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 <= S nStructural proof guide
Generated structural guide
Every natural number is below its successor.
Use the direct prerequisites zero_add, add_succ_left as previously established PA formulas.
The proof proceeds by certified simplification (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.
01Fix variables and assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro n
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
simp [add_succ_left, zero_add]