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. (a * c) * (b * d) = (a * d) * (b * c)Constructive proof overview
Generated structural guide
The cross products ac·bd and ad·bc coincide by associativity and commutativity.
The unchanged tactic script uses 2 declared prerequisites and contains 12 exact native proof lines.
dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged
Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.
Proof neighborhood
Direct dependencies
mul_shuffle_four Alpha theorem; checked-use authorized mul_comm Stable theorem; checked-use authorizedDirect dependents
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
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 assumptionsL1–4
02Calculate and transport equalitiesL5–6
03Use earlier factsL7–7
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L7
apply mul_shuffle_four