EI0032

eisenstein_natural_norm_coordinates

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The natural fundamental-parallelogram norm is exactly the signed-coordinate norm of the same nonnegative integers.

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.

Exact expanded first-order arithmetic statement

forall a b N. ((((a) * (a)) + ((b) * (b))) = (((a) * (b)) + (N))) -> (((((((((a) * (a))) + (((0) * (0))))) + (((((b) * (b))) + (((0) * (0))))))) + (((((a) * (0))) + (((0) * (b)))))) = ((((((((((a) * (0))) + (((0) * (a))))) + (((((b) * (0))) + (((0) * (b))))))) + (((((a) * (b))) + (((0) * (0))))))) + (N)))

Constructive proof overview

Generated structural guide

The natural fundamental-parallelogram norm is exactly the signed-coordinate norm of the same nonnegative integers.

The unchanged tactic script uses 2 declared prerequisites and contains 5 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mul_zero_left Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

EI0033 eisenstein_signed_euclidean_division_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

5 script commands · 2 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro N
  4. L4
    intro hnorm
02Calculate and transport equalitiesL5–5

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

  1. L5
    simp [mul_zero_left, zero_add]

Library-wide reading audit

Original exact command ledger · 5 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro N
  4. 0004intro hnorm
  5. 0005simp [mul_zero_left, zero_add]