EI0003

eisenstein_natural_norm_exists

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

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

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 9 declared prerequisites and contains 17 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_total Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized natural_mul_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

EI0009 eisenstein_normal_coordinate_norm_exists 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

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.

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 exact 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]