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 c a. ~(c = 0) -> exists r. r + S (c * a) = c * S aStructural proof guide
Multiplication by a nonzero left factor strictly increases across a successor step.
Direct prerequisites: add_comm. The authored body proceeds by structural induction (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)
01Induction on cL1–3
02Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
exfalso
03Use earlier factsL5–5
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L5
apply hc
04Calculate and transport equalitiesL6–6
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L6
refl
05Fix variables and assumptionsL7–8
06Construct an explicit witnessL9–9
Supply the displayed value, then prove that it has the required property.
- L9
exists c
07Calculate and transport equalitiesL10–10
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L10
trans S (c + S c * a)
08Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
apply PA4
09Calculate and transport equalitiesL12–13
10Use earlier factsL14–14
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L14
apply add_comm
11Calculate and transport equalitiesL15–16
12Use earlier factsL17–17
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L17
apply PA4
13Calculate and transport equalitiesL18–18
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L18
symm
14Use earlier factsL19–19
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L19
apply PA6