EI0003

eisenstein_natural_norm_exists

Every pair of natural coordinates has an actual natural Eisenstein norm, constructed by a decidable order comparison.

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. ∀ b. ∃ n. a · a + b · b = a · b + n

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

Definition DAG

none

Actual proof prerequisites

le_total · 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 b. exists n. ((((a) * (a)) + ((b) * (b))) = (((a) * (b)) + (n)))

Complete tactic proof in conservative notation

All 17 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

17 script commands · 8 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 assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–4

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

  1. L3
    specialize le_total a
  2. L4
    specialize le_total b
03Separate the logical casesL5–6

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

  1. L5
    cases le_total
  2. L6
    cases le_total_left
04Construct an explicit witnessL7–7

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

  1. L7
    exists a * a + a * x + x * x
05Calculate and transport equalitiesL8–11

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

  1. L8
    rewrite <- le_total_left_witness
  2. L9
    rewrite <- le_total_left_witness
  3. L10
    rewrite <- le_total_left_witness
  4. L11
    simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]
06Separate the logical casesL12–12

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

  1. L12
    cases le_total_right
07Construct an explicit witnessL13–13

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

  1. L13
    exists b * b + b * x + x * x
08Calculate and transport equalitiesL14–17

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

  1. L14
    rewrite <- le_total_right_witness
  2. L15
    rewrite <- le_total_right_witness
  3. L16
    rewrite <- le_total_right_witness
  4. L17
    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 · 17 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003specialize le_total a
  4. 0004specialize le_total b
  5. 0005cases le_total
  6. 0006cases le_total_left
  7. 0007exists a * a + a * x + x * x
  8. 0008rewrite <- le_total_left_witness
  9. 0009rewrite <- le_total_left_witness
  10. 0010rewrite <- le_total_left_witness
  11. 0011simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]
  12. 0012cases le_total_right
  13. 0013exists b * b + b * x + x * x
  14. 0014rewrite <- le_total_right_witness
  15. 0015rewrite <- le_total_right_witness
  16. 0016rewrite <- le_total_right_witness
  17. 0017simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]