EI0029

eisenstein_natural_scalar_product

Multiplication by a natural real Eisenstein scalar is actual coordinatewise scaling.

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

∀ N. ∀ e. ∀ f. ∀ g. ∀ h. EisensteinCoordinateProduct(N,0,0,0,e,f,g,h,N · e,N · f,N · g,N · h)

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

Definition DAG

EisensteinCoordinateProduct(ap,an,bp,bn,cp,cn,dp,dn,rp,rn,sp,sn) · 1

Actual proof prerequisites

mul_zero_left · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall N e f g h. (((((((((((N) * (e))) + (((0) * (f))))) + (((((0) * (h))) + (((0) * (g))))))) + (((N) * (f)))) = ((((N) * (e))) + (((((((N) * (f))) + (((0) * (e))))) + (((((0) * (g))) + (((0) * (h))))))))) /\ (((((((((((N) * (g))) + (((0) * (h))))) + (((((0) * (e))) + (((0) * (f))))))) + (((((0) * (h))) + (((0) * (g))))))) + (((N) * (h)))) = ((((N) * (g))) + (((((((((N) * (h))) + (((0) * (g))))) + (((((0) * (f))) + (((0) * (e))))))) + (((((0) * (g))) + (((0) * (h)))))))))))

Complete tactic proof in conservative notation

All 8 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

8 script commands · 3 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 assumptionsL1–5

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

  1. L1
    intro N
  2. L2
    intro e
  3. L3
    intro f
  4. L4
    intro g
  5. L5
    intro h
02Separate the logical casesL6–6

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

  1. L6
    split
03Calculate and transport equalitiesL7–8

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

  1. L7
    simp [mul_zero_left, zero_add]
  2. L8
    simp [mul_zero_left, zero_add]

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro N
  2. 0002intro e
  3. 0003intro f
  4. 0004intro g
  5. 0005intro h
  6. 0006split
  7. 0007simp [mul_zero_left, zero_add]
  8. 0008simp [mul_zero_left, zero_add]