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 b c k. ((k = 0 \/ exists pen_last_index_nonempty_list. k = S pen_last_index_nonempty_list /\ ((((exists fs_h_pen_nonempty_list_chain_initial. fs_h_pen_nonempty_list_chain_initial + S (2) = S ((S (0)) * c)) /\ exists fs_q_pen_nonempty_list_chain_initial. b = fs_q_pen_nonempty_list_chain_initial * S ((S (0)) * c) + (2))) /\ forall pen_index_nonempty_list_chain. (exists pc_lt_pen_nonempty_list_chain_bound. pc_lt_pen_nonempty_list_chain_bound + S (pen_index_nonempty_list_chain) = (pen_last_index_nonempty_list)) -> exists pen_previous_nonempty_list_chain pen_following_nonempty_list_chain. (((exists fs_h_pen_nonempty_list_chain_previous. fs_h_pen_nonempty_list_chain_previous + S (pen_previous_nonempty_list_chain) = S ((S (pen_index_nonempty_list_chain)) * c)) /\ exists fs_q_pen_nonempty_list_chain_previous. b = fs_q_pen_nonempty_list_chain_previous * S ((S (pen_index_nonempty_list_chain)) * c) + (pen_previous_nonempty_list_chain))) /\ ((((exists fs_h_pen_nonempty_list_chain_following. fs_h_pen_nonempty_list_chain_following + S (pen_following_nonempty_list_chain) = S ((S (S pen_index_nonempty_list_chain)) * c)) /\ exists fs_q_pen_nonempty_list_chain_following. b = fs_q_pen_nonempty_list_chain_following * S ((S (S pen_index_nonempty_list_chain)) * c) + (pen_following_nonempty_list_chain))) /\ (((~(pen_following_nonempty_list_chain = 1) /\ forall bpr_left_pc_pen_nonempty_list_chain_next_prime bpr_right_pc_pen_nonempty_list_chain_next_prime. pen_following_nonempty_list_chain = bpr_left_pc_pen_nonempty_list_chain_next_prime * bpr_right_pc_pen_nonempty_list_chain_next_prime -> bpr_left_pc_pen_nonempty_list_chain_next_prime = 1 \/ bpr_right_pc_pen_nonempty_list_chain_next_prime = 1)) /\ ((exists pc_lt_pen_nonempty_list_chain_next_greater. pc_lt_pen_nonempty_list_chain_next_greater + S (pen_previous_nonempty_list_chain) = (pen_following_nonempty_list_chain)) /\ forall pen_comparison_nonempty_list_chain_next. ((~(pen_comparison_nonempty_list_chain_next = 1) /\ forall bpr_left_pc_pen_nonempty_list_chain_next_comparison bpr_right_pc_pen_nonempty_list_chain_next_comparison. pen_comparison_nonempty_list_chain_next = bpr_left_pc_pen_nonempty_list_chain_next_comparison * bpr_right_pc_pen_nonempty_list_chain_next_comparison -> bpr_left_pc_pen_nonempty_list_chain_next_comparison = 1 \/ bpr_right_pc_pen_nonempty_list_chain_next_comparison = 1)) -> (exists pc_lt_pen_nonempty_list_chain_next_above. pc_lt_pen_nonempty_list_chain_next_above + S (pen_previous_nonempty_list_chain) = (pen_comparison_nonempty_list_chain_next)) -> (exists pc_le_pen_nonempty_list_chain_next_minimal. pc_le_pen_nonempty_list_chain_next_minimal + (pen_following_nonempty_list_chain) = (pen_comparison_nonempty_list_chain_next)))))))) -> ~(k = 0) -> exists j. k = S j /\ ((((exists fs_h_pen_nonempty_chain_initial. fs_h_pen_nonempty_chain_initial + S (2) = S ((S (0)) * c)) /\ exists fs_q_pen_nonempty_chain_initial. b = fs_q_pen_nonempty_chain_initial * S ((S (0)) * c) + (2))) /\ forall pen_index_nonempty_chain. (exists pc_lt_pen_nonempty_chain_bound. pc_lt_pen_nonempty_chain_bound + S (pen_index_nonempty_chain) = (j)) -> exists pen_previous_nonempty_chain pen_following_nonempty_chain. (((exists fs_h_pen_nonempty_chain_previous. fs_h_pen_nonempty_chain_previous + S (pen_previous_nonempty_chain) = S ((S (pen_index_nonempty_chain)) * c)) /\ exists fs_q_pen_nonempty_chain_previous. b = fs_q_pen_nonempty_chain_previous * S ((S (pen_index_nonempty_chain)) * c) + (pen_previous_nonempty_chain))) /\ ((((exists fs_h_pen_nonempty_chain_following. fs_h_pen_nonempty_chain_following + S (pen_following_nonempty_chain) = S ((S (S pen_index_nonempty_chain)) * c)) /\ exists fs_q_pen_nonempty_chain_following. b = fs_q_pen_nonempty_chain_following * S ((S (S pen_index_nonempty_chain)) * c) + (pen_following_nonempty_chain))) /\ (((~(pen_following_nonempty_chain = 1) /\ forall bpr_left_pc_pen_nonempty_chain_next_prime bpr_right_pc_pen_nonempty_chain_next_prime. pen_following_nonempty_chain = bpr_left_pc_pen_nonempty_chain_next_prime * bpr_right_pc_pen_nonempty_chain_next_prime -> bpr_left_pc_pen_nonempty_chain_next_prime = 1 \/ bpr_right_pc_pen_nonempty_chain_next_prime = 1)) /\ ((exists pc_lt_pen_nonempty_chain_next_greater. pc_lt_pen_nonempty_chain_next_greater + S (pen_previous_nonempty_chain) = (pen_following_nonempty_chain)) /\ forall pen_comparison_nonempty_chain_next. ((~(pen_comparison_nonempty_chain_next = 1) /\ forall bpr_left_pc_pen_nonempty_chain_next_comparison bpr_right_pc_pen_nonempty_chain_next_comparison. pen_comparison_nonempty_chain_next = bpr_left_pc_pen_nonempty_chain_next_comparison * bpr_right_pc_pen_nonempty_chain_next_comparison -> bpr_left_pc_pen_nonempty_chain_next_comparison = 1 \/ bpr_right_pc_pen_nonempty_chain_next_comparison = 1)) -> (exists pc_lt_pen_nonempty_chain_next_above. pc_lt_pen_nonempty_chain_next_above + S (pen_previous_nonempty_chain) = (pen_comparison_nonempty_chain_next)) -> (exists pc_le_pen_nonempty_chain_next_minimal. pc_le_pen_nonempty_chain_next_minimal + (pen_following_nonempty_chain) = (pen_comparison_nonempty_chain_next))))))Constructive proof overview
Generated structural guide
Every positive-length first-prime list exposes its actual final index and minimal-successor chain.
The unchanged tactic script uses 0 declared prerequisites and contains 10 exact native proof lines.
Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
Direct dependents
none
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.