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 b c a n qb qc r. (exists pfs_history_code_remainder_source pfs_history_scale_remainder_source. ((((exists pfa_gap_remainder_sourcetracebase. pfa_gap_remainder_sourcetracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_remainder_sourcetraceinitial. ff_h_pfp_remainder_sourcetraceinitial + S (0) = S ((S (0)) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetraceinitial. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetraceinitial * S ((S (0)) * pfs_history_scale_remainder_source) + (0))) /\ (((((exists ff_h_pfp_remainder_sourcetraceterminal. ff_h_pfp_remainder_sourcetraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetraceterminal. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetraceterminal * S ((S (S (n))) * pfs_history_scale_remainder_source) + (r))) /\ ((forall pfh_index_remainder_sourcetracesteps. (exists pfa_gap_remainder_sourcetracestepsindex. pfa_gap_remainder_sourcetracestepsindex + S (pfh_index_remainder_sourcetracesteps) = (S (n))) -> (exists pfh_coefficient_remainder_sourcetracestepsstep pfh_before_remainder_sourcetracestepsstep pfh_after_remainder_sourcetracestepsstep pfh_product_remainder_sourcetracestepsstep. ((((exists ff_h_pfp_remainder_sourcetracestepsstepcoefficient. ff_h_pfp_remainder_sourcetracestepsstepcoefficient + S (pfh_coefficient_remainder_sourcetracestepsstep) = S ((S (pfh_index_remainder_sourcetracesteps)) * c)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepcoefficient. b = ff_q_pfp_remainder_sourcetracestepsstepcoefficient * S ((S (pfh_index_remainder_sourcetracesteps)) * c) + (pfh_coefficient_remainder_sourcetracestepsstep))) /\ (((((exists ff_h_pfp_remainder_sourcetracestepsstepbefore. ff_h_pfp_remainder_sourcetracestepsstepbefore + S (pfh_before_remainder_sourcetracestepsstep) = S ((S (pfh_index_remainder_sourcetracesteps)) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepbefore. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetracestepsstepbefore * S ((S (pfh_index_remainder_sourcetracesteps)) * pfs_history_scale_remainder_source) + (pfh_before_remainder_sourcetracestepsstep))) /\ (((((exists ff_h_pfp_remainder_sourcetracestepsstepafter. ff_h_pfp_remainder_sourcetracestepsstepafter + S (pfh_after_remainder_sourcetracestepsstep) = S ((S (S (pfh_index_remainder_sourcetracesteps))) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepafter. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetracestepsstepafter * S ((S (S (pfh_index_remainder_sourcetracesteps))) * pfs_history_scale_remainder_source) + (pfh_after_remainder_sourcetracestepsstep))) /\ (((((exists pfa_gap_remainder_sourcetracestepsstepmultiplyleft. pfa_gap_remainder_sourcetracestepsstepmultiplyleft + S (pfh_before_remainder_sourcetracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_sourcetracestepsstepmultiplyright. pfa_gap_remainder_sourcetracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepmultiplyresultbound. pfa_gap_remainder_sourcetracestepsstepmultiplyresultbound + S (pfh_product_remainder_sourcetracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_sourcetracestepsstepmultiplyresultcongruence pfa_offset_right_remainder_sourcetracestepsstepmultiplyresultcongruence. ((pfh_before_remainder_sourcetracestepsstep) * (a)) + (p) * pfa_offset_left_remainder_sourcetracestepsstepmultiplyresultcongruence = (pfh_product_remainder_sourcetracestepsstep) + (p) * pfa_offset_right_remainder_sourcetracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepaddleft. pfa_gap_remainder_sourcetracestepsstepaddleft + S (pfh_product_remainder_sourcetracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_sourcetracestepsstepaddright. pfa_gap_remainder_sourcetracestepsstepaddright + S (pfh_coefficient_remainder_sourcetracestepsstep) = (p)) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepaddresultbound. pfa_gap_remainder_sourcetracestepsstepaddresultbound + S (pfh_after_remainder_sourcetracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_sourcetracestepsstepaddresultcongruence pfa_offset_right_remainder_sourcetracestepsstepaddresultcongruence. ((pfh_product_remainder_sourcetracestepsstep) + (pfh_coefficient_remainder_sourcetracestepsstep)) + (p) * pfa_offset_left_remainder_sourcetracestepsstepaddresultcongruence = (pfh_after_remainder_sourcetracestepsstep) + (p) * pfa_offset_right_remainder_sourcetracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_remainder_sourcequotient ff_source_mcp_pfs_remainder_sourcequotient ff_target_mcp_pfs_remainder_sourcequotient. (exists mcp_gap_pfs_remainder_sourcequotient_bound. mcp_gap_pfs_remainder_sourcequotient_bound + S (ff_index_mcp_pfs_remainder_sourcequotient) = (n)) -> (((exists fs_h_mcp_pfs_remainder_sourcequotient_source. fs_h_mcp_pfs_remainder_sourcequotient_source + S (ff_source_mcp_pfs_remainder_sourcequotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_remainder_sourcequotient)) * pfs_history_scale_remainder_source)) /\ exists fs_q_mcp_pfs_remainder_sourcequotient_source. pfs_history_code_remainder_source = fs_q_mcp_pfs_remainder_sourcequotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_remainder_sourcequotient)) * pfs_history_scale_remainder_source) + (ff_source_mcp_pfs_remainder_sourcequotient))) -> (((exists fs_h_mcp_pfs_remainder_sourcequotient_target. fs_h_mcp_pfs_remainder_sourcequotient_target + S (ff_target_mcp_pfs_remainder_sourcequotient) = S ((S (ff_index_mcp_pfs_remainder_sourcequotient)) * qc)) /\ exists fs_q_mcp_pfs_remainder_sourcequotient_target. qb = fs_q_mcp_pfs_remainder_sourcequotient_target * S ((S (ff_index_mcp_pfs_remainder_sourcequotient)) * qc) + (ff_target_mcp_pfs_remainder_sourcequotient))) -> ff_target_mcp_pfs_remainder_sourcequotient = ff_source_mcp_pfs_remainder_sourcequotient)))) -> (exists pfh_trace_code_remainder_execution pfh_trace_scale_remainder_execution. (((exists pfa_gap_remainder_executiontracebase. pfa_gap_remainder_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_remainder_executiontraceinitial. ff_h_pfp_remainder_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontraceinitial. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontraceinitial * S ((S (0)) * pfh_trace_scale_remainder_execution) + (0))) /\ (((((exists ff_h_pfp_remainder_executiontraceterminal. ff_h_pfp_remainder_executiontraceterminal + S (r) = S ((S (S n)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontraceterminal. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontraceterminal * S ((S (S n)) * pfh_trace_scale_remainder_execution) + (r))) /\ ((forall pfh_index_remainder_executiontracesteps. (exists pfa_gap_remainder_executiontracestepsindex. pfa_gap_remainder_executiontracestepsindex + S (pfh_index_remainder_executiontracesteps) = (S n)) -> (exists pfh_coefficient_remainder_executiontracestepsstep pfh_before_remainder_executiontracestepsstep pfh_after_remainder_executiontracestepsstep pfh_product_remainder_executiontracestepsstep. ((((exists ff_h_pfp_remainder_executiontracestepsstepcoefficient. ff_h_pfp_remainder_executiontracestepsstepcoefficient + S (pfh_coefficient_remainder_executiontracestepsstep) = S ((S (pfh_index_remainder_executiontracesteps)) * c)) /\ exists ff_q_pfp_remainder_executiontracestepsstepcoefficient. b = ff_q_pfp_remainder_executiontracestepsstepcoefficient * S ((S (pfh_index_remainder_executiontracesteps)) * c) + (pfh_coefficient_remainder_executiontracestepsstep))) /\ (((((exists ff_h_pfp_remainder_executiontracestepsstepbefore. ff_h_pfp_remainder_executiontracestepsstepbefore + S (pfh_before_remainder_executiontracestepsstep) = S ((S (pfh_index_remainder_executiontracesteps)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontracestepsstepbefore. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontracestepsstepbefore * S ((S (pfh_index_remainder_executiontracesteps)) * pfh_trace_scale_remainder_execution) + (pfh_before_remainder_executiontracestepsstep))) /\ (((((exists ff_h_pfp_remainder_executiontracestepsstepafter. ff_h_pfp_remainder_executiontracestepsstepafter + S (pfh_after_remainder_executiontracestepsstep) = S ((S (S (pfh_index_remainder_executiontracesteps))) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontracestepsstepafter. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontracestepsstepafter * S ((S (S (pfh_index_remainder_executiontracesteps))) * pfh_trace_scale_remainder_execution) + (pfh_after_remainder_executiontracestepsstep))) /\ (((((exists pfa_gap_remainder_executiontracestepsstepmultiplyleft. pfa_gap_remainder_executiontracestepsstepmultiplyleft + S (pfh_before_remainder_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_executiontracestepsstepmultiplyright. pfa_gap_remainder_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_remainder_executiontracestepsstepmultiplyresultbound. pfa_gap_remainder_executiontracestepsstepmultiplyresultbound + S (pfh_product_remainder_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_remainder_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_remainder_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_remainder_executiontracestepsstepmultiplyresultcongruence = (pfh_product_remainder_executiontracestepsstep) + (p) * pfa_offset_right_remainder_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_remainder_executiontracestepsstepaddleft. pfa_gap_remainder_executiontracestepsstepaddleft + S (pfh_product_remainder_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_executiontracestepsstepaddright. pfa_gap_remainder_executiontracestepsstepaddright + S (pfh_coefficient_remainder_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_remainder_executiontracestepsstepaddresultbound. pfa_gap_remainder_executiontracestepsstepaddresultbound + S (pfh_after_remainder_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_executiontracestepsstepaddresultcongruence pfa_offset_right_remainder_executiontracestepsstepaddresultcongruence. ((pfh_product_remainder_executiontracestepsstep) + (pfh_coefficient_remainder_executiontracestepsstep)) + (p) * pfa_offset_left_remainder_executiontracestepsstepaddresultcongruence = (pfh_after_remainder_executiontracestepsstep) + (p) * pfa_offset_right_remainder_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))Constructive proof overview
Generated structural guide
The synthetic remainder is the actual evaluation of the original input at the divisor root, not an assumed result certificate.
The unchanged tactic script uses 0 declared prerequisites and contains 15 exact native proof lines.
Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
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 assumptionsL1–9
02Separate the logical casesL10–12
03Construct an explicit witnessL13–14
04Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
exact hs_witness_witness_left