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. exists q. a = a * qStructural proof guide
Every natural number is a multiple of itself.
Direct prerequisites: mul_one. The authored body proceeds by direct introduction and elimination.
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 assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro a
02Construct an explicit witnessL2–2
Supply the displayed value, then prove that it has the required property.
- L2
exists 1
03Calculate and transport equalitiesL3–3
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L3
symm
04Use earlier factsL4–4
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L4
apply mul_one