EI0007

eisenstein_coordinate_norm_functional

The subtraction-free signed-coordinate norm has a unique natural value.

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. ∀ n. ∀ m. EisensteinCoordinateNorm(ap,an,bp,bn,n) → EisensteinCoordinateNorm(ap,an,bp,bn,m) → n = m

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

Definition DAG

EisensteinCoordinateNorm(ap,an,bp,bn,n) · 2

Actual proof prerequisites

add_left_cancel · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn n m. (((((((((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))) -> (((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp)))))) = ((((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))) + (m))) -> n = m

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.

01Fix variables and assumptionsL1–8

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 n
  6. L6
    intro m
  7. L7
    intro hn
  8. L8
    intro hm
02Use earlier factsL9–12

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

  1. L9
    specialize add_left_cancel ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))
  2. L10
    specialize add_left_cancel n
  3. L11
    specialize add_left_cancel m
  4. L12
    apply add_left_cancel
03Calculate and transport equalitiesL13–14

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

  1. L13
    trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L14
    symm
04Use earlier factsL15–16

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

  1. L15
    exact hn
  2. L16
    exact hm

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro n
  6. 0006intro m
  7. 0007intro hn
  8. 0008intro hm
  9. 0009specialize add_left_cancel ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))
  10. 0010specialize add_left_cancel n
  11. 0011specialize add_left_cancel m
  12. 0012apply add_left_cancel
  13. 0013trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  14. 0014symm
  15. 0015exact hn
  16. 0016exact hm