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 n. n = 0 \/ exists k. n = S kStructural proof guide
Generated structural guide
Every natural is either zero or the successor of a natural.
This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.
The proof proceeds by structural induction (1).
Referenced ingredients
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.
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.
01Induction on nL1–1
Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.
- L1
induction n
02Separate the logical casesL2–2
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L2
left
03Calculate and transport equalitiesL3–3
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L3
refl
04Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
right
05Construct an explicit witnessL5–5
Supply the displayed value, then prove that it has the required property.
- L5
exists n
06Calculate and transport equalitiesL6–6
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L6
refl