EI0002

eisenstein_natural_norm_gap_value

Ordering the coordinates gives the explicit subtraction-free norm a²+ad+d².

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

∀ a. ∀ d. ∀ n. a · a + (a + d) · (a + d) = a · (a + d) + n → n = a · a + a · d + d · d

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

Definition DAG

none

Actual proof prerequisites

add_left_cancel · checked external prerequisiteadd_mul · checked external prerequisitemul_add · checked external prerequisitemul_assoc · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitemul_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisitenatural_mul_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall a d n. ((((a) * (a)) + ((a + d) * (a + d))) = (((a) * (a + d)) + (n))) -> n = a * a + a * d + d * d

Complete tactic proof in conservative notation

All 12 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

12 script commands · 5 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro d
  3. L3
    intro n
  4. L4
    intro hnorm
02Use earlier factsL5–8

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

  1. L5
    specialize add_left_cancel (a * (a + d))
  2. L6
    specialize add_left_cancel n
  3. L7
    specialize add_left_cancel (a * a + a * d + d * d)
  4. L8
    apply add_left_cancel
03Calculate and transport equalitiesL9–10

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

  1. L9
    trans a * a + (a + d) * (a + d)
  2. L10
    symm
04Use earlier factsL11–11

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

  1. L11
    exact hnorm
05Calculate and transport equalitiesL12–12

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

  1. L12
    simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro a
  2. 0002intro d
  3. 0003intro n
  4. 0004intro hnorm
  5. 0005specialize add_left_cancel (a * (a + d))
  6. 0006specialize add_left_cancel n
  7. 0007specialize add_left_cancel (a * a + a * d + d * d)
  8. 0008apply add_left_cancel
  9. 0009trans a * a + (a + d) * (a + d)
  10. 0010symm
  11. 0011exact hnorm
  12. 0012simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]