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 first-order arithmetic statement
forall p n qb qc db dc. (((exists ff_h_lmd_initial_source. ff_h_lmd_initial_source + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_initial_source. qb = ff_q_lmd_initial_source * S ((S (0)) * qc) + (n))) -> (((((exists ff_h_lmd_empty_initial. ff_h_lmd_empty_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_empty_initial. qb = ff_q_lmd_empty_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_empty. (exists lmd_gap_empty_index. lmd_gap_empty_index + S (lmd_index_empty) = (0)) -> exists lmd_current_empty lmd_successor_empty lmd_digit_empty. ((((exists ff_h_lmd_empty_current. ff_h_lmd_empty_current + S (lmd_current_empty) = S ((S (lmd_index_empty)) * qc)) /\ exists ff_q_lmd_empty_current. qb = ff_q_lmd_empty_current * S ((S (lmd_index_empty)) * qc) + (lmd_current_empty))) /\ ((((exists ff_h_lmd_empty_successor. ff_h_lmd_empty_successor + S (lmd_successor_empty) = S ((S (S lmd_index_empty)) * qc)) /\ exists ff_q_lmd_empty_successor. qb = ff_q_lmd_empty_successor * S ((S (S lmd_index_empty)) * qc) + (lmd_successor_empty))) /\ ((((exists ff_h_lmd_empty_digit. ff_h_lmd_empty_digit + S (lmd_digit_empty) = S ((S (lmd_index_empty)) * dc)) /\ exists ff_q_lmd_empty_digit. db = ff_q_lmd_empty_digit * S ((S (lmd_index_empty)) * dc) + (lmd_digit_empty))) /\ ((lmd_current_empty = (p) * (lmd_successor_empty) + (lmd_digit_empty)) /\ (exists lmd_gap_empty_digit_bound. lmd_gap_empty_digit_bound + S (lmd_digit_empty) = (p))))))))Constructive proof overview
Generated structural guide
A decoded initial quotient constructively supplies the empty coherent digit chain.
The unchanged tactic script uses 0 declared prerequisites and contains 16 exact native proof lines.
dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged
Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
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 assumptionsL1–7
02Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
split
03Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
exact hinitial
04Fix variables and assumptionsL10–11
05Separate the logical casesL12–13
06Calculate and transport equalitiesL14–14
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L14
rewrite PA4 at hi_witness