EI0022

eisenstein_product_difference

Actual Eisenstein multiplication distributes over signed subtraction in the second argument.

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. ∀ e. ∀ f. ∀ g. ∀ h. ∀ i. ∀ j. ∀ k. ∀ l. EisensteinCoordinateProduct(a,b,c,d,e + j,f + i,g + l,h + k,a · e + b · f + (c · h + d · g) + (a · j + b · i + (c · k + d · l)),a · f + b · e + (c · g + d · h) + (a · i + b · j + (c · l + d · k)),a · g + b · h + (c · e + d · f) + (c · h + d · g) + (a · l + b · k + (c · j + d · i) + (c · k + d · l)),a · h + b · g + (c · f + d · e) + (c · g + d · h) + (a · k + b · l + (c · i + d · j) + (c · l + d · k)))

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

Definition DAG

EisensteinCoordinateProduct(ap,an,bp,bn,cp,cn,dp,dn,rp,rn,sp,sn) · 1

Actual proof prerequisites

eisenstein_product_difference_real_positiveeisenstein_product_difference_real_negativeeisenstein_product_difference_imaginary_positiveeisenstein_product_difference_imaginary_negative
Original expanded first-order statement 
forall a b c d e f g h i j k l. (((((((((((a) * (((e) + (j))))) + (((b) * (((f) + (i))))))) + (((((c) * (((h) + (k))))) + (((d) * (((g) + (l))))))))) + (((((((((a) * (f))) + (((b) * (e))))) + (((((c) * (g))) + (((d) * (h))))))) + (((((((a) * (i))) + (((b) * (j))))) + (((((c) * (l))) + (((d) * (k)))))))))) = ((((((((((a) * (e))) + (((b) * (f))))) + (((((c) * (h))) + (((d) * (g))))))) + (((((((a) * (j))) + (((b) * (i))))) + (((((c) * (k))) + (((d) * (l))))))))) + (((((((a) * (((f) + (i))))) + (((b) * (((e) + (j))))))) + (((((c) * (((g) + (l))))) + (((d) * (((h) + (k))))))))))) /\ (((((((((((a) * (((g) + (l))))) + (((b) * (((h) + (k))))))) + (((((c) * (((e) + (j))))) + (((d) * (((f) + (i))))))))) + (((((c) * (((h) + (k))))) + (((d) * (((g) + (l))))))))) + (((((((((((a) * (h))) + (((b) * (g))))) + (((((c) * (f))) + (((d) * (e))))))) + (((((c) * (g))) + (((d) * (h))))))) + (((((((((a) * (k))) + (((b) * (l))))) + (((((c) * (i))) + (((d) * (j))))))) + (((((c) * (l))) + (((d) * (k)))))))))) = ((((((((((((a) * (g))) + (((b) * (h))))) + (((((c) * (e))) + (((d) * (f))))))) + (((((c) * (h))) + (((d) * (g))))))) + (((((((((a) * (l))) + (((b) * (k))))) + (((((c) * (j))) + (((d) * (i))))))) + (((((c) * (k))) + (((d) * (l))))))))) + (((((((((a) * (((h) + (k))))) + (((b) * (((g) + (l))))))) + (((((c) * (((f) + (i))))) + (((d) * (((e) + (j))))))))) + (((((c) * (((g) + (l))))) + (((d) * (((h) + (k)))))))))))))

Complete tactic proof in conservative notation

All 21 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

21 script commands · 11 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 (4)
eisenstein product difference real positiveeisenstein product difference real negativeeisenstein product difference imaginary positiveeisenstein product difference imaginary negative
01Fix variables and assumptionsL1–10

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 e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
  9. L9
    intro i
  10. L10
    intro j
02Fix variables and assumptionsL11–12

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

  1. L11
    intro k
  2. L12
    intro l
03Separate the logical casesL13–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L13
    split
04Calculate and transport equalitiesL14–14

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

  1. L14
    congr
05Use earlier factsL15–15

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

  1. L15
    apply eisenstein_product_difference_real_positive
06Calculate and transport equalitiesL16–16

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

  1. L16
    symm
07Use earlier factsL17–17

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

  1. L17
    apply eisenstein_product_difference_real_negative
08Calculate and transport equalitiesL18–18

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

  1. L18
    congr
09Use earlier factsL19–19

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

  1. L19
    apply eisenstein_product_difference_imaginary_positive
10Calculate and transport equalitiesL20–20

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

  1. L20
    symm
11Use earlier factsL21–21

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

  1. L21
    apply eisenstein_product_difference_imaginary_negative

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009intro i
  10. 0010intro j
  11. 0011intro k
  12. 0012intro l
  13. 0013split
  14. 0014congr
  15. 0015apply eisenstein_product_difference_real_positive
  16. 0016symm
  17. 0017apply eisenstein_product_difference_real_negative
  18. 0018congr
  19. 0019apply eisenstein_product_difference_imaginary_positive
  20. 0020symm
  21. 0021apply eisenstein_product_difference_imaginary_negative