EI0011

eisenstein_weighted_norm_functional

The weighted norm is a genuine functional arithmetic relation, independent of chosen square witnesses.

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. ∀ M. WeightedSignedNormThree(ap,an,bp,bn,N) → WeightedSignedNormThree(ap,an,bp,bn,M) → N = M

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_functional · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn N M. (exists ee_real_square_first ee_imag_square_first. (((((ap) * (ap))) + (((an) * (an)))) = ((ee_real_square_first) + (((((ap) * (an))) + (((an) * (ap))))))) /\ ((((((bp) * (bp))) + (((bn) * (bn)))) = ((ee_imag_square_first) + (((((bp) * (bn))) + (((bn) * (bp))))))) /\ (N) = ee_real_square_first + 3 * ee_imag_square_first)) -> (exists ee_real_square_second ee_imag_square_second. (((((ap) * (ap))) + (((an) * (an)))) = ((ee_real_square_second) + (((((ap) * (an))) + (((an) * (ap))))))) /\ ((((((bp) * (bp))) + (((bn) * (bn)))) = ((ee_imag_square_second) + (((((bp) * (bn))) + (((bn) * (bp))))))) /\ (M) = ee_real_square_second + 3 * ee_imag_square_second)) -> N = M

Complete tactic proof in conservative notation

All 38 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

38 script commands · 6 reading checkpoints · 2 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 assumptionsL1–8

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 M
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Separate the logical casesL9–16

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

  1. L9
    cases hfirst
  2. L10
    cases hfirst_witness
  3. L11
    cases hfirst_witness_witness
  4. L12
    cases hfirst_witness_witness_right
  5. L13
    cases hsecond
  6. L14
    cases hsecond_witness
  7. L15
    cases hsecond_witness_witness
  8. L16
    cases hsecond_witness_witness_right
03Establish heqL17–24

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

  1. L17
    have heq : x = x2
  2. L18
    specialize gaussian_signed_square_functional ap
  3. L19
    specialize gaussian_signed_square_functional an
  4. L20
    specialize gaussian_signed_square_functional x
  5. L21
    specialize gaussian_signed_square_functional x2
  6. L22
    apply gaussian_signed_square_functional
  7. L23
    exact hfirst_witness_witness_left
  8. L24
    exact hsecond_witness_witness_left
04Establish heq2L25–34

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

  1. L25
    have heq2 : x1 = x3
  2. L26
    specialize gaussian_signed_square_functional bp
  3. L27
    specialize gaussian_signed_square_functional bn
  4. L28
    specialize gaussian_signed_square_functional x1
  5. L29
    specialize gaussian_signed_square_functional x3
  6. L30
    apply gaussian_signed_square_functional
  7. L31
    exact hfirst_witness_witness_right_left
  8. L32
    exact hsecond_witness_witness_right_left
  9. L33
    trans x + 3 * x1
  10. L34
    exact hfirst_witness_witness_right_right
05Calculate and transport equalitiesL35–37

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

  1. L35
    rewrite heq
  2. L36
    rewrite heq2
  3. L37
    symm
06Use earlier factsL38–38

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

  1. L38
    exact hsecond_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro N
  6. 0006intro M
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009cases hfirst
  10. 0010cases hfirst_witness
  11. 0011cases hfirst_witness_witness
  12. 0012cases hfirst_witness_witness_right
  13. 0013cases hsecond
  14. 0014cases hsecond_witness
  15. 0015cases hsecond_witness_witness
  16. 0016cases hsecond_witness_witness_right
  17. 0017have heq : x = x2
  18. 0018specialize gaussian_signed_square_functional ap
  19. 0019specialize gaussian_signed_square_functional an
  20. 0020specialize gaussian_signed_square_functional x
  21. 0021specialize gaussian_signed_square_functional x2
  22. 0022apply gaussian_signed_square_functional
  23. 0023exact hfirst_witness_witness_left
  24. 0024exact hsecond_witness_witness_left
  25. 0025have heq2 : x1 = x3
  26. 0026specialize gaussian_signed_square_functional bp
  27. 0027specialize gaussian_signed_square_functional bn
  28. 0028specialize gaussian_signed_square_functional x1
  29. 0029specialize gaussian_signed_square_functional x3
  30. 0030apply gaussian_signed_square_functional
  31. 0031exact hfirst_witness_witness_right_left
  32. 0032exact hsecond_witness_witness_right_left
  33. 0033trans x + 3 * x1
  34. 0034exact hfirst_witness_witness_right_right
  35. 0035rewrite heq
  36. 0036rewrite heq2
  37. 0037symm
  38. 0038exact hsecond_witness_witness_right_right