EI000D

eisenstein_norm_square_balance

Actual natural coordinate squares give the exact signed cross-term balance sa+sb=ENorm+ab.

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. ∀ sa. ∀ sb. ∀ N. SignedDifferenceSquare(ap,an,sa) → SignedDifferenceSquare(bp,bn,sb) → EisensteinCoordinateNorm(ap,an,bp,bn,N) → sa + sb + (ap · bn + an · bp) = ap · bp + an · bn + N

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

Definition DAG

SignedDifferenceSquare(p,n,s) · 2EisensteinCoordinateNorm(ap,an,bp,bn,n) · 1

Actual proof prerequisites

add_right_cancel · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall ap an bp bn sa sb N. (((((ap) * (ap))) + (((an) * (an)))) = ((sa) + (((((ap) * (an))) + (((an) * (ap))))))) -> (((((bp) * (bp))) + (((bn) * (bn)))) = ((sb) + (((((bp) * (bn))) + (((bn) * (bp))))))) -> (((((((((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))) -> (sa + sb) + (((((ap) * (bn))) + (((an) * (bp))))) = (((((ap) * (bp))) + (((an) * (bn))))) + N

Complete tactic proof in conservative notation

All 19 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

19 script commands · 5 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 assumptionsL1–10

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 sa
  6. L6
    intro sb
  7. L7
    intro N
  8. L8
    intro ha
  9. L9
    intro hb
  10. L10
    intro hnorm
02Use earlier factsL11–14

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

  1. L11
    specialize add_right_cancel ((sa + sb) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L12
    specialize add_right_cancel ((((((ap) * (bp))) + (((an) * (bn))))) + N)
  3. L13
    specialize add_right_cancel ((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))
  4. L14
    apply add_right_cancel
03Calculate and transport equalitiesL15–17

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

  1. L15
    trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L16
    simp [ha, hb, add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L17
    trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + N
04Use earlier factsL18–18

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

  1. L18
    exact hnorm
05Calculate and transport equalitiesL19–19

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

  1. L19
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro sa
  6. 0006intro sb
  7. 0007intro N
  8. 0008intro ha
  9. 0009intro hb
  10. 0010intro hnorm
  11. 0011specialize add_right_cancel ((sa + sb) + (((((ap) * (bn))) + (((an) * (bp))))))
  12. 0012specialize add_right_cancel ((((((ap) * (bp))) + (((an) * (bn))))) + N)
  13. 0013specialize add_right_cancel ((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))
  14. 0014apply add_right_cancel
  15. 0015trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  16. 0016simp [ha, hb, add_assoc, add_comm, four_square_add_swap_right_tail]
  17. 0017trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + N
  18. 0018exact hnorm
  19. 0019simp [add_assoc, add_comm, four_square_add_swap_right_tail]