EI0036

eisenstein_norm_exists

Every canonical Eisenstein integer has a genuinely constructed natural norm, including zero and all units.

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)

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

Definition DAG

ZPairValid(z) · 1EisensteinCoordinateNorm(ap,an,bp,bn,n) · 1ENorm(z,n) · 1

Actual proof prerequisites

eisenstein_coordinate_norm_existseisenstein_norm_of_representationgaussian_decode_representation · checked external prerequisite
Original expanded first-order statement 
forall z. (exists ge_real_positive_norm_input ge_real_negative_norm_input ge_imaginary_positive_norm_input ge_imaginary_negative_norm_input. (exists ge_real_code_norm_inputdecode ge_imaginary_code_norm_inputdecode. (((z) = ((ge_real_code_norm_inputdecode) + (ge_imaginary_code_norm_inputdecode)) * S ((ge_real_code_norm_inputdecode) + (ge_imaginary_code_norm_inputdecode)) + ((ge_imaginary_code_norm_inputdecode) + (ge_imaginary_code_norm_inputdecode))) /\ (((((ge_real_code_norm_inputdecode) = 2 * (ge_real_positive_norm_input) /\ (ge_real_negative_norm_input) = 0) \/ exists ge_signed_half_ge_norm_inputdecode_real. (((ge_real_code_norm_inputdecode) = 2 * ge_signed_half_ge_norm_inputdecode_real + 1 /\ (ge_real_positive_norm_input) = 0) /\ (ge_real_negative_norm_input) = S ge_signed_half_ge_norm_inputdecode_real))) /\ ((((ge_imaginary_code_norm_inputdecode) = 2 * (ge_imaginary_positive_norm_input) /\ (ge_imaginary_negative_norm_input) = 0) \/ exists ge_signed_half_ge_norm_inputdecode_imaginary. (((ge_imaginary_code_norm_inputdecode) = 2 * ge_signed_half_ge_norm_inputdecode_imaginary + 1 /\ (ge_imaginary_positive_norm_input) = 0) /\ (ge_imaginary_negative_norm_input) = S ge_signed_half_ge_norm_inputdecode_imaginary))))))) -> exists N. (exists ee_norm_rp_norm_output ee_norm_rn_norm_output ee_norm_ip_norm_output ee_norm_in_norm_output. ((exists ge_representation_real_code_norm_outputrepresentation ge_representation_imaginary_code_norm_outputrepresentation. (((z) = ((ge_representation_real_code_norm_outputrepresentation) + (ge_representation_imaginary_code_norm_outputrepresentation)) * S ((ge_representation_real_code_norm_outputrepresentation) + (ge_representation_imaginary_code_norm_outputrepresentation)) + ((ge_representation_imaginary_code_norm_outputrepresentation) + (ge_representation_imaginary_code_norm_outputrepresentation))) /\ ((exists ge_balance_positive_norm_outputrepresentationreal ge_balance_negative_norm_outputrepresentationreal. (((((ge_representation_real_code_norm_outputrepresentation) = 2 * (ge_balance_positive_norm_outputrepresentationreal) /\ (ge_balance_negative_norm_outputrepresentationreal) = 0) \/ exists ge_signed_half_norm_outputrepresentationrealdecode. (((ge_representation_real_code_norm_outputrepresentation) = 2 * ge_signed_half_norm_outputrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_outputrepresentationreal) = 0) /\ (ge_balance_negative_norm_outputrepresentationreal) = S ge_signed_half_norm_outputrepresentationrealdecode))) /\ ((ee_norm_rp_norm_output) + ge_balance_negative_norm_outputrepresentationreal = (ee_norm_rn_norm_output) + ge_balance_positive_norm_outputrepresentationreal))) /\ (exists ge_balance_positive_norm_outputrepresentationimaginary ge_balance_negative_norm_outputrepresentationimaginary. (((((ge_representation_imaginary_code_norm_outputrepresentation) = 2 * (ge_balance_positive_norm_outputrepresentationimaginary) /\ (ge_balance_negative_norm_outputrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_outputrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_outputrepresentation) = 2 * ge_signed_half_norm_outputrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_outputrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_outputrepresentationimaginary) = S ge_signed_half_norm_outputrepresentationimaginarydecode))) /\ ((ee_norm_ip_norm_output) + ge_balance_negative_norm_outputrepresentationimaginary = (ee_norm_in_norm_output) + ge_balance_positive_norm_outputrepresentationimaginary)))))) /\ (((((((((ee_norm_rp_norm_output) * (ee_norm_rp_norm_output))) + (((ee_norm_rn_norm_output) * (ee_norm_rn_norm_output))))) + (((((ee_norm_ip_norm_output) * (ee_norm_ip_norm_output))) + (((ee_norm_in_norm_output) * (ee_norm_in_norm_output))))))) + (((((ee_norm_rp_norm_output) * (ee_norm_in_norm_output))) + (((ee_norm_rn_norm_output) * (ee_norm_ip_norm_output)))))) = ((((((((((ee_norm_rp_norm_output) * (ee_norm_rn_norm_output))) + (((ee_norm_rn_norm_output) * (ee_norm_rp_norm_output))))) + (((((ee_norm_ip_norm_output) * (ee_norm_in_norm_output))) + (((ee_norm_in_norm_output) * (ee_norm_ip_norm_output))))))) + (((((ee_norm_rp_norm_output) * (ee_norm_ip_norm_output))) + (((ee_norm_rn_norm_output) * (ee_norm_in_norm_output))))))) + (N)))))

Complete tactic proof in conservative notation

All 29 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

29 script commands · 7 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 coordinate norm existseisenstein norm of representation
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro hvalid
02Separate the logical casesL3–6

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

  1. L3
    cases hvalid
  2. L4
    cases hvalid_witness
  3. L5
    cases hvalid_witness_witness
  4. L6
    cases hvalid_witness_witness_witness
03Establish hnormL7–12

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

  1. L7
    have hnorm : ∃ N. EisensteinCoordinateNorm(x,x1,x2,x3,N)Definitions: EisensteinCoordinateNorm(x,x1,x2,x3,N)Original native command in the exact edition
  2. L8
    specialize eisenstein_coordinate_norm_exists x
  3. L9
    specialize eisenstein_coordinate_norm_exists x1
  4. L10
    specialize eisenstein_coordinate_norm_exists x2
  5. L11
    specialize eisenstein_coordinate_norm_exists x3
  6. L12
    apply eisenstein_coordinate_norm_exists
04Separate the logical casesL13–13

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

  1. L13
    cases hnorm
05Construct an explicit witnessL14–14

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

  1. L14
    exists x4
06Use earlier factsL15–24

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

  1. L15
    specialize eisenstein_norm_of_representation z
  2. L16
    specialize eisenstein_norm_of_representation x
  3. L17
    specialize eisenstein_norm_of_representation x1
  4. L18
    specialize eisenstein_norm_of_representation x2
  5. L19
    specialize eisenstein_norm_of_representation x3
  6. L20
    specialize eisenstein_norm_of_representation x4
  7. L21
    apply eisenstein_norm_of_representation
  8. L22
    specialize gaussian_decode_representation z
  9. L23
    specialize gaussian_decode_representation x
  10. L24
    specialize gaussian_decode_representation x1
07Use earlier factsL25–29

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

  1. L25
    specialize gaussian_decode_representation x2
  2. L26
    specialize gaussian_decode_representation x3
  3. L27
    apply gaussian_decode_representation
  4. L28
    exact hvalid_witness_witness_witness_witness
  5. L29
    exact hnorm_witness

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro z
  2. 0002intro hvalid
  3. 0003cases hvalid
  4. 0004cases hvalid_witness
  5. 0005cases hvalid_witness_witness
  6. 0006cases hvalid_witness_witness_witness
  7. 0007have hnorm : ∃ N. EisensteinCoordinateNorm(x,x1,x2,x3,N)
  8. 0008specialize eisenstein_coordinate_norm_exists x
  9. 0009specialize eisenstein_coordinate_norm_exists x1
  10. 0010specialize eisenstein_coordinate_norm_exists x2
  11. 0011specialize eisenstein_coordinate_norm_exists x3
  12. 0012apply eisenstein_coordinate_norm_exists
  13. 0013cases hnorm
  14. 0014exists x4
  15. 0015specialize eisenstein_norm_of_representation z
  16. 0016specialize eisenstein_norm_of_representation x
  17. 0017specialize eisenstein_norm_of_representation x1
  18. 0018specialize eisenstein_norm_of_representation x2
  19. 0019specialize eisenstein_norm_of_representation x3
  20. 0020specialize eisenstein_norm_of_representation x4
  21. 0021apply eisenstein_norm_of_representation
  22. 0022specialize gaussian_decode_representation z
  23. 0023specialize gaussian_decode_representation x
  24. 0024specialize gaussian_decode_representation x1
  25. 0025specialize gaussian_decode_representation x2
  26. 0026specialize gaussian_decode_representation x3
  27. 0027apply gaussian_decode_representation
  28. 0028exact hvalid_witness_witness_witness_witness
  29. 0029exact hnorm_witness