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 k a z. exists u v. ((a + z) + k * z) + S k * u = a + S k * vStructural proof guide
Generated structural guide
The predecessor of a successor acts as minus one in balanced congruence.
Use the direct prerequisites add_assoc, add_comm, mul_succ_left as previously established PA formulas.
The proof proceeds by equality transport (3).
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.
Named ingredients (3)
01Fix variables and assumptionsL1–3
02Construct an explicit witnessL4–5
03Calculate and transport equalitiesL6–7
04Use earlier factsL8–9
05Calculate and transport equalitiesL10–11
06Use earlier factsL12–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L12
apply add_assoc
07Calculate and transport equalitiesL13–14
08Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
apply add_comm