EI0026

eisenstein_product_associate

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

Actual Eisenstein multiplication is associative on represented integer coordinates.

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

Constructive proof overview

Generated structural guide

Actual Eisenstein multiplication is associative on represented integer coordinates.

The unchanged tactic script uses 2 declared prerequisites and contains 39 exact native proof lines.

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

Proof neighborhood

Direct dependencies

EI001D eisenstein_product_associate_real EI0025 eisenstein_product_associate_imaginary

Direct dependents

EI002A eisenstein_adjoint_product_is_norm_scale EI002E eisenstein_product_shuffle

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

39 script commands · 6 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.

Named ingredients (2)
eisenstein product associate realeisenstein product associate imaginary
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
04Use earlier factsL14–23

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

  1. L14
    specialize eisenstein_product_associate_real a
  2. L15
    specialize eisenstein_product_associate_real b
  3. L16
    specialize eisenstein_product_associate_real c
  4. L17
    specialize eisenstein_product_associate_real d
  5. L18
    specialize eisenstein_product_associate_real e
  6. L19
    specialize eisenstein_product_associate_real f
  7. L20
    specialize eisenstein_product_associate_real g
  8. L21
    specialize eisenstein_product_associate_real h
  9. L22
    specialize eisenstein_product_associate_real i
  10. L23
    specialize eisenstein_product_associate_real j
05Use earlier factsL24–33

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

  1. L24
    specialize eisenstein_product_associate_real k
  2. L25
    specialize eisenstein_product_associate_real l
  3. L26
    apply eisenstein_product_associate_real
  4. L27
    specialize eisenstein_product_associate_imaginary a
  5. L28
    specialize eisenstein_product_associate_imaginary b
  6. L29
    specialize eisenstein_product_associate_imaginary c
  7. L30
    specialize eisenstein_product_associate_imaginary d
  8. L31
    specialize eisenstein_product_associate_imaginary e
  9. L32
    specialize eisenstein_product_associate_imaginary f
  10. L33
    specialize eisenstein_product_associate_imaginary g
06Use earlier factsL34–39

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

  1. L34
    specialize eisenstein_product_associate_imaginary h
  2. L35
    specialize eisenstein_product_associate_imaginary i
  3. L36
    specialize eisenstein_product_associate_imaginary j
  4. L37
    specialize eisenstein_product_associate_imaginary k
  5. L38
    specialize eisenstein_product_associate_imaginary l
  6. L39
    apply eisenstein_product_associate_imaginary

Library-wide reading audit

Original exact command ledger · 39 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. 0014specialize eisenstein_product_associate_real a
  15. 0015specialize eisenstein_product_associate_real b
  16. 0016specialize eisenstein_product_associate_real c
  17. 0017specialize eisenstein_product_associate_real d
  18. 0018specialize eisenstein_product_associate_real e
  19. 0019specialize eisenstein_product_associate_real f
  20. 0020specialize eisenstein_product_associate_real g
  21. 0021specialize eisenstein_product_associate_real h
  22. 0022specialize eisenstein_product_associate_real i
  23. 0023specialize eisenstein_product_associate_real j
  24. 0024specialize eisenstein_product_associate_real k
  25. 0025specialize eisenstein_product_associate_real l
  26. 0026apply eisenstein_product_associate_real
  27. 0027specialize eisenstein_product_associate_imaginary a
  28. 0028specialize eisenstein_product_associate_imaginary b
  29. 0029specialize eisenstein_product_associate_imaginary c
  30. 0030specialize eisenstein_product_associate_imaginary d
  31. 0031specialize eisenstein_product_associate_imaginary e
  32. 0032specialize eisenstein_product_associate_imaginary f
  33. 0033specialize eisenstein_product_associate_imaginary g
  34. 0034specialize eisenstein_product_associate_imaginary h
  35. 0035specialize eisenstein_product_associate_imaginary i
  36. 0036specialize eisenstein_product_associate_imaginary j
  37. 0037specialize eisenstein_product_associate_imaginary k
  38. 0038specialize eisenstein_product_associate_imaginary l
  39. 0039apply eisenstein_product_associate_imaginary