EI001E

eisenstein_product_difference_real_positive

Exact real positive contribution of Eisenstein multiplication distributes over an actual signed difference.

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

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

Definition DAG

none

Actual proof prerequisites

add_mul · checked external prerequisitemul_add · checked external prerequisitemul_assoc · checked external prerequisiteadd_assoc · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisite
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) * (e))) + (((b) * (f))))) + (((((c) * (h))) + (((d) * (g))))))) + (((((((a) * (j))) + (((b) * (i))))) + (((((c) * (k))) + (((d) * (l))))))))

Complete tactic proof in conservative notation

All 44 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

44 script commands · 9 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 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
03Calculate and transport equalitiesL13–18

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

  1. L13
    trans ((((a) * (e))) + ((((a) * (j))) + ((((b) * (f))) + ((((b) * (i))) + ((((c) * (h))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l))))))))))
  2. L14
    simp [add_mul, mul_add, mul_assoc, add_assoc]
  3. L15
    trans ((((a) * (e))) + ((((b) * (f))) + ((((c) * (h))) + ((((d) * (g))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (k))) + (((d) * (l))))))))))
  4. L16
    congr
  5. L17
    refl
  6. L18
    trans ((((b) * (f))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (h))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l)))))))))
04Use earlier factsL19–19

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

  1. L19
    apply four_square_add_swap_right_tail
05Calculate and transport equalitiesL20–25

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

  1. L20
    congr
  2. L21
    refl
  3. L22
    trans ((((c) * (h))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l))))))))
  4. L23
    trans ((((a) * (j))) + ((((c) * (h))) + ((((b) * (i))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l))))))))
  5. L24
    congr
  6. L25
    refl
06Use earlier factsL26–27

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

  1. L26
    apply four_square_add_swap_right_tail
  2. L27
    apply four_square_add_swap_right_tail
07Calculate and transport equalitiesL28–36

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

  1. L28
    congr
  2. L29
    refl
  3. L30
    trans ((((d) * (g))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (k))) + (((d) * (l)))))))
  4. L31
    trans ((((a) * (j))) + ((((d) * (g))) + ((((b) * (i))) + ((((c) * (k))) + (((d) * (l)))))))
  5. L32
    congr
  6. L33
    refl
  7. L34
    trans ((((b) * (i))) + ((((d) * (g))) + ((((c) * (k))) + (((d) * (l))))))
  8. L35
    congr
  9. L36
    refl
08Use earlier factsL37–39

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

  1. L37
    apply four_square_add_swap_right_tail
  2. L38
    apply four_square_add_swap_right_tail
  3. L39
    apply four_square_add_swap_right_tail
09Calculate and transport equalitiesL40–44

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

  1. L40
    congr
  2. L41
    refl
  3. L42
    refl
  4. L43
    symm
  5. L44
    simp [add_mul, mul_add, mul_assoc, add_assoc]

Library-wide reading audit

Original defined command ledger · 44 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. 0013trans ((((a) * (e))) + ((((a) * (j))) + ((((b) * (f))) + ((((b) * (i))) + ((((c) * (h))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l))))))))))
  14. 0014simp [add_mul, mul_add, mul_assoc, add_assoc]
  15. 0015trans ((((a) * (e))) + ((((b) * (f))) + ((((c) * (h))) + ((((d) * (g))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (k))) + (((d) * (l))))))))))
  16. 0016congr
  17. 0017refl
  18. 0018trans ((((b) * (f))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (h))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l)))))))))
  19. 0019apply four_square_add_swap_right_tail
  20. 0020congr
  21. 0021refl
  22. 0022trans ((((c) * (h))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l))))))))
  23. 0023trans ((((a) * (j))) + ((((c) * (h))) + ((((b) * (i))) + ((((c) * (k))) + ((((d) * (g))) + (((d) * (l))))))))
  24. 0024congr
  25. 0025refl
  26. 0026apply four_square_add_swap_right_tail
  27. 0027apply four_square_add_swap_right_tail
  28. 0028congr
  29. 0029refl
  30. 0030trans ((((d) * (g))) + ((((a) * (j))) + ((((b) * (i))) + ((((c) * (k))) + (((d) * (l)))))))
  31. 0031trans ((((a) * (j))) + ((((d) * (g))) + ((((b) * (i))) + ((((c) * (k))) + (((d) * (l)))))))
  32. 0032congr
  33. 0033refl
  34. 0034trans ((((b) * (i))) + ((((d) * (g))) + ((((c) * (k))) + (((d) * (l))))))
  35. 0035congr
  36. 0036refl
  37. 0037apply four_square_add_swap_right_tail
  38. 0038apply four_square_add_swap_right_tail
  39. 0039apply four_square_add_swap_right_tail
  40. 0040congr
  41. 0041refl
  42. 0042refl
  43. 0043symm
  44. 0044simp [add_mul, mul_add, mul_assoc, add_assoc]