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 expanded first-order arithmetic statement
forall p ab ac bb bc rb rc. forall pfs_index_subtract_empty. (exists pfa_gap_subtract_emptyindex. pfa_gap_subtract_emptyindex + S (pfs_index_subtract_empty) = (0)) -> exists pfs_left_subtract_empty pfs_right_subtract_empty pfs_result_subtract_empty. ((((exists ff_h_pfp_subtract_emptyleft. ff_h_pfp_subtract_emptyleft + S (pfs_left_subtract_empty) = S ((S (pfs_index_subtract_empty)) * ac)) /\ exists ff_q_pfp_subtract_emptyleft. ab = ff_q_pfp_subtract_emptyleft * S ((S (pfs_index_subtract_empty)) * ac) + (pfs_left_subtract_empty))) /\ (((((exists ff_h_pfp_subtract_emptyright. ff_h_pfp_subtract_emptyright + S (pfs_right_subtract_empty) = S ((S (pfs_index_subtract_empty)) * bc)) /\ exists ff_q_pfp_subtract_emptyright. bb = ff_q_pfp_subtract_emptyright * S ((S (pfs_index_subtract_empty)) * bc) + (pfs_right_subtract_empty))) /\ (((((exists ff_h_pfp_subtract_emptyresult. ff_h_pfp_subtract_emptyresult + S (pfs_result_subtract_empty) = S ((S (pfs_index_subtract_empty)) * rc)) /\ exists ff_q_pfp_subtract_emptyresult. rb = ff_q_pfp_subtract_emptyresult * S ((S (pfs_index_subtract_empty)) * rc) + (pfs_result_subtract_empty))) /\ ((((exists pfa_gap_subtract_emptyoperationleft. pfa_gap_subtract_emptyoperationleft + S (pfs_right_subtract_empty) = (p)) /\ (((exists pfa_gap_subtract_emptyoperationright. pfa_gap_subtract_emptyoperationright + S (pfs_result_subtract_empty) = (p)) /\ ((((exists pfa_gap_subtract_emptyoperationresultbound. pfa_gap_subtract_emptyoperationresultbound + S (pfs_left_subtract_empty) = (p)) /\ ((exists pfa_offset_left_subtract_emptyoperationresultcongruence pfa_offset_right_subtract_emptyoperationresultcongruence. ((pfs_right_subtract_empty) + (pfs_result_subtract_empty)) + (p) * pfa_offset_left_subtract_emptyoperationresultcongruence = (pfs_left_subtract_empty) + (p) * pfa_offset_right_subtract_emptyoperationresultcongruence)))))))))))))))Constructive proof overview
Generated structural guide
Every pair of empty coefficient prefixes satisfies the operation, including modulus zero and arbitrary encodings.
The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.
Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
lt_not_le Stable theorem; checked-use authorized zero_le Stable theorem; checked-use authorizedDirect 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 assumptionsL1–9
02Separate the logical casesL10–10
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L10
exfalso