EI0014

eisenstein_weighted_norm_scaled

Scaling both actual integer coordinates multiplies the weighted norm by the exact natural square of the scale.

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

∀ ap. ∀ an. ∀ bp. ∀ bn. ∀ N. ∀ k. WeightedSignedNormThree(ap,an,bp,bn,N) → WeightedSignedNormThree(k · ap,k · an,k · bp,k · bn,k · k · N)

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

Definition DAG

WeightedSignedNormThree(ap,an,bp,bn,n) · 2

Actual proof prerequisites

gaussian_signed_square_scaled · checked external prerequisitemul_add · checked external prerequisitemul_assoc · checked external prerequisitemul_comm · checked external prerequisitenatural_mul_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn N k. (exists ee_real_square_scaled_source ee_imag_square_scaled_source. (((((ap) * (ap))) + (((an) * (an)))) = ((ee_real_square_scaled_source) + (((((ap) * (an))) + (((an) * (ap))))))) /\ ((((((bp) * (bp))) + (((bn) * (bn)))) = ((ee_imag_square_scaled_source) + (((((bp) * (bn))) + (((bn) * (bp))))))) /\ (N) = ee_real_square_scaled_source + 3 * ee_imag_square_scaled_source)) -> (exists ee_real_square_scaled_target ee_imag_square_scaled_target. (((((((k) * (ap))) * (((k) * (ap))))) + (((((k) * (an))) * (((k) * (an)))))) = ((ee_real_square_scaled_target) + (((((((k) * (ap))) * (((k) * (an))))) + (((((k) * (an))) * (((k) * (ap))))))))) /\ ((((((((k) * (bp))) * (((k) * (bp))))) + (((((k) * (bn))) * (((k) * (bn)))))) = ((ee_imag_square_scaled_target) + (((((((k) * (bp))) * (((k) * (bn))))) + (((((k) * (bn))) * (((k) * (bp))))))))) /\ ((k * k) * N) = ee_real_square_scaled_target + 3 * ee_imag_square_scaled_target))

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 · 8 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro N
  6. L6
    intro k
  7. L7
    intro hnorm
02Separate the logical casesL8–11

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

  1. L8
    cases hnorm
  2. L9
    cases hnorm_witness
  3. L10
    cases hnorm_witness_witness
  4. L11
    cases hnorm_witness_witness_right
03Construct an explicit witnessL12–13

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

  1. L12
    exists (k * k) * x
  2. L13
    exists (k * k) * x1
04Separate the logical casesL14–14

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

  1. L14
    split
05Use earlier factsL15–20

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

  1. L15
    specialize gaussian_signed_square_scaled ap
  2. L16
    specialize gaussian_signed_square_scaled an
  3. L17
    specialize gaussian_signed_square_scaled x
  4. L18
    specialize gaussian_signed_square_scaled k
  5. L19
    apply gaussian_signed_square_scaled
  6. L20
    exact hnorm_witness_witness_left
06Separate the logical casesL21–21

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

  1. L21
    split
07Use earlier factsL22–27

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

  1. L22
    specialize gaussian_signed_square_scaled bp
  2. L23
    specialize gaussian_signed_square_scaled bn
  3. L24
    specialize gaussian_signed_square_scaled x1
  4. L25
    specialize gaussian_signed_square_scaled k
  5. L26
    apply gaussian_signed_square_scaled
  6. L27
    exact hnorm_witness_witness_right_left
08Calculate and transport equalitiesL28–29

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L28
    rewrite hnorm_witness_witness_right_right
  2. L29
    simp [mul_add, mul_assoc, mul_comm, natural_mul_swap_right_tail]

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro N
  6. 0006intro k
  7. 0007intro hnorm
  8. 0008cases hnorm
  9. 0009cases hnorm_witness
  10. 0010cases hnorm_witness_witness
  11. 0011cases hnorm_witness_witness_right
  12. 0012exists (k * k) * x
  13. 0013exists (k * k) * x1
  14. 0014split
  15. 0015specialize gaussian_signed_square_scaled ap
  16. 0016specialize gaussian_signed_square_scaled an
  17. 0017specialize gaussian_signed_square_scaled x
  18. 0018specialize gaussian_signed_square_scaled k
  19. 0019apply gaussian_signed_square_scaled
  20. 0020exact hnorm_witness_witness_left
  21. 0021split
  22. 0022specialize gaussian_signed_square_scaled bp
  23. 0023specialize gaussian_signed_square_scaled bn
  24. 0024specialize gaussian_signed_square_scaled x1
  25. 0025specialize gaussian_signed_square_scaled k
  26. 0026apply gaussian_signed_square_scaled
  27. 0027exact hnorm_witness_witness_right_left
  28. 0028rewrite hnorm_witness_witness_right_right
  29. 0029simp [mul_add, mul_assoc, mul_comm, natural_mul_swap_right_tail]