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 a b. (exists k. k + S a = b) -> exists r. r + a = bStructural proof guide
Generated structural guide
A witnessed strict inequality entails the corresponding weak inequality.
Use the direct prerequisites add_succ_left as previously established PA formulas.
The proof proceeds by case analysis (1).
Referenced ingredients
Proof neighborhood
Direct dependencies
Direct dependents
PA001U beta_exclusive_accumulated_product_step PA002G beta_exclusive_recode_congruence_step PA008B prime_mul_index_map_exists_up_to PA008S prime_scaled_inverse_prefix_exists_bounded PA009Q pair_index_left_below_double PA00A6 prime_inverse_prefix_exists_bounded PA00D9 distinct_odd_prime_half_cell_orientedFormal 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 (1)
01Fix variables and assumptionsL1–3
02Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
cases h
03Construct an explicit witnessL5–5
Supply the displayed value, then prove that it has the required property.
- L5
exists S x
04Calculate and transport equalitiesL6–7
05Use earlier factsL8–8
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L8
apply add_succ_left
06Calculate and transport equalitiesL9–9
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L9
symm