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. 3 * a = 2 * a + aStructural proof guide
Left multiplication by three is twice the input plus the input.
Direct prerequisites: mul_comm. The authored body proceeds by equality transport (1).
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
02Calculate and transport equalitiesL2–2
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L2
trans a * 3
03Use earlier factsL3–3
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L3
apply mul_comm
04Calculate and transport equalitiesL4–5
05Use earlier factsL6–6
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L6
apply mul_comm
06Calculate and transport equalitiesL7–7
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L7
refl