EI0010

eisenstein_weighted_norm_exists

Every genuine signed coordinate pair has a constructed positive-definite weight-three squared 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

∀ ap. ∀ an. ∀ bp. ∀ bn. ∃ N. WeightedSignedNormThree(ap,an,bp,bn,N)

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

Definition DAG

SignedDifferenceSquare(p,n,s) · 2WeightedSignedNormThree(ap,an,bp,bn,n) · 1

Actual proof prerequisites

gaussian_signed_square_exists · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn. exists N. (exists ee_real_square_total ee_imag_square_total. (((((ap) * (ap))) + (((an) * (an)))) = ((ee_real_square_total) + (((((ap) * (an))) + (((an) * (ap))))))) /\ ((((((bp) * (bp))) + (((bn) * (bn)))) = ((ee_imag_square_total) + (((((bp) * (bn))) + (((bn) * (bp))))))) /\ (N) = ee_real_square_total + 3 * ee_imag_square_total))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 11 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 assumptionsL1–4

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
02Establish haL5–8

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

  1. L5
    have ha : ∃ sa. SignedDifferenceSquare(ap,an,sa)Definitions: SignedDifferenceSquare(ap,an,sa)Original native command in the exact edition
  2. L6
    specialize gaussian_signed_square_exists ap
  3. L7
    specialize gaussian_signed_square_exists an
  4. L8
    apply gaussian_signed_square_exists
03Separate the logical casesL9–9

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

  1. L9
    cases ha
04Establish hbL10–13

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

  1. L10
    have hb : ∃ sb. SignedDifferenceSquare(bp,bn,sb)Definitions: SignedDifferenceSquare(bp,bn,sb)Original native command in the exact edition
  2. L11
    specialize gaussian_signed_square_exists bp
  3. L12
    specialize gaussian_signed_square_exists bn
  4. L13
    apply gaussian_signed_square_exists
05Separate the logical casesL14–14

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

  1. L14
    cases hb
06Construct an explicit witnessL15–17

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

  1. L15
    exists x + 3 * x1
  2. L16
    exists x
  3. L17
    exists x1
07Separate the logical casesL18–18

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

  1. L18
    split
08Use earlier factsL19–19

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

  1. L19
    exact ha_witness
09Separate the logical casesL20–20

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

  1. L20
    split
10Use earlier factsL21–21

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

  1. L21
    exact hb_witness
11Calculate and transport equalitiesL22–22

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

  1. L22
    refl

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005have ha : ∃ sa. SignedDifferenceSquare(ap,an,sa)
  6. 0006specialize gaussian_signed_square_exists ap
  7. 0007specialize gaussian_signed_square_exists an
  8. 0008apply gaussian_signed_square_exists
  9. 0009cases ha
  10. 0010have hb : ∃ sb. SignedDifferenceSquare(bp,bn,sb)
  11. 0011specialize gaussian_signed_square_exists bp
  12. 0012specialize gaussian_signed_square_exists bn
  13. 0013apply gaussian_signed_square_exists
  14. 0014cases hb
  15. 0015exists x + 3 * x1
  16. 0016exists x
  17. 0017exists x1
  18. 0018split
  19. 0019exact ha_witness
  20. 0020split
  21. 0021exact hb_witness
  22. 0022refl