EI0021

eisenstein_product_difference_imaginary_negative

Exact imaginary negative 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 · (h + k) + b · (g + l) + (c · (f + i) + d · (e + j)) + (c · (g + l) + d · (h + k)) = 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

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

Complete tactic proof in conservative notation

All 80 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

80 script commands · 15 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) * (h))) + ((((a) * (k))) + ((((b) * (g))) + ((((b) * (l))) + ((((c) * (f))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))))))
  2. L14
    simp [add_mul, mul_add, mul_assoc, add_assoc]
  3. L15
    trans ((((a) * (h))) + ((((b) * (g))) + ((((c) * (f))) + ((((d) * (e))) + ((((c) * (g))) + ((((d) * (h))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k))))))))))))))
  4. L16
    congr
  5. L17
    refl
  6. L18
    trans ((((b) * (g))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (f))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))))))
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) * (f))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))))
  4. L23
    trans ((((a) * (k))) + ((((c) * (f))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))))
  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) * (e))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))))
  4. L31
    trans ((((a) * (k))) + ((((d) * (e))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))))
  5. L32
    congr
  6. L33
    refl
  7. L34
    trans ((((b) * (l))) + ((((d) * (e))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  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 equalitiesL40–49

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
    trans ((((c) * (g))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  4. L43
    trans ((((a) * (k))) + ((((c) * (g))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  5. L44
    congr
  6. L45
    refl
  7. L46
    trans ((((b) * (l))) + ((((c) * (g))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))
  8. L47
    congr
  9. L48
    refl
  10. L49
    trans ((((c) * (i))) + ((((c) * (g))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))
10Calculate and transport equalitiesL50–51

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

  1. L50
    congr
  2. L51
    refl
11Use earlier factsL52–55

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

  1. L52
    apply four_square_add_swap_right_tail
  2. L53
    apply four_square_add_swap_right_tail
  3. L54
    apply four_square_add_swap_right_tail
  4. L55
    apply four_square_add_swap_right_tail
12Calculate and transport equalitiesL56–65

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

  1. L56
    congr
  2. L57
    refl
  3. L58
    trans ((((d) * (h))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k)))))))))
  4. L59
    trans ((((a) * (k))) + ((((d) * (h))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k)))))))))
  5. L60
    congr
  6. L61
    refl
  7. L62
    trans ((((b) * (l))) + ((((d) * (h))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k))))))))
  8. L63
    congr
  9. L64
    refl
  10. L65
    trans ((((c) * (i))) + ((((d) * (h))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k)))))))
13Calculate and transport equalitiesL66–70

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

  1. L66
    congr
  2. L67
    refl
  3. L68
    trans ((((d) * (j))) + ((((d) * (h))) + ((((c) * (l))) + (((d) * (k))))))
  4. L69
    congr
  5. L70
    refl
14Use earlier factsL71–75

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

  1. L71
    apply four_square_add_swap_right_tail
  2. L72
    apply four_square_add_swap_right_tail
  3. L73
    apply four_square_add_swap_right_tail
  4. L74
    apply four_square_add_swap_right_tail
  5. L75
    apply four_square_add_swap_right_tail
15Calculate and transport equalitiesL76–80

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

  1. L76
    congr
  2. L77
    refl
  3. L78
    refl
  4. L79
    symm
  5. L80
    simp [add_mul, mul_add, mul_assoc, add_assoc]

Library-wide reading audit

Original defined command ledger · 80 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) * (h))) + ((((a) * (k))) + ((((b) * (g))) + ((((b) * (l))) + ((((c) * (f))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))))))
  14. 0014simp [add_mul, mul_add, mul_assoc, add_assoc]
  15. 0015trans ((((a) * (h))) + ((((b) * (g))) + ((((c) * (f))) + ((((d) * (e))) + ((((c) * (g))) + ((((d) * (h))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k))))))))))))))
  16. 0016congr
  17. 0017refl
  18. 0018trans ((((b) * (g))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (f))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))))))
  19. 0019apply four_square_add_swap_right_tail
  20. 0020congr
  21. 0021refl
  22. 0022trans ((((c) * (f))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))))
  23. 0023trans ((((a) * (k))) + ((((c) * (f))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (e))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))))
  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) * (e))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))))
  31. 0031trans ((((a) * (k))) + ((((d) * (e))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))))
  32. 0032congr
  33. 0033refl
  34. 0034trans ((((b) * (l))) + ((((d) * (e))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  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. 0042trans ((((c) * (g))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  43. 0043trans ((((a) * (k))) + ((((c) * (g))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  44. 0044congr
  45. 0045refl
  46. 0046trans ((((b) * (l))) + ((((c) * (g))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))
  47. 0047congr
  48. 0048refl
  49. 0049trans ((((c) * (i))) + ((((c) * (g))) + ((((d) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))
  50. 0050congr
  51. 0051refl
  52. 0052apply four_square_add_swap_right_tail
  53. 0053apply four_square_add_swap_right_tail
  54. 0054apply four_square_add_swap_right_tail
  55. 0055apply four_square_add_swap_right_tail
  56. 0056congr
  57. 0057refl
  58. 0058trans ((((d) * (h))) + ((((a) * (k))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k)))))))))
  59. 0059trans ((((a) * (k))) + ((((d) * (h))) + ((((b) * (l))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k)))))))))
  60. 0060congr
  61. 0061refl
  62. 0062trans ((((b) * (l))) + ((((d) * (h))) + ((((c) * (i))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k))))))))
  63. 0063congr
  64. 0064refl
  65. 0065trans ((((c) * (i))) + ((((d) * (h))) + ((((d) * (j))) + ((((c) * (l))) + (((d) * (k)))))))
  66. 0066congr
  67. 0067refl
  68. 0068trans ((((d) * (j))) + ((((d) * (h))) + ((((c) * (l))) + (((d) * (k))))))
  69. 0069congr
  70. 0070refl
  71. 0071apply four_square_add_swap_right_tail
  72. 0072apply four_square_add_swap_right_tail
  73. 0073apply four_square_add_swap_right_tail
  74. 0074apply four_square_add_swap_right_tail
  75. 0075apply four_square_add_swap_right_tail
  76. 0076congr
  77. 0077refl
  78. 0078refl
  79. 0079symm
  80. 0080simp [add_mul, mul_add, mul_assoc, add_assoc]