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 x y z. x * (y * z) = y * (x * z)Constructive proof overview
Generated structural guide
Associativity and commutativity swap the first two factors of a right-associated natural product.
The unchanged tactic script uses 2 declared prerequisites and contains 11 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
Proof neighborhood
Direct dependencies
mul_assoc Stable 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 dependency-curried candidate body does not grant checked theorem use or 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 assumptionsL1–3
02Calculate and transport equalitiesL4–5
03Use earlier factsL6–6
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L6
apply mul_assoc
04Calculate and transport equalitiesL7–8
05Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
apply mul_comm
06Calculate and transport equalitiesL10–10
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L10
refl
07Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
apply mul_assoc