EI001D — eisenstein_product_associate_real

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))))))))

Constructive proof overview

Generated structural guide

Real Eisenstein associativity follows from checked Gaussian real associativity and associative scalar triple products, with all natural contribution equations explicit.

The unchanged tactic script uses 7 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

EI0019 eisenstein_real_associate_left_positive EI001A eisenstein_real_associate_left_negative EI001B eisenstein_real_associate_right_positive EI001C eisenstein_real_associate_right_negative signed_pair_mul_components_associate Alpha theorem; checked-use authorized gaussian_product_associate_real_positive Alpha theorem; checked-use authorized gaussian_product_associate_real_negative Alpha theorem; checked-use authorized

Direct dependents

EI0025 eisenstein_product_associate_imaginary EI0026 eisenstein_product_associate

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 · 16 reading checkpoints · 1 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 (4)

eisenstein real associate left positive eisenstein real associate left negative eisenstein real associate right positive eisenstein real associate right negative

01Fix variables and assumptionsL1–10

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

L1
intro a
L2
intro b
L3
intro c
L4
intro d
L5
intro e
L6
intro f
L7
intro g
L8
intro h
L9
intro i
L10
intro j

02Fix variables and assumptionsL11–12

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

L11
intro k
L12
intro l

03Establish htailL13–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed pair mul components associate.

L13
have htail : (((((((((c) * (g))) + (((d) * (h))))) * (k))) + (((((((c) * (h))) + (((d) * (g))))) * (l)))) = ((((c) * (((((g) * (k))) + (((h) * (l))))))) + (((d) * (((((g) * (l))) + (((h) * (k))))))))) /\ (((((((((c) * (g))) + (((d) * (h))))) * (l))) + (((((((c) * (h))) + (((d) * (g))))) * (k)))) = ((((c) * (((((g) * (l))) + (((h) * (k))))))) + (((d) * (((((g) * (k))) + (((h) * (l)))))))))
L14
specialize signed_pair_mul_components_associate c
L15
specialize signed_pair_mul_components_associate d
L16
specialize signed_pair_mul_components_associate g
L17
specialize signed_pair_mul_components_associate h
L18
specialize signed_pair_mul_components_associate k
L19
specialize signed_pair_mul_components_associate l
L20
apply signed_pair_mul_components_associate

04Separate the logical casesL21–21

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

L21
cases htail

05Calculate and transport equalitiesL22–23

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

L22
congr
L23
trans ((((((((((((((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))))))) * (l))) + (((((((((a) * (h))) + (((b) * (g))))) + (((((c) * (f))) + (((d) * (e))))))) * (k))))))) + (((((((((c) * (g))) + (((d) * (h))))) * (k))) + (((((((c) * (h))) + (((d) * (g))))) * (l))))))

06Use earlier factsL24–24