EI0038

eisenstein_norm_exists_unique

Every valid canonical Eisenstein code has one actually computed and uniquely determined natural 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.

A floor quotient in the fundamental parallelogram already gives the required strict norm decrease; global nearest-point optimality is not asserted. The shared carrier is identical to the Gaussian carrier, but the multiplication law and norm are different. Eisenstein gcd, factorization, and prime classification remain separate targets.

Exact theorem in conservative defined notation

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

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

Definition DAG

ZPairValid(z) · 1ENorm(z,n) · 3

Actual proof prerequisites

eisenstein_norm_existseisenstein_norm_functional
Original expanded first-order statement 
forall z. (exists ge_real_positive_norm_unique_input ge_real_negative_norm_unique_input ge_imaginary_positive_norm_unique_input ge_imaginary_negative_norm_unique_input. (exists ge_real_code_norm_unique_inputdecode ge_imaginary_code_norm_unique_inputdecode. (((z) = ((ge_real_code_norm_unique_inputdecode) + (ge_imaginary_code_norm_unique_inputdecode)) * S ((ge_real_code_norm_unique_inputdecode) + (ge_imaginary_code_norm_unique_inputdecode)) + ((ge_imaginary_code_norm_unique_inputdecode) + (ge_imaginary_code_norm_unique_inputdecode))) /\ (((((ge_real_code_norm_unique_inputdecode) = 2 * (ge_real_positive_norm_unique_input) /\ (ge_real_negative_norm_unique_input) = 0) \/ exists ge_signed_half_ge_norm_unique_inputdecode_real. (((ge_real_code_norm_unique_inputdecode) = 2 * ge_signed_half_ge_norm_unique_inputdecode_real + 1 /\ (ge_real_positive_norm_unique_input) = 0) /\ (ge_real_negative_norm_unique_input) = S ge_signed_half_ge_norm_unique_inputdecode_real))) /\ ((((ge_imaginary_code_norm_unique_inputdecode) = 2 * (ge_imaginary_positive_norm_unique_input) /\ (ge_imaginary_negative_norm_unique_input) = 0) \/ exists ge_signed_half_ge_norm_unique_inputdecode_imaginary. (((ge_imaginary_code_norm_unique_inputdecode) = 2 * ge_signed_half_ge_norm_unique_inputdecode_imaginary + 1 /\ (ge_imaginary_positive_norm_unique_input) = 0) /\ (ge_imaginary_negative_norm_unique_input) = S ge_signed_half_ge_norm_unique_inputdecode_imaginary))))))) -> exists N. (((exists ee_norm_rp_unique_witness ee_norm_rn_unique_witness ee_norm_ip_unique_witness ee_norm_in_unique_witness. ((exists ge_representation_real_code_unique_witnessrepresentation ge_representation_imaginary_code_unique_witnessrepresentation. (((z) = ((ge_representation_real_code_unique_witnessrepresentation) + (ge_representation_imaginary_code_unique_witnessrepresentation)) * S ((ge_representation_real_code_unique_witnessrepresentation) + (ge_representation_imaginary_code_unique_witnessrepresentation)) + ((ge_representation_imaginary_code_unique_witnessrepresentation) + (ge_representation_imaginary_code_unique_witnessrepresentation))) /\ ((exists ge_balance_positive_unique_witnessrepresentationreal ge_balance_negative_unique_witnessrepresentationreal. (((((ge_representation_real_code_unique_witnessrepresentation) = 2 * (ge_balance_positive_unique_witnessrepresentationreal) /\ (ge_balance_negative_unique_witnessrepresentationreal) = 0) \/ exists ge_signed_half_unique_witnessrepresentationrealdecode. (((ge_representation_real_code_unique_witnessrepresentation) = 2 * ge_signed_half_unique_witnessrepresentationrealdecode + 1 /\ (ge_balance_positive_unique_witnessrepresentationreal) = 0) /\ (ge_balance_negative_unique_witnessrepresentationreal) = S ge_signed_half_unique_witnessrepresentationrealdecode))) /\ ((ee_norm_rp_unique_witness) + ge_balance_negative_unique_witnessrepresentationreal = (ee_norm_rn_unique_witness) + ge_balance_positive_unique_witnessrepresentationreal))) /\ (exists ge_balance_positive_unique_witnessrepresentationimaginary ge_balance_negative_unique_witnessrepresentationimaginary. (((((ge_representation_imaginary_code_unique_witnessrepresentation) = 2 * (ge_balance_positive_unique_witnessrepresentationimaginary) /\ (ge_balance_negative_unique_witnessrepresentationimaginary) = 0) \/ exists ge_signed_half_unique_witnessrepresentationimaginarydecode. (((ge_representation_imaginary_code_unique_witnessrepresentation) = 2 * ge_signed_half_unique_witnessrepresentationimaginarydecode + 1 /\ (ge_balance_positive_unique_witnessrepresentationimaginary) = 0) /\ (ge_balance_negative_unique_witnessrepresentationimaginary) = S ge_signed_half_unique_witnessrepresentationimaginarydecode))) /\ ((ee_norm_ip_unique_witness) + ge_balance_negative_unique_witnessrepresentationimaginary = (ee_norm_in_unique_witness) + ge_balance_positive_unique_witnessrepresentationimaginary)))))) /\ (((((((((ee_norm_rp_unique_witness) * (ee_norm_rp_unique_witness))) + (((ee_norm_rn_unique_witness) * (ee_norm_rn_unique_witness))))) + (((((ee_norm_ip_unique_witness) * (ee_norm_ip_unique_witness))) + (((ee_norm_in_unique_witness) * (ee_norm_in_unique_witness))))))) + (((((ee_norm_rp_unique_witness) * (ee_norm_in_unique_witness))) + (((ee_norm_rn_unique_witness) * (ee_norm_ip_unique_witness)))))) = ((((((((((ee_norm_rp_unique_witness) * (ee_norm_rn_unique_witness))) + (((ee_norm_rn_unique_witness) * (ee_norm_rp_unique_witness))))) + (((((ee_norm_ip_unique_witness) * (ee_norm_in_unique_witness))) + (((ee_norm_in_unique_witness) * (ee_norm_ip_unique_witness))))))) + (((((ee_norm_rp_unique_witness) * (ee_norm_ip_unique_witness))) + (((ee_norm_rn_unique_witness) * (ee_norm_in_unique_witness))))))) + (N))))) /\ (forall M. (exists ee_norm_rp_unique_compare ee_norm_rn_unique_compare ee_norm_ip_unique_compare ee_norm_in_unique_compare. ((exists ge_representation_real_code_unique_comparerepresentation ge_representation_imaginary_code_unique_comparerepresentation. (((z) = ((ge_representation_real_code_unique_comparerepresentation) + (ge_representation_imaginary_code_unique_comparerepresentation)) * S ((ge_representation_real_code_unique_comparerepresentation) + (ge_representation_imaginary_code_unique_comparerepresentation)) + ((ge_representation_imaginary_code_unique_comparerepresentation) + (ge_representation_imaginary_code_unique_comparerepresentation))) /\ ((exists ge_balance_positive_unique_comparerepresentationreal ge_balance_negative_unique_comparerepresentationreal. (((((ge_representation_real_code_unique_comparerepresentation) = 2 * (ge_balance_positive_unique_comparerepresentationreal) /\ (ge_balance_negative_unique_comparerepresentationreal) = 0) \/ exists ge_signed_half_unique_comparerepresentationrealdecode. (((ge_representation_real_code_unique_comparerepresentation) = 2 * ge_signed_half_unique_comparerepresentationrealdecode + 1 /\ (ge_balance_positive_unique_comparerepresentationreal) = 0) /\ (ge_balance_negative_unique_comparerepresentationreal) = S ge_signed_half_unique_comparerepresentationrealdecode))) /\ ((ee_norm_rp_unique_compare) + ge_balance_negative_unique_comparerepresentationreal = (ee_norm_rn_unique_compare) + ge_balance_positive_unique_comparerepresentationreal))) /\ (exists ge_balance_positive_unique_comparerepresentationimaginary ge_balance_negative_unique_comparerepresentationimaginary. (((((ge_representation_imaginary_code_unique_comparerepresentation) = 2 * (ge_balance_positive_unique_comparerepresentationimaginary) /\ (ge_balance_negative_unique_comparerepresentationimaginary) = 0) \/ exists ge_signed_half_unique_comparerepresentationimaginarydecode. (((ge_representation_imaginary_code_unique_comparerepresentation) = 2 * ge_signed_half_unique_comparerepresentationimaginarydecode + 1 /\ (ge_balance_positive_unique_comparerepresentationimaginary) = 0) /\ (ge_balance_negative_unique_comparerepresentationimaginary) = S ge_signed_half_unique_comparerepresentationimaginarydecode))) /\ ((ee_norm_ip_unique_compare) + ge_balance_negative_unique_comparerepresentationimaginary = (ee_norm_in_unique_compare) + ge_balance_positive_unique_comparerepresentationimaginary)))))) /\ (((((((((ee_norm_rp_unique_compare) * (ee_norm_rp_unique_compare))) + (((ee_norm_rn_unique_compare) * (ee_norm_rn_unique_compare))))) + (((((ee_norm_ip_unique_compare) * (ee_norm_ip_unique_compare))) + (((ee_norm_in_unique_compare) * (ee_norm_in_unique_compare))))))) + (((((ee_norm_rp_unique_compare) * (ee_norm_in_unique_compare))) + (((ee_norm_rn_unique_compare) * (ee_norm_ip_unique_compare)))))) = ((((((((((ee_norm_rp_unique_compare) * (ee_norm_rn_unique_compare))) + (((ee_norm_rn_unique_compare) * (ee_norm_rp_unique_compare))))) + (((((ee_norm_ip_unique_compare) * (ee_norm_in_unique_compare))) + (((ee_norm_in_unique_compare) * (ee_norm_ip_unique_compare))))))) + (((((ee_norm_rp_unique_compare) * (ee_norm_ip_unique_compare))) + (((ee_norm_rn_unique_compare) * (ee_norm_in_unique_compare))))))) + (M))))) -> 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)
eisenstein norm existseisenstein 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 eisenstein norm exists.

  1. L3
    have hnorm : ∃ N. ENorm(z,N)Definitions: ENorm(z,N)Original native command in the exact edition
  2. L4
    specialize eisenstein_norm_exists z
  3. L5
    apply eisenstein_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 eisenstein_norm_functional z
  2. L14
    specialize eisenstein_norm_functional M
  3. L15
    specialize eisenstein_norm_functional x
  4. L16
    apply eisenstein_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. ENorm(z,N)
  4. 0004specialize eisenstein_norm_exists z
  5. 0005apply eisenstein_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 eisenstein_norm_functional z
  14. 0014specialize eisenstein_norm_functional M
  15. 0015specialize eisenstein_norm_functional x
  16. 0016apply eisenstein_norm_functional
  17. 0017exact hother
  18. 0018exact hnorm_witness