EI0008

eisenstein_coordinate_norm_negation

Negating both genuine signed coordinates leaves the Eisenstein norm unchanged.

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. EisensteinCoordinateNorm(ap,an,bp,bn,n) → EisensteinCoordinateNorm(an,ap,bn,bp,n)

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

Definition DAG

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

Actual proof prerequisites

add_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order statement 
forall ap an bp 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))))))) + (n))) -> (((((((((an) * (an))) + (((ap) * (ap))))) + (((((bn) * (bn))) + (((bp) * (bp))))))) + (((((an) * (bp))) + (((ap) * (bn)))))) = ((((((((((an) * (ap))) + (((ap) * (an))))) + (((((bn) * (bp))) + (((bp) * (bn))))))) + (((((an) * (bn))) + (((ap) * (bp))))))) + (n)))

Complete tactic proof in conservative notation

All 11 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

11 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 assumptionsL1–6

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 n
  6. L6
    intro hnorm
02Calculate and transport equalitiesL7–9

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

  1. L7
    trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L8
    simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]
  3. L9
    trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + n
03Use earlier factsL10–10

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

  1. L10
    exact hnorm
04Calculate and transport equalitiesL11–11

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

  1. L11
    simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro n
  6. 0006intro hnorm
  7. 0007trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  8. 0008simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]
  9. 0009trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + n
  10. 0010exact hnorm
  11. 0011simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]