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 PA statement
forall a b n. (exists q. n = a * q) -> (exists r. a = b * r) -> exists s. n = b * sStructural proof guide
The multiple relation is transitive.
Direct prerequisites: mul_assoc. The authored body proceeds by case analysis (2), equality transport (2).
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing 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.
Named ingredients (1)
01Fix variables and assumptionsL1–5
02Separate the logical casesL6–7
03Construct an explicit witnessL8–8
Supply the displayed value, then prove that it has the required property.
- L8
exists x1 * x
04Calculate and transport equalitiesL9–10
05Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
apply mul_assoc