EI0009

eisenstein_normal_coordinate_norm_exists

Every normalized signed-coordinate pair has an actual natural norm, in all four sign quadrants.

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

∀ ap. ∀ an. ∀ bp. ∀ bn. ap = 0 ∨ an = 0 → bp = 0 ∨ bn = 0 → ∃ x. EisensteinCoordinateNorm(ap,an,bp,bn,x)

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

Definition DAG

EisensteinCoordinateNorm(ap,an,bp,bn,n) · 1

Actual proof prerequisites

eisenstein_natural_norm_existsmul_zero_left · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn. (ap = 0 \/ an = 0) -> (bp = 0 \/ bn = 0) -> exists n. (((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp)))))) = ((((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))) + (n)))

Complete tactic proof in conservative notation

All 27 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

27 script commands · 15 reading checkpoints · 2 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.

Named ingredients (1)
eisenstein natural norm exists
01Fix variables and assumptionsL1–6

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

  1. L1
    intro ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro ha
  6. L6
    intro hb
02Separate the logical casesL7–8

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

  1. L7
    cases ha
  2. L8
    cases hb
03Establish hnaturalL9–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eisenstein natural norm exists.

  1. L9
    have hnatural : exists n. (((an) * (an)) + ((bn) * (bn))) = (((an) * (bn)) + (n))
  2. L10
    specialize eisenstein_natural_norm_exists an
  3. L11
    specialize eisenstein_natural_norm_exists bn
  4. L12
    apply eisenstein_natural_norm_exists
04Separate the logical casesL13–13

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

  1. L13
    cases hnatural
05Construct an explicit witnessL14–14

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

  1. L14
    exists x
06Calculate and transport equalitiesL15–15

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

  1. L15
    simp [ha_left, hb_left, mul_zero_left, zero_add]
07Construct an explicit witnessL16–16

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

  1. L16
    exists an * an + bp * bp + an * bp
08Calculate and transport equalitiesL17–17

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

  1. L17
    simp [ha_left, hb_right, mul_zero_left, zero_add]
09Separate the logical casesL18–18

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

  1. L18
    cases hb
10Construct an explicit witnessL19–19

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

  1. L19
    exists ap * ap + bn * bn + ap * bn
11Calculate and transport equalitiesL20–20

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

  1. L20
    simp [ha_right, hb_left, mul_zero_left, zero_add]
12Establish hnaturalL21–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eisenstein natural norm exists.

  1. L21
    have hnatural : exists n. (((ap) * (ap)) + ((bp) * (bp))) = (((ap) * (bp)) + (n))
  2. L22
    specialize eisenstein_natural_norm_exists ap
  3. L23
    specialize eisenstein_natural_norm_exists bp
  4. L24
    apply eisenstein_natural_norm_exists
13Separate the logical casesL25–25

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

  1. L25
    cases hnatural
14Construct an explicit witnessL26–26

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

  1. L26
    exists x
15Calculate and transport equalitiesL27–27

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

  1. L27
    simp [ha_right, hb_right, mul_zero_left, zero_add]

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro ha
  6. 0006intro hb
  7. 0007cases ha
  8. 0008cases hb
  9. 0009have hnatural : exists n. (((an) * (an)) + ((bn) * (bn))) = (((an) * (bn)) + (n))
  10. 0010specialize eisenstein_natural_norm_exists an
  11. 0011specialize eisenstein_natural_norm_exists bn
  12. 0012apply eisenstein_natural_norm_exists
  13. 0013cases hnatural
  14. 0014exists x
  15. 0015simp [ha_left, hb_left, mul_zero_left, zero_add]
  16. 0016exists an * an + bp * bp + an * bp
  17. 0017simp [ha_left, hb_right, mul_zero_left, zero_add]
  18. 0018cases hb
  19. 0019exists ap * ap + bn * bn + ap * bn
  20. 0020simp [ha_right, hb_left, mul_zero_left, zero_add]
  21. 0021have hnatural : exists n. (((ap) * (ap)) + ((bp) * (bp))) = (((ap) * (bp)) + (n))
  22. 0022specialize eisenstein_natural_norm_exists ap
  23. 0023specialize eisenstein_natural_norm_exists bp
  24. 0024apply eisenstein_natural_norm_exists
  25. 0025cases hnatural
  26. 0026exists x
  27. 0027simp [ha_right, hb_right, mul_zero_left, zero_add]