EI0037

eisenstein_norm_functional

The canonical natural Eisenstein norm is unique, independently of all signed-coordinate 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.

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

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

Definition DAG

ENorm(z,n) · 2

Actual proof prerequisites

eisenstein_coordinate_norm_functionaleisenstein_norm_for_representation
Original expanded first-order statement 
forall z N M. (exists ee_norm_rp_norm_first ee_norm_rn_norm_first ee_norm_ip_norm_first ee_norm_in_norm_first. ((exists ge_representation_real_code_norm_firstrepresentation ge_representation_imaginary_code_norm_firstrepresentation. (((z) = ((ge_representation_real_code_norm_firstrepresentation) + (ge_representation_imaginary_code_norm_firstrepresentation)) * S ((ge_representation_real_code_norm_firstrepresentation) + (ge_representation_imaginary_code_norm_firstrepresentation)) + ((ge_representation_imaginary_code_norm_firstrepresentation) + (ge_representation_imaginary_code_norm_firstrepresentation))) /\ ((exists ge_balance_positive_norm_firstrepresentationreal ge_balance_negative_norm_firstrepresentationreal. (((((ge_representation_real_code_norm_firstrepresentation) = 2 * (ge_balance_positive_norm_firstrepresentationreal) /\ (ge_balance_negative_norm_firstrepresentationreal) = 0) \/ exists ge_signed_half_norm_firstrepresentationrealdecode. (((ge_representation_real_code_norm_firstrepresentation) = 2 * ge_signed_half_norm_firstrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_firstrepresentationreal) = 0) /\ (ge_balance_negative_norm_firstrepresentationreal) = S ge_signed_half_norm_firstrepresentationrealdecode))) /\ ((ee_norm_rp_norm_first) + ge_balance_negative_norm_firstrepresentationreal = (ee_norm_rn_norm_first) + ge_balance_positive_norm_firstrepresentationreal))) /\ (exists ge_balance_positive_norm_firstrepresentationimaginary ge_balance_negative_norm_firstrepresentationimaginary. (((((ge_representation_imaginary_code_norm_firstrepresentation) = 2 * (ge_balance_positive_norm_firstrepresentationimaginary) /\ (ge_balance_negative_norm_firstrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_firstrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_firstrepresentation) = 2 * ge_signed_half_norm_firstrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_firstrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_firstrepresentationimaginary) = S ge_signed_half_norm_firstrepresentationimaginarydecode))) /\ ((ee_norm_ip_norm_first) + ge_balance_negative_norm_firstrepresentationimaginary = (ee_norm_in_norm_first) + ge_balance_positive_norm_firstrepresentationimaginary)))))) /\ (((((((((ee_norm_rp_norm_first) * (ee_norm_rp_norm_first))) + (((ee_norm_rn_norm_first) * (ee_norm_rn_norm_first))))) + (((((ee_norm_ip_norm_first) * (ee_norm_ip_norm_first))) + (((ee_norm_in_norm_first) * (ee_norm_in_norm_first))))))) + (((((ee_norm_rp_norm_first) * (ee_norm_in_norm_first))) + (((ee_norm_rn_norm_first) * (ee_norm_ip_norm_first)))))) = ((((((((((ee_norm_rp_norm_first) * (ee_norm_rn_norm_first))) + (((ee_norm_rn_norm_first) * (ee_norm_rp_norm_first))))) + (((((ee_norm_ip_norm_first) * (ee_norm_in_norm_first))) + (((ee_norm_in_norm_first) * (ee_norm_ip_norm_first))))))) + (((((ee_norm_rp_norm_first) * (ee_norm_ip_norm_first))) + (((ee_norm_rn_norm_first) * (ee_norm_in_norm_first))))))) + (N))))) -> (exists ee_norm_rp_norm_second ee_norm_rn_norm_second ee_norm_ip_norm_second ee_norm_in_norm_second. ((exists ge_representation_real_code_norm_secondrepresentation ge_representation_imaginary_code_norm_secondrepresentation. (((z) = ((ge_representation_real_code_norm_secondrepresentation) + (ge_representation_imaginary_code_norm_secondrepresentation)) * S ((ge_representation_real_code_norm_secondrepresentation) + (ge_representation_imaginary_code_norm_secondrepresentation)) + ((ge_representation_imaginary_code_norm_secondrepresentation) + (ge_representation_imaginary_code_norm_secondrepresentation))) /\ ((exists ge_balance_positive_norm_secondrepresentationreal ge_balance_negative_norm_secondrepresentationreal. (((((ge_representation_real_code_norm_secondrepresentation) = 2 * (ge_balance_positive_norm_secondrepresentationreal) /\ (ge_balance_negative_norm_secondrepresentationreal) = 0) \/ exists ge_signed_half_norm_secondrepresentationrealdecode. (((ge_representation_real_code_norm_secondrepresentation) = 2 * ge_signed_half_norm_secondrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_secondrepresentationreal) = 0) /\ (ge_balance_negative_norm_secondrepresentationreal) = S ge_signed_half_norm_secondrepresentationrealdecode))) /\ ((ee_norm_rp_norm_second) + ge_balance_negative_norm_secondrepresentationreal = (ee_norm_rn_norm_second) + ge_balance_positive_norm_secondrepresentationreal))) /\ (exists ge_balance_positive_norm_secondrepresentationimaginary ge_balance_negative_norm_secondrepresentationimaginary. (((((ge_representation_imaginary_code_norm_secondrepresentation) = 2 * (ge_balance_positive_norm_secondrepresentationimaginary) /\ (ge_balance_negative_norm_secondrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_secondrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_secondrepresentation) = 2 * ge_signed_half_norm_secondrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_secondrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_secondrepresentationimaginary) = S ge_signed_half_norm_secondrepresentationimaginarydecode))) /\ ((ee_norm_ip_norm_second) + ge_balance_negative_norm_secondrepresentationimaginary = (ee_norm_in_norm_second) + ge_balance_positive_norm_secondrepresentationimaginary)))))) /\ (((((((((ee_norm_rp_norm_second) * (ee_norm_rp_norm_second))) + (((ee_norm_rn_norm_second) * (ee_norm_rn_norm_second))))) + (((((ee_norm_ip_norm_second) * (ee_norm_ip_norm_second))) + (((ee_norm_in_norm_second) * (ee_norm_in_norm_second))))))) + (((((ee_norm_rp_norm_second) * (ee_norm_in_norm_second))) + (((ee_norm_rn_norm_second) * (ee_norm_ip_norm_second)))))) = ((((((((((ee_norm_rp_norm_second) * (ee_norm_rn_norm_second))) + (((ee_norm_rn_norm_second) * (ee_norm_rp_norm_second))))) + (((((ee_norm_ip_norm_second) * (ee_norm_in_norm_second))) + (((ee_norm_in_norm_second) * (ee_norm_ip_norm_second))))))) + (((((ee_norm_rp_norm_second) * (ee_norm_ip_norm_second))) + (((ee_norm_rn_norm_second) * (ee_norm_in_norm_second))))))) + (M))))) -> 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)
eisenstein coordinate norm functionaleisenstein 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 eisenstein_coordinate_norm_functional x
  2. L12
    specialize eisenstein_coordinate_norm_functional x1
  3. L13
    specialize eisenstein_coordinate_norm_functional x2
  4. L14
    specialize eisenstein_coordinate_norm_functional x3
  5. L15
    specialize eisenstein_coordinate_norm_functional N
  6. L16
    specialize eisenstein_coordinate_norm_functional M
  7. L17
    apply eisenstein_coordinate_norm_functional
  8. L18
    exact hfirst_witness_witness_witness_witness_right
  9. L19
    specialize eisenstein_norm_for_representation z
  10. L20
    specialize eisenstein_norm_for_representation x
04Use earlier factsL21–27

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

  1. L21
    specialize eisenstein_norm_for_representation x1
  2. L22
    specialize eisenstein_norm_for_representation x2
  3. L23
    specialize eisenstein_norm_for_representation x3
  4. L24
    specialize eisenstein_norm_for_representation M
  5. L25
    apply eisenstein_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 eisenstein_coordinate_norm_functional x
  12. 0012specialize eisenstein_coordinate_norm_functional x1
  13. 0013specialize eisenstein_coordinate_norm_functional x2
  14. 0014specialize eisenstein_coordinate_norm_functional x3
  15. 0015specialize eisenstein_coordinate_norm_functional N
  16. 0016specialize eisenstein_coordinate_norm_functional M
  17. 0017apply eisenstein_coordinate_norm_functional
  18. 0018exact hfirst_witness_witness_witness_witness_right
  19. 0019specialize eisenstein_norm_for_representation z
  20. 0020specialize eisenstein_norm_for_representation x
  21. 0021specialize eisenstein_norm_for_representation x1
  22. 0022specialize eisenstein_norm_for_representation x2
  23. 0023specialize eisenstein_norm_for_representation x3
  24. 0024specialize eisenstein_norm_for_representation M
  25. 0025apply eisenstein_norm_for_representation
  26. 0026exact hfirst_witness_witness_witness_witness_left
  27. 0027exact hsecond