EI0031

eisenstein_coordinate_norm_nonzero

Every nonzero represented Eisenstein integer has a strictly positive natural norm.

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. ∀ c. ∀ d. ∀ N. EisensteinCoordinateNorm(a,b,c,d,N) → ¬(a = b ∧ c = d) → ¬N = 0

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

Definition DAG

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

Actual proof prerequisites

eisenstein_coordinate_norm_zero
Original expanded first-order statement 
forall a b c d N. (((((((((a) * (a))) + (((b) * (b))))) + (((((c) * (c))) + (((d) * (d))))))) + (((((a) * (d))) + (((b) * (c)))))) = ((((((((((a) * (b))) + (((b) * (a))))) + (((((c) * (d))) + (((d) * (c))))))) + (((((a) * (c))) + (((b) * (d))))))) + (N))) -> ~(a = b /\ c = d) -> ~(N = 0)

Complete tactic proof in conservative notation

All 16 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

16 script commands · 4 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.

Named ingredients (1)
eisenstein coordinate norm zero
01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro N
  6. L6
    intro hnorm
  7. L7
    intro hnonzero
  8. L8
    intro hzero
02Use earlier factsL9–14

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

  1. L9
    apply hnonzero
  2. L10
    specialize eisenstein_coordinate_norm_zero a
  3. L11
    specialize eisenstein_coordinate_norm_zero b
  4. L12
    specialize eisenstein_coordinate_norm_zero c
  5. L13
    specialize eisenstein_coordinate_norm_zero d
  6. L14
    apply eisenstein_coordinate_norm_zero
03Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite hzero at hnorm
04Use earlier factsL16–16

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

  1. L16
    exact hnorm

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro N
  6. 0006intro hnorm
  7. 0007intro hnonzero
  8. 0008intro hzero
  9. 0009apply hnonzero
  10. 0010specialize eisenstein_coordinate_norm_zero a
  11. 0011specialize eisenstein_coordinate_norm_zero b
  12. 0012specialize eisenstein_coordinate_norm_zero c
  13. 0013specialize eisenstein_coordinate_norm_zero d
  14. 0014apply eisenstein_coordinate_norm_zero
  15. 0015rewrite hzero at hnorm
  16. 0016exact hnorm