EI0031

eisenstein_coordinate_norm_nonzero

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

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

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 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 16 exact native proof lines.

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

Proof neighborhood

Direct dependencies

EI0030 eisenstein_coordinate_norm_zero

Direct dependents

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

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.

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 exact 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