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. (exists pmt_odd_n. n = 2 * pmt_odd_n + 1) -> (exists pmt_u_n_one pmt_v_n_one. n + 2 * pmt_u_n_one = 1 + 2 * pmt_v_n_one)Structural proof guide
Generated structural guide
Every odd natural is congruent to one modulo two.
Use the direct prerequisites add_comm as previously established PA formulas.
The proof proceeds by case analysis (1), equality transport (1), certified simplification (2).
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 Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.
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 assumptionsL1–2
02Separate the logical casesL3–3
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L3
cases hodd
03Construct an explicit witnessL4–5
04Calculate and transport equalitiesL6–9
05Use earlier factsL10–10
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L10
apply add_comm
06Calculate and transport equalitiesL11–11
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L11
simp