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))))) = (((((c) * (a))) + (((d) * (b)))))) /\ ((((((a) * (d))) + (((b) * (c))))) = (((((c) * (b))) + (((d) * (a)))))))Constructive proof overview
Generated structural guide
The two actual signed product components are commutative, before any quotient normalization.
The unchanged tactic script uses 2 declared prerequisites and contains 7 exact native proof lines.
Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
mul_comm Stable theorem; checked-use authorized add_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
02Separate the logical casesL5–5
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L5
split