earlier factsL10–10

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

  1. L10
    exact hnorm_witness
07Fix variables and assumptionsL11–12

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

  1. L11
    intro M
  2. L12
    intro hother
08Use earlier factsL13–18

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

  1. L13
    specialize gaussian_norm_functional z
  2. L14
    specialize gaussian_norm_functional M
  3. L15
    specialize gaussian_norm_functional x
  4. L16
    apply gaussian_norm_functional
  5. L17
    exact hother
  6. L18
    exact hnorm_witness

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro z
  2. 0002intro hvalid
  3. 0003have hnorm : ∃ N. GNorm(z,N)
  4. 0004specialize gaussian_norm_exists z
  5. 0005apply gaussian_norm_exists
  6. 0006exact hvalid
  7. 0007cases hnorm
  8. 0008exists x
  9. 0009split
  10. 0010exact hnorm_witness
  11. 0011intro M
  12. 0012intro hother
  13. 0013specialize gaussian_norm_functional z
  14. 0014specialize gaussian_norm_functional M
  15. 0015specialize gaussian_norm_functional x
  16. 0016apply gaussian_norm_functional
  17. 0017exact hother
  18. 0018exact hnorm_witness