EI001F

eisenstein_product_difference_real_negative

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic statement

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 5 declared prerequisites and contains 44 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

EI0022 eisenstein_product_difference

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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) * (f))) + ((((a) * (i))) + ((((b) * (e))) + ((((b) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  2. L14
    simp [add_mul, mul_add, mul_assoc, add_assoc]
  3. L15
    trans ((((a) * (f))) + ((((b) * (e))) + ((((c) * (g))) + ((((d) * (h))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (l))) + (((d) * (k))))))))))
  4. L16
    congr
  5. L17
    refl
  6. L18
    trans ((((b) * (e))) + ((((a) * (i))) + ((((b) * (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) * (g))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))
  4. L23
    trans ((((a) * (i))) + ((((c) * (g))) + ((((b) * (j))) + ((((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) * (h))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (l))) + (((d) * (k)))))))
  4. L31
    trans ((((a) * (i))) + ((((d) * (h))) + ((((b) * (j))) + ((((c) * (l))) + (((d) * (k)))))))
  5. L32
    congr
  6. L33
    refl
  7. L34
    trans ((((b) * (j))) + ((((d) * (h))) + ((((c) * (l))) + (((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 equalitiesL40–44

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 exact 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) * (f))) + ((((a) * (i))) + ((((b) * (e))) + ((((b) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))))
  14. 0014simp [add_mul, mul_add, mul_assoc, add_assoc]
  15. 0015trans ((((a) * (f))) + ((((b) * (e))) + ((((c) * (g))) + ((((d) * (h))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (l))) + (((d) * (k))))))))))
  16. 0016congr
  17. 0017refl
  18. 0018trans ((((b) * (e))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (g))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k)))))))))
  19. 0019apply four_square_add_swap_right_tail
  20. 0020congr
  21. 0021refl
  22. 0022trans ((((c) * (g))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (l))) + ((((d) * (h))) + (((d) * (k))))))))
  23. 0023trans ((((a) * (i))) + ((((c) * (g))) + ((((b) * (j))) + ((((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) * (h))) + ((((a) * (i))) + ((((b) * (j))) + ((((c) * (l))) + (((d) * (k)))))))
  31. 0031trans ((((a) * (i))) + ((((d) * (h))) + ((((b) * (j))) + ((((c) * (l))) + (((d) * (k)))))))
  32. 0032congr
  33. 0033refl
  34. 0034trans ((((b) * (j))) + ((((d) * (h))) + ((((c) * (l))) + (((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. 0042refl
  43. 0043symm
  44. 0044simp [add_mul, mul_add, mul_assoc, add_assoc]