Exact research checkpoint receipt

forall p b c qb qc rb rc l. (forall fdp_index_pfp_division. (exists gsp_lt_gap_pfp_division_index_bound. gsp_lt_gap_pfp_division_index_bound + S fdp_index_pfp_division = l) -> exists fdp_value_pfp_division fdp_quotient_pfp_division fdp_remainder_pfp_division. (((exists ff_h_fdp_pfp_division_source. ff_h_fdp_pfp_division_source + S (fdp_value_pfp_division) = S ((S (fdp_index_pfp_division)) * c)) /\ exists ff_q_fdp_pfp_division_source. b = ff_q_fdp_pfp_division_source * S ((S (fdp_index_pfp_division)) * c) + (fdp_value_pfp_division))) /\ ((((exists ff_h_fdp_pfp_division_quotient_entry. ff_h_fdp_pfp_division_quotient_entry + S (fdp_quotient_pfp_division) = S ((S (fdp_index_pfp_division)) * qc)) /\ exists ff_q_fdp_pfp_division_quotient_entry. qb = ff_q_fdp_pfp_division_quotient_entry * S ((S (fdp_index_pfp_division)) * qc) + (fdp_quotient_pfp_division))) /\ ((((exists ff_h_fdp_pfp_division_remainder_entry. ff_h_fdp_pfp_division_remainder_entry + S (fdp_remainder_pfp_division) = S ((S (fdp_index_pfp_division)) * rc)) /\ exists ff_q_fdp_pfp_division_remainder_entry. rb = ff_q_fdp_pfp_division_remainder_entry * S ((S (fdp_index_pfp_division)) * rc) + (fdp_remainder_pfp_division))) /\ (fdp_value_pfp_division = p * fdp_quotient_pfp_division + fdp_remainder_pfp_division /\ (exists gsp_lt_gap_pfp_division_remainder_bound. gsp_lt_gap_pfp_division_remainder_bound + S fdp_remainder_pfp_division = p))))) -> (forall pfp_index_division_result. (exists pfa_gap_division_resultindex. pfa_gap_division_resultindex + S (pfp_index_division_result) = (l)) -> exists pfp_source_division_result pfp_residue_division_result. ((((exists ff_h_pfp_division_resultsource. ff_h_pfp_division_resultsource + S (pfp_source_division_result) = S ((S (pfp_index_division_result)) * c)) /\ exists ff_q_pfp_division_resultsource. b = ff_q_pfp_division_resultsource * S ((S (pfp_index_division_result)) * c) + (pfp_source_division_result))) /\ (((((exists ff_h_pfp_division_resulttarget. ff_h_pfp_division_resulttarget + S (pfp_residue_division_result) = S ((S (pfp_index_division_result)) * rc)) /\ exists ff_q_pfp_division_resulttarget. rb = ff_q_pfp_division_resulttarget * S ((S (pfp_index_division_result)) * rc) + (pfp_residue_division_result))) /\ ((((exists pfa_gap_division_resultresiduebound. pfa_gap_division_resultresiduebound + S (pfp_residue_division_result) = (p)) /\ ((exists pfa_offset_left_division_resultresiduecongruence pfa_offset_right_division_resultresiduecongruence. (pfp_source_division_result) + (p) * pfa_offset_left_division_resultresiduecongruence = (pfp_residue_division_result) + (p) * pfa_offset_right_division_resultresiduecongruence)))))))))

forall p b c l. ~(p=0) -> exists d e. (forall pfp_index_exists. (exists pfa_gap_existsindex. pfa_gap_existsindex + S (pfp_index_exists) = (l)) -> exists pfp_source_exists pfp_residue_exists. ((((exists ff_h_pfp_existssource. ff_h_pfp_existssource + S (pfp_source_exists) = S ((S (pfp_index_exists)) * c)) /\ exists ff_q_pfp_existssource. b = ff_q_pfp_existssource * S ((S (pfp_index_exists)) * c) + (pfp_source_exists))) /\ (((((exists ff_h_pfp_existstarget. ff_h_pfp_existstarget + S (pfp_residue_exists) = S ((S (pfp_index_exists)) * e)) /\ exists ff_q_pfp_existstarget. d = ff_q_pfp_existstarget * S ((S (pfp_index_exists)) * e) + (pfp_residue_exists))) /\ ((((exists pfa_gap_existsresiduebound. pfa_gap_existsresiduebound + S (pfp_residue_exists) = (p)) /\ ((exists pfa_offset_left_existsresiduecongruence pfa_offset_right_existsresiduecongruence. (pfp_source_exists) + (p) * pfa_offset_left_existsresiduecongruence = (pfp_residue_exists) + (p) * pfa_offset_right_existsresiduecongruence)))))))))

forall p b c d e l i a r. (forall pfp_index_entry_table. (exists pfa_gap_entry_tableindex. pfa_gap_entry_tableindex + S (pfp_index_entry_table) = (l)) -> exists pfp_source_entry_table pfp_residue_entry_table. ((((exists ff_h_pfp_entry_tablesource. ff_h_pfp_entry_tablesource + S (pfp_source_entry_table) = S ((S (pfp_index_entry_table)) * c)) /\ exists ff_q_pfp_entry_tablesource. b = ff_q_pfp_entry_tablesource * S ((S (pfp_index_entry_table)) * c) + (pfp_source_entry_table))) /\ (((((exists ff_h_pfp_entry_tabletarget. ff_h_pfp_entry_tabletarget + S (pfp_residue_entry_table) = S ((S (pfp_index_entry_table)) * e)) /\ exists ff_q_pfp_entry_tabletarget. d = ff_q_pfp_entry_tabletarget * S ((S (pfp_index_entry_table)) * e) + (pfp_residue_entry_table))) /\ ((((exists pfa_gap_entry_tableresiduebound. pfa_gap_entry_tableresiduebound + S (pfp_residue_entry_table) = (p)) /\ ((exists pfa_offset_left_entry_tableresiduecongruence pfa_offset_right_entry_tableresiduecongruence. (pfp_source_entry_table) + (p) * pfa_offset_left_entry_tableresiduecongruence = (pfp_residue_entry_table) + (p) * pfa_offset_right_entry_tableresiduecongruence))))))))) -> (exists pfa_gap_entry_index. pfa_gap_entry_index + S (i) = (l)) -> (((exists ff_h_pfp_entry_source. ff_h_pfp_entry_source + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_entry_source. b = ff_q_pfp_entry_source * S ((S (i)) * c) + (a))) -> (((exists ff_h_pfp_entry_target. ff_h_pfp_entry_target + S (r) = S ((S (i)) * e)) /\ exists ff_q_pfp_entry_target. d = ff_q_pfp_entry_target * S ((S (i)) * e) + (r))) -> (((exists pfa_gap_entry_valuebound. pfa_gap_entry_valuebound + S (r) = (p)) /\ ((exists pfa_offset_left_entry_valuecongruence pfa_offset_right_entry_valuecongruence. (a) + (p) * pfa_offset_left_entry_valuecongruence = (r) + (p) * pfa_offset_right_entry_valuecongruence))))

forall p b c d e l. (forall pfp_index_bounded_source. (exists pfa_gap_bounded_sourceindex. pfa_gap_bounded_sourceindex + S (pfp_index_bounded_source) = (l)) -> exists pfp_source_bounded_source pfp_residue_bounded_source. ((((exists ff_h_pfp_bounded_sourcesource. ff_h_pfp_bounded_sourcesource + S (pfp_source_bounded_source) = S ((S (pfp_index_bounded_source)) * c)) /\ exists ff_q_pfp_bounded_sourcesource. b = ff_q_pfp_bounded_sourcesource * S ((S (pfp_index_bounded_source)) * c) + (pfp_source_bounded_source))) /\ (((((exists ff_h_pfp_bounded_sourcetarget. ff_h_pfp_bounded_sourcetarget + S (pfp_residue_bounded_source) = S ((S (pfp_index_bounded_source)) * e)) /\ exists ff_q_pfp_bounded_sourcetarget. d = ff_q_pfp_bounded_sourcetarget * S ((S (pfp_index_bounded_source)) * e) + (pfp_residue_bounded_source))) /\ ((((exists pfa_gap_bounded_sourceresiduebound. pfa_gap_bounded_sourceresiduebound + S (pfp_residue_bounded_source) = (p)) /\ ((exists pfa_offset_left_bounded_sourceresiduecongruence pfa_offset_right_bounded_sourceresiduecongruence. (pfp_source_bounded_source) + (p) * pfa_offset_left_bounded_sourceresiduecongruence = (pfp_residue_bounded_source) + (p) * pfa_offset_right_bounded_sourceresiduecongruence))))))))) -> (forall fom_index_pfp_bounded_result. (exists fom_gap_pfp_bounded_result_index_bound. fom_gap_pfp_bounded_result_index_bound + S (fom_index_pfp_bounded_result) = l) -> exists fom_value_pfp_bounded_result. ((((exists fom_beta_height_pfp_bounded_result_entry. fom_beta_height_pfp_bounded_result_entry + S (fom_value_pfp_bounded_result) = S ((S (fom_index_pfp_bounded_result)) * e)) /\ exists fom_beta_quotient_pfp_bounded_result_entry. d = fom_beta_quotient_pfp_bounded_result_entry * S ((S (fom_index_pfp_bounded_result)) * e) + (fom_value_pfp_bounded_result))) /\ (exists fom_gap_pfp_bounded_result_value_bound. fom_gap_pfp_bounded_result_value_bound + S (fom_value_pfp_bounded_result) = p)))

forall p b c d e f g l. (forall pfp_index_functional_first. (exists pfa_gap_functional_firstindex. pfa_gap_functional_firstindex + S (pfp_index_functional_first) = (l)) -> exists pfp_source_functional_first pfp_residue_functional_first. ((((exists ff_h_pfp_functional_firstsource. ff_h_pfp_functional_firstsource + S (pfp_source_functional_first) = S ((S (pfp_index_functional_first)) * c)) /\ exists ff_q_pfp_functional_firstsource. b = ff_q_pfp_functional_firstsource * S ((S (pfp_index_functional_first)) * c) + (pfp_source_functional_first))) /\ (((((exists ff_h_pfp_functional_firsttarget. ff_h_pfp_functional_firsttarget + S (pfp_residue_functional_first) = S ((S (pfp_index_functional_first)) * e)) /\ exists ff_q_pfp_functional_firsttarget. d = ff_q_pfp_functional_firsttarget * S ((S (pfp_index_functional_first)) * e) + (pfp_residue_functional_first))) /\ ((((exists pfa_gap_functional_firstresiduebound. pfa_gap_functional_firstresiduebound + S (pfp_residue_functional_first) = (p)) /\ ((exists pfa_offset_left_functional_firstresiduecongruence pfa_offset_right_functional_firstresiduecongruence. (pfp_source_functional_first) + (p) * pfa_offset_left_functional_firstresiduecongruence = (pfp_residue_functional_first) + (p) * pfa_offset_right_functional_firstresiduecongruence))))))))) -> (forall pfp_index_functional_second. (exists pfa_gap_functional_secondindex. pfa_gap_functional_secondindex + S (pfp_index_functional_second) = (l)) -> exists pfp_source_functional_second pfp_residue_functional_second. ((((exists ff_h_pfp_functional_secondsource. ff_h_pfp_functional_secondsource + S (pfp_source_functional_second) = S ((S (pfp_index_functional_second)) * c)) /\ exists ff_q_pfp_functional_secondsource. b = ff_q_pfp_functional_secondsource * S ((S (pfp_index_functional_second)) * c) + (pfp_source_functional_second))) /\ (((((exists ff_h_pfp_functional_secondtarget. ff_h_pfp_functional_secondtarget + S (pfp_residue_functional_second) = S ((S (pfp_index_functional_second)) * g)) /\ exists ff_q_pfp_functional_secondtarget. f = ff_q_pfp_functional_secondtarget * S ((S (pfp_index_functional_second)) * g) + (pfp_residue_functional_second))) /\ ((((exists pfa_gap_functional_secondresiduebound. pfa_gap_functional_secondresiduebound + S (pfp_residue_functional_second) = (p)) /\ ((exists pfa_offset_left_functional_secondresiduecongruence pfa_offset_right_functional_secondresiduecongruence. (pfp_source_functional_second) + (p) * pfa_offset_left_functional_secondresiduecongruence = (pfp_residue_functional_second) + (p) * pfa_offset_right_functional_secondresiduecongruence))))))))) -> (forall mdr_i_pfp_functional_result mdr_a_pfp_functional_result. (exists mdr_gap_pfp_functional_resultb. mdr_gap_pfp_functional_resultb + S (mdr_i_pfp_functional_result) = (l)) -> (((exists ff_h_mdr_pfp_functional_resulto. ff_h_mdr_pfp_functional_resulto + S (mdr_a_pfp_functional_result) = S ((S (mdr_i_pfp_functional_result)) * e)) /\ exists ff_q_mdr_pfp_functional_resulto. d = ff_q_mdr_pfp_functional_resulto * S ((S (mdr_i_pfp_functional_result)) * e) + (mdr_a_pfp_functional_result))) -> (((exists ff_h_mdr_pfp_functional_resultn. ff_h_mdr_pfp_functional_resultn + S (mdr_a_pfp_functional_result) = S ((S (mdr_i_pfp_functional_result)) * g)) /\ exists ff_q_mdr_pfp_functional_resultn. f = ff_q_mdr_pfp_functional_resultn * S ((S (mdr_i_pfp_functional_result)) * g) + (mdr_a_pfp_functional_result))))

forall p b c l. (forall fom_index_pfp_reflexive_source. (exists fom_gap_pfp_reflexive_source_index_bound. fom_gap_pfp_reflexive_source_index_bound + S (fom_index_pfp_reflexive_source) = l) -> exists fom_value_pfp_reflexive_source. ((((exists fom_beta_height_pfp_reflexive_source_entry. fom_beta_height_pfp_reflexive_source_entry + S (fom_value_pfp_reflexive_source) = S ((S (fom_index_pfp_reflexive_source)) * c)) /\ exists fom_beta_quotient_pfp_reflexive_source_entry. b = fom_beta_quotient_pfp_reflexive_source_entry * S ((S (fom_index_pfp_reflexive_source)) * c) + (fom_value_pfp_reflexive_source))) /\ (exists fom_gap_pfp_reflexive_source_value_bound. fom_gap_pfp_reflexive_source_value_bound + S (fom_value_pfp_reflexive_source) = p))) -> (forall pfp_index_reflexive_result. (exists pfa_gap_reflexive_resultindex. pfa_gap_reflexive_resultindex + S (pfp_index_reflexive_result) = (l)) -> exists pfp_source_reflexive_result pfp_residue_reflexive_result. ((((exists ff_h_pfp_reflexive_resultsource. ff_h_pfp_reflexive_resultsource + S (pfp_source_reflexive_result) = S ((S (pfp_index_reflexive_result)) * c)) /\ exists ff_q_pfp_reflexive_resultsource. b = ff_q_pfp_reflexive_resultsource * S ((S (pfp_index_reflexive_result)) * c) + (pfp_source_reflexive_result))) /\ (((((exists ff_h_pfp_reflexive_resulttarget. ff_h_pfp_reflexive_resulttarget + S (pfp_residue_reflexive_result) = S ((S (pfp_index_reflexive_result)) * c)) /\ exists ff_q_pfp_reflexive_resulttarget. b = ff_q_pfp_reflexive_resulttarget * S ((S (pfp_index_reflexive_result)) * c) + (pfp_residue_reflexive_result))) /\ ((((exists pfa_gap_reflexive_resultresiduebound. pfa_gap_reflexive_resultresiduebound + S (pfp_residue_reflexive_result) = (p)) /\ ((exists pfa_offset_left_reflexive_resultresiduecongruence pfa_offset_right_reflexive_resultresiduecongruence. (pfp_source_reflexive_result) + (p) * pfa_offset_left_reflexive_resultresiduecongruence = (pfp_residue_reflexive_result) + (p) * pfa_offset_right_reflexive_resultresiduecongruence)))))))))

forall p b c d e B C D E l. (forall mdr_i_pfp_transport_source mdr_a_pfp_transport_source. (exists mdr_gap_pfp_transport_sourceb. mdr_gap_pfp_transport_sourceb + S (mdr_i_pfp_transport_source) = (l)) -> (((exists ff_h_mdr_pfp_transport_sourceo. ff_h_mdr_pfp_transport_sourceo + S (mdr_a_pfp_transport_source) = S ((S (mdr_i_pfp_transport_source)) * c)) /\ exists ff_q_mdr_pfp_transport_sourceo. b = ff_q_mdr_pfp_transport_sourceo * S ((S (mdr_i_pfp_transport_source)) * c) + (mdr_a_pfp_transport_source))) -> (((exists ff_h_mdr_pfp_transport_sourcen. ff_h_mdr_pfp_transport_sourcen + S (mdr_a_pfp_transport_source) = S ((S (mdr_i_pfp_transport_source)) * C)) /\ exists ff_q_mdr_pfp_transport_sourcen. B = ff_q_mdr_pfp_transport_sourcen * S ((S (mdr_i_pfp_transport_source)) * C) + (mdr_a_pfp_transport_source)))) -> (forall mdr_i_pfp_transport_target mdr_a_pfp_transport_target. (exists mdr_gap_pfp_transport_targetb. mdr_gap_pfp_transport_targetb + S (mdr_i_pfp_transport_target) = (l)) -> (((exists ff_h_mdr_pfp_transport_targeto. ff_h_mdr_pfp_transport_targeto + S (mdr_a_pfp_transport_target) = S ((S (mdr_i_pfp_transport_target)) * e)) /\ exists ff_q_mdr_pfp_transport_targeto. d = ff_q_mdr_pfp_transport_targeto * S ((S (mdr_i_pfp_transport_target)) * e) + (mdr_a_pfp_transport_target))) -> (((exists ff_h_mdr_pfp_transport_targetn. ff_h_mdr_pfp_transport_targetn + S (mdr_a_pfp_transport_target) = S ((S (mdr_i_pfp_transport_target)) * E)) /\ exists ff_q_mdr_pfp_transport_targetn. D = ff_q_mdr_pfp_transport_targetn * S ((S (mdr_i_pfp_transport_target)) * E) + (mdr_a_pfp_transport_target)))) -> (forall pfp_index_transport_old. (exists pfa_gap_transport_oldindex. pfa_gap_transport_oldindex + S (pfp_index_transport_old) = (l)) -> exists pfp_source_transport_old pfp_residue_transport_old. ((((exists ff_h_pfp_transport_oldsource. ff_h_pfp_transport_oldsource + S (pfp_source_transport_old) = S ((S (pfp_index_transport_old)) * c)) /\ exists ff_q_pfp_transport_oldsource. b = ff_q_pfp_transport_oldsource * S ((S (pfp_index_transport_old)) * c) + (pfp_source_transport_old))) /\ (((((exists ff_h_pfp_transport_oldtarget. ff_h_pfp_transport_oldtarget + S (pfp_residue_transport_old) = S ((S (pfp_index_transport_old)) * e)) /\ exists ff_q_pfp_transport_oldtarget. d = ff_q_pfp_transport_oldtarget * S ((S (pfp_index_transport_old)) * e) + (pfp_residue_transport_old))) /\ ((((exists pfa_gap_transport_oldresiduebound. pfa_gap_transport_oldresiduebound + S (pfp_residue_transport_old) = (p)) /\ ((exists pfa_offset_left_transport_oldresiduecongruence pfa_offset_right_transport_oldresiduecongruence. (pfp_source_transport_old) + (p) * pfa_offset_left_transport_oldresiduecongruence = (pfp_residue_transport_old) + (p) * pfa_offset_right_transport_oldresiduecongruence))))))))) -> (forall pfp_index_transport_new. (exists pfa_gap_transport_newindex. pfa_gap_transport_newindex + S (pfp_index_transport_new) = (l)) -> exists pfp_source_transport_new pfp_residue_transport_new. ((((exists ff_h_pfp_transport_newsource. ff_h_pfp_transport_newsource + S (pfp_source_transport_new) = S ((S (pfp_index_transport_new)) * C)) /\ exists ff_q_pfp_transport_newsource. B = ff_q_pfp_transport_newsource * S ((S (pfp_index_transport_new)) * C) + (pfp_source_transport_new))) /\ (((((exists ff_h_pfp_transport_newtarget. ff_h_pfp_transport_newtarget + S (pfp_residue_transport_new) = S ((S (pfp_index_transport_new)) * E)) /\ exists ff_q_pfp_transport_newtarget. D = ff_q_pfp_transport_newtarget * S ((S (pfp_index_transport_new)) * E) + (pfp_residue_transport_new))) /\ ((((exists pfa_gap_transport_newresiduebound. pfa_gap_transport_newresiduebound + S (pfp_residue_transport_new) = (p)) /\ ((exists pfa_offset_left_transport_newresiduecongruence pfa_offset_right_transport_newresiduecongruence. (pfp_source_transport_new) + (p) * pfa_offset_left_transport_newresiduecongruence = (pfp_residue_transport_new) + (p) * pfa_offset_right_transport_newresiduecongruence)))))))))

forall p b c d e f g l. (forall pfp_index_idempotent_first. (exists pfa_gap_idempotent_firstindex. pfa_gap_idempotent_firstindex + S (pfp_index_idempotent_first) = (l)) -> exists pfp_source_idempotent_first pfp_residue_idempotent_first. ((((exists ff_h_pfp_idempotent_firstsource. ff_h_pfp_idempotent_firstsource + S (pfp_source_idempotent_first) = S ((S (pfp_index_idempotent_first)) * c)) /\ exists ff_q_pfp_idempotent_firstsource. b = ff_q_pfp_idempotent_firstsource * S ((S (pfp_index_idempotent_first)) * c) + (pfp_source_idempotent_first))) /\ (((((exists ff_h_pfp_idempotent_firsttarget. ff_h_pfp_idempotent_firsttarget + S (pfp_residue_idempotent_first) = S ((S (pfp_index_idempotent_first)) * e)) /\ exists ff_q_pfp_idempotent_firsttarget. d = ff_q_pfp_idempotent_firsttarget * S ((S (pfp_index_idempotent_first)) * e) + (pfp_residue_idempotent_first))) /\ ((((exists pfa_gap_idempotent_firstresiduebound. pfa_gap_idempotent_firstresiduebound + S (pfp_residue_idempotent_first) = (p)) /\ ((exists pfa_offset_left_idempotent_firstresiduecongruence pfa_offset_right_idempotent_firstresiduecongruence. (pfp_source_idempotent_first) + (p) * pfa_offset_left_idempotent_firstresiduecongruence = (pfp_residue_idempotent_first) + (p) * pfa_offset_right_idempotent_firstresiduecongruence))))))))) -> (forall pfp_index_idempotent_second. (exists pfa_gap_idempotent_secondindex. pfa_gap_idempotent_secondindex + S (pfp_index_idempotent_second) = (l)) -> exists pfp_source_idempotent_second pfp_residue_idempotent_second. ((((exists ff_h_pfp_idempotent_secondsource. ff_h_pfp_idempotent_secondsource + S (pfp_source_idempotent_second) = S ((S (pfp_index_idempotent_second)) * e)) /\ exists ff_q_pfp_idempotent_secondsource. d = ff_q_pfp_idempotent_secondsource * S ((S (pfp_index_idempotent_second)) * e) + (pfp_source_idempotent_second))) /\ (((((exists ff_h_pfp_idempotent_secondtarget. ff_h_pfp_idempotent_secondtarget + S (pfp_residue_idempotent_second) = S ((S (pfp_index_idempotent_second)) * g)) /\ exists ff_q_pfp_idempotent_secondtarget. f = ff_q_pfp_idempotent_secondtarget * S ((S (pfp_index_idempotent_second)) * g) + (pfp_residue_idempotent_second))) /\ ((((exists pfa_gap_idempotent_secondresiduebound. pfa_gap_idempotent_secondresiduebound + S (pfp_residue_idempotent_second) = (p)) /\ ((exists pfa_offset_left_idempotent_secondresiduecongruence pfa_offset_right_idempotent_secondresiduecongruence. (pfp_source_idempotent_second) + (p) * pfa_offset_left_idempotent_secondresiduecongruence = (pfp_residue_idempotent_second) + (p) * pfa_offset_right_idempotent_secondresiduecongruence))))))))) -> (forall mdr_i_pfp_idempotent_result mdr_a_pfp_idempotent_result. (exists mdr_gap_pfp_idempotent_resultb. mdr_gap_pfp_idempotent_resultb + S (mdr_i_pfp_idempotent_result) = (l)) -> (((exists ff_h_mdr_pfp_idempotent_resulto. ff_h_mdr_pfp_idempotent_resulto + S (mdr_a_pfp_idempotent_result) = S ((S (mdr_i_pfp_idempotent_result)) * e)) /\ exists ff_q_mdr_pfp_idempotent_resulto. d = ff_q_mdr_pfp_idempotent_resulto * S ((S (mdr_i_pfp_idempotent_result)) * e) + (mdr_a_pfp_idempotent_result))) -> (((exists ff_h_mdr_pfp_idempotent_resultn. ff_h_mdr_pfp_idempotent_resultn + S (mdr_a_pfp_idempotent_result) = S ((S (mdr_i_pfp_idempotent_result)) * g)) /\ exists ff_q_mdr_pfp_idempotent_resultn. f = ff_q_mdr_pfp_idempotent_resultn * S ((S (mdr_i_pfp_idempotent_result)) * g) + (mdr_a_pfp_idempotent_result))))

forall p b c a l. (exists pfa_gap_constant_bound. pfa_gap_constant_bound + S (a) = (p)) -> (forall pfp_repeat_index_constant_repeat. (exists pfa_gap_constant_repeatindex. pfa_gap_constant_repeatindex + S (pfp_repeat_index_constant_repeat) = (l)) -> (((exists ff_h_pfp_constant_repeatentry. ff_h_pfp_constant_repeatentry + S (a) = S ((S (pfp_repeat_index_constant_repeat)) * c)) /\ exists ff_q_pfp_constant_repeatentry. b = ff_q_pfp_constant_repeatentry * S ((S (pfp_repeat_index_constant_repeat)) * c) + (a)))) -> (forall fom_index_pfp_constant_result. (exists fom_gap_pfp_constant_result_index_bound. fom_gap_pfp_constant_result_index_bound + S (fom_index_pfp_constant_result) = l) -> exists fom_value_pfp_constant_result. ((((exists fom_beta_height_pfp_constant_result_entry. fom_beta_height_pfp_constant_result_entry + S (fom_value_pfp_constant_result) = S ((S (fom_index_pfp_constant_result)) * c)) /\ exists fom_beta_quotient_pfp_constant_result_entry. b = fom_beta_quotient_pfp_constant_result_entry * S ((S (fom_index_pfp_constant_result)) * c) + (fom_value_pfp_constant_result))) /\ (exists fom_gap_pfp_constant_result_value_bound. fom_gap_pfp_constant_result_value_bound + S (fom_value_pfp_constant_result) = p)))

forall p a l. (exists pfa_gap_repeat_domain. pfa_gap_repeat_domain + S (a) = (p)) -> exists b c. ((forall fom_index_pfp_repeat_table. (exists fom_gap_pfp_repeat_table_index_bound. fom_gap_pfp_repeat_table_index_bound + S (fom_index_pfp_repeat_table) = l) -> exists fom_value_pfp_repeat_table. ((((exists fom_beta_height_pfp_repeat_table_entry. fom_beta_height_pfp_repeat_table_entry + S (fom_value_pfp_repeat_table) = S ((S (fom_index_pfp_repeat_table)) * c)) /\ exists fom_beta_quotient_pfp_repeat_table_entry. b = fom_beta_quotient_pfp_repeat_table_entry * S ((S (fom_index_pfp_repeat_table)) * c) + (fom_value_pfp_repeat_table))) /\ (exists fom_gap_pfp_repeat_table_value_bound. fom_gap_pfp_repeat_table_value_bound + S (fom_value_pfp_repeat_table) = p))) /\ ((forall pfp_repeat_index_repeat_value. (exists pfa_gap_repeat_valueindex. pfa_gap_repeat_valueindex + S (pfp_repeat_index_repeat_value) = (l)) -> (((exists ff_h_pfp_repeat_valueentry. ff_h_pfp_repeat_valueentry + S (a) = S ((S (pfp_repeat_index_repeat_value)) * c)) /\ exists ff_q_pfp_repeat_valueentry. b = ff_q_pfp_repeat_valueentry * S ((S (pfp_repeat_index_repeat_value)) * c) + (a))))))

forall p l. (~((p) = 1) /\ forall pfa_factor_left_zero_domain pfa_factor_right_zero_domain. (p) = pfa_factor_left_zero_domain * pfa_factor_right_zero_domain -> pfa_factor_left_zero_domain = 1 \/ pfa_factor_right_zero_domain = 1) -> exists b c. ((forall fom_index_pfp_zero_table. (exists fom_gap_pfp_zero_table_index_bound. fom_gap_pfp_zero_table_index_bound + S (fom_index_pfp_zero_table) = l) -> exists fom_value_pfp_zero_table. ((((exists fom_beta_height_pfp_zero_table_entry. fom_beta_height_pfp_zero_table_entry + S (fom_value_pfp_zero_table) = S ((S (fom_index_pfp_zero_table)) * c)) /\ exists fom_beta_quotient_pfp_zero_table_entry. b = fom_beta_quotient_pfp_zero_table_entry * S ((S (fom_index_pfp_zero_table)) * c) + (fom_value_pfp_zero_table))) /\ (exists fom_gap_pfp_zero_table_value_bound. fom_gap_pfp_zero_table_value_bound + S (fom_value_pfp_zero_table) = p))) /\ ((forall pfp_repeat_index_zero_value. (exists pfa_gap_zero_valueindex. pfa_gap_zero_valueindex + S (pfp_repeat_index_zero_value) = (l)) -> (((exists ff_h_pfp_zero_valueentry. ff_h_pfp_zero_valueentry + S (0) = S ((S (pfp_repeat_index_zero_value)) * c)) /\ exists ff_q_pfp_zero_valueentry. b = ff_q_pfp_zero_valueentry * S ((S (pfp_repeat_index_zero_value)) * c) + (0))))))

forall p ab ac bb bc rb rc cb cc l. (forall fom_index_pfp_add_source_a. (exists fom_gap_pfp_add_source_a_index_bound. fom_gap_pfp_add_source_a_index_bound + S (fom_index_pfp_add_source_a) = l) -> exists fom_value_pfp_add_source_a. ((((exists fom_beta_height_pfp_add_source_a_entry. fom_beta_height_pfp_add_source_a_entry + S (fom_value_pfp_add_source_a) = S ((S (fom_index_pfp_add_source_a)) * ac)) /\ exists fom_beta_quotient_pfp_add_source_a_entry. ab = fom_beta_quotient_pfp_add_source_a_entry * S ((S (fom_index_pfp_add_source_a)) * ac) + (fom_value_pfp_add_source_a))) /\ (exists fom_gap_pfp_add_source_a_value_bound. fom_gap_pfp_add_source_a_value_bound + S (fom_value_pfp_add_source_a) = p))) -> (forall fom_index_pfp_add_source_b. (exists fom_gap_pfp_add_source_b_index_bound. fom_gap_pfp_add_source_b_index_bound + S (fom_index_pfp_add_source_b) = l) -> exists fom_value_pfp_add_source_b. ((((exists fom_beta_height_pfp_add_source_b_entry. fom_beta_height_pfp_add_source_b_entry + S (fom_value_pfp_add_source_b) = S ((S (fom_index_pfp_add_source_b)) * bc)) /\ exists fom_beta_quotient_pfp_add_source_b_entry. bb = fom_beta_quotient_pfp_add_source_b_entry * S ((S (fom_index_pfp_add_source_b)) * bc) + (fom_value_pfp_add_source_b))) /\ (exists fom_gap_pfp_add_source_b_value_bound. fom_gap_pfp_add_source_b_value_bound + S (fom_value_pfp_add_source_b) = p))) -> (forall ff_index_mcp_add_pfp_add_raw ff_left_mcp_add_pfp_add_raw ff_right_mcp_add_pfp_add_raw ff_target_mcp_add_pfp_add_raw. (exists mcp_gap_pfp_add_raw_bound. mcp_gap_pfp_add_raw_bound + S (ff_index_mcp_add_pfp_add_raw) = (l)) -> (((exists fs_h_mcp_pfp_add_raw_left. fs_h_mcp_pfp_add_raw_left + S (ff_left_mcp_add_pfp_add_raw) = S ((S (ff_index_mcp_add_pfp_add_raw)) * ac)) /\ exists fs_q_mcp_pfp_add_raw_left. ab = fs_q_mcp_pfp_add_raw_left * S ((S (ff_index_mcp_add_pfp_add_raw)) * ac) + (ff_left_mcp_add_pfp_add_raw))) -> (((exists fs_h_mcp_pfp_add_raw_right. fs_h_mcp_pfp_add_raw_right + S (ff_right_mcp_add_pfp_add_raw) = S ((S (ff_index_mcp_add_pfp_add_raw)) * bc)) /\ exists fs_q_mcp_pfp_add_raw_right. bb = fs_q_mcp_pfp_add_raw_right * S ((S (ff_index_mcp_add_pfp_add_raw)) * bc) + (ff_right_mcp_add_pfp_add_raw))) -> (((exists fs_h_mcp_pfp_add_raw_target. fs_h_mcp_pfp_add_raw_target + S (ff_target_mcp_add_pfp_add_raw) = S ((S (ff_index_mcp_add_pfp_add_raw)) * rc)) /\ exists fs_q_mcp_pfp_add_raw_target. rb = fs_q_mcp_pfp_add_raw_target * S ((S (ff_index_mcp_add_pfp_add_raw)) * rc) + (ff_target_mcp_add_pfp_add_raw))) -> ff_target_mcp_add_pfp_add_raw = ff_left_mcp_add_pfp_add_raw + ff_right_mcp_add_pfp_add_raw) -> (forall pfp_index_add_normalize. (exists pfa_gap_add_normalizeindex. pfa_gap_add_normalizeindex + S (pfp_index_add_normalize) = (l)) -> exists pfp_source_add_normalize pfp_residue_add_normalize. ((((exists ff_h_pfp_add_normalizesource. ff_h_pfp_add_normalizesource + S (pfp_source_add_normalize) = S ((S (pfp_index_add_normalize)) * rc)) /\ exists ff_q_pfp_add_normalizesource. rb = ff_q_pfp_add_normalizesource * S ((S (pfp_index_add_normalize)) * rc) + (pfp_source_add_normalize))) /\ (((((exists ff_h_pfp_add_normalizetarget. ff_h_pfp_add_normalizetarget + S (pfp_residue_add_normalize) = S ((S (pfp_index_add_normalize)) * cc)) /\ exists ff_q_pfp_add_normalizetarget. cb = ff_q_pfp_add_normalizetarget * S ((S (pfp_index_add_normalize)) * cc) + (pfp_residue_add_normalize))) /\ ((((exists pfa_gap_add_normalizeresiduebound. pfa_gap_add_normalizeresiduebound + S (pfp_residue_add_normalize) = (p)) /\ ((exists pfa_offset_left_add_normalizeresiduecongruence pfa_offset_right_add_normalizeresiduecongruence. (pfp_source_add_normalize) + (p) * pfa_offset_left_add_normalizeresiduecongruence = (pfp_residue_add_normalize) + (p) * pfa_offset_right_add_normalizeresiduecongruence))))))))) -> (forall pfp_index_add_result. (exists pfa_gap_add_resultindex. pfa_gap_add_resultindex + S (pfp_index_add_result) = (l)) -> exists pfp_left_add_result pfp_right_add_result pfp_value_add_result. ((((exists ff_h_pfp_add_resultleft. ff_h_pfp_add_resultleft + S (pfp_left_add_result) = S ((S (pfp_index_add_result)) * ac)) /\ exists ff_q_pfp_add_resultleft. ab = ff_q_pfp_add_resultleft * S ((S (pfp_index_add_result)) * ac) + (pfp_left_add_result))) /\ (((((exists ff_h_pfp_add_resultright. ff_h_pfp_add_resultright + S (pfp_right_add_result) = S ((S (pfp_index_add_result)) * bc)) /\ exists ff_q_pfp_add_resultright. bb = ff_q_pfp_add_resultright * S ((S (pfp_index_add_result)) * bc) + (pfp_right_add_result))) /\ (((((exists ff_h_pfp_add_resulttarget. ff_h_pfp_add_resulttarget + S (pfp_value_add_result) = S ((S (pfp_index_add_result)) * cc)) /\ exists ff_q_pfp_add_resulttarget. cb = ff_q_pfp_add_resulttarget * S ((S (pfp_index_add_result)) * cc) + (pfp_value_add_result))) /\ ((((exists pfa_gap_add_resultoperationleft. pfa_gap_add_resultoperationleft + S (pfp_left_add_result) = (p)) /\ (((exists pfa_gap_add_resultoperationright. pfa_gap_add_resultoperationright + S (pfp_right_add_result) = (p)) /\ ((((exists pfa_gap_add_resultoperationresultbound. pfa_gap_add_resultoperationresultbound + S (pfp_value_add_result) = (p)) /\ ((exists pfa_offset_left_add_resultoperationresultcongruence pfa_offset_right_add_resultoperationresultcongruence. ((pfp_left_add_result) + (pfp_right_add_result)) + (p) * pfa_offset_left_add_resultoperationresultcongruence = (pfp_value_add_result) + (p) * pfa_offset_right_add_resultoperationresultcongruence))))))))))))))))

forall p ab ac bb bc l. ~(p=0) -> (forall fom_index_pfp_add_exists_a. (exists fom_gap_pfp_add_exists_a_index_bound. fom_gap_pfp_add_exists_a_index_bound + S (fom_index_pfp_add_exists_a) = l) -> exists fom_value_pfp_add_exists_a. ((((exists fom_beta_height_pfp_add_exists_a_entry. fom_beta_height_pfp_add_exists_a_entry + S (fom_value_pfp_add_exists_a) = S ((S (fom_index_pfp_add_exists_a)) * ac)) /\ exists fom_beta_quotient_pfp_add_exists_a_entry. ab = fom_beta_quotient_pfp_add_exists_a_entry * S ((S (fom_index_pfp_add_exists_a)) * ac) + (fom_value_pfp_add_exists_a))) /\ (exists fom_gap_pfp_add_exists_a_value_bound. fom_gap_pfp_add_exists_a_value_bound + S (fom_value_pfp_add_exists_a) = p))) -> (forall fom_index_pfp_add_exists_b. (exists fom_gap_pfp_add_exists_b_index_bound. fom_gap_pfp_add_exists_b_index_bound + S (fom_index_pfp_add_exists_b) = l) -> exists fom_value_pfp_add_exists_b. ((((exists fom_beta_height_pfp_add_exists_b_entry. fom_beta_height_pfp_add_exists_b_entry + S (fom_value_pfp_add_exists_b) = S ((S (fom_index_pfp_add_exists_b)) * bc)) /\ exists fom_beta_quotient_pfp_add_exists_b_entry. bb = fom_beta_quotient_pfp_add_exists_b_entry * S ((S (fom_index_pfp_add_exists_b)) * bc) + (fom_value_pfp_add_exists_b))) /\ (exists fom_gap_pfp_add_exists_b_value_bound. fom_gap_pfp_add_exists_b_value_bound + S (fom_value_pfp_add_exists_b) = p))) -> exists cb cc. (forall pfp_index_add_exists_result. (exists pfa_gap_add_exists_resultindex. pfa_gap_add_exists_resultindex + S (pfp_index_add_exists_result) = (l)) -> exists pfp_left_add_exists_result pfp_right_add_exists_result pfp_value_add_exists_result. ((((exists ff_h_pfp_add_exists_resultleft. ff_h_pfp_add_exists_resultleft + S (pfp_left_add_exists_result) = S ((S (pfp_index_add_exists_result)) * ac)) /\ exists ff_q_pfp_add_exists_resultleft. ab = ff_q_pfp_add_exists_resultleft * S ((S (pfp_index_add_exists_result)) * ac) + (pfp_left_add_exists_result))) /\ (((((exists ff_h_pfp_add_exists_resultright. ff_h_pfp_add_exists_resultright + S (pfp_right_add_exists_result) = S ((S (pfp_index_add_exists_result)) * bc)) /\ exists ff_q_pfp_add_exists_resultright. bb = ff_q_pfp_add_exists_resultright * S ((S (pfp_index_add_exists_result)) * bc) + (pfp_right_add_exists_result))) /\ (((((exists ff_h_pfp_add_exists_resulttarget. ff_h_pfp_add_exists_resulttarget + S (pfp_value_add_exists_result) = S ((S (pfp_index_add_exists_result)) * cc)) /\ exists ff_q_pfp_add_exists_resulttarget. cb = ff_q_pfp_add_exists_resulttarget * S ((S (pfp_index_add_exists_result)) * cc) + (pfp_value_add_exists_result))) /\ ((((exists pfa_gap_add_exists_resultoperationleft. pfa_gap_add_exists_resultoperationleft + S (pfp_left_add_exists_result) = (p)) /\ (((exists pfa_gap_add_exists_resultoperationright. pfa_gap_add_exists_resultoperationright + S (pfp_right_add_exists_result) = (p)) /\ ((((exists pfa_gap_add_exists_resultoperationresultbound. pfa_gap_add_exists_resultoperationresultbound + S (pfp_value_add_exists_result) = (p)) /\ ((exists pfa_offset_left_add_exists_resultoperationresultcongruence pfa_offset_right_add_exists_resultoperationresultcongruence. ((pfp_left_add_exists_result) + (pfp_right_add_exists_result)) + (p) * pfa_offset_left_add_exists_resultoperationresultcongruence = (pfp_value_add_exists_result) + (p) * pfa_offset_right_add_exists_resultoperationresultcongruence))))))))))))))))

forall p ab ac bb bc cb cc l i a b r. (forall pfp_index_add_entry_table. (exists pfa_gap_add_entry_tableindex. pfa_gap_add_entry_tableindex + S (pfp_index_add_entry_table) = (l)) -> exists pfp_left_add_entry_table pfp_right_add_entry_table pfp_value_add_entry_table. ((((exists ff_h_pfp_add_entry_tableleft. ff_h_pfp_add_entry_tableleft + S (pfp_left_add_entry_table) = S ((S (pfp_index_add_entry_table)) * ac)) /\ exists ff_q_pfp_add_entry_tableleft. ab = ff_q_pfp_add_entry_tableleft * S ((S (pfp_index_add_entry_table)) * ac) + (pfp_left_add_entry_table))) /\ (((((exists ff_h_pfp_add_entry_tableright. ff_h_pfp_add_entry_tableright + S (pfp_right_add_entry_table) = S ((S (pfp_index_add_entry_table)) * bc)) /\ exists ff_q_pfp_add_entry_tableright. bb = ff_q_pfp_add_entry_tableright * S ((S (pfp_index_add_entry_table)) * bc) + (pfp_right_add_entry_table))) /\ (((((exists ff_h_pfp_add_entry_tabletarget. ff_h_pfp_add_entry_tabletarget + S (pfp_value_add_entry_table) = S ((S (pfp_index_add_entry_table)) * cc)) /\ exists ff_q_pfp_add_entry_tabletarget. cb = ff_q_pfp_add_entry_tabletarget * S ((S (pfp_index_add_entry_table)) * cc) + (pfp_value_add_entry_table))) /\ ((((exists pfa_gap_add_entry_tableoperationleft. pfa_gap_add_entry_tableoperationleft + S (pfp_left_add_entry_table) = (p)) /\ (((exists pfa_gap_add_entry_tableoperationright. pfa_gap_add_entry_tableoperationright + S (pfp_right_add_entry_table) = (p)) /\ ((((exists pfa_gap_add_entry_tableoperationresultbound. pfa_gap_add_entry_tableoperationresultbound + S (pfp_value_add_entry_table) = (p)) /\ ((exists pfa_offset_left_add_entry_tableoperationresultcongruence pfa_offset_right_add_entry_tableoperationresultcongruence. ((pfp_left_add_entry_table) + (pfp_right_add_entry_table)) + (p) * pfa_offset_left_add_entry_tableoperationresultcongruence = (pfp_value_add_entry_table) + (p) * pfa_offset_right_add_entry_tableoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_add_entry_index. pfa_gap_add_entry_index + S (i) = (l)) -> (((exists ff_h_pfp_add_entry_input_a. ff_h_pfp_add_entry_input_a + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_add_entry_input_a. ab = ff_q_pfp_add_entry_input_a * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_add_entry_input_b. ff_h_pfp_add_entry_input_b + S (b) = S ((S (i)) * bc)) /\ exists ff_q_pfp_add_entry_input_b. bb = ff_q_pfp_add_entry_input_b * S ((S (i)) * bc) + (b))) -> (((exists ff_h_pfp_add_entry_output. ff_h_pfp_add_entry_output + S (r) = S ((S (i)) * cc)) /\ exists ff_q_pfp_add_entry_output. cb = ff_q_pfp_add_entry_output * S ((S (i)) * cc) + (r))) -> (((exists pfa_gap_add_entry_valueleft. pfa_gap_add_entry_valueleft + S (a) = (p)) /\ (((exists pfa_gap_add_entry_valueright. pfa_gap_add_entry_valueright + S (b) = (p)) /\ ((((exists pfa_gap_add_entry_valueresultbound. pfa_gap_add_entry_valueresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_add_entry_valueresultcongruence pfa_offset_right_add_entry_valueresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_add_entry_valueresultcongruence = (r) + (p) * pfa_offset_right_add_entry_valueresultcongruence)))))))))

forall p ab ac bb bc cb cc l. (forall pfp_index_add_bounded_table. (exists pfa_gap_add_bounded_tableindex. pfa_gap_add_bounded_tableindex + S (pfp_index_add_bounded_table) = (l)) -> exists pfp_left_add_bounded_table pfp_right_add_bounded_table pfp_value_add_bounded_table. ((((exists ff_h_pfp_add_bounded_tableleft. ff_h_pfp_add_bounded_tableleft + S (pfp_left_add_bounded_table) = S ((S (pfp_index_add_bounded_table)) * ac)) /\ exists ff_q_pfp_add_bounded_tableleft. ab = ff_q_pfp_add_bounded_tableleft * S ((S (pfp_index_add_bounded_table)) * ac) + (pfp_left_add_bounded_table))) /\ (((((exists ff_h_pfp_add_bounded_tableright. ff_h_pfp_add_bounded_tableright + S (pfp_right_add_bounded_table) = S ((S (pfp_index_add_bounded_table)) * bc)) /\ exists ff_q_pfp_add_bounded_tableright. bb = ff_q_pfp_add_bounded_tableright * S ((S (pfp_index_add_bounded_table)) * bc) + (pfp_right_add_bounded_table))) /\ (((((exists ff_h_pfp_add_bounded_tabletarget. ff_h_pfp_add_bounded_tabletarget + S (pfp_value_add_bounded_table) = S ((S (pfp_index_add_bounded_table)) * cc)) /\ exists ff_q_pfp_add_bounded_tabletarget. cb = ff_q_pfp_add_bounded_tabletarget * S ((S (pfp_index_add_bounded_table)) * cc) + (pfp_value_add_bounded_table))) /\ ((((exists pfa_gap_add_bounded_tableoperationleft. pfa_gap_add_bounded_tableoperationleft + S (pfp_left_add_bounded_table) = (p)) /\ (((exists pfa_gap_add_bounded_tableoperationright. pfa_gap_add_bounded_tableoperationright + S (pfp_right_add_bounded_table) = (p)) /\ ((((exists pfa_gap_add_bounded_tableoperationresultbound. pfa_gap_add_bounded_tableoperationresultbound + S (pfp_value_add_bounded_table) = (p)) /\ ((exists pfa_offset_left_add_bounded_tableoperationresultcongruence pfa_offset_right_add_bounded_tableoperationresultcongruence. ((pfp_left_add_bounded_table) + (pfp_right_add_bounded_table)) + (p) * pfa_offset_left_add_bounded_tableoperationresultcongruence = (pfp_value_add_bounded_table) + (p) * pfa_offset_right_add_bounded_tableoperationresultcongruence)))))))))))))))) -> ((forall fom_index_pfp_add_bounded_ab. (exists fom_gap_pfp_add_bounded_ab_index_bound. fom_gap_pfp_add_bounded_ab_index_bound + S (fom_index_pfp_add_bounded_ab) = l) -> exists fom_value_pfp_add_bounded_ab. ((((exists fom_beta_height_pfp_add_bounded_ab_entry. fom_beta_height_pfp_add_bounded_ab_entry + S (fom_value_pfp_add_bounded_ab) = S ((S (fom_index_pfp_add_bounded_ab)) * ac)) /\ exists fom_beta_quotient_pfp_add_bounded_ab_entry. ab = fom_beta_quotient_pfp_add_bounded_ab_entry * S ((S (fom_index_pfp_add_bounded_ab)) * ac) + (fom_value_pfp_add_bounded_ab))) /\ (exists fom_gap_pfp_add_bounded_ab_value_bound. fom_gap_pfp_add_bounded_ab_value_bound + S (fom_value_pfp_add_bounded_ab) = p))) /\ (((forall fom_index_pfp_add_bounded_bb. (exists fom_gap_pfp_add_bounded_bb_index_bound. fom_gap_pfp_add_bounded_bb_index_bound + S (fom_index_pfp_add_bounded_bb) = l) -> exists fom_value_pfp_add_bounded_bb. ((((exists fom_beta_height_pfp_add_bounded_bb_entry. fom_beta_height_pfp_add_bounded_bb_entry + S (fom_value_pfp_add_bounded_bb) = S ((S (fom_index_pfp_add_bounded_bb)) * bc)) /\ exists fom_beta_quotient_pfp_add_bounded_bb_entry. bb = fom_beta_quotient_pfp_add_bounded_bb_entry * S ((S (fom_index_pfp_add_bounded_bb)) * bc) + (fom_value_pfp_add_bounded_bb))) /\ (exists fom_gap_pfp_add_bounded_bb_value_bound. fom_gap_pfp_add_bounded_bb_value_bound + S (fom_value_pfp_add_bounded_bb) = p))) /\ ((forall fom_index_pfp_add_bounded_cb. (exists fom_gap_pfp_add_bounded_cb_index_bound. fom_gap_pfp_add_bounded_cb_index_bound + S (fom_index_pfp_add_bounded_cb) = l) -> exists fom_value_pfp_add_bounded_cb. ((((exists fom_beta_height_pfp_add_bounded_cb_entry. fom_beta_height_pfp_add_bounded_cb_entry + S (fom_value_pfp_add_bounded_cb) = S ((S (fom_index_pfp_add_bounded_cb)) * cc)) /\ exists fom_beta_quotient_pfp_add_bounded_cb_entry. cb = fom_beta_quotient_pfp_add_bounded_cb_entry * S ((S (fom_index_pfp_add_bounded_cb)) * cc) + (fom_value_pfp_add_bounded_cb))) /\ (exists fom_gap_pfp_add_bounded_cb_value_bound. fom_gap_pfp_add_bounded_cb_value_bound + S (fom_value_pfp_add_bounded_cb) = p)))))))

forall p ab ac bb bc cb cc db dc l. (forall pfp_index_add_functional_first. (exists pfa_gap_add_functional_firstindex. pfa_gap_add_functional_firstindex + S (pfp_index_add_functional_first) = (l)) -> exists pfp_left_add_functional_first pfp_right_add_functional_first pfp_value_add_functional_first. ((((exists ff_h_pfp_add_functional_firstleft. ff_h_pfp_add_functional_firstleft + S (pfp_left_add_functional_first) = S ((S (pfp_index_add_functional_first)) * ac)) /\ exists ff_q_pfp_add_functional_firstleft. ab = ff_q_pfp_add_functional_firstleft * S ((S (pfp_index_add_functional_first)) * ac) + (pfp_left_add_functional_first))) /\ (((((exists ff_h_pfp_add_functional_firstright. ff_h_pfp_add_functional_firstright + S (pfp_right_add_functional_first) = S ((S (pfp_index_add_functional_first)) * bc)) /\ exists ff_q_pfp_add_functional_firstright. bb = ff_q_pfp_add_functional_firstright * S ((S (pfp_index_add_functional_first)) * bc) + (pfp_right_add_functional_first))) /\ (((((exists ff_h_pfp_add_functional_firsttarget. ff_h_pfp_add_functional_firsttarget + S (pfp_value_add_functional_first) = S ((S (pfp_index_add_functional_first)) * cc)) /\ exists ff_q_pfp_add_functional_firsttarget. cb = ff_q_pfp_add_functional_firsttarget * S ((S (pfp_index_add_functional_first)) * cc) + (pfp_value_add_functional_first))) /\ ((((exists pfa_gap_add_functional_firstoperationleft. pfa_gap_add_functional_firstoperationleft + S (pfp_left_add_functional_first) = (p)) /\ (((exists pfa_gap_add_functional_firstoperationright. pfa_gap_add_functional_firstoperationright + S (pfp_right_add_functional_first) = (p)) /\ ((((exists pfa_gap_add_functional_firstoperationresultbound. pfa_gap_add_functional_firstoperationresultbound + S (pfp_value_add_functional_first) = (p)) /\ ((exists pfa_offset_left_add_functional_firstoperationresultcongruence pfa_offset_right_add_functional_firstoperationresultcongruence. ((pfp_left_add_functional_first) + (pfp_right_add_functional_first)) + (p) * pfa_offset_left_add_functional_firstoperationresultcongruence = (pfp_value_add_functional_first) + (p) * pfa_offset_right_add_functional_firstoperationresultcongruence)))))))))))))))) -> (forall pfp_index_add_functional_second. (exists pfa_gap_add_functional_secondindex. pfa_gap_add_functional_secondindex + S (pfp_index_add_functional_second) = (l)) -> exists pfp_left_add_functional_second pfp_right_add_functional_second pfp_value_add_functional_second. ((((exists ff_h_pfp_add_functional_secondleft. ff_h_pfp_add_functional_secondleft + S (pfp_left_add_functional_second) = S ((S (pfp_index_add_functional_second)) * ac)) /\ exists ff_q_pfp_add_functional_secondleft. ab = ff_q_pfp_add_functional_secondleft * S ((S (pfp_index_add_functional_second)) * ac) + (pfp_left_add_functional_second))) /\ (((((exists ff_h_pfp_add_functional_secondright. ff_h_pfp_add_functional_secondright + S (pfp_right_add_functional_second) = S ((S (pfp_index_add_functional_second)) * bc)) /\ exists ff_q_pfp_add_functional_secondright. bb = ff_q_pfp_add_functional_secondright * S ((S (pfp_index_add_functional_second)) * bc) + (pfp_right_add_functional_second))) /\ (((((exists ff_h_pfp_add_functional_secondtarget. ff_h_pfp_add_functional_secondtarget + S (pfp_value_add_functional_second) = S ((S (pfp_index_add_functional_second)) * dc)) /\ exists ff_q_pfp_add_functional_secondtarget. db = ff_q_pfp_add_functional_secondtarget * S ((S (pfp_index_add_functional_second)) * dc) + (pfp_value_add_functional_second))) /\ ((((exists pfa_gap_add_functional_secondoperationleft. pfa_gap_add_functional_secondoperationleft + S (pfp_left_add_functional_second) = (p)) /\ (((exists pfa_gap_add_functional_secondoperationright. pfa_gap_add_functional_secondoperationright + S (pfp_right_add_functional_second) = (p)) /\ ((((exists pfa_gap_add_functional_secondoperationresultbound. pfa_gap_add_functional_secondoperationresultbound + S (pfp_value_add_functional_second) = (p)) /\ ((exists pfa_offset_left_add_functional_secondoperationresultcongruence pfa_offset_right_add_functional_secondoperationresultcongruence. ((pfp_left_add_functional_second) + (pfp_right_add_functional_second)) + (p) * pfa_offset_left_add_functional_secondoperationresultcongruence = (pfp_value_add_functional_second) + (p) * pfa_offset_right_add_functional_secondoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_add_functional_result mdr_a_pfp_add_functional_result. (exists mdr_gap_pfp_add_functional_resultb. mdr_gap_pfp_add_functional_resultb + S (mdr_i_pfp_add_functional_result) = (l)) -> (((exists ff_h_mdr_pfp_add_functional_resulto. ff_h_mdr_pfp_add_functional_resulto + S (mdr_a_pfp_add_functional_result) = S ((S (mdr_i_pfp_add_functional_result)) * cc)) /\ exists ff_q_mdr_pfp_add_functional_resulto. cb = ff_q_mdr_pfp_add_functional_resulto * S ((S (mdr_i_pfp_add_functional_result)) * cc) + (mdr_a_pfp_add_functional_result))) -> (((exists ff_h_mdr_pfp_add_functional_resultn. ff_h_mdr_pfp_add_functional_resultn + S (mdr_a_pfp_add_functional_result) = S ((S (mdr_i_pfp_add_functional_result)) * dc)) /\ exists ff_q_mdr_pfp_add_functional_resultn. db = ff_q_mdr_pfp_add_functional_resultn * S ((S (mdr_i_pfp_add_functional_result)) * dc) + (mdr_a_pfp_add_functional_result))))

forall p ab ac bb bc cb cc AB AC BB BC CB CC l. (forall mdr_i_pfp_add_transport_a mdr_a_pfp_add_transport_a. (exists mdr_gap_pfp_add_transport_ab. mdr_gap_pfp_add_transport_ab + S (mdr_i_pfp_add_transport_a) = (l)) -> (((exists ff_h_mdr_pfp_add_transport_ao. ff_h_mdr_pfp_add_transport_ao + S (mdr_a_pfp_add_transport_a) = S ((S (mdr_i_pfp_add_transport_a)) * ac)) /\ exists ff_q_mdr_pfp_add_transport_ao. ab = ff_q_mdr_pfp_add_transport_ao * S ((S (mdr_i_pfp_add_transport_a)) * ac) + (mdr_a_pfp_add_transport_a))) -> (((exists ff_h_mdr_pfp_add_transport_an. ff_h_mdr_pfp_add_transport_an + S (mdr_a_pfp_add_transport_a) = S ((S (mdr_i_pfp_add_transport_a)) * AC)) /\ exists ff_q_mdr_pfp_add_transport_an. AB = ff_q_mdr_pfp_add_transport_an * S ((S (mdr_i_pfp_add_transport_a)) * AC) + (mdr_a_pfp_add_transport_a)))) -> (forall mdr_i_pfp_add_transport_b mdr_a_pfp_add_transport_b. (exists mdr_gap_pfp_add_transport_bb. mdr_gap_pfp_add_transport_bb + S (mdr_i_pfp_add_transport_b) = (l)) -> (((exists ff_h_mdr_pfp_add_transport_bo. ff_h_mdr_pfp_add_transport_bo + S (mdr_a_pfp_add_transport_b) = S ((S (mdr_i_pfp_add_transport_b)) * bc)) /\ exists ff_q_mdr_pfp_add_transport_bo. bb = ff_q_mdr_pfp_add_transport_bo * S ((S (mdr_i_pfp_add_transport_b)) * bc) + (mdr_a_pfp_add_transport_b))) -> (((exists ff_h_mdr_pfp_add_transport_bn. ff_h_mdr_pfp_add_transport_bn + S (mdr_a_pfp_add_transport_b) = S ((S (mdr_i_pfp_add_transport_b)) * BC)) /\ exists ff_q_mdr_pfp_add_transport_bn. BB = ff_q_mdr_pfp_add_transport_bn * S ((S (mdr_i_pfp_add_transport_b)) * BC) + (mdr_a_pfp_add_transport_b)))) -> (forall mdr_i_pfp_add_transport_c mdr_a_pfp_add_transport_c. (exists mdr_gap_pfp_add_transport_cb. mdr_gap_pfp_add_transport_cb + S (mdr_i_pfp_add_transport_c) = (l)) -> (((exists ff_h_mdr_pfp_add_transport_co. ff_h_mdr_pfp_add_transport_co + S (mdr_a_pfp_add_transport_c) = S ((S (mdr_i_pfp_add_transport_c)) * cc)) /\ exists ff_q_mdr_pfp_add_transport_co. cb = ff_q_mdr_pfp_add_transport_co * S ((S (mdr_i_pfp_add_transport_c)) * cc) + (mdr_a_pfp_add_transport_c))) -> (((exists ff_h_mdr_pfp_add_transport_cn. ff_h_mdr_pfp_add_transport_cn + S (mdr_a_pfp_add_transport_c) = S ((S (mdr_i_pfp_add_transport_c)) * CC)) /\ exists ff_q_mdr_pfp_add_transport_cn. CB = ff_q_mdr_pfp_add_transport_cn * S ((S (mdr_i_pfp_add_transport_c)) * CC) + (mdr_a_pfp_add_transport_c)))) -> (forall pfp_index_add_transport_old. (exists pfa_gap_add_transport_oldindex. pfa_gap_add_transport_oldindex + S (pfp_index_add_transport_old) = (l)) -> exists pfp_left_add_transport_old pfp_right_add_transport_old pfp_value_add_transport_old. ((((exists ff_h_pfp_add_transport_oldleft. ff_h_pfp_add_transport_oldleft + S (pfp_left_add_transport_old) = S ((S (pfp_index_add_transport_old)) * ac)) /\ exists ff_q_pfp_add_transport_oldleft. ab = ff_q_pfp_add_transport_oldleft * S ((S (pfp_index_add_transport_old)) * ac) + (pfp_left_add_transport_old))) /\ (((((exists ff_h_pfp_add_transport_oldright. ff_h_pfp_add_transport_oldright + S (pfp_right_add_transport_old) = S ((S (pfp_index_add_transport_old)) * bc)) /\ exists ff_q_pfp_add_transport_oldright. bb = ff_q_pfp_add_transport_oldright * S ((S (pfp_index_add_transport_old)) * bc) + (pfp_right_add_transport_old))) /\ (((((exists ff_h_pfp_add_transport_oldtarget. ff_h_pfp_add_transport_oldtarget + S (pfp_value_add_transport_old) = S ((S (pfp_index_add_transport_old)) * cc)) /\ exists ff_q_pfp_add_transport_oldtarget. cb = ff_q_pfp_add_transport_oldtarget * S ((S (pfp_index_add_transport_old)) * cc) + (pfp_value_add_transport_old))) /\ ((((exists pfa_gap_add_transport_oldoperationleft. pfa_gap_add_transport_oldoperationleft + S (pfp_left_add_transport_old) = (p)) /\ (((exists pfa_gap_add_transport_oldoperationright. pfa_gap_add_transport_oldoperationright + S (pfp_right_add_transport_old) = (p)) /\ ((((exists pfa_gap_add_transport_oldoperationresultbound. pfa_gap_add_transport_oldoperationresultbound + S (pfp_value_add_transport_old) = (p)) /\ ((exists pfa_offset_left_add_transport_oldoperationresultcongruence pfa_offset_right_add_transport_oldoperationresultcongruence. ((pfp_left_add_transport_old) + (pfp_right_add_transport_old)) + (p) * pfa_offset_left_add_transport_oldoperationresultcongruence = (pfp_value_add_transport_old) + (p) * pfa_offset_right_add_transport_oldoperationresultcongruence)))))))))))))))) -> (forall pfp_index_add_transport_new. (exists pfa_gap_add_transport_newindex. pfa_gap_add_transport_newindex + S (pfp_index_add_transport_new) = (l)) -> exists pfp_left_add_transport_new pfp_right_add_transport_new pfp_value_add_transport_new. ((((exists ff_h_pfp_add_transport_newleft. ff_h_pfp_add_transport_newleft + S (pfp_left_add_transport_new) = S ((S (pfp_index_add_transport_new)) * AC)) /\ exists ff_q_pfp_add_transport_newleft. AB = ff_q_pfp_add_transport_newleft * S ((S (pfp_index_add_transport_new)) * AC) + (pfp_left_add_transport_new))) /\ (((((exists ff_h_pfp_add_transport_newright. ff_h_pfp_add_transport_newright + S (pfp_right_add_transport_new) = S ((S (pfp_index_add_transport_new)) * BC)) /\ exists ff_q_pfp_add_transport_newright. BB = ff_q_pfp_add_transport_newright * S ((S (pfp_index_add_transport_new)) * BC) + (pfp_right_add_transport_new))) /\ (((((exists ff_h_pfp_add_transport_newtarget. ff_h_pfp_add_transport_newtarget + S (pfp_value_add_transport_new) = S ((S (pfp_index_add_transport_new)) * CC)) /\ exists ff_q_pfp_add_transport_newtarget. CB = ff_q_pfp_add_transport_newtarget * S ((S (pfp_index_add_transport_new)) * CC) + (pfp_value_add_transport_new))) /\ ((((exists pfa_gap_add_transport_newoperationleft. pfa_gap_add_transport_newoperationleft + S (pfp_left_add_transport_new) = (p)) /\ (((exists pfa_gap_add_transport_newoperationright. pfa_gap_add_transport_newoperationright + S (pfp_right_add_transport_new) = (p)) /\ ((((exists pfa_gap_add_transport_newoperationresultbound. pfa_gap_add_transport_newoperationresultbound + S (pfp_value_add_transport_new) = (p)) /\ ((exists pfa_offset_left_add_transport_newoperationresultcongruence pfa_offset_right_add_transport_newoperationresultcongruence. ((pfp_left_add_transport_new) + (pfp_right_add_transport_new)) + (p) * pfa_offset_left_add_transport_newoperationresultcongruence = (pfp_value_add_transport_new) + (p) * pfa_offset_right_add_transport_newoperationresultcongruence))))))))))))))))

forall p ab ac bb bc cb cc l. (forall pfp_index_add_comm_old. (exists pfa_gap_add_comm_oldindex. pfa_gap_add_comm_oldindex + S (pfp_index_add_comm_old) = (l)) -> exists pfp_left_add_comm_old pfp_right_add_comm_old pfp_value_add_comm_old. ((((exists ff_h_pfp_add_comm_oldleft. ff_h_pfp_add_comm_oldleft + S (pfp_left_add_comm_old) = S ((S (pfp_index_add_comm_old)) * ac)) /\ exists ff_q_pfp_add_comm_oldleft. ab = ff_q_pfp_add_comm_oldleft * S ((S (pfp_index_add_comm_old)) * ac) + (pfp_left_add_comm_old))) /\ (((((exists ff_h_pfp_add_comm_oldright. ff_h_pfp_add_comm_oldright + S (pfp_right_add_comm_old) = S ((S (pfp_index_add_comm_old)) * bc)) /\ exists ff_q_pfp_add_comm_oldright. bb = ff_q_pfp_add_comm_oldright * S ((S (pfp_index_add_comm_old)) * bc) + (pfp_right_add_comm_old))) /\ (((((exists ff_h_pfp_add_comm_oldtarget. ff_h_pfp_add_comm_oldtarget + S (pfp_value_add_comm_old) = S ((S (pfp_index_add_comm_old)) * cc)) /\ exists ff_q_pfp_add_comm_oldtarget. cb = ff_q_pfp_add_comm_oldtarget * S ((S (pfp_index_add_comm_old)) * cc) + (pfp_value_add_comm_old))) /\ ((((exists pfa_gap_add_comm_oldoperationleft. pfa_gap_add_comm_oldoperationleft + S (pfp_left_add_comm_old) = (p)) /\ (((exists pfa_gap_add_comm_oldoperationright. pfa_gap_add_comm_oldoperationright + S (pfp_right_add_comm_old) = (p)) /\ ((((exists pfa_gap_add_comm_oldoperationresultbound. pfa_gap_add_comm_oldoperationresultbound + S (pfp_value_add_comm_old) = (p)) /\ ((exists pfa_offset_left_add_comm_oldoperationresultcongruence pfa_offset_right_add_comm_oldoperationresultcongruence. ((pfp_left_add_comm_old) + (pfp_right_add_comm_old)) + (p) * pfa_offset_left_add_comm_oldoperationresultcongruence = (pfp_value_add_comm_old) + (p) * pfa_offset_right_add_comm_oldoperationresultcongruence)))))))))))))))) -> (forall pfp_index_add_comm_new. (exists pfa_gap_add_comm_newindex. pfa_gap_add_comm_newindex + S (pfp_index_add_comm_new) = (l)) -> exists pfp_left_add_comm_new pfp_right_add_comm_new pfp_value_add_comm_new. ((((exists ff_h_pfp_add_comm_newleft. ff_h_pfp_add_comm_newleft + S (pfp_left_add_comm_new) = S ((S (pfp_index_add_comm_new)) * bc)) /\ exists ff_q_pfp_add_comm_newleft. bb = ff_q_pfp_add_comm_newleft * S ((S (pfp_index_add_comm_new)) * bc) + (pfp_left_add_comm_new))) /\ (((((exists ff_h_pfp_add_comm_newright. ff_h_pfp_add_comm_newright + S (pfp_right_add_comm_new) = S ((S (pfp_index_add_comm_new)) * ac)) /\ exists ff_q_pfp_add_comm_newright. ab = ff_q_pfp_add_comm_newright * S ((S (pfp_index_add_comm_new)) * ac) + (pfp_right_add_comm_new))) /\ (((((exists ff_h_pfp_add_comm_newtarget. ff_h_pfp_add_comm_newtarget + S (pfp_value_add_comm_new) = S ((S (pfp_index_add_comm_new)) * cc)) /\ exists ff_q_pfp_add_comm_newtarget. cb = ff_q_pfp_add_comm_newtarget * S ((S (pfp_index_add_comm_new)) * cc) + (pfp_value_add_comm_new))) /\ ((((exists pfa_gap_add_comm_newoperationleft. pfa_gap_add_comm_newoperationleft + S (pfp_left_add_comm_new) = (p)) /\ (((exists pfa_gap_add_comm_newoperationright. pfa_gap_add_comm_newoperationright + S (pfp_right_add_comm_new) = (p)) /\ ((((exists pfa_gap_add_comm_newoperationresultbound. pfa_gap_add_comm_newoperationresultbound + S (pfp_value_add_comm_new) = (p)) /\ ((exists pfa_offset_left_add_comm_newoperationresultcongruence pfa_offset_right_add_comm_newoperationresultcongruence. ((pfp_left_add_comm_new) + (pfp_right_add_comm_new)) + (p) * pfa_offset_left_add_comm_newoperationresultcongruence = (pfp_value_add_comm_new) + (p) * pfa_offset_right_add_comm_newoperationresultcongruence))))))))))))))))

forall p b c zb zc l. (~((p) = 1) /\ forall pfa_factor_left_add_zero_prime pfa_factor_right_add_zero_prime. (p) = pfa_factor_left_add_zero_prime * pfa_factor_right_add_zero_prime -> pfa_factor_left_add_zero_prime = 1 \/ pfa_factor_right_add_zero_prime = 1) -> (forall fom_index_pfp_add_zero_coefficients. (exists fom_gap_pfp_add_zero_coefficients_index_bound. fom_gap_pfp_add_zero_coefficients_index_bound + S (fom_index_pfp_add_zero_coefficients) = l) -> exists fom_value_pfp_add_zero_coefficients. ((((exists fom_beta_height_pfp_add_zero_coefficients_entry. fom_beta_height_pfp_add_zero_coefficients_entry + S (fom_value_pfp_add_zero_coefficients) = S ((S (fom_index_pfp_add_zero_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_add_zero_coefficients_entry. b = fom_beta_quotient_pfp_add_zero_coefficients_entry * S ((S (fom_index_pfp_add_zero_coefficients)) * c) + (fom_value_pfp_add_zero_coefficients))) /\ (exists fom_gap_pfp_add_zero_coefficients_value_bound. fom_gap_pfp_add_zero_coefficients_value_bound + S (fom_value_pfp_add_zero_coefficients) = p))) -> (forall pfp_repeat_index_add_zero_table. (exists pfa_gap_add_zero_tableindex. pfa_gap_add_zero_tableindex + S (pfp_repeat_index_add_zero_table) = (l)) -> (((exists ff_h_pfp_add_zero_tableentry. ff_h_pfp_add_zero_tableentry + S (0) = S ((S (pfp_repeat_index_add_zero_table)) * zc)) /\ exists ff_q_pfp_add_zero_tableentry. zb = ff_q_pfp_add_zero_tableentry * S ((S (pfp_repeat_index_add_zero_table)) * zc) + (0)))) -> (forall pfp_index_add_zero_result. (exists pfa_gap_add_zero_resultindex. pfa_gap_add_zero_resultindex + S (pfp_index_add_zero_result) = (l)) -> exists pfp_left_add_zero_result pfp_right_add_zero_result pfp_value_add_zero_result. ((((exists ff_h_pfp_add_zero_resultleft. ff_h_pfp_add_zero_resultleft + S (pfp_left_add_zero_result) = S ((S (pfp_index_add_zero_result)) * c)) /\ exists ff_q_pfp_add_zero_resultleft. b = ff_q_pfp_add_zero_resultleft * S ((S (pfp_index_add_zero_result)) * c) + (pfp_left_add_zero_result))) /\ (((((exists ff_h_pfp_add_zero_resultright. ff_h_pfp_add_zero_resultright + S (pfp_right_add_zero_result) = S ((S (pfp_index_add_zero_result)) * zc)) /\ exists ff_q_pfp_add_zero_resultright. zb = ff_q_pfp_add_zero_resultright * S ((S (pfp_index_add_zero_result)) * zc) + (pfp_right_add_zero_result))) /\ (((((exists ff_h_pfp_add_zero_resulttarget. ff_h_pfp_add_zero_resulttarget + S (pfp_value_add_zero_result) = S ((S (pfp_index_add_zero_result)) * c)) /\ exists ff_q_pfp_add_zero_resulttarget. b = ff_q_pfp_add_zero_resulttarget * S ((S (pfp_index_add_zero_result)) * c) + (pfp_value_add_zero_result))) /\ ((((exists pfa_gap_add_zero_resultoperationleft. pfa_gap_add_zero_resultoperationleft + S (pfp_left_add_zero_result) = (p)) /\ (((exists pfa_gap_add_zero_resultoperationright. pfa_gap_add_zero_resultoperationright + S (pfp_right_add_zero_result) = (p)) /\ ((((exists pfa_gap_add_zero_resultoperationresultbound. pfa_gap_add_zero_resultoperationresultbound + S (pfp_value_add_zero_result) = (p)) /\ ((exists pfa_offset_left_add_zero_resultoperationresultcongruence pfa_offset_right_add_zero_resultoperationresultcongruence. ((pfp_left_add_zero_result) + (pfp_right_add_zero_result)) + (p) * pfa_offset_left_add_zero_resultoperationresultcongruence = (pfp_value_add_zero_result) + (p) * pfa_offset_right_add_zero_resultoperationresultcongruence))))))))))))))))

forall p k kb kc ab ac rb rc bb bc l. (exists pfa_gap_scale_scalar. pfa_gap_scale_scalar + S (k) = (p)) -> (forall fom_index_pfp_scale_source. (exists fom_gap_pfp_scale_source_index_bound. fom_gap_pfp_scale_source_index_bound + S (fom_index_pfp_scale_source) = l) -> exists fom_value_pfp_scale_source. ((((exists fom_beta_height_pfp_scale_source_entry. fom_beta_height_pfp_scale_source_entry + S (fom_value_pfp_scale_source) = S ((S (fom_index_pfp_scale_source)) * ac)) /\ exists fom_beta_quotient_pfp_scale_source_entry. ab = fom_beta_quotient_pfp_scale_source_entry * S ((S (fom_index_pfp_scale_source)) * ac) + (fom_value_pfp_scale_source))) /\ (exists fom_gap_pfp_scale_source_value_bound. fom_gap_pfp_scale_source_value_bound + S (fom_value_pfp_scale_source) = p))) -> (forall pfp_repeat_index_scale_repeated. (exists pfa_gap_scale_repeatedindex. pfa_gap_scale_repeatedindex + S (pfp_repeat_index_scale_repeated) = (l)) -> (((exists ff_h_pfp_scale_repeatedentry. ff_h_pfp_scale_repeatedentry + S (k) = S ((S (pfp_repeat_index_scale_repeated)) * kc)) /\ exists ff_q_pfp_scale_repeatedentry. kb = ff_q_pfp_scale_repeatedentry * S ((S (pfp_repeat_index_scale_repeated)) * kc) + (k)))) -> (forall fpmp_index_pfp_scale_raw fpmp_left_pfp_scale_raw fpmp_right_pfp_scale_raw fpmp_target_pfp_scale_raw. (exists fpmp_gap_pfp_scale_raw. fpmp_gap_pfp_scale_raw + S fpmp_index_pfp_scale_raw = l) -> (((exists ff_h_fpmp_pfp_scale_raw_left. ff_h_fpmp_pfp_scale_raw_left + S (fpmp_left_pfp_scale_raw) = S ((S (fpmp_index_pfp_scale_raw)) * kc)) /\ exists ff_q_fpmp_pfp_scale_raw_left. kb = ff_q_fpmp_pfp_scale_raw_left * S ((S (fpmp_index_pfp_scale_raw)) * kc) + (fpmp_left_pfp_scale_raw))) -> (((exists ff_h_fpmp_pfp_scale_raw_right. ff_h_fpmp_pfp_scale_raw_right + S (fpmp_right_pfp_scale_raw) = S ((S (fpmp_index_pfp_scale_raw)) * ac)) /\ exists ff_q_fpmp_pfp_scale_raw_right. ab = ff_q_fpmp_pfp_scale_raw_right * S ((S (fpmp_index_pfp_scale_raw)) * ac) + (fpmp_right_pfp_scale_raw))) -> (((exists ff_h_fpmp_pfp_scale_raw_target. ff_h_fpmp_pfp_scale_raw_target + S (fpmp_target_pfp_scale_raw) = S ((S (fpmp_index_pfp_scale_raw)) * rc)) /\ exists ff_q_fpmp_pfp_scale_raw_target. rb = ff_q_fpmp_pfp_scale_raw_target * S ((S (fpmp_index_pfp_scale_raw)) * rc) + (fpmp_target_pfp_scale_raw))) -> fpmp_target_pfp_scale_raw = fpmp_left_pfp_scale_raw * fpmp_right_pfp_scale_raw) -> (forall pfp_index_scale_normalize. (exists pfa_gap_scale_normalizeindex. pfa_gap_scale_normalizeindex + S (pfp_index_scale_normalize) = (l)) -> exists pfp_source_scale_normalize pfp_residue_scale_normalize. ((((exists ff_h_pfp_scale_normalizesource. ff_h_pfp_scale_normalizesource + S (pfp_source_scale_normalize) = S ((S (pfp_index_scale_normalize)) * rc)) /\ exists ff_q_pfp_scale_normalizesource. rb = ff_q_pfp_scale_normalizesource * S ((S (pfp_index_scale_normalize)) * rc) + (pfp_source_scale_normalize))) /\ (((((exists ff_h_pfp_scale_normalizetarget. ff_h_pfp_scale_normalizetarget + S (pfp_residue_scale_normalize) = S ((S (pfp_index_scale_normalize)) * bc)) /\ exists ff_q_pfp_scale_normalizetarget. bb = ff_q_pfp_scale_normalizetarget * S ((S (pfp_index_scale_normalize)) * bc) + (pfp_residue_scale_normalize))) /\ ((((exists pfa_gap_scale_normalizeresiduebound. pfa_gap_scale_normalizeresiduebound + S (pfp_residue_scale_normalize) = (p)) /\ ((exists pfa_offset_left_scale_normalizeresiduecongruence pfa_offset_right_scale_normalizeresiduecongruence. (pfp_source_scale_normalize) + (p) * pfa_offset_left_scale_normalizeresiduecongruence = (pfp_residue_scale_normalize) + (p) * pfa_offset_right_scale_normalizeresiduecongruence))))))))) -> (((exists pfa_gap_scale_resultscalar. pfa_gap_scale_resultscalar + S (k) = (p)) /\ ((forall pfp_index_scale_result. (exists pfa_gap_scale_resultindex. pfa_gap_scale_resultindex + S (pfp_index_scale_result) = (l)) -> exists pfp_source_scale_result pfp_value_scale_result. ((((exists ff_h_pfp_scale_resultsource. ff_h_pfp_scale_resultsource + S (pfp_source_scale_result) = S ((S (pfp_index_scale_result)) * ac)) /\ exists ff_q_pfp_scale_resultsource. ab = ff_q_pfp_scale_resultsource * S ((S (pfp_index_scale_result)) * ac) + (pfp_source_scale_result))) /\ (((((exists ff_h_pfp_scale_resulttarget. ff_h_pfp_scale_resulttarget + S (pfp_value_scale_result) = S ((S (pfp_index_scale_result)) * bc)) /\ exists ff_q_pfp_scale_resulttarget. bb = ff_q_pfp_scale_resulttarget * S ((S (pfp_index_scale_result)) * bc) + (pfp_value_scale_result))) /\ ((((exists pfa_gap_scale_resultoperationleft. pfa_gap_scale_resultoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_resultoperationright. pfa_gap_scale_resultoperationright + S (pfp_source_scale_result) = (p)) /\ ((((exists pfa_gap_scale_resultoperationresultbound. pfa_gap_scale_resultoperationresultbound + S (pfp_value_scale_result) = (p)) /\ ((exists pfa_offset_left_scale_resultoperationresultcongruence pfa_offset_right_scale_resultoperationresultcongruence. ((k) * (pfp_source_scale_result)) + (p) * pfa_offset_left_scale_resultoperationresultcongruence = (pfp_value_scale_result) + (p) * pfa_offset_right_scale_resultoperationresultcongruence)))))))))))))))))

forall p k ab ac l. ~(p=0) -> (exists pfa_gap_scale_exists_scalar. pfa_gap_scale_exists_scalar + S (k) = (p)) -> (forall fom_index_pfp_scale_exists_source. (exists fom_gap_pfp_scale_exists_source_index_bound. fom_gap_pfp_scale_exists_source_index_bound + S (fom_index_pfp_scale_exists_source) = l) -> exists fom_value_pfp_scale_exists_source. ((((exists fom_beta_height_pfp_scale_exists_source_entry. fom_beta_height_pfp_scale_exists_source_entry + S (fom_value_pfp_scale_exists_source) = S ((S (fom_index_pfp_scale_exists_source)) * ac)) /\ exists fom_beta_quotient_pfp_scale_exists_source_entry. ab = fom_beta_quotient_pfp_scale_exists_source_entry * S ((S (fom_index_pfp_scale_exists_source)) * ac) + (fom_value_pfp_scale_exists_source))) /\ (exists fom_gap_pfp_scale_exists_source_value_bound. fom_gap_pfp_scale_exists_source_value_bound + S (fom_value_pfp_scale_exists_source) = p))) -> exists bb bc. (((exists pfa_gap_scale_exists_resultscalar. pfa_gap_scale_exists_resultscalar + S (k) = (p)) /\ ((forall pfp_index_scale_exists_result. (exists pfa_gap_scale_exists_resultindex. pfa_gap_scale_exists_resultindex + S (pfp_index_scale_exists_result) = (l)) -> exists pfp_source_scale_exists_result pfp_value_scale_exists_result. ((((exists ff_h_pfp_scale_exists_resultsource. ff_h_pfp_scale_exists_resultsource + S (pfp_source_scale_exists_result) = S ((S (pfp_index_scale_exists_result)) * ac)) /\ exists ff_q_pfp_scale_exists_resultsource. ab = ff_q_pfp_scale_exists_resultsource * S ((S (pfp_index_scale_exists_result)) * ac) + (pfp_source_scale_exists_result))) /\ (((((exists ff_h_pfp_scale_exists_resulttarget. ff_h_pfp_scale_exists_resulttarget + S (pfp_value_scale_exists_result) = S ((S (pfp_index_scale_exists_result)) * bc)) /\ exists ff_q_pfp_scale_exists_resulttarget. bb = ff_q_pfp_scale_exists_resulttarget * S ((S (pfp_index_scale_exists_result)) * bc) + (pfp_value_scale_exists_result))) /\ ((((exists pfa_gap_scale_exists_resultoperationleft. pfa_gap_scale_exists_resultoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_exists_resultoperationright. pfa_gap_scale_exists_resultoperationright + S (pfp_source_scale_exists_result) = (p)) /\ ((((exists pfa_gap_scale_exists_resultoperationresultbound. pfa_gap_scale_exists_resultoperationresultbound + S (pfp_value_scale_exists_result) = (p)) /\ ((exists pfa_offset_left_scale_exists_resultoperationresultcongruence pfa_offset_right_scale_exists_resultoperationresultcongruence. ((k) * (pfp_source_scale_exists_result)) + (p) * pfa_offset_left_scale_exists_resultoperationresultcongruence = (pfp_value_scale_exists_result) + (p) * pfa_offset_right_scale_exists_resultoperationresultcongruence)))))))))))))))))

forall p k ab ac bb bc l i a r. (((exists pfa_gap_scale_entry_tablescalar. pfa_gap_scale_entry_tablescalar + S (k) = (p)) /\ ((forall pfp_index_scale_entry_table. (exists pfa_gap_scale_entry_tableindex. pfa_gap_scale_entry_tableindex + S (pfp_index_scale_entry_table) = (l)) -> exists pfp_source_scale_entry_table pfp_value_scale_entry_table. ((((exists ff_h_pfp_scale_entry_tablesource. ff_h_pfp_scale_entry_tablesource + S (pfp_source_scale_entry_table) = S ((S (pfp_index_scale_entry_table)) * ac)) /\ exists ff_q_pfp_scale_entry_tablesource. ab = ff_q_pfp_scale_entry_tablesource * S ((S (pfp_index_scale_entry_table)) * ac) + (pfp_source_scale_entry_table))) /\ (((((exists ff_h_pfp_scale_entry_tabletarget. ff_h_pfp_scale_entry_tabletarget + S (pfp_value_scale_entry_table) = S ((S (pfp_index_scale_entry_table)) * bc)) /\ exists ff_q_pfp_scale_entry_tabletarget. bb = ff_q_pfp_scale_entry_tabletarget * S ((S (pfp_index_scale_entry_table)) * bc) + (pfp_value_scale_entry_table))) /\ ((((exists pfa_gap_scale_entry_tableoperationleft. pfa_gap_scale_entry_tableoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_entry_tableoperationright. pfa_gap_scale_entry_tableoperationright + S (pfp_source_scale_entry_table) = (p)) /\ ((((exists pfa_gap_scale_entry_tableoperationresultbound. pfa_gap_scale_entry_tableoperationresultbound + S (pfp_value_scale_entry_table) = (p)) /\ ((exists pfa_offset_left_scale_entry_tableoperationresultcongruence pfa_offset_right_scale_entry_tableoperationresultcongruence. ((k) * (pfp_source_scale_entry_table)) + (p) * pfa_offset_left_scale_entry_tableoperationresultcongruence = (pfp_value_scale_entry_table) + (p) * pfa_offset_right_scale_entry_tableoperationresultcongruence))))))))))))))))) -> (exists pfa_gap_scale_entry_index. pfa_gap_scale_entry_index + S (i) = (l)) -> (((exists ff_h_pfp_scale_entry_input. ff_h_pfp_scale_entry_input + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_scale_entry_input. ab = ff_q_pfp_scale_entry_input * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_scale_entry_output. ff_h_pfp_scale_entry_output + S (r) = S ((S (i)) * bc)) /\ exists ff_q_pfp_scale_entry_output. bb = ff_q_pfp_scale_entry_output * S ((S (i)) * bc) + (r))) -> (((exists pfa_gap_scale_entry_valueleft. pfa_gap_scale_entry_valueleft + S (k) = (p)) /\ (((exists pfa_gap_scale_entry_valueright. pfa_gap_scale_entry_valueright + S (a) = (p)) /\ ((((exists pfa_gap_scale_entry_valueresultbound. pfa_gap_scale_entry_valueresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_scale_entry_valueresultcongruence pfa_offset_right_scale_entry_valueresultcongruence. ((k) * (a)) + (p) * pfa_offset_left_scale_entry_valueresultcongruence = (r) + (p) * pfa_offset_right_scale_entry_valueresultcongruence)))))))))

forall p k ab ac bb bc l. (((exists pfa_gap_scale_bounded_operationscalar. pfa_gap_scale_bounded_operationscalar + S (k) = (p)) /\ ((forall pfp_index_scale_bounded_operation. (exists pfa_gap_scale_bounded_operationindex. pfa_gap_scale_bounded_operationindex + S (pfp_index_scale_bounded_operation) = (l)) -> exists pfp_source_scale_bounded_operation pfp_value_scale_bounded_operation. ((((exists ff_h_pfp_scale_bounded_operationsource. ff_h_pfp_scale_bounded_operationsource + S (pfp_source_scale_bounded_operation) = S ((S (pfp_index_scale_bounded_operation)) * ac)) /\ exists ff_q_pfp_scale_bounded_operationsource. ab = ff_q_pfp_scale_bounded_operationsource * S ((S (pfp_index_scale_bounded_operation)) * ac) + (pfp_source_scale_bounded_operation))) /\ (((((exists ff_h_pfp_scale_bounded_operationtarget. ff_h_pfp_scale_bounded_operationtarget + S (pfp_value_scale_bounded_operation) = S ((S (pfp_index_scale_bounded_operation)) * bc)) /\ exists ff_q_pfp_scale_bounded_operationtarget. bb = ff_q_pfp_scale_bounded_operationtarget * S ((S (pfp_index_scale_bounded_operation)) * bc) + (pfp_value_scale_bounded_operation))) /\ ((((exists pfa_gap_scale_bounded_operationoperationleft. pfa_gap_scale_bounded_operationoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_bounded_operationoperationright. pfa_gap_scale_bounded_operationoperationright + S (pfp_source_scale_bounded_operation) = (p)) /\ ((((exists pfa_gap_scale_bounded_operationoperationresultbound. pfa_gap_scale_bounded_operationoperationresultbound + S (pfp_value_scale_bounded_operation) = (p)) /\ ((exists pfa_offset_left_scale_bounded_operationoperationresultcongruence pfa_offset_right_scale_bounded_operationoperationresultcongruence. ((k) * (pfp_source_scale_bounded_operation)) + (p) * pfa_offset_left_scale_bounded_operationoperationresultcongruence = (pfp_value_scale_bounded_operation) + (p) * pfa_offset_right_scale_bounded_operationoperationresultcongruence))))))))))))))))) -> ((forall fom_index_pfp_scale_bounded_source. (exists fom_gap_pfp_scale_bounded_source_index_bound. fom_gap_pfp_scale_bounded_source_index_bound + S (fom_index_pfp_scale_bounded_source) = l) -> exists fom_value_pfp_scale_bounded_source. ((((exists fom_beta_height_pfp_scale_bounded_source_entry. fom_beta_height_pfp_scale_bounded_source_entry + S (fom_value_pfp_scale_bounded_source) = S ((S (fom_index_pfp_scale_bounded_source)) * ac)) /\ exists fom_beta_quotient_pfp_scale_bounded_source_entry. ab = fom_beta_quotient_pfp_scale_bounded_source_entry * S ((S (fom_index_pfp_scale_bounded_source)) * ac) + (fom_value_pfp_scale_bounded_source))) /\ (exists fom_gap_pfp_scale_bounded_source_value_bound. fom_gap_pfp_scale_bounded_source_value_bound + S (fom_value_pfp_scale_bounded_source) = p))) /\ ((forall fom_index_pfp_scale_bounded_target. (exists fom_gap_pfp_scale_bounded_target_index_bound. fom_gap_pfp_scale_bounded_target_index_bound + S (fom_index_pfp_scale_bounded_target) = l) -> exists fom_value_pfp_scale_bounded_target. ((((exists fom_beta_height_pfp_scale_bounded_target_entry. fom_beta_height_pfp_scale_bounded_target_entry + S (fom_value_pfp_scale_bounded_target) = S ((S (fom_index_pfp_scale_bounded_target)) * bc)) /\ exists fom_beta_quotient_pfp_scale_bounded_target_entry. bb = fom_beta_quotient_pfp_scale_bounded_target_entry * S ((S (fom_index_pfp_scale_bounded_target)) * bc) + (fom_value_pfp_scale_bounded_target))) /\ (exists fom_gap_pfp_scale_bounded_target_value_bound. fom_gap_pfp_scale_bounded_target_value_bound + S (fom_value_pfp_scale_bounded_target) = p)))))

forall p k ab ac bb bc cb cc l. (((exists pfa_gap_scale_functional_firstscalar. pfa_gap_scale_functional_firstscalar + S (k) = (p)) /\ ((forall pfp_index_scale_functional_first. (exists pfa_gap_scale_functional_firstindex. pfa_gap_scale_functional_firstindex + S (pfp_index_scale_functional_first) = (l)) -> exists pfp_source_scale_functional_first pfp_value_scale_functional_first. ((((exists ff_h_pfp_scale_functional_firstsource. ff_h_pfp_scale_functional_firstsource + S (pfp_source_scale_functional_first) = S ((S (pfp_index_scale_functional_first)) * ac)) /\ exists ff_q_pfp_scale_functional_firstsource. ab = ff_q_pfp_scale_functional_firstsource * S ((S (pfp_index_scale_functional_first)) * ac) + (pfp_source_scale_functional_first))) /\ (((((exists ff_h_pfp_scale_functional_firsttarget. ff_h_pfp_scale_functional_firsttarget + S (pfp_value_scale_functional_first) = S ((S (pfp_index_scale_functional_first)) * bc)) /\ exists ff_q_pfp_scale_functional_firsttarget. bb = ff_q_pfp_scale_functional_firsttarget * S ((S (pfp_index_scale_functional_first)) * bc) + (pfp_value_scale_functional_first))) /\ ((((exists pfa_gap_scale_functional_firstoperationleft. pfa_gap_scale_functional_firstoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_functional_firstoperationright. pfa_gap_scale_functional_firstoperationright + S (pfp_source_scale_functional_first) = (p)) /\ ((((exists pfa_gap_scale_functional_firstoperationresultbound. pfa_gap_scale_functional_firstoperationresultbound + S (pfp_value_scale_functional_first) = (p)) /\ ((exists pfa_offset_left_scale_functional_firstoperationresultcongruence pfa_offset_right_scale_functional_firstoperationresultcongruence. ((k) * (pfp_source_scale_functional_first)) + (p) * pfa_offset_left_scale_functional_firstoperationresultcongruence = (pfp_value_scale_functional_first) + (p) * pfa_offset_right_scale_functional_firstoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_scale_functional_secondscalar. pfa_gap_scale_functional_secondscalar + S (k) = (p)) /\ ((forall pfp_index_scale_functional_second. (exists pfa_gap_scale_functional_secondindex. pfa_gap_scale_functional_secondindex + S (pfp_index_scale_functional_second) = (l)) -> exists pfp_source_scale_functional_second pfp_value_scale_functional_second. ((((exists ff_h_pfp_scale_functional_secondsource. ff_h_pfp_scale_functional_secondsource + S (pfp_source_scale_functional_second) = S ((S (pfp_index_scale_functional_second)) * ac)) /\ exists ff_q_pfp_scale_functional_secondsource. ab = ff_q_pfp_scale_functional_secondsource * S ((S (pfp_index_scale_functional_second)) * ac) + (pfp_source_scale_functional_second))) /\ (((((exists ff_h_pfp_scale_functional_secondtarget. ff_h_pfp_scale_functional_secondtarget + S (pfp_value_scale_functional_second) = S ((S (pfp_index_scale_functional_second)) * cc)) /\ exists ff_q_pfp_scale_functional_secondtarget. cb = ff_q_pfp_scale_functional_secondtarget * S ((S (pfp_index_scale_functional_second)) * cc) + (pfp_value_scale_functional_second))) /\ ((((exists pfa_gap_scale_functional_secondoperationleft. pfa_gap_scale_functional_secondoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_functional_secondoperationright. pfa_gap_scale_functional_secondoperationright + S (pfp_source_scale_functional_second) = (p)) /\ ((((exists pfa_gap_scale_functional_secondoperationresultbound. pfa_gap_scale_functional_secondoperationresultbound + S (pfp_value_scale_functional_second) = (p)) /\ ((exists pfa_offset_left_scale_functional_secondoperationresultcongruence pfa_offset_right_scale_functional_secondoperationresultcongruence. ((k) * (pfp_source_scale_functional_second)) + (p) * pfa_offset_left_scale_functional_secondoperationresultcongruence = (pfp_value_scale_functional_second) + (p) * pfa_offset_right_scale_functional_secondoperationresultcongruence))))))))))))))))) -> (forall mdr_i_pfp_scale_functional_result mdr_a_pfp_scale_functional_result. (exists mdr_gap_pfp_scale_functional_resultb. mdr_gap_pfp_scale_functional_resultb + S (mdr_i_pfp_scale_functional_result) = (l)) -> (((exists ff_h_mdr_pfp_scale_functional_resulto. ff_h_mdr_pfp_scale_functional_resulto + S (mdr_a_pfp_scale_functional_result) = S ((S (mdr_i_pfp_scale_functional_result)) * bc)) /\ exists ff_q_mdr_pfp_scale_functional_resulto. bb = ff_q_mdr_pfp_scale_functional_resulto * S ((S (mdr_i_pfp_scale_functional_result)) * bc) + (mdr_a_pfp_scale_functional_result))) -> (((exists ff_h_mdr_pfp_scale_functional_resultn. ff_h_mdr_pfp_scale_functional_resultn + S (mdr_a_pfp_scale_functional_result) = S ((S (mdr_i_pfp_scale_functional_result)) * cc)) /\ exists ff_q_mdr_pfp_scale_functional_resultn. cb = ff_q_mdr_pfp_scale_functional_resultn * S ((S (mdr_i_pfp_scale_functional_result)) * cc) + (mdr_a_pfp_scale_functional_result))))

forall p k ab ac bb bc AB AC BB BC l. (forall mdr_i_pfp_scale_transport_source mdr_a_pfp_scale_transport_source. (exists mdr_gap_pfp_scale_transport_sourceb. mdr_gap_pfp_scale_transport_sourceb + S (mdr_i_pfp_scale_transport_source) = (l)) -> (((exists ff_h_mdr_pfp_scale_transport_sourceo. ff_h_mdr_pfp_scale_transport_sourceo + S (mdr_a_pfp_scale_transport_source) = S ((S (mdr_i_pfp_scale_transport_source)) * ac)) /\ exists ff_q_mdr_pfp_scale_transport_sourceo. ab = ff_q_mdr_pfp_scale_transport_sourceo * S ((S (mdr_i_pfp_scale_transport_source)) * ac) + (mdr_a_pfp_scale_transport_source))) -> (((exists ff_h_mdr_pfp_scale_transport_sourcen. ff_h_mdr_pfp_scale_transport_sourcen + S (mdr_a_pfp_scale_transport_source) = S ((S (mdr_i_pfp_scale_transport_source)) * AC)) /\ exists ff_q_mdr_pfp_scale_transport_sourcen. AB = ff_q_mdr_pfp_scale_transport_sourcen * S ((S (mdr_i_pfp_scale_transport_source)) * AC) + (mdr_a_pfp_scale_transport_source)))) -> (forall mdr_i_pfp_scale_transport_target mdr_a_pfp_scale_transport_target. (exists mdr_gap_pfp_scale_transport_targetb. mdr_gap_pfp_scale_transport_targetb + S (mdr_i_pfp_scale_transport_target) = (l)) -> (((exists ff_h_mdr_pfp_scale_transport_targeto. ff_h_mdr_pfp_scale_transport_targeto + S (mdr_a_pfp_scale_transport_target) = S ((S (mdr_i_pfp_scale_transport_target)) * bc)) /\ exists ff_q_mdr_pfp_scale_transport_targeto. bb = ff_q_mdr_pfp_scale_transport_targeto * S ((S (mdr_i_pfp_scale_transport_target)) * bc) + (mdr_a_pfp_scale_transport_target))) -> (((exists ff_h_mdr_pfp_scale_transport_targetn. ff_h_mdr_pfp_scale_transport_targetn + S (mdr_a_pfp_scale_transport_target) = S ((S (mdr_i_pfp_scale_transport_target)) * BC)) /\ exists ff_q_mdr_pfp_scale_transport_targetn. BB = ff_q_mdr_pfp_scale_transport_targetn * S ((S (mdr_i_pfp_scale_transport_target)) * BC) + (mdr_a_pfp_scale_transport_target)))) -> (((exists pfa_gap_scale_transport_oldscalar. pfa_gap_scale_transport_oldscalar + S (k) = (p)) /\ ((forall pfp_index_scale_transport_old. (exists pfa_gap_scale_transport_oldindex. pfa_gap_scale_transport_oldindex + S (pfp_index_scale_transport_old) = (l)) -> exists pfp_source_scale_transport_old pfp_value_scale_transport_old. ((((exists ff_h_pfp_scale_transport_oldsource. ff_h_pfp_scale_transport_oldsource + S (pfp_source_scale_transport_old) = S ((S (pfp_index_scale_transport_old)) * ac)) /\ exists ff_q_pfp_scale_transport_oldsource. ab = ff_q_pfp_scale_transport_oldsource * S ((S (pfp_index_scale_transport_old)) * ac) + (pfp_source_scale_transport_old))) /\ (((((exists ff_h_pfp_scale_transport_oldtarget. ff_h_pfp_scale_transport_oldtarget + S (pfp_value_scale_transport_old) = S ((S (pfp_index_scale_transport_old)) * bc)) /\ exists ff_q_pfp_scale_transport_oldtarget. bb = ff_q_pfp_scale_transport_oldtarget * S ((S (pfp_index_scale_transport_old)) * bc) + (pfp_value_scale_transport_old))) /\ ((((exists pfa_gap_scale_transport_oldoperationleft. pfa_gap_scale_transport_oldoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_transport_oldoperationright. pfa_gap_scale_transport_oldoperationright + S (pfp_source_scale_transport_old) = (p)) /\ ((((exists pfa_gap_scale_transport_oldoperationresultbound. pfa_gap_scale_transport_oldoperationresultbound + S (pfp_value_scale_transport_old) = (p)) /\ ((exists pfa_offset_left_scale_transport_oldoperationresultcongruence pfa_offset_right_scale_transport_oldoperationresultcongruence. ((k) * (pfp_source_scale_transport_old)) + (p) * pfa_offset_left_scale_transport_oldoperationresultcongruence = (pfp_value_scale_transport_old) + (p) * pfa_offset_right_scale_transport_oldoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_scale_transport_newscalar. pfa_gap_scale_transport_newscalar + S (k) = (p)) /\ ((forall pfp_index_scale_transport_new. (exists pfa_gap_scale_transport_newindex. pfa_gap_scale_transport_newindex + S (pfp_index_scale_transport_new) = (l)) -> exists pfp_source_scale_transport_new pfp_value_scale_transport_new. ((((exists ff_h_pfp_scale_transport_newsource. ff_h_pfp_scale_transport_newsource + S (pfp_source_scale_transport_new) = S ((S (pfp_index_scale_transport_new)) * AC)) /\ exists ff_q_pfp_scale_transport_newsource. AB = ff_q_pfp_scale_transport_newsource * S ((S (pfp_index_scale_transport_new)) * AC) + (pfp_source_scale_transport_new))) /\ (((((exists ff_h_pfp_scale_transport_newtarget. ff_h_pfp_scale_transport_newtarget + S (pfp_value_scale_transport_new) = S ((S (pfp_index_scale_transport_new)) * BC)) /\ exists ff_q_pfp_scale_transport_newtarget. BB = ff_q_pfp_scale_transport_newtarget * S ((S (pfp_index_scale_transport_new)) * BC) + (pfp_value_scale_transport_new))) /\ ((((exists pfa_gap_scale_transport_newoperationleft. pfa_gap_scale_transport_newoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_transport_newoperationright. pfa_gap_scale_transport_newoperationright + S (pfp_source_scale_transport_new) = (p)) /\ ((((exists pfa_gap_scale_transport_newoperationresultbound. pfa_gap_scale_transport_newoperationresultbound + S (pfp_value_scale_transport_new) = (p)) /\ ((exists pfa_offset_left_scale_transport_newoperationresultcongruence pfa_offset_right_scale_transport_newoperationresultcongruence. ((k) * (pfp_source_scale_transport_new)) + (p) * pfa_offset_left_scale_transport_newoperationresultcongruence = (pfp_value_scale_transport_new) + (p) * pfa_offset_right_scale_transport_newoperationresultcongruence)))))))))))))))))

forall p b c l. (~((p) = 1) /\ forall pfa_factor_left_scale_one_prime pfa_factor_right_scale_one_prime. (p) = pfa_factor_left_scale_one_prime * pfa_factor_right_scale_one_prime -> pfa_factor_left_scale_one_prime = 1 \/ pfa_factor_right_scale_one_prime = 1) -> (forall fom_index_pfp_scale_one_coefficients. (exists fom_gap_pfp_scale_one_coefficients_index_bound. fom_gap_pfp_scale_one_coefficients_index_bound + S (fom_index_pfp_scale_one_coefficients) = l) -> exists fom_value_pfp_scale_one_coefficients. ((((exists fom_beta_height_pfp_scale_one_coefficients_entry. fom_beta_height_pfp_scale_one_coefficients_entry + S (fom_value_pfp_scale_one_coefficients) = S ((S (fom_index_pfp_scale_one_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_scale_one_coefficients_entry. b = fom_beta_quotient_pfp_scale_one_coefficients_entry * S ((S (fom_index_pfp_scale_one_coefficients)) * c) + (fom_value_pfp_scale_one_coefficients))) /\ (exists fom_gap_pfp_scale_one_coefficients_value_bound. fom_gap_pfp_scale_one_coefficients_value_bound + S (fom_value_pfp_scale_one_coefficients) = p))) -> (((exists pfa_gap_scale_one_resultscalar. pfa_gap_scale_one_resultscalar + S (1) = (p)) /\ ((forall pfp_index_scale_one_result. (exists pfa_gap_scale_one_resultindex. pfa_gap_scale_one_resultindex + S (pfp_index_scale_one_result) = (l)) -> exists pfp_source_scale_one_result pfp_value_scale_one_result. ((((exists ff_h_pfp_scale_one_resultsource. ff_h_pfp_scale_one_resultsource + S (pfp_source_scale_one_result) = S ((S (pfp_index_scale_one_result)) * c)) /\ exists ff_q_pfp_scale_one_resultsource. b = ff_q_pfp_scale_one_resultsource * S ((S (pfp_index_scale_one_result)) * c) + (pfp_source_scale_one_result))) /\ (((((exists ff_h_pfp_scale_one_resulttarget. ff_h_pfp_scale_one_resulttarget + S (pfp_value_scale_one_result) = S ((S (pfp_index_scale_one_result)) * c)) /\ exists ff_q_pfp_scale_one_resulttarget. b = ff_q_pfp_scale_one_resulttarget * S ((S (pfp_index_scale_one_result)) * c) + (pfp_value_scale_one_result))) /\ ((((exists pfa_gap_scale_one_resultoperationleft. pfa_gap_scale_one_resultoperationleft + S (1) = (p)) /\ (((exists pfa_gap_scale_one_resultoperationright. pfa_gap_scale_one_resultoperationright + S (pfp_source_scale_one_result) = (p)) /\ ((((exists pfa_gap_scale_one_resultoperationresultbound. pfa_gap_scale_one_resultoperationresultbound + S (pfp_value_scale_one_result) = (p)) /\ ((exists pfa_offset_left_scale_one_resultoperationresultcongruence pfa_offset_right_scale_one_resultoperationresultcongruence. ((1) * (pfp_source_scale_one_result)) + (p) * pfa_offset_left_scale_one_resultoperationresultcongruence = (pfp_value_scale_one_result) + (p) * pfa_offset_right_scale_one_resultoperationresultcongruence)))))))))))))))))

forall p b c zb zc l. (~((p) = 1) /\ forall pfa_factor_left_scale_zero_prime pfa_factor_right_scale_zero_prime. (p) = pfa_factor_left_scale_zero_prime * pfa_factor_right_scale_zero_prime -> pfa_factor_left_scale_zero_prime = 1 \/ pfa_factor_right_scale_zero_prime = 1) -> (forall fom_index_pfp_scale_zero_coefficients. (exists fom_gap_pfp_scale_zero_coefficients_index_bound. fom_gap_pfp_scale_zero_coefficients_index_bound + S (fom_index_pfp_scale_zero_coefficients) = l) -> exists fom_value_pfp_scale_zero_coefficients. ((((exists fom_beta_height_pfp_scale_zero_coefficients_entry. fom_beta_height_pfp_scale_zero_coefficients_entry + S (fom_value_pfp_scale_zero_coefficients) = S ((S (fom_index_pfp_scale_zero_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_scale_zero_coefficients_entry. b = fom_beta_quotient_pfp_scale_zero_coefficients_entry * S ((S (fom_index_pfp_scale_zero_coefficients)) * c) + (fom_value_pfp_scale_zero_coefficients))) /\ (exists fom_gap_pfp_scale_zero_coefficients_value_bound. fom_gap_pfp_scale_zero_coefficients_value_bound + S (fom_value_pfp_scale_zero_coefficients) = p))) -> (forall pfp_repeat_index_scale_zero_table. (exists pfa_gap_scale_zero_tableindex. pfa_gap_scale_zero_tableindex + S (pfp_repeat_index_scale_zero_table) = (l)) -> (((exists ff_h_pfp_scale_zero_tableentry. ff_h_pfp_scale_zero_tableentry + S (0) = S ((S (pfp_repeat_index_scale_zero_table)) * zc)) /\ exists ff_q_pfp_scale_zero_tableentry. zb = ff_q_pfp_scale_zero_tableentry * S ((S (pfp_repeat_index_scale_zero_table)) * zc) + (0)))) -> (((exists pfa_gap_scale_zero_resultscalar. pfa_gap_scale_zero_resultscalar + S (0) = (p)) /\ ((forall pfp_index_scale_zero_result. (exists pfa_gap_scale_zero_resultindex. pfa_gap_scale_zero_resultindex + S (pfp_index_scale_zero_result) = (l)) -> exists pfp_source_scale_zero_result pfp_value_scale_zero_result. ((((exists ff_h_pfp_scale_zero_resultsource. ff_h_pfp_scale_zero_resultsource + S (pfp_source_scale_zero_result) = S ((S (pfp_index_scale_zero_result)) * c)) /\ exists ff_q_pfp_scale_zero_resultsource. b = ff_q_pfp_scale_zero_resultsource * S ((S (pfp_index_scale_zero_result)) * c) + (pfp_source_scale_zero_result))) /\ (((((exists ff_h_pfp_scale_zero_resulttarget. ff_h_pfp_scale_zero_resulttarget + S (pfp_value_scale_zero_result) = S ((S (pfp_index_scale_zero_result)) * zc)) /\ exists ff_q_pfp_scale_zero_resulttarget. zb = ff_q_pfp_scale_zero_resulttarget * S ((S (pfp_index_scale_zero_result)) * zc) + (pfp_value_scale_zero_result))) /\ ((((exists pfa_gap_scale_zero_resultoperationleft. pfa_gap_scale_zero_resultoperationleft + S (0) = (p)) /\ (((exists pfa_gap_scale_zero_resultoperationright. pfa_gap_scale_zero_resultoperationright + S (pfp_source_scale_zero_result) = (p)) /\ ((((exists pfa_gap_scale_zero_resultoperationresultbound. pfa_gap_scale_zero_resultoperationresultbound + S (pfp_value_scale_zero_result) = (p)) /\ ((exists pfa_offset_left_scale_zero_resultoperationresultcongruence pfa_offset_right_scale_zero_resultoperationresultcongruence. ((0) * (pfp_source_scale_zero_result)) + (p) * pfa_offset_left_scale_zero_resultoperationresultcongruence = (pfp_value_scale_zero_result) + (p) * pfa_offset_right_scale_zero_resultoperationresultcongruence)))))))))))))))))

forall p ab ac bb bc cb cc xb xc yb yc ub uc vb vc l. (forall pfp_index_assoc_ab. (exists pfa_gap_assoc_abindex. pfa_gap_assoc_abindex + S (pfp_index_assoc_ab) = (l)) -> exists pfp_left_assoc_ab pfp_right_assoc_ab pfp_value_assoc_ab. ((((exists ff_h_pfp_assoc_ableft. ff_h_pfp_assoc_ableft + S (pfp_left_assoc_ab) = S ((S (pfp_index_assoc_ab)) * ac)) /\ exists ff_q_pfp_assoc_ableft. ab = ff_q_pfp_assoc_ableft * S ((S (pfp_index_assoc_ab)) * ac) + (pfp_left_assoc_ab))) /\ (((((exists ff_h_pfp_assoc_abright. ff_h_pfp_assoc_abright + S (pfp_right_assoc_ab) = S ((S (pfp_index_assoc_ab)) * bc)) /\ exists ff_q_pfp_assoc_abright. bb = ff_q_pfp_assoc_abright * S ((S (pfp_index_assoc_ab)) * bc) + (pfp_right_assoc_ab))) /\ (((((exists ff_h_pfp_assoc_abtarget. ff_h_pfp_assoc_abtarget + S (pfp_value_assoc_ab) = S ((S (pfp_index_assoc_ab)) * xc)) /\ exists ff_q_pfp_assoc_abtarget. xb = ff_q_pfp_assoc_abtarget * S ((S (pfp_index_assoc_ab)) * xc) + (pfp_value_assoc_ab))) /\ ((((exists pfa_gap_assoc_aboperationleft. pfa_gap_assoc_aboperationleft + S (pfp_left_assoc_ab) = (p)) /\ (((exists pfa_gap_assoc_aboperationright. pfa_gap_assoc_aboperationright + S (pfp_right_assoc_ab) = (p)) /\ ((((exists pfa_gap_assoc_aboperationresultbound. pfa_gap_assoc_aboperationresultbound + S (pfp_value_assoc_ab) = (p)) /\ ((exists pfa_offset_left_assoc_aboperationresultcongruence pfa_offset_right_assoc_aboperationresultcongruence. ((pfp_left_assoc_ab) + (pfp_right_assoc_ab)) + (p) * pfa_offset_left_assoc_aboperationresultcongruence = (pfp_value_assoc_ab) + (p) * pfa_offset_right_assoc_aboperationresultcongruence)))))))))))))))) -> (forall pfp_index_assoc_left. (exists pfa_gap_assoc_leftindex. pfa_gap_assoc_leftindex + S (pfp_index_assoc_left) = (l)) -> exists pfp_left_assoc_left pfp_right_assoc_left pfp_value_assoc_left. ((((exists ff_h_pfp_assoc_leftleft. ff_h_pfp_assoc_leftleft + S (pfp_left_assoc_left) = S ((S (pfp_index_assoc_left)) * xc)) /\ exists ff_q_pfp_assoc_leftleft. xb = ff_q_pfp_assoc_leftleft * S ((S (pfp_index_assoc_left)) * xc) + (pfp_left_assoc_left))) /\ (((((exists ff_h_pfp_assoc_leftright. ff_h_pfp_assoc_leftright + S (pfp_right_assoc_left) = S ((S (pfp_index_assoc_left)) * cc)) /\ exists ff_q_pfp_assoc_leftright. cb = ff_q_pfp_assoc_leftright * S ((S (pfp_index_assoc_left)) * cc) + (pfp_right_assoc_left))) /\ (((((exists ff_h_pfp_assoc_lefttarget. ff_h_pfp_assoc_lefttarget + S (pfp_value_assoc_left) = S ((S (pfp_index_assoc_left)) * uc)) /\ exists ff_q_pfp_assoc_lefttarget. ub = ff_q_pfp_assoc_lefttarget * S ((S (pfp_index_assoc_left)) * uc) + (pfp_value_assoc_left))) /\ ((((exists pfa_gap_assoc_leftoperationleft. pfa_gap_assoc_leftoperationleft + S (pfp_left_assoc_left) = (p)) /\ (((exists pfa_gap_assoc_leftoperationright. pfa_gap_assoc_leftoperationright + S (pfp_right_assoc_left) = (p)) /\ ((((exists pfa_gap_assoc_leftoperationresultbound. pfa_gap_assoc_leftoperationresultbound + S (pfp_value_assoc_left) = (p)) /\ ((exists pfa_offset_left_assoc_leftoperationresultcongruence pfa_offset_right_assoc_leftoperationresultcongruence. ((pfp_left_assoc_left) + (pfp_right_assoc_left)) + (p) * pfa_offset_left_assoc_leftoperationresultcongruence = (pfp_value_assoc_left) + (p) * pfa_offset_right_assoc_leftoperationresultcongruence)))))))))))))))) -> (forall pfp_index_assoc_bc. (exists pfa_gap_assoc_bcindex. pfa_gap_assoc_bcindex + S (pfp_index_assoc_bc) = (l)) -> exists pfp_left_assoc_bc pfp_right_assoc_bc pfp_value_assoc_bc. ((((exists ff_h_pfp_assoc_bcleft. ff_h_pfp_assoc_bcleft + S (pfp_left_assoc_bc) = S ((S (pfp_index_assoc_bc)) * bc)) /\ exists ff_q_pfp_assoc_bcleft. bb = ff_q_pfp_assoc_bcleft * S ((S (pfp_index_assoc_bc)) * bc) + (pfp_left_assoc_bc))) /\ (((((exists ff_h_pfp_assoc_bcright. ff_h_pfp_assoc_bcright + S (pfp_right_assoc_bc) = S ((S (pfp_index_assoc_bc)) * cc)) /\ exists ff_q_pfp_assoc_bcright. cb = ff_q_pfp_assoc_bcright * S ((S (pfp_index_assoc_bc)) * cc) + (pfp_right_assoc_bc))) /\ (((((exists ff_h_pfp_assoc_bctarget. ff_h_pfp_assoc_bctarget + S (pfp_value_assoc_bc) = S ((S (pfp_index_assoc_bc)) * yc)) /\ exists ff_q_pfp_assoc_bctarget. yb = ff_q_pfp_assoc_bctarget * S ((S (pfp_index_assoc_bc)) * yc) + (pfp_value_assoc_bc))) /\ ((((exists pfa_gap_assoc_bcoperationleft. pfa_gap_assoc_bcoperationleft + S (pfp_left_assoc_bc) = (p)) /\ (((exists pfa_gap_assoc_bcoperationright. pfa_gap_assoc_bcoperationright + S (pfp_right_assoc_bc) = (p)) /\ ((((exists pfa_gap_assoc_bcoperationresultbound. pfa_gap_assoc_bcoperationresultbound + S (pfp_value_assoc_bc) = (p)) /\ ((exists pfa_offset_left_assoc_bcoperationresultcongruence pfa_offset_right_assoc_bcoperationresultcongruence. ((pfp_left_assoc_bc) + (pfp_right_assoc_bc)) + (p) * pfa_offset_left_assoc_bcoperationresultcongruence = (pfp_value_assoc_bc) + (p) * pfa_offset_right_assoc_bcoperationresultcongruence)))))))))))))))) -> (forall pfp_index_assoc_right. (exists pfa_gap_assoc_rightindex. pfa_gap_assoc_rightindex + S (pfp_index_assoc_right) = (l)) -> exists pfp_left_assoc_right pfp_right_assoc_right pfp_value_assoc_right. ((((exists ff_h_pfp_assoc_rightleft. ff_h_pfp_assoc_rightleft + S (pfp_left_assoc_right) = S ((S (pfp_index_assoc_right)) * ac)) /\ exists ff_q_pfp_assoc_rightleft. ab = ff_q_pfp_assoc_rightleft * S ((S (pfp_index_assoc_right)) * ac) + (pfp_left_assoc_right))) /\ (((((exists ff_h_pfp_assoc_rightright. ff_h_pfp_assoc_rightright + S (pfp_right_assoc_right) = S ((S (pfp_index_assoc_right)) * yc)) /\ exists ff_q_pfp_assoc_rightright. yb = ff_q_pfp_assoc_rightright * S ((S (pfp_index_assoc_right)) * yc) + (pfp_right_assoc_right))) /\ (((((exists ff_h_pfp_assoc_righttarget. ff_h_pfp_assoc_righttarget + S (pfp_value_assoc_right) = S ((S (pfp_index_assoc_right)) * vc)) /\ exists ff_q_pfp_assoc_righttarget. vb = ff_q_pfp_assoc_righttarget * S ((S (pfp_index_assoc_right)) * vc) + (pfp_value_assoc_right))) /\ ((((exists pfa_gap_assoc_rightoperationleft. pfa_gap_assoc_rightoperationleft + S (pfp_left_assoc_right) = (p)) /\ (((exists pfa_gap_assoc_rightoperationright. pfa_gap_assoc_rightoperationright + S (pfp_right_assoc_right) = (p)) /\ ((((exists pfa_gap_assoc_rightoperationresultbound. pfa_gap_assoc_rightoperationresultbound + S (pfp_value_assoc_right) = (p)) /\ ((exists pfa_offset_left_assoc_rightoperationresultcongruence pfa_offset_right_assoc_rightoperationresultcongruence. ((pfp_left_assoc_right) + (pfp_right_assoc_right)) + (p) * pfa_offset_left_assoc_rightoperationresultcongruence = (pfp_value_assoc_right) + (p) * pfa_offset_right_assoc_rightoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_assoc_result mdr_a_pfp_assoc_result. (exists mdr_gap_pfp_assoc_resultb. mdr_gap_pfp_assoc_resultb + S (mdr_i_pfp_assoc_result) = (l)) -> (((exists ff_h_mdr_pfp_assoc_resulto. ff_h_mdr_pfp_assoc_resulto + S (mdr_a_pfp_assoc_result) = S ((S (mdr_i_pfp_assoc_result)) * uc)) /\ exists ff_q_mdr_pfp_assoc_resulto. ub = ff_q_mdr_pfp_assoc_resulto * S ((S (mdr_i_pfp_assoc_result)) * uc) + (mdr_a_pfp_assoc_result))) -> (((exists ff_h_mdr_pfp_assoc_resultn. ff_h_mdr_pfp_assoc_resultn + S (mdr_a_pfp_assoc_result) = S ((S (mdr_i_pfp_assoc_result)) * vc)) /\ exists ff_q_mdr_pfp_assoc_resultn. vb = ff_q_mdr_pfp_assoc_resultn * S ((S (mdr_i_pfp_assoc_result)) * vc) + (mdr_a_pfp_assoc_result))))

forall p a b k ab ac bb bc ub uc vb vc l. (((exists pfa_gap_scale_assoc_scalarsleft. pfa_gap_scale_assoc_scalarsleft + S (a) = (p)) /\ (((exists pfa_gap_scale_assoc_scalarsright. pfa_gap_scale_assoc_scalarsright + S (b) = (p)) /\ ((((exists pfa_gap_scale_assoc_scalarsresultbound. pfa_gap_scale_assoc_scalarsresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_scale_assoc_scalarsresultcongruence pfa_offset_right_scale_assoc_scalarsresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_scale_assoc_scalarsresultcongruence = (k) + (p) * pfa_offset_right_scale_assoc_scalarsresultcongruence))))))))) -> (((exists pfa_gap_scale_assoc_firstscalar. pfa_gap_scale_assoc_firstscalar + S (b) = (p)) /\ ((forall pfp_index_scale_assoc_first. (exists pfa_gap_scale_assoc_firstindex. pfa_gap_scale_assoc_firstindex + S (pfp_index_scale_assoc_first) = (l)) -> exists pfp_source_scale_assoc_first pfp_value_scale_assoc_first. ((((exists ff_h_pfp_scale_assoc_firstsource. ff_h_pfp_scale_assoc_firstsource + S (pfp_source_scale_assoc_first) = S ((S (pfp_index_scale_assoc_first)) * ac)) /\ exists ff_q_pfp_scale_assoc_firstsource. ab = ff_q_pfp_scale_assoc_firstsource * S ((S (pfp_index_scale_assoc_first)) * ac) + (pfp_source_scale_assoc_first))) /\ (((((exists ff_h_pfp_scale_assoc_firsttarget. ff_h_pfp_scale_assoc_firsttarget + S (pfp_value_scale_assoc_first) = S ((S (pfp_index_scale_assoc_first)) * bc)) /\ exists ff_q_pfp_scale_assoc_firsttarget. bb = ff_q_pfp_scale_assoc_firsttarget * S ((S (pfp_index_scale_assoc_first)) * bc) + (pfp_value_scale_assoc_first))) /\ ((((exists pfa_gap_scale_assoc_firstoperationleft. pfa_gap_scale_assoc_firstoperationleft + S (b) = (p)) /\ (((exists pfa_gap_scale_assoc_firstoperationright. pfa_gap_scale_assoc_firstoperationright + S (pfp_source_scale_assoc_first) = (p)) /\ ((((exists pfa_gap_scale_assoc_firstoperationresultbound. pfa_gap_scale_assoc_firstoperationresultbound + S (pfp_value_scale_assoc_first) = (p)) /\ ((exists pfa_offset_left_scale_assoc_firstoperationresultcongruence pfa_offset_right_scale_assoc_firstoperationresultcongruence. ((b) * (pfp_source_scale_assoc_first)) + (p) * pfa_offset_left_scale_assoc_firstoperationresultcongruence = (pfp_value_scale_assoc_first) + (p) * pfa_offset_right_scale_assoc_firstoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_scale_assoc_secondscalar. pfa_gap_scale_assoc_secondscalar + S (a) = (p)) /\ ((forall pfp_index_scale_assoc_second. (exists pfa_gap_scale_assoc_secondindex. pfa_gap_scale_assoc_secondindex + S (pfp_index_scale_assoc_second) = (l)) -> exists pfp_source_scale_assoc_second pfp_value_scale_assoc_second. ((((exists ff_h_pfp_scale_assoc_secondsource. ff_h_pfp_scale_assoc_secondsource + S (pfp_source_scale_assoc_second) = S ((S (pfp_index_scale_assoc_second)) * bc)) /\ exists ff_q_pfp_scale_assoc_secondsource. bb = ff_q_pfp_scale_assoc_secondsource * S ((S (pfp_index_scale_assoc_second)) * bc) + (pfp_source_scale_assoc_second))) /\ (((((exists ff_h_pfp_scale_assoc_secondtarget. ff_h_pfp_scale_assoc_secondtarget + S (pfp_value_scale_assoc_second) = S ((S (pfp_index_scale_assoc_second)) * uc)) /\ exists ff_q_pfp_scale_assoc_secondtarget. ub = ff_q_pfp_scale_assoc_secondtarget * S ((S (pfp_index_scale_assoc_second)) * uc) + (pfp_value_scale_assoc_second))) /\ ((((exists pfa_gap_scale_assoc_secondoperationleft. pfa_gap_scale_assoc_secondoperationleft + S (a) = (p)) /\ (((exists pfa_gap_scale_assoc_secondoperationright. pfa_gap_scale_assoc_secondoperationright + S (pfp_source_scale_assoc_second) = (p)) /\ ((((exists pfa_gap_scale_assoc_secondoperationresultbound. pfa_gap_scale_assoc_secondoperationresultbound + S (pfp_value_scale_assoc_second) = (p)) /\ ((exists pfa_offset_left_scale_assoc_secondoperationresultcongruence pfa_offset_right_scale_assoc_secondoperationresultcongruence. ((a) * (pfp_source_scale_assoc_second)) + (p) * pfa_offset_left_scale_assoc_secondoperationresultcongruence = (pfp_value_scale_assoc_second) + (p) * pfa_offset_right_scale_assoc_secondoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_scale_assoc_productscalar. pfa_gap_scale_assoc_productscalar + S (k) = (p)) /\ ((forall pfp_index_scale_assoc_product. (exists pfa_gap_scale_assoc_productindex. pfa_gap_scale_assoc_productindex + S (pfp_index_scale_assoc_product) = (l)) -> exists pfp_source_scale_assoc_product pfp_value_scale_assoc_product. ((((exists ff_h_pfp_scale_assoc_productsource. ff_h_pfp_scale_assoc_productsource + S (pfp_source_scale_assoc_product) = S ((S (pfp_index_scale_assoc_product)) * ac)) /\ exists ff_q_pfp_scale_assoc_productsource. ab = ff_q_pfp_scale_assoc_productsource * S ((S (pfp_index_scale_assoc_product)) * ac) + (pfp_source_scale_assoc_product))) /\ (((((exists ff_h_pfp_scale_assoc_producttarget. ff_h_pfp_scale_assoc_producttarget + S (pfp_value_scale_assoc_product) = S ((S (pfp_index_scale_assoc_product)) * vc)) /\ exists ff_q_pfp_scale_assoc_producttarget. vb = ff_q_pfp_scale_assoc_producttarget * S ((S (pfp_index_scale_assoc_product)) * vc) + (pfp_value_scale_assoc_product))) /\ ((((exists pfa_gap_scale_assoc_productoperationleft. pfa_gap_scale_assoc_productoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_assoc_productoperationright. pfa_gap_scale_assoc_productoperationright + S (pfp_source_scale_assoc_product) = (p)) /\ ((((exists pfa_gap_scale_assoc_productoperationresultbound. pfa_gap_scale_assoc_productoperationresultbound + S (pfp_value_scale_assoc_product) = (p)) /\ ((exists pfa_offset_left_scale_assoc_productoperationresultcongruence pfa_offset_right_scale_assoc_productoperationresultcongruence. ((k) * (pfp_source_scale_assoc_product)) + (p) * pfa_offset_left_scale_assoc_productoperationresultcongruence = (pfp_value_scale_assoc_product) + (p) * pfa_offset_right_scale_assoc_productoperationresultcongruence))))))))))))))))) -> (forall mdr_i_pfp_scale_assoc_result mdr_a_pfp_scale_assoc_result. (exists mdr_gap_pfp_scale_assoc_resultb. mdr_gap_pfp_scale_assoc_resultb + S (mdr_i_pfp_scale_assoc_result) = (l)) -> (((exists ff_h_mdr_pfp_scale_assoc_resulto. ff_h_mdr_pfp_scale_assoc_resulto + S (mdr_a_pfp_scale_assoc_result) = S ((S (mdr_i_pfp_scale_assoc_result)) * uc)) /\ exists ff_q_mdr_pfp_scale_assoc_resulto. ub = ff_q_mdr_pfp_scale_assoc_resulto * S ((S (mdr_i_pfp_scale_assoc_result)) * uc) + (mdr_a_pfp_scale_assoc_result))) -> (((exists ff_h_mdr_pfp_scale_assoc_resultn. ff_h_mdr_pfp_scale_assoc_resultn + S (mdr_a_pfp_scale_assoc_result) = S ((S (mdr_i_pfp_scale_assoc_result)) * vc)) /\ exists ff_q_mdr_pfp_scale_assoc_resultn. vb = ff_q_mdr_pfp_scale_assoc_resultn * S ((S (mdr_i_pfp_scale_assoc_result)) * vc) + (mdr_a_pfp_scale_assoc_result))))

forall p k ab ac bb bc sb sc xb xc yb yc ub uc vb vc l. (forall pfp_index_distribute_sum. (exists pfa_gap_distribute_sumindex. pfa_gap_distribute_sumindex + S (pfp_index_distribute_sum) = (l)) -> exists pfp_left_distribute_sum pfp_right_distribute_sum pfp_value_distribute_sum. ((((exists ff_h_pfp_distribute_sumleft. ff_h_pfp_distribute_sumleft + S (pfp_left_distribute_sum) = S ((S (pfp_index_distribute_sum)) * ac)) /\ exists ff_q_pfp_distribute_sumleft. ab = ff_q_pfp_distribute_sumleft * S ((S (pfp_index_distribute_sum)) * ac) + (pfp_left_distribute_sum))) /\ (((((exists ff_h_pfp_distribute_sumright. ff_h_pfp_distribute_sumright + S (pfp_right_distribute_sum) = S ((S (pfp_index_distribute_sum)) * bc)) /\ exists ff_q_pfp_distribute_sumright. bb = ff_q_pfp_distribute_sumright * S ((S (pfp_index_distribute_sum)) * bc) + (pfp_right_distribute_sum))) /\ (((((exists ff_h_pfp_distribute_sumtarget. ff_h_pfp_distribute_sumtarget + S (pfp_value_distribute_sum) = S ((S (pfp_index_distribute_sum)) * sc)) /\ exists ff_q_pfp_distribute_sumtarget. sb = ff_q_pfp_distribute_sumtarget * S ((S (pfp_index_distribute_sum)) * sc) + (pfp_value_distribute_sum))) /\ ((((exists pfa_gap_distribute_sumoperationleft. pfa_gap_distribute_sumoperationleft + S (pfp_left_distribute_sum) = (p)) /\ (((exists pfa_gap_distribute_sumoperationright. pfa_gap_distribute_sumoperationright + S (pfp_right_distribute_sum) = (p)) /\ ((((exists pfa_gap_distribute_sumoperationresultbound. pfa_gap_distribute_sumoperationresultbound + S (pfp_value_distribute_sum) = (p)) /\ ((exists pfa_offset_left_distribute_sumoperationresultcongruence pfa_offset_right_distribute_sumoperationresultcongruence. ((pfp_left_distribute_sum) + (pfp_right_distribute_sum)) + (p) * pfa_offset_left_distribute_sumoperationresultcongruence = (pfp_value_distribute_sum) + (p) * pfa_offset_right_distribute_sumoperationresultcongruence)))))))))))))))) -> (((exists pfa_gap_distribute_leftscalar. pfa_gap_distribute_leftscalar + S (k) = (p)) /\ ((forall pfp_index_distribute_left. (exists pfa_gap_distribute_leftindex. pfa_gap_distribute_leftindex + S (pfp_index_distribute_left) = (l)) -> exists pfp_source_distribute_left pfp_value_distribute_left. ((((exists ff_h_pfp_distribute_leftsource. ff_h_pfp_distribute_leftsource + S (pfp_source_distribute_left) = S ((S (pfp_index_distribute_left)) * sc)) /\ exists ff_q_pfp_distribute_leftsource. sb = ff_q_pfp_distribute_leftsource * S ((S (pfp_index_distribute_left)) * sc) + (pfp_source_distribute_left))) /\ (((((exists ff_h_pfp_distribute_lefttarget. ff_h_pfp_distribute_lefttarget + S (pfp_value_distribute_left) = S ((S (pfp_index_distribute_left)) * uc)) /\ exists ff_q_pfp_distribute_lefttarget. ub = ff_q_pfp_distribute_lefttarget * S ((S (pfp_index_distribute_left)) * uc) + (pfp_value_distribute_left))) /\ ((((exists pfa_gap_distribute_leftoperationleft. pfa_gap_distribute_leftoperationleft + S (k) = (p)) /\ (((exists pfa_gap_distribute_leftoperationright. pfa_gap_distribute_leftoperationright + S (pfp_source_distribute_left) = (p)) /\ ((((exists pfa_gap_distribute_leftoperationresultbound. pfa_gap_distribute_leftoperationresultbound + S (pfp_value_distribute_left) = (p)) /\ ((exists pfa_offset_left_distribute_leftoperationresultcongruence pfa_offset_right_distribute_leftoperationresultcongruence. ((k) * (pfp_source_distribute_left)) + (p) * pfa_offset_left_distribute_leftoperationresultcongruence = (pfp_value_distribute_left) + (p) * pfa_offset_right_distribute_leftoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_distribute_ascalar. pfa_gap_distribute_ascalar + S (k) = (p)) /\ ((forall pfp_index_distribute_a. (exists pfa_gap_distribute_aindex. pfa_gap_distribute_aindex + S (pfp_index_distribute_a) = (l)) -> exists pfp_source_distribute_a pfp_value_distribute_a. ((((exists ff_h_pfp_distribute_asource. ff_h_pfp_distribute_asource + S (pfp_source_distribute_a) = S ((S (pfp_index_distribute_a)) * ac)) /\ exists ff_q_pfp_distribute_asource. ab = ff_q_pfp_distribute_asource * S ((S (pfp_index_distribute_a)) * ac) + (pfp_source_distribute_a))) /\ (((((exists ff_h_pfp_distribute_atarget. ff_h_pfp_distribute_atarget + S (pfp_value_distribute_a) = S ((S (pfp_index_distribute_a)) * xc)) /\ exists ff_q_pfp_distribute_atarget. xb = ff_q_pfp_distribute_atarget * S ((S (pfp_index_distribute_a)) * xc) + (pfp_value_distribute_a))) /\ ((((exists pfa_gap_distribute_aoperationleft. pfa_gap_distribute_aoperationleft + S (k) = (p)) /\ (((exists pfa_gap_distribute_aoperationright. pfa_gap_distribute_aoperationright + S (pfp_source_distribute_a) = (p)) /\ ((((exists pfa_gap_distribute_aoperationresultbound. pfa_gap_distribute_aoperationresultbound + S (pfp_value_distribute_a) = (p)) /\ ((exists pfa_offset_left_distribute_aoperationresultcongruence pfa_offset_right_distribute_aoperationresultcongruence. ((k) * (pfp_source_distribute_a)) + (p) * pfa_offset_left_distribute_aoperationresultcongruence = (pfp_value_distribute_a) + (p) * pfa_offset_right_distribute_aoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_distribute_bscalar. pfa_gap_distribute_bscalar + S (k) = (p)) /\ ((forall pfp_index_distribute_b. (exists pfa_gap_distribute_bindex. pfa_gap_distribute_bindex + S (pfp_index_distribute_b) = (l)) -> exists pfp_source_distribute_b pfp_value_distribute_b. ((((exists ff_h_pfp_distribute_bsource. ff_h_pfp_distribute_bsource + S (pfp_source_distribute_b) = S ((S (pfp_index_distribute_b)) * bc)) /\ exists ff_q_pfp_distribute_bsource. bb = ff_q_pfp_distribute_bsource * S ((S (pfp_index_distribute_b)) * bc) + (pfp_source_distribute_b))) /\ (((((exists ff_h_pfp_distribute_btarget. ff_h_pfp_distribute_btarget + S (pfp_value_distribute_b) = S ((S (pfp_index_distribute_b)) * yc)) /\ exists ff_q_pfp_distribute_btarget. yb = ff_q_pfp_distribute_btarget * S ((S (pfp_index_distribute_b)) * yc) + (pfp_value_distribute_b))) /\ ((((exists pfa_gap_distribute_boperationleft. pfa_gap_distribute_boperationleft + S (k) = (p)) /\ (((exists pfa_gap_distribute_boperationright. pfa_gap_distribute_boperationright + S (pfp_source_distribute_b) = (p)) /\ ((((exists pfa_gap_distribute_boperationresultbound. pfa_gap_distribute_boperationresultbound + S (pfp_value_distribute_b) = (p)) /\ ((exists pfa_offset_left_distribute_boperationresultcongruence pfa_offset_right_distribute_boperationresultcongruence. ((k) * (pfp_source_distribute_b)) + (p) * pfa_offset_left_distribute_boperationresultcongruence = (pfp_value_distribute_b) + (p) * pfa_offset_right_distribute_boperationresultcongruence))))))))))))))))) -> (forall pfp_index_distribute_right. (exists pfa_gap_distribute_rightindex. pfa_gap_distribute_rightindex + S (pfp_index_distribute_right) = (l)) -> exists pfp_left_distribute_right pfp_right_distribute_right pfp_value_distribute_right. ((((exists ff_h_pfp_distribute_rightleft. ff_h_pfp_distribute_rightleft + S (pfp_left_distribute_right) = S ((S (pfp_index_distribute_right)) * xc)) /\ exists ff_q_pfp_distribute_rightleft. xb = ff_q_pfp_distribute_rightleft * S ((S (pfp_index_distribute_right)) * xc) + (pfp_left_distribute_right))) /\ (((((exists ff_h_pfp_distribute_rightright. ff_h_pfp_distribute_rightright + S (pfp_right_distribute_right) = S ((S (pfp_index_distribute_right)) * yc)) /\ exists ff_q_pfp_distribute_rightright. yb = ff_q_pfp_distribute_rightright * S ((S (pfp_index_distribute_right)) * yc) + (pfp_right_distribute_right))) /\ (((((exists ff_h_pfp_distribute_righttarget. ff_h_pfp_distribute_righttarget + S (pfp_value_distribute_right) = S ((S (pfp_index_distribute_right)) * vc)) /\ exists ff_q_pfp_distribute_righttarget. vb = ff_q_pfp_distribute_righttarget * S ((S (pfp_index_distribute_right)) * vc) + (pfp_value_distribute_right))) /\ ((((exists pfa_gap_distribute_rightoperationleft. pfa_gap_distribute_rightoperationleft + S (pfp_left_distribute_right) = (p)) /\ (((exists pfa_gap_distribute_rightoperationright. pfa_gap_distribute_rightoperationright + S (pfp_right_distribute_right) = (p)) /\ ((((exists pfa_gap_distribute_rightoperationresultbound. pfa_gap_distribute_rightoperationresultbound + S (pfp_value_distribute_right) = (p)) /\ ((exists pfa_offset_left_distribute_rightoperationresultcongruence pfa_offset_right_distribute_rightoperationresultcongruence. ((pfp_left_distribute_right) + (pfp_right_distribute_right)) + (p) * pfa_offset_left_distribute_rightoperationresultcongruence = (pfp_value_distribute_right) + (p) * pfa_offset_right_distribute_rightoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_distribute_result mdr_a_pfp_distribute_result. (exists mdr_gap_pfp_distribute_resultb. mdr_gap_pfp_distribute_resultb + S (mdr_i_pfp_distribute_result) = (l)) -> (((exists ff_h_mdr_pfp_distribute_resulto. ff_h_mdr_pfp_distribute_resulto + S (mdr_a_pfp_distribute_result) = S ((S (mdr_i_pfp_distribute_result)) * uc)) /\ exists ff_q_mdr_pfp_distribute_resulto. ub = ff_q_mdr_pfp_distribute_resulto * S ((S (mdr_i_pfp_distribute_result)) * uc) + (mdr_a_pfp_distribute_result))) -> (((exists ff_h_mdr_pfp_distribute_resultn. ff_h_mdr_pfp_distribute_resultn + S (mdr_a_pfp_distribute_result) = S ((S (mdr_i_pfp_distribute_result)) * vc)) /\ exists ff_q_mdr_pfp_distribute_resultn. vb = ff_q_mdr_pfp_distribute_resultn * S ((S (mdr_i_pfp_distribute_result)) * vc) + (mdr_a_pfp_distribute_result))))

forall p a b k ab ac xb xc yb yc ub uc vb vc l. (((exists pfa_gap_scalar_distribute_sumleft. pfa_gap_scalar_distribute_sumleft + S (a) = (p)) /\ (((exists pfa_gap_scalar_distribute_sumright. pfa_gap_scalar_distribute_sumright + S (b) = (p)) /\ ((((exists pfa_gap_scalar_distribute_sumresultbound. pfa_gap_scalar_distribute_sumresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_scalar_distribute_sumresultcongruence pfa_offset_right_scalar_distribute_sumresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_scalar_distribute_sumresultcongruence = (k) + (p) * pfa_offset_right_scalar_distribute_sumresultcongruence))))))))) -> (((exists pfa_gap_scalar_distribute_leftscalar. pfa_gap_scalar_distribute_leftscalar + S (k) = (p)) /\ ((forall pfp_index_scalar_distribute_left. (exists pfa_gap_scalar_distribute_leftindex. pfa_gap_scalar_distribute_leftindex + S (pfp_index_scalar_distribute_left) = (l)) -> exists pfp_source_scalar_distribute_left pfp_value_scalar_distribute_left. ((((exists ff_h_pfp_scalar_distribute_leftsource. ff_h_pfp_scalar_distribute_leftsource + S (pfp_source_scalar_distribute_left) = S ((S (pfp_index_scalar_distribute_left)) * ac)) /\ exists ff_q_pfp_scalar_distribute_leftsource. ab = ff_q_pfp_scalar_distribute_leftsource * S ((S (pfp_index_scalar_distribute_left)) * ac) + (pfp_source_scalar_distribute_left))) /\ (((((exists ff_h_pfp_scalar_distribute_lefttarget. ff_h_pfp_scalar_distribute_lefttarget + S (pfp_value_scalar_distribute_left) = S ((S (pfp_index_scalar_distribute_left)) * uc)) /\ exists ff_q_pfp_scalar_distribute_lefttarget. ub = ff_q_pfp_scalar_distribute_lefttarget * S ((S (pfp_index_scalar_distribute_left)) * uc) + (pfp_value_scalar_distribute_left))) /\ ((((exists pfa_gap_scalar_distribute_leftoperationleft. pfa_gap_scalar_distribute_leftoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scalar_distribute_leftoperationright. pfa_gap_scalar_distribute_leftoperationright + S (pfp_source_scalar_distribute_left) = (p)) /\ ((((exists pfa_gap_scalar_distribute_leftoperationresultbound. pfa_gap_scalar_distribute_leftoperationresultbound + S (pfp_value_scalar_distribute_left) = (p)) /\ ((exists pfa_offset_left_scalar_distribute_leftoperationresultcongruence pfa_offset_right_scalar_distribute_leftoperationresultcongruence. ((k) * (pfp_source_scalar_distribute_left)) + (p) * pfa_offset_left_scalar_distribute_leftoperationresultcongruence = (pfp_value_scalar_distribute_left) + (p) * pfa_offset_right_scalar_distribute_leftoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_scalar_distribute_ascalar. pfa_gap_scalar_distribute_ascalar + S (a) = (p)) /\ ((forall pfp_index_scalar_distribute_a. (exists pfa_gap_scalar_distribute_aindex. pfa_gap_scalar_distribute_aindex + S (pfp_index_scalar_distribute_a) = (l)) -> exists pfp_source_scalar_distribute_a pfp_value_scalar_distribute_a. ((((exists ff_h_pfp_scalar_distribute_asource. ff_h_pfp_scalar_distribute_asource + S (pfp_source_scalar_distribute_a) = S ((S (pfp_index_scalar_distribute_a)) * ac)) /\ exists ff_q_pfp_scalar_distribute_asource. ab = ff_q_pfp_scalar_distribute_asource * S ((S (pfp_index_scalar_distribute_a)) * ac) + (pfp_source_scalar_distribute_a))) /\ (((((exists ff_h_pfp_scalar_distribute_atarget. ff_h_pfp_scalar_distribute_atarget + S (pfp_value_scalar_distribute_a) = S ((S (pfp_index_scalar_distribute_a)) * xc)) /\ exists ff_q_pfp_scalar_distribute_atarget. xb = ff_q_pfp_scalar_distribute_atarget * S ((S (pfp_index_scalar_distribute_a)) * xc) + (pfp_value_scalar_distribute_a))) /\ ((((exists pfa_gap_scalar_distribute_aoperationleft. pfa_gap_scalar_distribute_aoperationleft + S (a) = (p)) /\ (((exists pfa_gap_scalar_distribute_aoperationright. pfa_gap_scalar_distribute_aoperationright + S (pfp_source_scalar_distribute_a) = (p)) /\ ((((exists pfa_gap_scalar_distribute_aoperationresultbound. pfa_gap_scalar_distribute_aoperationresultbound + S (pfp_value_scalar_distribute_a) = (p)) /\ ((exists pfa_offset_left_scalar_distribute_aoperationresultcongruence pfa_offset_right_scalar_distribute_aoperationresultcongruence. ((a) * (pfp_source_scalar_distribute_a)) + (p) * pfa_offset_left_scalar_distribute_aoperationresultcongruence = (pfp_value_scalar_distribute_a) + (p) * pfa_offset_right_scalar_distribute_aoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_scalar_distribute_bscalar. pfa_gap_scalar_distribute_bscalar + S (b) = (p)) /\ ((forall pfp_index_scalar_distribute_b. (exists pfa_gap_scalar_distribute_bindex. pfa_gap_scalar_distribute_bindex + S (pfp_index_scalar_distribute_b) = (l)) -> exists pfp_source_scalar_distribute_b pfp_value_scalar_distribute_b. ((((exists ff_h_pfp_scalar_distribute_bsource. ff_h_pfp_scalar_distribute_bsource + S (pfp_source_scalar_distribute_b) = S ((S (pfp_index_scalar_distribute_b)) * ac)) /\ exists ff_q_pfp_scalar_distribute_bsource. ab = ff_q_pfp_scalar_distribute_bsource * S ((S (pfp_index_scalar_distribute_b)) * ac) + (pfp_source_scalar_distribute_b))) /\ (((((exists ff_h_pfp_scalar_distribute_btarget. ff_h_pfp_scalar_distribute_btarget + S (pfp_value_scalar_distribute_b) = S ((S (pfp_index_scalar_distribute_b)) * yc)) /\ exists ff_q_pfp_scalar_distribute_btarget. yb = ff_q_pfp_scalar_distribute_btarget * S ((S (pfp_index_scalar_distribute_b)) * yc) + (pfp_value_scalar_distribute_b))) /\ ((((exists pfa_gap_scalar_distribute_boperationleft. pfa_gap_scalar_distribute_boperationleft + S (b) = (p)) /\ (((exists pfa_gap_scalar_distribute_boperationright. pfa_gap_scalar_distribute_boperationright + S (pfp_source_scalar_distribute_b) = (p)) /\ ((((exists pfa_gap_scalar_distribute_boperationresultbound. pfa_gap_scalar_distribute_boperationresultbound + S (pfp_value_scalar_distribute_b) = (p)) /\ ((exists pfa_offset_left_scalar_distribute_boperationresultcongruence pfa_offset_right_scalar_distribute_boperationresultcongruence. ((b) * (pfp_source_scalar_distribute_b)) + (p) * pfa_offset_left_scalar_distribute_boperationresultcongruence = (pfp_value_scalar_distribute_b) + (p) * pfa_offset_right_scalar_distribute_boperationresultcongruence))))))))))))))))) -> (forall pfp_index_scalar_distribute_right. (exists pfa_gap_scalar_distribute_rightindex. pfa_gap_scalar_distribute_rightindex + S (pfp_index_scalar_distribute_right) = (l)) -> exists pfp_left_scalar_distribute_right pfp_right_scalar_distribute_right pfp_value_scalar_distribute_right. ((((exists ff_h_pfp_scalar_distribute_rightleft. ff_h_pfp_scalar_distribute_rightleft + S (pfp_left_scalar_distribute_right) = S ((S (pfp_index_scalar_distribute_right)) * xc)) /\ exists ff_q_pfp_scalar_distribute_rightleft. xb = ff_q_pfp_scalar_distribute_rightleft * S ((S (pfp_index_scalar_distribute_right)) * xc) + (pfp_left_scalar_distribute_right))) /\ (((((exists ff_h_pfp_scalar_distribute_rightright. ff_h_pfp_scalar_distribute_rightright + S (pfp_right_scalar_distribute_right) = S ((S (pfp_index_scalar_distribute_right)) * yc)) /\ exists ff_q_pfp_scalar_distribute_rightright. yb = ff_q_pfp_scalar_distribute_rightright * S ((S (pfp_index_scalar_distribute_right)) * yc) + (pfp_right_scalar_distribute_right))) /\ (((((exists ff_h_pfp_scalar_distribute_righttarget. ff_h_pfp_scalar_distribute_righttarget + S (pfp_value_scalar_distribute_right) = S ((S (pfp_index_scalar_distribute_right)) * vc)) /\ exists ff_q_pfp_scalar_distribute_righttarget. vb = ff_q_pfp_scalar_distribute_righttarget * S ((S (pfp_index_scalar_distribute_right)) * vc) + (pfp_value_scalar_distribute_right))) /\ ((((exists pfa_gap_scalar_distribute_rightoperationleft. pfa_gap_scalar_distribute_rightoperationleft + S (pfp_left_scalar_distribute_right) = (p)) /\ (((exists pfa_gap_scalar_distribute_rightoperationright. pfa_gap_scalar_distribute_rightoperationright + S (pfp_right_scalar_distribute_right) = (p)) /\ ((((exists pfa_gap_scalar_distribute_rightoperationresultbound. pfa_gap_scalar_distribute_rightoperationresultbound + S (pfp_value_scalar_distribute_right) = (p)) /\ ((exists pfa_offset_left_scalar_distribute_rightoperationresultcongruence pfa_offset_right_scalar_distribute_rightoperationresultcongruence. ((pfp_left_scalar_distribute_right) + (pfp_right_scalar_distribute_right)) + (p) * pfa_offset_left_scalar_distribute_rightoperationresultcongruence = (pfp_value_scalar_distribute_right) + (p) * pfa_offset_right_scalar_distribute_rightoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_scalar_distribute_result mdr_a_pfp_scalar_distribute_result. (exists mdr_gap_pfp_scalar_distribute_resultb. mdr_gap_pfp_scalar_distribute_resultb + S (mdr_i_pfp_scalar_distribute_result) = (l)) -> (((exists ff_h_mdr_pfp_scalar_distribute_resulto. ff_h_mdr_pfp_scalar_distribute_resulto + S (mdr_a_pfp_scalar_distribute_result) = S ((S (mdr_i_pfp_scalar_distribute_result)) * uc)) /\ exists ff_q_mdr_pfp_scalar_distribute_resulto. ub = ff_q_mdr_pfp_scalar_distribute_resulto * S ((S (mdr_i_pfp_scalar_distribute_result)) * uc) + (mdr_a_pfp_scalar_distribute_result))) -> (((exists ff_h_mdr_pfp_scalar_distribute_resultn. ff_h_mdr_pfp_scalar_distribute_resultn + S (mdr_a_pfp_scalar_distribute_result) = S ((S (mdr_i_pfp_scalar_distribute_result)) * vc)) /\ exists ff_q_mdr_pfp_scalar_distribute_resultn. vb = ff_q_mdr_pfp_scalar_distribute_resultn * S ((S (mdr_i_pfp_scalar_distribute_result)) * vc) + (mdr_a_pfp_scalar_distribute_result))))

forall p n t a r s. (~((p) = 1) /\ forall pfa_factor_left_step_prime pfa_factor_right_step_prime. (p) = pfa_factor_left_step_prime * pfa_factor_right_step_prime -> pfa_factor_left_step_prime = 1 \/ pfa_factor_right_step_prime = 1) -> (exists pfa_gap_step_base. pfa_gap_step_base + S (t) = (p)) -> (exists pfa_gap_step_coefficient. pfa_gap_step_coefficient + S (a) = (p)) -> (((exists pfa_gap_step_previousbound. pfa_gap_step_previousbound + S (r) = (p)) /\ ((exists pfa_offset_left_step_previouscongruence pfa_offset_right_step_previouscongruence. (n) + (p) * pfa_offset_left_step_previouscongruence = (r) + (p) * pfa_offset_right_step_previouscongruence)))) -> (((exists pfa_gap_step_nextbound. pfa_gap_step_nextbound + S (s) = (p)) /\ ((exists pfa_offset_left_step_nextcongruence pfa_offset_right_step_nextcongruence. (n*t+a) + (p) * pfa_offset_left_step_nextcongruence = (s) + (p) * pfa_offset_right_step_nextcongruence)))) -> exists k. ((((exists pfa_gap_step_productleft. pfa_gap_step_productleft + S (r) = (p)) /\ (((exists pfa_gap_step_productright. pfa_gap_step_productright + S (t) = (p)) /\ ((((exists pfa_gap_step_productresultbound. pfa_gap_step_productresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_step_productresultcongruence pfa_offset_right_step_productresultcongruence. ((r) * (t)) + (p) * pfa_offset_left_step_productresultcongruence = (k) + (p) * pfa_offset_right_step_productresultcongruence))))))))) /\ ((((exists pfa_gap_step_sumleft. pfa_gap_step_sumleft + S (k) = (p)) /\ (((exists pfa_gap_step_sumright. pfa_gap_step_sumright + S (a) = (p)) /\ ((((exists pfa_gap_step_sumresultbound. pfa_gap_step_sumresultbound + S (s) = (p)) /\ ((exists pfa_offset_left_step_sumresultcongruence pfa_offset_right_step_sumresultcongruence. ((k) + (a)) + (p) * pfa_offset_left_step_sumresultcongruence = (s) + (p) * pfa_offset_right_step_sumresultcongruence)))))))))))

forall p b c t l n u v U V. (~((p) = 1) /\ forall pfa_factor_left_trace_construct_prime pfa_factor_right_trace_construct_prime. (p) = pfa_factor_left_trace_construct_prime * pfa_factor_right_trace_construct_prime -> pfa_factor_left_trace_construct_prime = 1 \/ pfa_factor_right_trace_construct_prime = 1) -> (forall fom_index_pfp_trace_construct_coefficients. (exists fom_gap_pfp_trace_construct_coefficients_index_bound. fom_gap_pfp_trace_construct_coefficients_index_bound + S (fom_index_pfp_trace_construct_coefficients) = l) -> exists fom_value_pfp_trace_construct_coefficients. ((((exists fom_beta_height_pfp_trace_construct_coefficients_entry. fom_beta_height_pfp_trace_construct_coefficients_entry + S (fom_value_pfp_trace_construct_coefficients) = S ((S (fom_index_pfp_trace_construct_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_trace_construct_coefficients_entry. b = fom_beta_quotient_pfp_trace_construct_coefficients_entry * S ((S (fom_index_pfp_trace_construct_coefficients)) * c) + (fom_value_pfp_trace_construct_coefficients))) /\ (exists fom_gap_pfp_trace_construct_coefficients_value_bound. fom_gap_pfp_trace_construct_coefficients_value_bound + S (fom_value_pfp_trace_construct_coefficients) = p))) -> (exists pfa_gap_trace_construct_base. pfa_gap_trace_construct_base + S (t) = (p)) -> (((((exists fs_h_ph_pfh_trace_construct_natural_start. fs_h_ph_pfh_trace_construct_natural_start + S (0) = S ((S (0)) * v)) /\ exists fs_q_ph_pfh_trace_construct_natural_start. u = fs_q_ph_pfh_trace_construct_natural_start * S ((S (0)) * v) + (0))) /\ ((((exists fs_h_ph_pfh_trace_construct_natural_terminal. fs_h_ph_pfh_trace_construct_natural_terminal + S (n) = S ((S (l)) * v)) /\ exists fs_q_ph_pfh_trace_construct_natural_terminal. u = fs_q_ph_pfh_trace_construct_natural_terminal * S ((S (l)) * v) + (n))) /\ forall ff_i_ph_pfh_trace_construct_natural_steps. (exists ph_bound_pfh_trace_construct_natural_steps. ph_bound_pfh_trace_construct_natural_steps + S ff_i_ph_pfh_trace_construct_natural_steps = l) -> exists ff_coefficient_ph_pfh_trace_construct_natural_steps ff_previous_ph_pfh_trace_construct_natural_steps ff_current_ph_pfh_trace_construct_natural_steps. ((((exists fs_h_ph_pfh_trace_construct_natural_steps_coefficient. fs_h_ph_pfh_trace_construct_natural_steps_coefficient + S (ff_coefficient_ph_pfh_trace_construct_natural_steps) = S ((S (ff_i_ph_pfh_trace_construct_natural_steps)) * c)) /\ exists fs_q_ph_pfh_trace_construct_natural_steps_coefficient. b = fs_q_ph_pfh_trace_construct_natural_steps_coefficient * S ((S (ff_i_ph_pfh_trace_construct_natural_steps)) * c) + (ff_coefficient_ph_pfh_trace_construct_natural_steps))) /\ ((((exists fs_h_ph_pfh_trace_construct_natural_steps_before. fs_h_ph_pfh_trace_construct_natural_steps_before + S (ff_previous_ph_pfh_trace_construct_natural_steps) = S ((S (ff_i_ph_pfh_trace_construct_natural_steps)) * v)) /\ exists fs_q_ph_pfh_trace_construct_natural_steps_before. u = fs_q_ph_pfh_trace_construct_natural_steps_before * S ((S (ff_i_ph_pfh_trace_construct_natural_steps)) * v) + (ff_previous_ph_pfh_trace_construct_natural_steps))) /\ ((((exists fs_h_ph_pfh_trace_construct_natural_steps_after. fs_h_ph_pfh_trace_construct_natural_steps_after + S (ff_current_ph_pfh_trace_construct_natural_steps) = S ((S (S ff_i_ph_pfh_trace_construct_natural_steps)) * v)) /\ exists fs_q_ph_pfh_trace_construct_natural_steps_after. u = fs_q_ph_pfh_trace_construct_natural_steps_after * S ((S (S ff_i_ph_pfh_trace_construct_natural_steps)) * v) + (ff_current_ph_pfh_trace_construct_natural_steps))) /\ ff_current_ph_pfh_trace_construct_natural_steps = ff_previous_ph_pfh_trace_construct_natural_steps * t + ff_coefficient_ph_pfh_trace_construct_natural_steps)))))) -> (forall pfp_index_trace_construct_normalization. (exists pfa_gap_trace_construct_normalizationindex. pfa_gap_trace_construct_normalizationindex + S (pfp_index_trace_construct_normalization) = (S l)) -> exists pfp_source_trace_construct_normalization pfp_residue_trace_construct_normalization. ((((exists ff_h_pfp_trace_construct_normalizationsource. ff_h_pfp_trace_construct_normalizationsource + S (pfp_source_trace_construct_normalization) = S ((S (pfp_index_trace_construct_normalization)) * v)) /\ exists ff_q_pfp_trace_construct_normalizationsource. u = ff_q_pfp_trace_construct_normalizationsource * S ((S (pfp_index_trace_construct_normalization)) * v) + (pfp_source_trace_construct_normalization))) /\ (((((exists ff_h_pfp_trace_construct_normalizationtarget. ff_h_pfp_trace_construct_normalizationtarget + S (pfp_residue_trace_construct_normalization) = S ((S (pfp_index_trace_construct_normalization)) * V)) /\ exists ff_q_pfp_trace_construct_normalizationtarget. U = ff_q_pfp_trace_construct_normalizationtarget * S ((S (pfp_index_trace_construct_normalization)) * V) + (pfp_residue_trace_construct_normalization))) /\ ((((exists pfa_gap_trace_construct_normalizationresiduebound. pfa_gap_trace_construct_normalizationresiduebound + S (pfp_residue_trace_construct_normalization) = (p)) /\ ((exists pfa_offset_left_trace_construct_normalizationresiduecongruence pfa_offset_right_trace_construct_normalizationresiduecongruence. (pfp_source_trace_construct_normalization) + (p) * pfa_offset_left_trace_construct_normalizationresiduecongruence = (pfp_residue_trace_construct_normalization) + (p) * pfa_offset_right_trace_construct_normalizationresiduecongruence))))))))) -> exists r. ((((exists pfa_gap_trace_construct_executionbase. pfa_gap_trace_construct_executionbase + S (t) = (p)) /\ (((((exists ff_h_pfp_trace_construct_executioninitial. ff_h_pfp_trace_construct_executioninitial + S (0) = S ((S (0)) * V)) /\ exists ff_q_pfp_trace_construct_executioninitial. U = ff_q_pfp_trace_construct_executioninitial * S ((S (0)) * V) + (0))) /\ (((((exists ff_h_pfp_trace_construct_executionterminal. ff_h_pfp_trace_construct_executionterminal + S (r) = S ((S (l)) * V)) /\ exists ff_q_pfp_trace_construct_executionterminal. U = ff_q_pfp_trace_construct_executionterminal * S ((S (l)) * V) + (r))) /\ ((forall pfh_index_trace_construct_executionsteps. (exists pfa_gap_trace_construct_executionstepsindex. pfa_gap_trace_construct_executionstepsindex + S (pfh_index_trace_construct_executionsteps) = (l)) -> (exists pfh_coefficient_trace_construct_executionstepsstep pfh_before_trace_construct_executionstepsstep pfh_after_trace_construct_executionstepsstep pfh_product_trace_construct_executionstepsstep. ((((exists ff_h_pfp_trace_construct_executionstepsstepcoefficient. ff_h_pfp_trace_construct_executionstepsstepcoefficient + S (pfh_coefficient_trace_construct_executionstepsstep) = S ((S (pfh_index_trace_construct_executionsteps)) * c)) /\ exists ff_q_pfp_trace_construct_executionstepsstepcoefficient. b = ff_q_pfp_trace_construct_executionstepsstepcoefficient * S ((S (pfh_index_trace_construct_executionsteps)) * c) + (pfh_coefficient_trace_construct_executionstepsstep))) /\ (((((exists ff_h_pfp_trace_construct_executionstepsstepbefore. ff_h_pfp_trace_construct_executionstepsstepbefore + S (pfh_before_trace_construct_executionstepsstep) = S ((S (pfh_index_trace_construct_executionsteps)) * V)) /\ exists ff_q_pfp_trace_construct_executionstepsstepbefore. U = ff_q_pfp_trace_construct_executionstepsstepbefore * S ((S (pfh_index_trace_construct_executionsteps)) * V) + (pfh_before_trace_construct_executionstepsstep))) /\ (((((exists ff_h_pfp_trace_construct_executionstepsstepafter. ff_h_pfp_trace_construct_executionstepsstepafter + S (pfh_after_trace_construct_executionstepsstep) = S ((S (S (pfh_index_trace_construct_executionsteps))) * V)) /\ exists ff_q_pfp_trace_construct_executionstepsstepafter. U = ff_q_pfp_trace_construct_executionstepsstepafter * S ((S (S (pfh_index_trace_construct_executionsteps))) * V) + (pfh_after_trace_construct_executionstepsstep))) /\ (((((exists pfa_gap_trace_construct_executionstepsstepmultiplyleft. pfa_gap_trace_construct_executionstepsstepmultiplyleft + S (pfh_before_trace_construct_executionstepsstep) = (p)) /\ (((exists pfa_gap_trace_construct_executionstepsstepmultiplyright. pfa_gap_trace_construct_executionstepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_trace_construct_executionstepsstepmultiplyresultbound. pfa_gap_trace_construct_executionstepsstepmultiplyresultbound + S (pfh_product_trace_construct_executionstepsstep) = (p)) /\ ((exists pfa_offset_left_trace_construct_executionstepsstepmultiplyresultcongruence pfa_offset_right_trace_construct_executionstepsstepmultiplyresultcongruence. ((pfh_before_trace_construct_executionstepsstep) * (t)) + (p) * pfa_offset_left_trace_construct_executionstepsstepmultiplyresultcongruence = (pfh_product_trace_construct_executionstepsstep) + (p) * pfa_offset_right_trace_construct_executionstepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_trace_construct_executionstepsstepaddleft. pfa_gap_trace_construct_executionstepsstepaddleft + S (pfh_product_trace_construct_executionstepsstep) = (p)) /\ (((exists pfa_gap_trace_construct_executionstepsstepaddright. pfa_gap_trace_construct_executionstepsstepaddright + S (pfh_coefficient_trace_construct_executionstepsstep) = (p)) /\ ((((exists pfa_gap_trace_construct_executionstepsstepaddresultbound. pfa_gap_trace_construct_executionstepsstepaddresultbound + S (pfh_after_trace_construct_executionstepsstep) = (p)) /\ ((exists pfa_offset_left_trace_construct_executionstepsstepaddresultcongruence pfa_offset_right_trace_construct_executionstepsstepaddresultcongruence. ((pfh_product_trace_construct_executionstepsstep) + (pfh_coefficient_trace_construct_executionstepsstep)) + (p) * pfa_offset_left_trace_construct_executionstepsstepaddresultcongruence = (pfh_after_trace_construct_executionstepsstep) + (p) * pfa_offset_right_trace_construct_executionstepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((((exists pfa_gap_trace_construct_final_residuebound. pfa_gap_trace_construct_final_residuebound + S (r) = (p)) /\ ((exists pfa_offset_left_trace_construct_final_residuecongruence pfa_offset_right_trace_construct_final_residuecongruence. (n) + (p) * pfa_offset_left_trace_construct_final_residuecongruence = (r) + (p) * pfa_offset_right_trace_construct_final_residuecongruence))))))

forall p b c t l. (~((p) = 1) /\ forall pfa_factor_left_exists_prime pfa_factor_right_exists_prime. (p) = pfa_factor_left_exists_prime * pfa_factor_right_exists_prime -> pfa_factor_left_exists_prime = 1 \/ pfa_factor_right_exists_prime = 1) -> (forall fom_index_pfp_exists_coefficients. (exists fom_gap_pfp_exists_coefficients_index_bound. fom_gap_pfp_exists_coefficients_index_bound + S (fom_index_pfp_exists_coefficients) = l) -> exists fom_value_pfp_exists_coefficients. ((((exists fom_beta_height_pfp_exists_coefficients_entry. fom_beta_height_pfp_exists_coefficients_entry + S (fom_value_pfp_exists_coefficients) = S ((S (fom_index_pfp_exists_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_exists_coefficients_entry. b = fom_beta_quotient_pfp_exists_coefficients_entry * S ((S (fom_index_pfp_exists_coefficients)) * c) + (fom_value_pfp_exists_coefficients))) /\ (exists fom_gap_pfp_exists_coefficients_value_bound. fom_gap_pfp_exists_coefficients_value_bound + S (fom_value_pfp_exists_coefficients) = p))) -> (exists pfa_gap_exists_base. pfa_gap_exists_base + S (t) = (p)) -> exists r. (exists pfh_trace_code_exists_execution pfh_trace_scale_exists_execution. (((exists pfa_gap_exists_executiontracebase. pfa_gap_exists_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_exists_executiontraceinitial. ff_h_pfp_exists_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_exists_execution)) /\ exists ff_q_pfp_exists_executiontraceinitial. pfh_trace_code_exists_execution = ff_q_pfp_exists_executiontraceinitial * S ((S (0)) * pfh_trace_scale_exists_execution) + (0))) /\ (((((exists ff_h_pfp_exists_executiontraceterminal. ff_h_pfp_exists_executiontraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_exists_execution)) /\ exists ff_q_pfp_exists_executiontraceterminal. pfh_trace_code_exists_execution = ff_q_pfp_exists_executiontraceterminal * S ((S (l)) * pfh_trace_scale_exists_execution) + (r))) /\ ((forall pfh_index_exists_executiontracesteps. (exists pfa_gap_exists_executiontracestepsindex. pfa_gap_exists_executiontracestepsindex + S (pfh_index_exists_executiontracesteps) = (l)) -> (exists pfh_coefficient_exists_executiontracestepsstep pfh_before_exists_executiontracestepsstep pfh_after_exists_executiontracestepsstep pfh_product_exists_executiontracestepsstep. ((((exists ff_h_pfp_exists_executiontracestepsstepcoefficient. ff_h_pfp_exists_executiontracestepsstepcoefficient + S (pfh_coefficient_exists_executiontracestepsstep) = S ((S (pfh_index_exists_executiontracesteps)) * c)) /\ exists ff_q_pfp_exists_executiontracestepsstepcoefficient. b = ff_q_pfp_exists_executiontracestepsstepcoefficient * S ((S (pfh_index_exists_executiontracesteps)) * c) + (pfh_coefficient_exists_executiontracestepsstep))) /\ (((((exists ff_h_pfp_exists_executiontracestepsstepbefore. ff_h_pfp_exists_executiontracestepsstepbefore + S (pfh_before_exists_executiontracestepsstep) = S ((S (pfh_index_exists_executiontracesteps)) * pfh_trace_scale_exists_execution)) /\ exists ff_q_pfp_exists_executiontracestepsstepbefore. pfh_trace_code_exists_execution = ff_q_pfp_exists_executiontracestepsstepbefore * S ((S (pfh_index_exists_executiontracesteps)) * pfh_trace_scale_exists_execution) + (pfh_before_exists_executiontracestepsstep))) /\ (((((exists ff_h_pfp_exists_executiontracestepsstepafter. ff_h_pfp_exists_executiontracestepsstepafter + S (pfh_after_exists_executiontracestepsstep) = S ((S (S (pfh_index_exists_executiontracesteps))) * pfh_trace_scale_exists_execution)) /\ exists ff_q_pfp_exists_executiontracestepsstepafter. pfh_trace_code_exists_execution = ff_q_pfp_exists_executiontracestepsstepafter * S ((S (S (pfh_index_exists_executiontracesteps))) * pfh_trace_scale_exists_execution) + (pfh_after_exists_executiontracestepsstep))) /\ (((((exists pfa_gap_exists_executiontracestepsstepmultiplyleft. pfa_gap_exists_executiontracestepsstepmultiplyleft + S (pfh_before_exists_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_exists_executiontracestepsstepmultiplyright. pfa_gap_exists_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_exists_executiontracestepsstepmultiplyresultbound. pfa_gap_exists_executiontracestepsstepmultiplyresultbound + S (pfh_product_exists_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_exists_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_exists_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_exists_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_exists_executiontracestepsstepmultiplyresultcongruence = (pfh_product_exists_executiontracestepsstep) + (p) * pfa_offset_right_exists_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_exists_executiontracestepsstepaddleft. pfa_gap_exists_executiontracestepsstepaddleft + S (pfh_product_exists_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_exists_executiontracestepsstepaddright. pfa_gap_exists_executiontracestepsstepaddright + S (pfh_coefficient_exists_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_exists_executiontracestepsstepaddresultbound. pfa_gap_exists_executiontracestepsstepaddresultbound + S (pfh_after_exists_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_exists_executiontracestepsstepaddresultcongruence pfa_offset_right_exists_executiontracestepsstepaddresultcongruence. ((pfh_product_exists_executiontracestepsstep) + (pfh_coefficient_exists_executiontracestepsstep)) + (p) * pfa_offset_left_exists_executiontracestepsstepaddresultcongruence = (pfh_after_exists_executiontracestepsstep) + (p) * pfa_offset_right_exists_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

forall p b c t l r. (exists pfh_trace_code_input_bounds_execution pfh_trace_scale_input_bounds_execution. (((exists pfa_gap_input_bounds_executiontracebase. pfa_gap_input_bounds_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_input_bounds_executiontraceinitial. ff_h_pfp_input_bounds_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_input_bounds_execution)) /\ exists ff_q_pfp_input_bounds_executiontraceinitial. pfh_trace_code_input_bounds_execution = ff_q_pfp_input_bounds_executiontraceinitial * S ((S (0)) * pfh_trace_scale_input_bounds_execution) + (0))) /\ (((((exists ff_h_pfp_input_bounds_executiontraceterminal. ff_h_pfp_input_bounds_executiontraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_input_bounds_execution)) /\ exists ff_q_pfp_input_bounds_executiontraceterminal. pfh_trace_code_input_bounds_execution = ff_q_pfp_input_bounds_executiontraceterminal * S ((S (l)) * pfh_trace_scale_input_bounds_execution) + (r))) /\ ((forall pfh_index_input_bounds_executiontracesteps. (exists pfa_gap_input_bounds_executiontracestepsindex. pfa_gap_input_bounds_executiontracestepsindex + S (pfh_index_input_bounds_executiontracesteps) = (l)) -> (exists pfh_coefficient_input_bounds_executiontracestepsstep pfh_before_input_bounds_executiontracestepsstep pfh_after_input_bounds_executiontracestepsstep pfh_product_input_bounds_executiontracestepsstep. ((((exists ff_h_pfp_input_bounds_executiontracestepsstepcoefficient. ff_h_pfp_input_bounds_executiontracestepsstepcoefficient + S (pfh_coefficient_input_bounds_executiontracestepsstep) = S ((S (pfh_index_input_bounds_executiontracesteps)) * c)) /\ exists ff_q_pfp_input_bounds_executiontracestepsstepcoefficient. b = ff_q_pfp_input_bounds_executiontracestepsstepcoefficient * S ((S (pfh_index_input_bounds_executiontracesteps)) * c) + (pfh_coefficient_input_bounds_executiontracestepsstep))) /\ (((((exists ff_h_pfp_input_bounds_executiontracestepsstepbefore. ff_h_pfp_input_bounds_executiontracestepsstepbefore + S (pfh_before_input_bounds_executiontracestepsstep) = S ((S (pfh_index_input_bounds_executiontracesteps)) * pfh_trace_scale_input_bounds_execution)) /\ exists ff_q_pfp_input_bounds_executiontracestepsstepbefore. pfh_trace_code_input_bounds_execution = ff_q_pfp_input_bounds_executiontracestepsstepbefore * S ((S (pfh_index_input_bounds_executiontracesteps)) * pfh_trace_scale_input_bounds_execution) + (pfh_before_input_bounds_executiontracestepsstep))) /\ (((((exists ff_h_pfp_input_bounds_executiontracestepsstepafter. ff_h_pfp_input_bounds_executiontracestepsstepafter + S (pfh_after_input_bounds_executiontracestepsstep) = S ((S (S (pfh_index_input_bounds_executiontracesteps))) * pfh_trace_scale_input_bounds_execution)) /\ exists ff_q_pfp_input_bounds_executiontracestepsstepafter. pfh_trace_code_input_bounds_execution = ff_q_pfp_input_bounds_executiontracestepsstepafter * S ((S (S (pfh_index_input_bounds_executiontracesteps))) * pfh_trace_scale_input_bounds_execution) + (pfh_after_input_bounds_executiontracestepsstep))) /\ (((((exists pfa_gap_input_bounds_executiontracestepsstepmultiplyleft. pfa_gap_input_bounds_executiontracestepsstepmultiplyleft + S (pfh_before_input_bounds_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_input_bounds_executiontracestepsstepmultiplyright. pfa_gap_input_bounds_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_input_bounds_executiontracestepsstepmultiplyresultbound. pfa_gap_input_bounds_executiontracestepsstepmultiplyresultbound + S (pfh_product_input_bounds_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_input_bounds_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_input_bounds_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_input_bounds_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_input_bounds_executiontracestepsstepmultiplyresultcongruence = (pfh_product_input_bounds_executiontracestepsstep) + (p) * pfa_offset_right_input_bounds_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_input_bounds_executiontracestepsstepaddleft. pfa_gap_input_bounds_executiontracestepsstepaddleft + S (pfh_product_input_bounds_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_input_bounds_executiontracestepsstepaddright. pfa_gap_input_bounds_executiontracestepsstepaddright + S (pfh_coefficient_input_bounds_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_input_bounds_executiontracestepsstepaddresultbound. pfa_gap_input_bounds_executiontracestepsstepaddresultbound + S (pfh_after_input_bounds_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_input_bounds_executiontracestepsstepaddresultcongruence pfa_offset_right_input_bounds_executiontracestepsstepaddresultcongruence. ((pfh_product_input_bounds_executiontracestepsstep) + (pfh_coefficient_input_bounds_executiontracestepsstep)) + (p) * pfa_offset_left_input_bounds_executiontracestepsstepaddresultcongruence = (pfh_after_input_bounds_executiontracestepsstep) + (p) * pfa_offset_right_input_bounds_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> ((exists pfa_gap_input_bounds_base. pfa_gap_input_bounds_base + S (t) = (p)) /\ ((forall fom_index_pfp_input_bounds_coefficients. (exists fom_gap_pfp_input_bounds_coefficients_index_bound. fom_gap_pfp_input_bounds_coefficients_index_bound + S (fom_index_pfp_input_bounds_coefficients) = l) -> exists fom_value_pfp_input_bounds_coefficients. ((((exists fom_beta_height_pfp_input_bounds_coefficients_entry. fom_beta_height_pfp_input_bounds_coefficients_entry + S (fom_value_pfp_input_bounds_coefficients) = S ((S (fom_index_pfp_input_bounds_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_input_bounds_coefficients_entry. b = fom_beta_quotient_pfp_input_bounds_coefficients_entry * S ((S (fom_index_pfp_input_bounds_coefficients)) * c) + (fom_value_pfp_input_bounds_coefficients))) /\ (exists fom_gap_pfp_input_bounds_coefficients_value_bound. fom_gap_pfp_input_bounds_coefficients_value_bound + S (fom_value_pfp_input_bounds_coefficients) = p)))))

forall p b c t r. (exists pfh_trace_code_empty_execution pfh_trace_scale_empty_execution. (((exists pfa_gap_empty_executiontracebase. pfa_gap_empty_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_empty_executiontraceinitial. ff_h_pfp_empty_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontraceinitial. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontraceinitial * S ((S (0)) * pfh_trace_scale_empty_execution) + (0))) /\ (((((exists ff_h_pfp_empty_executiontraceterminal. ff_h_pfp_empty_executiontraceterminal + S (r) = S ((S (0)) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontraceterminal. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontraceterminal * S ((S (0)) * pfh_trace_scale_empty_execution) + (r))) /\ ((forall pfh_index_empty_executiontracesteps. (exists pfa_gap_empty_executiontracestepsindex. pfa_gap_empty_executiontracestepsindex + S (pfh_index_empty_executiontracesteps) = (0)) -> (exists pfh_coefficient_empty_executiontracestepsstep pfh_before_empty_executiontracestepsstep pfh_after_empty_executiontracestepsstep pfh_product_empty_executiontracestepsstep. ((((exists ff_h_pfp_empty_executiontracestepsstepcoefficient. ff_h_pfp_empty_executiontracestepsstepcoefficient + S (pfh_coefficient_empty_executiontracestepsstep) = S ((S (pfh_index_empty_executiontracesteps)) * c)) /\ exists ff_q_pfp_empty_executiontracestepsstepcoefficient. b = ff_q_pfp_empty_executiontracestepsstepcoefficient * S ((S (pfh_index_empty_executiontracesteps)) * c) + (pfh_coefficient_empty_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_executiontracestepsstepbefore. ff_h_pfp_empty_executiontracestepsstepbefore + S (pfh_before_empty_executiontracestepsstep) = S ((S (pfh_index_empty_executiontracesteps)) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontracestepsstepbefore. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontracestepsstepbefore * S ((S (pfh_index_empty_executiontracesteps)) * pfh_trace_scale_empty_execution) + (pfh_before_empty_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_executiontracestepsstepafter. ff_h_pfp_empty_executiontracestepsstepafter + S (pfh_after_empty_executiontracestepsstep) = S ((S (S (pfh_index_empty_executiontracesteps))) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontracestepsstepafter. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontracestepsstepafter * S ((S (S (pfh_index_empty_executiontracesteps))) * pfh_trace_scale_empty_execution) + (pfh_after_empty_executiontracestepsstep))) /\ (((((exists pfa_gap_empty_executiontracestepsstepmultiplyleft. pfa_gap_empty_executiontracestepsstepmultiplyleft + S (pfh_before_empty_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_executiontracestepsstepmultiplyright. pfa_gap_empty_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_empty_executiontracestepsstepmultiplyresultbound. pfa_gap_empty_executiontracestepsstepmultiplyresultbound + S (pfh_product_empty_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_empty_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_empty_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_empty_executiontracestepsstepmultiplyresultcongruence = (pfh_product_empty_executiontracestepsstep) + (p) * pfa_offset_right_empty_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_empty_executiontracestepsstepaddleft. pfa_gap_empty_executiontracestepsstepaddleft + S (pfh_product_empty_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_executiontracestepsstepaddright. pfa_gap_empty_executiontracestepsstepaddright + S (pfh_coefficient_empty_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_empty_executiontracestepsstepaddresultbound. pfa_gap_empty_executiontracestepsstepaddresultbound + S (pfh_after_empty_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_executiontracestepsstepaddresultcongruence pfa_offset_right_empty_executiontracestepsstepaddresultcongruence. ((pfh_product_empty_executiontracestepsstep) + (pfh_coefficient_empty_executiontracestepsstep)) + (p) * pfa_offset_left_empty_executiontracestepsstepaddresultcongruence = (pfh_after_empty_executiontracestepsstep) + (p) * pfa_offset_right_empty_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> r=0

forall p b c t l r. (exists pfh_trace_code_successor_execution pfh_trace_scale_successor_execution. (((exists pfa_gap_successor_executiontracebase. pfa_gap_successor_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_successor_executiontraceinitial. ff_h_pfp_successor_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_successor_execution)) /\ exists ff_q_pfp_successor_executiontraceinitial. pfh_trace_code_successor_execution = ff_q_pfp_successor_executiontraceinitial * S ((S (0)) * pfh_trace_scale_successor_execution) + (0))) /\ (((((exists ff_h_pfp_successor_executiontraceterminal. ff_h_pfp_successor_executiontraceterminal + S (r) = S ((S (S l)) * pfh_trace_scale_successor_execution)) /\ exists ff_q_pfp_successor_executiontraceterminal. pfh_trace_code_successor_execution = ff_q_pfp_successor_executiontraceterminal * S ((S (S l)) * pfh_trace_scale_successor_execution) + (r))) /\ ((forall pfh_index_successor_executiontracesteps. (exists pfa_gap_successor_executiontracestepsindex. pfa_gap_successor_executiontracestepsindex + S (pfh_index_successor_executiontracesteps) = (S l)) -> (exists pfh_coefficient_successor_executiontracestepsstep pfh_before_successor_executiontracestepsstep pfh_after_successor_executiontracestepsstep pfh_product_successor_executiontracestepsstep. ((((exists ff_h_pfp_successor_executiontracestepsstepcoefficient. ff_h_pfp_successor_executiontracestepsstepcoefficient + S (pfh_coefficient_successor_executiontracestepsstep) = S ((S (pfh_index_successor_executiontracesteps)) * c)) /\ exists ff_q_pfp_successor_executiontracestepsstepcoefficient. b = ff_q_pfp_successor_executiontracestepsstepcoefficient * S ((S (pfh_index_successor_executiontracesteps)) * c) + (pfh_coefficient_successor_executiontracestepsstep))) /\ (((((exists ff_h_pfp_successor_executiontracestepsstepbefore. ff_h_pfp_successor_executiontracestepsstepbefore + S (pfh_before_successor_executiontracestepsstep) = S ((S (pfh_index_successor_executiontracesteps)) * pfh_trace_scale_successor_execution)) /\ exists ff_q_pfp_successor_executiontracestepsstepbefore. pfh_trace_code_successor_execution = ff_q_pfp_successor_executiontracestepsstepbefore * S ((S (pfh_index_successor_executiontracesteps)) * pfh_trace_scale_successor_execution) + (pfh_before_successor_executiontracestepsstep))) /\ (((((exists ff_h_pfp_successor_executiontracestepsstepafter. ff_h_pfp_successor_executiontracestepsstepafter + S (pfh_after_successor_executiontracestepsstep) = S ((S (S (pfh_index_successor_executiontracesteps))) * pfh_trace_scale_successor_execution)) /\ exists ff_q_pfp_successor_executiontracestepsstepafter. pfh_trace_code_successor_execution = ff_q_pfp_successor_executiontracestepsstepafter * S ((S (S (pfh_index_successor_executiontracesteps))) * pfh_trace_scale_successor_execution) + (pfh_after_successor_executiontracestepsstep))) /\ (((((exists pfa_gap_successor_executiontracestepsstepmultiplyleft. pfa_gap_successor_executiontracestepsstepmultiplyleft + S (pfh_before_successor_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_successor_executiontracestepsstepmultiplyright. pfa_gap_successor_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_successor_executiontracestepsstepmultiplyresultbound. pfa_gap_successor_executiontracestepsstepmultiplyresultbound + S (pfh_product_successor_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_successor_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_successor_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_successor_executiontracestepsstepmultiplyresultcongruence = (pfh_product_successor_executiontracestepsstep) + (p) * pfa_offset_right_successor_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_successor_executiontracestepsstepaddleft. pfa_gap_successor_executiontracestepsstepaddleft + S (pfh_product_successor_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_successor_executiontracestepsstepaddright. pfa_gap_successor_executiontracestepsstepaddright + S (pfh_coefficient_successor_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_successor_executiontracestepsstepaddresultbound. pfa_gap_successor_executiontracestepsstepaddresultbound + S (pfh_after_successor_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_executiontracestepsstepaddresultcongruence pfa_offset_right_successor_executiontracestepsstepaddresultcongruence. ((pfh_product_successor_executiontracestepsstep) + (pfh_coefficient_successor_executiontracestepsstep)) + (p) * pfa_offset_left_successor_executiontracestepsstepaddresultcongruence = (pfh_after_successor_executiontracestepsstep) + (p) * pfa_offset_right_successor_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> exists a h k. ((((exists ff_h_pfp_successor_coefficient. ff_h_pfp_successor_coefficient + S (a) = S ((S (l)) * c)) /\ exists ff_q_pfp_successor_coefficient. b = ff_q_pfp_successor_coefficient * S ((S (l)) * c) + (a))) /\ (((exists pfh_trace_code_successor_prefix pfh_trace_scale_successor_prefix. (((exists pfa_gap_successor_prefixtracebase. pfa_gap_successor_prefixtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_successor_prefixtraceinitial. ff_h_pfp_successor_prefixtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_successor_prefix)) /\ exists ff_q_pfp_successor_prefixtraceinitial. pfh_trace_code_successor_prefix = ff_q_pfp_successor_prefixtraceinitial * S ((S (0)) * pfh_trace_scale_successor_prefix) + (0))) /\ (((((exists ff_h_pfp_successor_prefixtraceterminal. ff_h_pfp_successor_prefixtraceterminal + S (h) = S ((S (l)) * pfh_trace_scale_successor_prefix)) /\ exists ff_q_pfp_successor_prefixtraceterminal. pfh_trace_code_successor_prefix = ff_q_pfp_successor_prefixtraceterminal * S ((S (l)) * pfh_trace_scale_successor_prefix) + (h))) /\ ((forall pfh_index_successor_prefixtracesteps. (exists pfa_gap_successor_prefixtracestepsindex. pfa_gap_successor_prefixtracestepsindex + S (pfh_index_successor_prefixtracesteps) = (l)) -> (exists pfh_coefficient_successor_prefixtracestepsstep pfh_before_successor_prefixtracestepsstep pfh_after_successor_prefixtracestepsstep pfh_product_successor_prefixtracestepsstep. ((((exists ff_h_pfp_successor_prefixtracestepsstepcoefficient. ff_h_pfp_successor_prefixtracestepsstepcoefficient + S (pfh_coefficient_successor_prefixtracestepsstep) = S ((S (pfh_index_successor_prefixtracesteps)) * c)) /\ exists ff_q_pfp_successor_prefixtracestepsstepcoefficient. b = ff_q_pfp_successor_prefixtracestepsstepcoefficient * S ((S (pfh_index_successor_prefixtracesteps)) * c) + (pfh_coefficient_successor_prefixtracestepsstep))) /\ (((((exists ff_h_pfp_successor_prefixtracestepsstepbefore. ff_h_pfp_successor_prefixtracestepsstepbefore + S (pfh_before_successor_prefixtracestepsstep) = S ((S (pfh_index_successor_prefixtracesteps)) * pfh_trace_scale_successor_prefix)) /\ exists ff_q_pfp_successor_prefixtracestepsstepbefore. pfh_trace_code_successor_prefix = ff_q_pfp_successor_prefixtracestepsstepbefore * S ((S (pfh_index_successor_prefixtracesteps)) * pfh_trace_scale_successor_prefix) + (pfh_before_successor_prefixtracestepsstep))) /\ (((((exists ff_h_pfp_successor_prefixtracestepsstepafter. ff_h_pfp_successor_prefixtracestepsstepafter + S (pfh_after_successor_prefixtracestepsstep) = S ((S (S (pfh_index_successor_prefixtracesteps))) * pfh_trace_scale_successor_prefix)) /\ exists ff_q_pfp_successor_prefixtracestepsstepafter. pfh_trace_code_successor_prefix = ff_q_pfp_successor_prefixtracestepsstepafter * S ((S (S (pfh_index_successor_prefixtracesteps))) * pfh_trace_scale_successor_prefix) + (pfh_after_successor_prefixtracestepsstep))) /\ (((((exists pfa_gap_successor_prefixtracestepsstepmultiplyleft. pfa_gap_successor_prefixtracestepsstepmultiplyleft + S (pfh_before_successor_prefixtracestepsstep) = (p)) /\ (((exists pfa_gap_successor_prefixtracestepsstepmultiplyright. pfa_gap_successor_prefixtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_successor_prefixtracestepsstepmultiplyresultbound. pfa_gap_successor_prefixtracestepsstepmultiplyresultbound + S (pfh_product_successor_prefixtracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_prefixtracestepsstepmultiplyresultcongruence pfa_offset_right_successor_prefixtracestepsstepmultiplyresultcongruence. ((pfh_before_successor_prefixtracestepsstep) * (t)) + (p) * pfa_offset_left_successor_prefixtracestepsstepmultiplyresultcongruence = (pfh_product_successor_prefixtracestepsstep) + (p) * pfa_offset_right_successor_prefixtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_successor_prefixtracestepsstepaddleft. pfa_gap_successor_prefixtracestepsstepaddleft + S (pfh_product_successor_prefixtracestepsstep) = (p)) /\ (((exists pfa_gap_successor_prefixtracestepsstepaddright. pfa_gap_successor_prefixtracestepsstepaddright + S (pfh_coefficient_successor_prefixtracestepsstep) = (p)) /\ ((((exists pfa_gap_successor_prefixtracestepsstepaddresultbound. pfa_gap_successor_prefixtracestepsstepaddresultbound + S (pfh_after_successor_prefixtracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_prefixtracestepsstepaddresultcongruence pfa_offset_right_successor_prefixtracestepsstepaddresultcongruence. ((pfh_product_successor_prefixtracestepsstep) + (pfh_coefficient_successor_prefixtracestepsstep)) + (p) * pfa_offset_left_successor_prefixtracestepsstepaddresultcongruence = (pfh_after_successor_prefixtracestepsstep) + (p) * pfa_offset_right_successor_prefixtracestepsstepaddresultcongruence))))))))))))))))))))))))))) /\ (((((exists pfa_gap_successor_multiplyleft. pfa_gap_successor_multiplyleft + S (h) = (p)) /\ (((exists pfa_gap_successor_multiplyright. pfa_gap_successor_multiplyright + S (t) = (p)) /\ ((((exists pfa_gap_successor_multiplyresultbound. pfa_gap_successor_multiplyresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_successor_multiplyresultcongruence pfa_offset_right_successor_multiplyresultcongruence. ((h) * (t)) + (p) * pfa_offset_left_successor_multiplyresultcongruence = (k) + (p) * pfa_offset_right_successor_multiplyresultcongruence))))))))) /\ ((((exists pfa_gap_successor_addleft. pfa_gap_successor_addleft + S (k) = (p)) /\ (((exists pfa_gap_successor_addright. pfa_gap_successor_addright + S (a) = (p)) /\ ((((exists pfa_gap_successor_addresultbound. pfa_gap_successor_addresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_successor_addresultcongruence pfa_offset_right_successor_addresultcongruence. ((k) + (a)) + (p) * pfa_offset_left_successor_addresultcongruence = (r) + (p) * pfa_offset_right_successor_addresultcongruence)))))))))))))))

forall p b c B C t l r. (forall mdr_i_pfp_transport_equal mdr_a_pfp_transport_equal. (exists mdr_gap_pfp_transport_equalb. mdr_gap_pfp_transport_equalb + S (mdr_i_pfp_transport_equal) = (l)) -> (((exists ff_h_mdr_pfp_transport_equalo. ff_h_mdr_pfp_transport_equalo + S (mdr_a_pfp_transport_equal) = S ((S (mdr_i_pfp_transport_equal)) * c)) /\ exists ff_q_mdr_pfp_transport_equalo. b = ff_q_mdr_pfp_transport_equalo * S ((S (mdr_i_pfp_transport_equal)) * c) + (mdr_a_pfp_transport_equal))) -> (((exists ff_h_mdr_pfp_transport_equaln. ff_h_mdr_pfp_transport_equaln + S (mdr_a_pfp_transport_equal) = S ((S (mdr_i_pfp_transport_equal)) * C)) /\ exists ff_q_mdr_pfp_transport_equaln. B = ff_q_mdr_pfp_transport_equaln * S ((S (mdr_i_pfp_transport_equal)) * C) + (mdr_a_pfp_transport_equal)))) -> (exists pfh_trace_code_transport_old pfh_trace_scale_transport_old. (((exists pfa_gap_transport_oldtracebase. pfa_gap_transport_oldtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_transport_oldtraceinitial. ff_h_pfp_transport_oldtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_transport_old)) /\ exists ff_q_pfp_transport_oldtraceinitial. pfh_trace_code_transport_old = ff_q_pfp_transport_oldtraceinitial * S ((S (0)) * pfh_trace_scale_transport_old) + (0))) /\ (((((exists ff_h_pfp_transport_oldtraceterminal. ff_h_pfp_transport_oldtraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_transport_old)) /\ exists ff_q_pfp_transport_oldtraceterminal. pfh_trace_code_transport_old = ff_q_pfp_transport_oldtraceterminal * S ((S (l)) * pfh_trace_scale_transport_old) + (r))) /\ ((forall pfh_index_transport_oldtracesteps. (exists pfa_gap_transport_oldtracestepsindex. pfa_gap_transport_oldtracestepsindex + S (pfh_index_transport_oldtracesteps) = (l)) -> (exists pfh_coefficient_transport_oldtracestepsstep pfh_before_transport_oldtracestepsstep pfh_after_transport_oldtracestepsstep pfh_product_transport_oldtracestepsstep. ((((exists ff_h_pfp_transport_oldtracestepsstepcoefficient. ff_h_pfp_transport_oldtracestepsstepcoefficient + S (pfh_coefficient_transport_oldtracestepsstep) = S ((S (pfh_index_transport_oldtracesteps)) * c)) /\ exists ff_q_pfp_transport_oldtracestepsstepcoefficient. b = ff_q_pfp_transport_oldtracestepsstepcoefficient * S ((S (pfh_index_transport_oldtracesteps)) * c) + (pfh_coefficient_transport_oldtracestepsstep))) /\ (((((exists ff_h_pfp_transport_oldtracestepsstepbefore. ff_h_pfp_transport_oldtracestepsstepbefore + S (pfh_before_transport_oldtracestepsstep) = S ((S (pfh_index_transport_oldtracesteps)) * pfh_trace_scale_transport_old)) /\ exists ff_q_pfp_transport_oldtracestepsstepbefore. pfh_trace_code_transport_old = ff_q_pfp_transport_oldtracestepsstepbefore * S ((S (pfh_index_transport_oldtracesteps)) * pfh_trace_scale_transport_old) + (pfh_before_transport_oldtracestepsstep))) /\ (((((exists ff_h_pfp_transport_oldtracestepsstepafter. ff_h_pfp_transport_oldtracestepsstepafter + S (pfh_after_transport_oldtracestepsstep) = S ((S (S (pfh_index_transport_oldtracesteps))) * pfh_trace_scale_transport_old)) /\ exists ff_q_pfp_transport_oldtracestepsstepafter. pfh_trace_code_transport_old = ff_q_pfp_transport_oldtracestepsstepafter * S ((S (S (pfh_index_transport_oldtracesteps))) * pfh_trace_scale_transport_old) + (pfh_after_transport_oldtracestepsstep))) /\ (((((exists pfa_gap_transport_oldtracestepsstepmultiplyleft. pfa_gap_transport_oldtracestepsstepmultiplyleft + S (pfh_before_transport_oldtracestepsstep) = (p)) /\ (((exists pfa_gap_transport_oldtracestepsstepmultiplyright. pfa_gap_transport_oldtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_transport_oldtracestepsstepmultiplyresultbound. pfa_gap_transport_oldtracestepsstepmultiplyresultbound + S (pfh_product_transport_oldtracestepsstep) = (p)) /\ ((exists pfa_offset_left_transport_oldtracestepsstepmultiplyresultcongruence pfa_offset_right_transport_oldtracestepsstepmultiplyresultcongruence. ((pfh_before_transport_oldtracestepsstep) * (t)) + (p) * pfa_offset_left_transport_oldtracestepsstepmultiplyresultcongruence = (pfh_product_transport_oldtracestepsstep) + (p) * pfa_offset_right_transport_oldtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_transport_oldtracestepsstepaddleft. pfa_gap_transport_oldtracestepsstepaddleft + S (pfh_product_transport_oldtracestepsstep) = (p)) /\ (((exists pfa_gap_transport_oldtracestepsstepaddright. pfa_gap_transport_oldtracestepsstepaddright + S (pfh_coefficient_transport_oldtracestepsstep) = (p)) /\ ((((exists pfa_gap_transport_oldtracestepsstepaddresultbound. pfa_gap_transport_oldtracestepsstepaddresultbound + S (pfh_after_transport_oldtracestepsstep) = (p)) /\ ((exists pfa_offset_left_transport_oldtracestepsstepaddresultcongruence pfa_offset_right_transport_oldtracestepsstepaddresultcongruence. ((pfh_product_transport_oldtracestepsstep) + (pfh_coefficient_transport_oldtracestepsstep)) + (p) * pfa_offset_left_transport_oldtracestepsstepaddresultcongruence = (pfh_after_transport_oldtracestepsstep) + (p) * pfa_offset_right_transport_oldtracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (exists pfh_trace_code_transport_new pfh_trace_scale_transport_new. (((exists pfa_gap_transport_newtracebase. pfa_gap_transport_newtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_transport_newtraceinitial. ff_h_pfp_transport_newtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_transport_new)) /\ exists ff_q_pfp_transport_newtraceinitial. pfh_trace_code_transport_new = ff_q_pfp_transport_newtraceinitial * S ((S (0)) * pfh_trace_scale_transport_new) + (0))) /\ (((((exists ff_h_pfp_transport_newtraceterminal. ff_h_pfp_transport_newtraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_transport_new)) /\ exists ff_q_pfp_transport_newtraceterminal. pfh_trace_code_transport_new = ff_q_pfp_transport_newtraceterminal * S ((S (l)) * pfh_trace_scale_transport_new) + (r))) /\ ((forall pfh_index_transport_newtracesteps. (exists pfa_gap_transport_newtracestepsindex. pfa_gap_transport_newtracestepsindex + S (pfh_index_transport_newtracesteps) = (l)) -> (exists pfh_coefficient_transport_newtracestepsstep pfh_before_transport_newtracestepsstep pfh_after_transport_newtracestepsstep pfh_product_transport_newtracestepsstep. ((((exists ff_h_pfp_transport_newtracestepsstepcoefficient. ff_h_pfp_transport_newtracestepsstepcoefficient + S (pfh_coefficient_transport_newtracestepsstep) = S ((S (pfh_index_transport_newtracesteps)) * C)) /\ exists ff_q_pfp_transport_newtracestepsstepcoefficient. B = ff_q_pfp_transport_newtracestepsstepcoefficient * S ((S (pfh_index_transport_newtracesteps)) * C) + (pfh_coefficient_transport_newtracestepsstep))) /\ (((((exists ff_h_pfp_transport_newtracestepsstepbefore. ff_h_pfp_transport_newtracestepsstepbefore + S (pfh_before_transport_newtracestepsstep) = S ((S (pfh_index_transport_newtracesteps)) * pfh_trace_scale_transport_new)) /\ exists ff_q_pfp_transport_newtracestepsstepbefore. pfh_trace_code_transport_new = ff_q_pfp_transport_newtracestepsstepbefore * S ((S (pfh_index_transport_newtracesteps)) * pfh_trace_scale_transport_new) + (pfh_before_transport_newtracestepsstep))) /\ (((((exists ff_h_pfp_transport_newtracestepsstepafter. ff_h_pfp_transport_newtracestepsstepafter + S (pfh_after_transport_newtracestepsstep) = S ((S (S (pfh_index_transport_newtracesteps))) * pfh_trace_scale_transport_new)) /\ exists ff_q_pfp_transport_newtracestepsstepafter. pfh_trace_code_transport_new = ff_q_pfp_transport_newtracestepsstepafter * S ((S (S (pfh_index_transport_newtracesteps))) * pfh_trace_scale_transport_new) + (pfh_after_transport_newtracestepsstep))) /\ (((((exists pfa_gap_transport_newtracestepsstepmultiplyleft. pfa_gap_transport_newtracestepsstepmultiplyleft + S (pfh_before_transport_newtracestepsstep) = (p)) /\ (((exists pfa_gap_transport_newtracestepsstepmultiplyright. pfa_gap_transport_newtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_transport_newtracestepsstepmultiplyresultbound. pfa_gap_transport_newtracestepsstepmultiplyresultbound + S (pfh_product_transport_newtracestepsstep) = (p)) /\ ((exists pfa_offset_left_transport_newtracestepsstepmultiplyresultcongruence pfa_offset_right_transport_newtracestepsstepmultiplyresultcongruence. ((pfh_before_transport_newtracestepsstep) * (t)) + (p) * pfa_offset_left_transport_newtracestepsstepmultiplyresultcongruence = (pfh_product_transport_newtracestepsstep) + (p) * pfa_offset_right_transport_newtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_transport_newtracestepsstepaddleft. pfa_gap_transport_newtracestepsstepaddleft + S (pfh_product_transport_newtracestepsstep) = (p)) /\ (((exists pfa_gap_transport_newtracestepsstepaddright. pfa_gap_transport_newtracestepsstepaddright + S (pfh_coefficient_transport_newtracestepsstep) = (p)) /\ ((((exists pfa_gap_transport_newtracestepsstepaddresultbound. pfa_gap_transport_newtracestepsstepaddresultbound + S (pfh_after_transport_newtracestepsstep) = (p)) /\ ((exists pfa_offset_left_transport_newtracestepsstepaddresultcongruence pfa_offset_right_transport_newtracestepsstepaddresultcongruence. ((pfh_product_transport_newtracestepsstep) + (pfh_coefficient_transport_newtracestepsstep)) + (p) * pfa_offset_left_transport_newtracestepsstepaddresultcongruence = (pfh_after_transport_newtracestepsstep) + (p) * pfa_offset_right_transport_newtracestepsstepaddresultcongruence)))))))))))))))))))))))))))

forall p b c d e t l n r. (~((p) = 1) /\ forall pfa_factor_left_invariant_prime pfa_factor_right_invariant_prime. (p) = pfa_factor_left_invariant_prime * pfa_factor_right_invariant_prime -> pfa_factor_left_invariant_prime = 1 \/ pfa_factor_right_invariant_prime = 1) -> (forall pfp_index_invariant_coefficients_reduced. (exists pfa_gap_invariant_coefficients_reducedindex. pfa_gap_invariant_coefficients_reducedindex + S (pfp_index_invariant_coefficients_reduced) = (l)) -> exists pfp_source_invariant_coefficients_reduced pfp_residue_invariant_coefficients_reduced. ((((exists ff_h_pfp_invariant_coefficients_reducedsource. ff_h_pfp_invariant_coefficients_reducedsource + S (pfp_source_invariant_coefficients_reduced) = S ((S (pfp_index_invariant_coefficients_reduced)) * c)) /\ exists ff_q_pfp_invariant_coefficients_reducedsource. b = ff_q_pfp_invariant_coefficients_reducedsource * S ((S (pfp_index_invariant_coefficients_reduced)) * c) + (pfp_source_invariant_coefficients_reduced))) /\ (((((exists ff_h_pfp_invariant_coefficients_reducedtarget. ff_h_pfp_invariant_coefficients_reducedtarget + S (pfp_residue_invariant_coefficients_reduced) = S ((S (pfp_index_invariant_coefficients_reduced)) * e)) /\ exists ff_q_pfp_invariant_coefficients_reducedtarget. d = ff_q_pfp_invariant_coefficients_reducedtarget * S ((S (pfp_index_invariant_coefficients_reduced)) * e) + (pfp_residue_invariant_coefficients_reduced))) /\ ((((exists pfa_gap_invariant_coefficients_reducedresiduebound. pfa_gap_invariant_coefficients_reducedresiduebound + S (pfp_residue_invariant_coefficients_reduced) = (p)) /\ ((exists pfa_offset_left_invariant_coefficients_reducedresiduecongruence pfa_offset_right_invariant_coefficients_reducedresiduecongruence. (pfp_source_invariant_coefficients_reduced) + (p) * pfa_offset_left_invariant_coefficients_reducedresiduecongruence = (pfp_residue_invariant_coefficients_reduced) + (p) * pfa_offset_right_invariant_coefficients_reducedresiduecongruence))))))))) -> (exists ff_u_ph_pfh_invariant_natural ff_v_ph_pfh_invariant_natural. ((((exists fs_h_ph_pfh_invariant_natural_body_start. fs_h_ph_pfh_invariant_natural_body_start + S (0) = S ((S (0)) * ff_v_ph_pfh_invariant_natural)) /\ exists fs_q_ph_pfh_invariant_natural_body_start. ff_u_ph_pfh_invariant_natural = fs_q_ph_pfh_invariant_natural_body_start * S ((S (0)) * ff_v_ph_pfh_invariant_natural) + (0))) /\ ((((exists fs_h_ph_pfh_invariant_natural_body_terminal. fs_h_ph_pfh_invariant_natural_body_terminal + S (n) = S ((S (l)) * ff_v_ph_pfh_invariant_natural)) /\ exists fs_q_ph_pfh_invariant_natural_body_terminal. ff_u_ph_pfh_invariant_natural = fs_q_ph_pfh_invariant_natural_body_terminal * S ((S (l)) * ff_v_ph_pfh_invariant_natural) + (n))) /\ forall ff_i_ph_pfh_invariant_natural_body_steps. (exists ph_bound_pfh_invariant_natural_body_steps. ph_bound_pfh_invariant_natural_body_steps + S ff_i_ph_pfh_invariant_natural_body_steps = l) -> exists ff_coefficient_ph_pfh_invariant_natural_body_steps ff_previous_ph_pfh_invariant_natural_body_steps ff_current_ph_pfh_invariant_natural_body_steps. ((((exists fs_h_ph_pfh_invariant_natural_body_steps_coefficient. fs_h_ph_pfh_invariant_natural_body_steps_coefficient + S (ff_coefficient_ph_pfh_invariant_natural_body_steps) = S ((S (ff_i_ph_pfh_invariant_natural_body_steps)) * c)) /\ exists fs_q_ph_pfh_invariant_natural_body_steps_coefficient. b = fs_q_ph_pfh_invariant_natural_body_steps_coefficient * S ((S (ff_i_ph_pfh_invariant_natural_body_steps)) * c) + (ff_coefficient_ph_pfh_invariant_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_invariant_natural_body_steps_before. fs_h_ph_pfh_invariant_natural_body_steps_before + S (ff_previous_ph_pfh_invariant_natural_body_steps) = S ((S (ff_i_ph_pfh_invariant_natural_body_steps)) * ff_v_ph_pfh_invariant_natural)) /\ exists fs_q_ph_pfh_invariant_natural_body_steps_before. ff_u_ph_pfh_invariant_natural = fs_q_ph_pfh_invariant_natural_body_steps_before * S ((S (ff_i_ph_pfh_invariant_natural_body_steps)) * ff_v_ph_pfh_invariant_natural) + (ff_previous_ph_pfh_invariant_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_invariant_natural_body_steps_after. fs_h_ph_pfh_invariant_natural_body_steps_after + S (ff_current_ph_pfh_invariant_natural_body_steps) = S ((S (S ff_i_ph_pfh_invariant_natural_body_steps)) * ff_v_ph_pfh_invariant_natural)) /\ exists fs_q_ph_pfh_invariant_natural_body_steps_after. ff_u_ph_pfh_invariant_natural = fs_q_ph_pfh_invariant_natural_body_steps_after * S ((S (S ff_i_ph_pfh_invariant_natural_body_steps)) * ff_v_ph_pfh_invariant_natural) + (ff_current_ph_pfh_invariant_natural_body_steps))) /\ ff_current_ph_pfh_invariant_natural_body_steps = ff_previous_ph_pfh_invariant_natural_body_steps * t + ff_coefficient_ph_pfh_invariant_natural_body_steps)))))) -> (exists pfh_trace_code_invariant_execution pfh_trace_scale_invariant_execution. (((exists pfa_gap_invariant_executiontracebase. pfa_gap_invariant_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_invariant_executiontraceinitial. ff_h_pfp_invariant_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_invariant_execution)) /\ exists ff_q_pfp_invariant_executiontraceinitial. pfh_trace_code_invariant_execution = ff_q_pfp_invariant_executiontraceinitial * S ((S (0)) * pfh_trace_scale_invariant_execution) + (0))) /\ (((((exists ff_h_pfp_invariant_executiontraceterminal. ff_h_pfp_invariant_executiontraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_invariant_execution)) /\ exists ff_q_pfp_invariant_executiontraceterminal. pfh_trace_code_invariant_execution = ff_q_pfp_invariant_executiontraceterminal * S ((S (l)) * pfh_trace_scale_invariant_execution) + (r))) /\ ((forall pfh_index_invariant_executiontracesteps. (exists pfa_gap_invariant_executiontracestepsindex. pfa_gap_invariant_executiontracestepsindex + S (pfh_index_invariant_executiontracesteps) = (l)) -> (exists pfh_coefficient_invariant_executiontracestepsstep pfh_before_invariant_executiontracestepsstep pfh_after_invariant_executiontracestepsstep pfh_product_invariant_executiontracestepsstep. ((((exists ff_h_pfp_invariant_executiontracestepsstepcoefficient. ff_h_pfp_invariant_executiontracestepsstepcoefficient + S (pfh_coefficient_invariant_executiontracestepsstep) = S ((S (pfh_index_invariant_executiontracesteps)) * e)) /\ exists ff_q_pfp_invariant_executiontracestepsstepcoefficient. d = ff_q_pfp_invariant_executiontracestepsstepcoefficient * S ((S (pfh_index_invariant_executiontracesteps)) * e) + (pfh_coefficient_invariant_executiontracestepsstep))) /\ (((((exists ff_h_pfp_invariant_executiontracestepsstepbefore. ff_h_pfp_invariant_executiontracestepsstepbefore + S (pfh_before_invariant_executiontracestepsstep) = S ((S (pfh_index_invariant_executiontracesteps)) * pfh_trace_scale_invariant_execution)) /\ exists ff_q_pfp_invariant_executiontracestepsstepbefore. pfh_trace_code_invariant_execution = ff_q_pfp_invariant_executiontracestepsstepbefore * S ((S (pfh_index_invariant_executiontracesteps)) * pfh_trace_scale_invariant_execution) + (pfh_before_invariant_executiontracestepsstep))) /\ (((((exists ff_h_pfp_invariant_executiontracestepsstepafter. ff_h_pfp_invariant_executiontracestepsstepafter + S (pfh_after_invariant_executiontracestepsstep) = S ((S (S (pfh_index_invariant_executiontracesteps))) * pfh_trace_scale_invariant_execution)) /\ exists ff_q_pfp_invariant_executiontracestepsstepafter. pfh_trace_code_invariant_execution = ff_q_pfp_invariant_executiontracestepsstepafter * S ((S (S (pfh_index_invariant_executiontracesteps))) * pfh_trace_scale_invariant_execution) + (pfh_after_invariant_executiontracestepsstep))) /\ (((((exists pfa_gap_invariant_executiontracestepsstepmultiplyleft. pfa_gap_invariant_executiontracestepsstepmultiplyleft + S (pfh_before_invariant_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_invariant_executiontracestepsstepmultiplyright. pfa_gap_invariant_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_invariant_executiontracestepsstepmultiplyresultbound. pfa_gap_invariant_executiontracestepsstepmultiplyresultbound + S (pfh_product_invariant_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_invariant_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_invariant_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_invariant_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_invariant_executiontracestepsstepmultiplyresultcongruence = (pfh_product_invariant_executiontracestepsstep) + (p) * pfa_offset_right_invariant_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_invariant_executiontracestepsstepaddleft. pfa_gap_invariant_executiontracestepsstepaddleft + S (pfh_product_invariant_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_invariant_executiontracestepsstepaddright. pfa_gap_invariant_executiontracestepsstepaddright + S (pfh_coefficient_invariant_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_invariant_executiontracestepsstepaddresultbound. pfa_gap_invariant_executiontracestepsstepaddresultbound + S (pfh_after_invariant_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_invariant_executiontracestepsstepaddresultcongruence pfa_offset_right_invariant_executiontracestepsstepaddresultcongruence. ((pfh_product_invariant_executiontracestepsstep) + (pfh_coefficient_invariant_executiontracestepsstep)) + (p) * pfa_offset_left_invariant_executiontracestepsstepaddresultcongruence = (pfh_after_invariant_executiontracestepsstep) + (p) * pfa_offset_right_invariant_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (((exists pfa_gap_invariant_resultbound. pfa_gap_invariant_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_invariant_resultcongruence pfa_offset_right_invariant_resultcongruence. (n) + (p) * pfa_offset_left_invariant_resultcongruence = (r) + (p) * pfa_offset_right_invariant_resultcongruence))))

forall p b c t l n r. (~((p) = 1) /\ forall pfa_factor_left_residue_prime pfa_factor_right_residue_prime. (p) = pfa_factor_left_residue_prime * pfa_factor_right_residue_prime -> pfa_factor_left_residue_prime = 1 \/ pfa_factor_right_residue_prime = 1) -> (exists ff_u_ph_pfh_residue_natural ff_v_ph_pfh_residue_natural. ((((exists fs_h_ph_pfh_residue_natural_body_start. fs_h_ph_pfh_residue_natural_body_start + S (0) = S ((S (0)) * ff_v_ph_pfh_residue_natural)) /\ exists fs_q_ph_pfh_residue_natural_body_start. ff_u_ph_pfh_residue_natural = fs_q_ph_pfh_residue_natural_body_start * S ((S (0)) * ff_v_ph_pfh_residue_natural) + (0))) /\ ((((exists fs_h_ph_pfh_residue_natural_body_terminal. fs_h_ph_pfh_residue_natural_body_terminal + S (n) = S ((S (l)) * ff_v_ph_pfh_residue_natural)) /\ exists fs_q_ph_pfh_residue_natural_body_terminal. ff_u_ph_pfh_residue_natural = fs_q_ph_pfh_residue_natural_body_terminal * S ((S (l)) * ff_v_ph_pfh_residue_natural) + (n))) /\ forall ff_i_ph_pfh_residue_natural_body_steps. (exists ph_bound_pfh_residue_natural_body_steps. ph_bound_pfh_residue_natural_body_steps + S ff_i_ph_pfh_residue_natural_body_steps = l) -> exists ff_coefficient_ph_pfh_residue_natural_body_steps ff_previous_ph_pfh_residue_natural_body_steps ff_current_ph_pfh_residue_natural_body_steps. ((((exists fs_h_ph_pfh_residue_natural_body_steps_coefficient. fs_h_ph_pfh_residue_natural_body_steps_coefficient + S (ff_coefficient_ph_pfh_residue_natural_body_steps) = S ((S (ff_i_ph_pfh_residue_natural_body_steps)) * c)) /\ exists fs_q_ph_pfh_residue_natural_body_steps_coefficient. b = fs_q_ph_pfh_residue_natural_body_steps_coefficient * S ((S (ff_i_ph_pfh_residue_natural_body_steps)) * c) + (ff_coefficient_ph_pfh_residue_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_residue_natural_body_steps_before. fs_h_ph_pfh_residue_natural_body_steps_before + S (ff_previous_ph_pfh_residue_natural_body_steps) = S ((S (ff_i_ph_pfh_residue_natural_body_steps)) * ff_v_ph_pfh_residue_natural)) /\ exists fs_q_ph_pfh_residue_natural_body_steps_before. ff_u_ph_pfh_residue_natural = fs_q_ph_pfh_residue_natural_body_steps_before * S ((S (ff_i_ph_pfh_residue_natural_body_steps)) * ff_v_ph_pfh_residue_natural) + (ff_previous_ph_pfh_residue_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_residue_natural_body_steps_after. fs_h_ph_pfh_residue_natural_body_steps_after + S (ff_current_ph_pfh_residue_natural_body_steps) = S ((S (S ff_i_ph_pfh_residue_natural_body_steps)) * ff_v_ph_pfh_residue_natural)) /\ exists fs_q_ph_pfh_residue_natural_body_steps_after. ff_u_ph_pfh_residue_natural = fs_q_ph_pfh_residue_natural_body_steps_after * S ((S (S ff_i_ph_pfh_residue_natural_body_steps)) * ff_v_ph_pfh_residue_natural) + (ff_current_ph_pfh_residue_natural_body_steps))) /\ ff_current_ph_pfh_residue_natural_body_steps = ff_previous_ph_pfh_residue_natural_body_steps * t + ff_coefficient_ph_pfh_residue_natural_body_steps)))))) -> (exists pfh_trace_code_residue_execution pfh_trace_scale_residue_execution. (((exists pfa_gap_residue_executiontracebase. pfa_gap_residue_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_residue_executiontraceinitial. ff_h_pfp_residue_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_residue_execution)) /\ exists ff_q_pfp_residue_executiontraceinitial. pfh_trace_code_residue_execution = ff_q_pfp_residue_executiontraceinitial * S ((S (0)) * pfh_trace_scale_residue_execution) + (0))) /\ (((((exists ff_h_pfp_residue_executiontraceterminal. ff_h_pfp_residue_executiontraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_residue_execution)) /\ exists ff_q_pfp_residue_executiontraceterminal. pfh_trace_code_residue_execution = ff_q_pfp_residue_executiontraceterminal * S ((S (l)) * pfh_trace_scale_residue_execution) + (r))) /\ ((forall pfh_index_residue_executiontracesteps. (exists pfa_gap_residue_executiontracestepsindex. pfa_gap_residue_executiontracestepsindex + S (pfh_index_residue_executiontracesteps) = (l)) -> (exists pfh_coefficient_residue_executiontracestepsstep pfh_before_residue_executiontracestepsstep pfh_after_residue_executiontracestepsstep pfh_product_residue_executiontracestepsstep. ((((exists ff_h_pfp_residue_executiontracestepsstepcoefficient. ff_h_pfp_residue_executiontracestepsstepcoefficient + S (pfh_coefficient_residue_executiontracestepsstep) = S ((S (pfh_index_residue_executiontracesteps)) * c)) /\ exists ff_q_pfp_residue_executiontracestepsstepcoefficient. b = ff_q_pfp_residue_executiontracestepsstepcoefficient * S ((S (pfh_index_residue_executiontracesteps)) * c) + (pfh_coefficient_residue_executiontracestepsstep))) /\ (((((exists ff_h_pfp_residue_executiontracestepsstepbefore. ff_h_pfp_residue_executiontracestepsstepbefore + S (pfh_before_residue_executiontracestepsstep) = S ((S (pfh_index_residue_executiontracesteps)) * pfh_trace_scale_residue_execution)) /\ exists ff_q_pfp_residue_executiontracestepsstepbefore. pfh_trace_code_residue_execution = ff_q_pfp_residue_executiontracestepsstepbefore * S ((S (pfh_index_residue_executiontracesteps)) * pfh_trace_scale_residue_execution) + (pfh_before_residue_executiontracestepsstep))) /\ (((((exists ff_h_pfp_residue_executiontracestepsstepafter. ff_h_pfp_residue_executiontracestepsstepafter + S (pfh_after_residue_executiontracestepsstep) = S ((S (S (pfh_index_residue_executiontracesteps))) * pfh_trace_scale_residue_execution)) /\ exists ff_q_pfp_residue_executiontracestepsstepafter. pfh_trace_code_residue_execution = ff_q_pfp_residue_executiontracestepsstepafter * S ((S (S (pfh_index_residue_executiontracesteps))) * pfh_trace_scale_residue_execution) + (pfh_after_residue_executiontracestepsstep))) /\ (((((exists pfa_gap_residue_executiontracestepsstepmultiplyleft. pfa_gap_residue_executiontracestepsstepmultiplyleft + S (pfh_before_residue_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_residue_executiontracestepsstepmultiplyright. pfa_gap_residue_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_residue_executiontracestepsstepmultiplyresultbound. pfa_gap_residue_executiontracestepsstepmultiplyresultbound + S (pfh_product_residue_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_residue_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_residue_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_residue_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_residue_executiontracestepsstepmultiplyresultcongruence = (pfh_product_residue_executiontracestepsstep) + (p) * pfa_offset_right_residue_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_residue_executiontracestepsstepaddleft. pfa_gap_residue_executiontracestepsstepaddleft + S (pfh_product_residue_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_residue_executiontracestepsstepaddright. pfa_gap_residue_executiontracestepsstepaddright + S (pfh_coefficient_residue_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_residue_executiontracestepsstepaddresultbound. pfa_gap_residue_executiontracestepsstepaddresultbound + S (pfh_after_residue_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_residue_executiontracestepsstepaddresultcongruence pfa_offset_right_residue_executiontracestepsstepaddresultcongruence. ((pfh_product_residue_executiontracestepsstep) + (pfh_coefficient_residue_executiontracestepsstep)) + (p) * pfa_offset_left_residue_executiontracestepsstepaddresultcongruence = (pfh_after_residue_executiontracestepsstep) + (p) * pfa_offset_right_residue_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (((exists pfa_gap_residue_resultbound. pfa_gap_residue_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_residue_resultcongruence pfa_offset_right_residue_resultcongruence. (n) + (p) * pfa_offset_left_residue_resultcongruence = (r) + (p) * pfa_offset_right_residue_resultcongruence))))

forall p b c t l r s. (~((p) = 1) /\ forall pfa_factor_left_functional_prime pfa_factor_right_functional_prime. (p) = pfa_factor_left_functional_prime * pfa_factor_right_functional_prime -> pfa_factor_left_functional_prime = 1 \/ pfa_factor_right_functional_prime = 1) -> (exists pfh_trace_code_functional_first pfh_trace_scale_functional_first. (((exists pfa_gap_functional_firsttracebase. pfa_gap_functional_firsttracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_functional_firsttraceinitial. ff_h_pfp_functional_firsttraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttraceinitial. pfh_trace_code_functional_first = ff_q_pfp_functional_firsttraceinitial * S ((S (0)) * pfh_trace_scale_functional_first) + (0))) /\ (((((exists ff_h_pfp_functional_firsttraceterminal. ff_h_pfp_functional_firsttraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttraceterminal. pfh_trace_code_functional_first = ff_q_pfp_functional_firsttraceterminal * S ((S (l)) * pfh_trace_scale_functional_first) + (r))) /\ ((forall pfh_index_functional_firsttracesteps. (exists pfa_gap_functional_firsttracestepsindex. pfa_gap_functional_firsttracestepsindex + S (pfh_index_functional_firsttracesteps) = (l)) -> (exists pfh_coefficient_functional_firsttracestepsstep pfh_before_functional_firsttracestepsstep pfh_after_functional_firsttracestepsstep pfh_product_functional_firsttracestepsstep. ((((exists ff_h_pfp_functional_firsttracestepsstepcoefficient. ff_h_pfp_functional_firsttracestepsstepcoefficient + S (pfh_coefficient_functional_firsttracestepsstep) = S ((S (pfh_index_functional_firsttracesteps)) * c)) /\ exists ff_q_pfp_functional_firsttracestepsstepcoefficient. b = ff_q_pfp_functional_firsttracestepsstepcoefficient * S ((S (pfh_index_functional_firsttracesteps)) * c) + (pfh_coefficient_functional_firsttracestepsstep))) /\ (((((exists ff_h_pfp_functional_firsttracestepsstepbefore. ff_h_pfp_functional_firsttracestepsstepbefore + S (pfh_before_functional_firsttracestepsstep) = S ((S (pfh_index_functional_firsttracesteps)) * pfh_trace_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttracestepsstepbefore. pfh_trace_code_functional_first = ff_q_pfp_functional_firsttracestepsstepbefore * S ((S (pfh_index_functional_firsttracesteps)) * pfh_trace_scale_functional_first) + (pfh_before_functional_firsttracestepsstep))) /\ (((((exists ff_h_pfp_functional_firsttracestepsstepafter. ff_h_pfp_functional_firsttracestepsstepafter + S (pfh_after_functional_firsttracestepsstep) = S ((S (S (pfh_index_functional_firsttracesteps))) * pfh_trace_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttracestepsstepafter. pfh_trace_code_functional_first = ff_q_pfp_functional_firsttracestepsstepafter * S ((S (S (pfh_index_functional_firsttracesteps))) * pfh_trace_scale_functional_first) + (pfh_after_functional_firsttracestepsstep))) /\ (((((exists pfa_gap_functional_firsttracestepsstepmultiplyleft. pfa_gap_functional_firsttracestepsstepmultiplyleft + S (pfh_before_functional_firsttracestepsstep) = (p)) /\ (((exists pfa_gap_functional_firsttracestepsstepmultiplyright. pfa_gap_functional_firsttracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_functional_firsttracestepsstepmultiplyresultbound. pfa_gap_functional_firsttracestepsstepmultiplyresultbound + S (pfh_product_functional_firsttracestepsstep) = (p)) /\ ((exists pfa_offset_left_functional_firsttracestepsstepmultiplyresultcongruence pfa_offset_right_functional_firsttracestepsstepmultiplyresultcongruence. ((pfh_before_functional_firsttracestepsstep) * (t)) + (p) * pfa_offset_left_functional_firsttracestepsstepmultiplyresultcongruence = (pfh_product_functional_firsttracestepsstep) + (p) * pfa_offset_right_functional_firsttracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_functional_firsttracestepsstepaddleft. pfa_gap_functional_firsttracestepsstepaddleft + S (pfh_product_functional_firsttracestepsstep) = (p)) /\ (((exists pfa_gap_functional_firsttracestepsstepaddright. pfa_gap_functional_firsttracestepsstepaddright + S (pfh_coefficient_functional_firsttracestepsstep) = (p)) /\ ((((exists pfa_gap_functional_firsttracestepsstepaddresultbound. pfa_gap_functional_firsttracestepsstepaddresultbound + S (pfh_after_functional_firsttracestepsstep) = (p)) /\ ((exists pfa_offset_left_functional_firsttracestepsstepaddresultcongruence pfa_offset_right_functional_firsttracestepsstepaddresultcongruence. ((pfh_product_functional_firsttracestepsstep) + (pfh_coefficient_functional_firsttracestepsstep)) + (p) * pfa_offset_left_functional_firsttracestepsstepaddresultcongruence = (pfh_after_functional_firsttracestepsstep) + (p) * pfa_offset_right_functional_firsttracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (exists pfh_trace_code_functional_second pfh_trace_scale_functional_second. (((exists pfa_gap_functional_secondtracebase. pfa_gap_functional_secondtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_functional_secondtraceinitial. ff_h_pfp_functional_secondtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtraceinitial. pfh_trace_code_functional_second = ff_q_pfp_functional_secondtraceinitial * S ((S (0)) * pfh_trace_scale_functional_second) + (0))) /\ (((((exists ff_h_pfp_functional_secondtraceterminal. ff_h_pfp_functional_secondtraceterminal + S (s) = S ((S (l)) * pfh_trace_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtraceterminal. pfh_trace_code_functional_second = ff_q_pfp_functional_secondtraceterminal * S ((S (l)) * pfh_trace_scale_functional_second) + (s))) /\ ((forall pfh_index_functional_secondtracesteps. (exists pfa_gap_functional_secondtracestepsindex. pfa_gap_functional_secondtracestepsindex + S (pfh_index_functional_secondtracesteps) = (l)) -> (exists pfh_coefficient_functional_secondtracestepsstep pfh_before_functional_secondtracestepsstep pfh_after_functional_secondtracestepsstep pfh_product_functional_secondtracestepsstep. ((((exists ff_h_pfp_functional_secondtracestepsstepcoefficient. ff_h_pfp_functional_secondtracestepsstepcoefficient + S (pfh_coefficient_functional_secondtracestepsstep) = S ((S (pfh_index_functional_secondtracesteps)) * c)) /\ exists ff_q_pfp_functional_secondtracestepsstepcoefficient. b = ff_q_pfp_functional_secondtracestepsstepcoefficient * S ((S (pfh_index_functional_secondtracesteps)) * c) + (pfh_coefficient_functional_secondtracestepsstep))) /\ (((((exists ff_h_pfp_functional_secondtracestepsstepbefore. ff_h_pfp_functional_secondtracestepsstepbefore + S (pfh_before_functional_secondtracestepsstep) = S ((S (pfh_index_functional_secondtracesteps)) * pfh_trace_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtracestepsstepbefore. pfh_trace_code_functional_second = ff_q_pfp_functional_secondtracestepsstepbefore * S ((S (pfh_index_functional_secondtracesteps)) * pfh_trace_scale_functional_second) + (pfh_before_functional_secondtracestepsstep))) /\ (((((exists ff_h_pfp_functional_secondtracestepsstepafter. ff_h_pfp_functional_secondtracestepsstepafter + S (pfh_after_functional_secondtracestepsstep) = S ((S (S (pfh_index_functional_secondtracesteps))) * pfh_trace_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtracestepsstepafter. pfh_trace_code_functional_second = ff_q_pfp_functional_secondtracestepsstepafter * S ((S (S (pfh_index_functional_secondtracesteps))) * pfh_trace_scale_functional_second) + (pfh_after_functional_secondtracestepsstep))) /\ (((((exists pfa_gap_functional_secondtracestepsstepmultiplyleft. pfa_gap_functional_secondtracestepsstepmultiplyleft + S (pfh_before_functional_secondtracestepsstep) = (p)) /\ (((exists pfa_gap_functional_secondtracestepsstepmultiplyright. pfa_gap_functional_secondtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_functional_secondtracestepsstepmultiplyresultbound. pfa_gap_functional_secondtracestepsstepmultiplyresultbound + S (pfh_product_functional_secondtracestepsstep) = (p)) /\ ((exists pfa_offset_left_functional_secondtracestepsstepmultiplyresultcongruence pfa_offset_right_functional_secondtracestepsstepmultiplyresultcongruence. ((pfh_before_functional_secondtracestepsstep) * (t)) + (p) * pfa_offset_left_functional_secondtracestepsstepmultiplyresultcongruence = (pfh_product_functional_secondtracestepsstep) + (p) * pfa_offset_right_functional_secondtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_functional_secondtracestepsstepaddleft. pfa_gap_functional_secondtracestepsstepaddleft + S (pfh_product_functional_secondtracestepsstep) = (p)) /\ (((exists pfa_gap_functional_secondtracestepsstepaddright. pfa_gap_functional_secondtracestepsstepaddright + S (pfh_coefficient_functional_secondtracestepsstep) = (p)) /\ ((((exists pfa_gap_functional_secondtracestepsstepaddresultbound. pfa_gap_functional_secondtracestepsstepaddresultbound + S (pfh_after_functional_secondtracestepsstep) = (p)) /\ ((exists pfa_offset_left_functional_secondtracestepsstepaddresultcongruence pfa_offset_right_functional_secondtracestepsstepaddresultcongruence. ((pfh_product_functional_secondtracestepsstep) + (pfh_coefficient_functional_secondtracestepsstep)) + (p) * pfa_offset_left_functional_secondtracestepsstepaddresultcongruence = (pfh_after_functional_secondtracestepsstep) + (p) * pfa_offset_right_functional_secondtracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> r=s

forall p b c t l. (~((p) = 1) /\ forall pfa_factor_left_unique_prime pfa_factor_right_unique_prime. (p) = pfa_factor_left_unique_prime * pfa_factor_right_unique_prime -> pfa_factor_left_unique_prime = 1 \/ pfa_factor_right_unique_prime = 1) -> (forall fom_index_pfp_unique_coefficients. (exists fom_gap_pfp_unique_coefficients_index_bound. fom_gap_pfp_unique_coefficients_index_bound + S (fom_index_pfp_unique_coefficients) = l) -> exists fom_value_pfp_unique_coefficients. ((((exists fom_beta_height_pfp_unique_coefficients_entry. fom_beta_height_pfp_unique_coefficients_entry + S (fom_value_pfp_unique_coefficients) = S ((S (fom_index_pfp_unique_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_unique_coefficients_entry. b = fom_beta_quotient_pfp_unique_coefficients_entry * S ((S (fom_index_pfp_unique_coefficients)) * c) + (fom_value_pfp_unique_coefficients))) /\ (exists fom_gap_pfp_unique_coefficients_value_bound. fom_gap_pfp_unique_coefficients_value_bound + S (fom_value_pfp_unique_coefficients) = p))) -> (exists pfa_gap_unique_base. pfa_gap_unique_base + S (t) = (p)) -> exists r. (exists pfh_trace_code_unique_chosen pfh_trace_scale_unique_chosen. (((exists pfa_gap_unique_chosentracebase. pfa_gap_unique_chosentracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_unique_chosentraceinitial. ff_h_pfp_unique_chosentraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_unique_chosen)) /\ exists ff_q_pfp_unique_chosentraceinitial. pfh_trace_code_unique_chosen = ff_q_pfp_unique_chosentraceinitial * S ((S (0)) * pfh_trace_scale_unique_chosen) + (0))) /\ (((((exists ff_h_pfp_unique_chosentraceterminal. ff_h_pfp_unique_chosentraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_unique_chosen)) /\ exists ff_q_pfp_unique_chosentraceterminal. pfh_trace_code_unique_chosen = ff_q_pfp_unique_chosentraceterminal * S ((S (l)) * pfh_trace_scale_unique_chosen) + (r))) /\ ((forall pfh_index_unique_chosentracesteps. (exists pfa_gap_unique_chosentracestepsindex. pfa_gap_unique_chosentracestepsindex + S (pfh_index_unique_chosentracesteps) = (l)) -> (exists pfh_coefficient_unique_chosentracestepsstep pfh_before_unique_chosentracestepsstep pfh_after_unique_chosentracestepsstep pfh_product_unique_chosentracestepsstep. ((((exists ff_h_pfp_unique_chosentracestepsstepcoefficient. ff_h_pfp_unique_chosentracestepsstepcoefficient + S (pfh_coefficient_unique_chosentracestepsstep) = S ((S (pfh_index_unique_chosentracesteps)) * c)) /\ exists ff_q_pfp_unique_chosentracestepsstepcoefficient. b = ff_q_pfp_unique_chosentracestepsstepcoefficient * S ((S (pfh_index_unique_chosentracesteps)) * c) + (pfh_coefficient_unique_chosentracestepsstep))) /\ (((((exists ff_h_pfp_unique_chosentracestepsstepbefore. ff_h_pfp_unique_chosentracestepsstepbefore + S (pfh_before_unique_chosentracestepsstep) = S ((S (pfh_index_unique_chosentracesteps)) * pfh_trace_scale_unique_chosen)) /\ exists ff_q_pfp_unique_chosentracestepsstepbefore. pfh_trace_code_unique_chosen = ff_q_pfp_unique_chosentracestepsstepbefore * S ((S (pfh_index_unique_chosentracesteps)) * pfh_trace_scale_unique_chosen) + (pfh_before_unique_chosentracestepsstep))) /\ (((((exists ff_h_pfp_unique_chosentracestepsstepafter. ff_h_pfp_unique_chosentracestepsstepafter + S (pfh_after_unique_chosentracestepsstep) = S ((S (S (pfh_index_unique_chosentracesteps))) * pfh_trace_scale_unique_chosen)) /\ exists ff_q_pfp_unique_chosentracestepsstepafter. pfh_trace_code_unique_chosen = ff_q_pfp_unique_chosentracestepsstepafter * S ((S (S (pfh_index_unique_chosentracesteps))) * pfh_trace_scale_unique_chosen) + (pfh_after_unique_chosentracestepsstep))) /\ (((((exists pfa_gap_unique_chosentracestepsstepmultiplyleft. pfa_gap_unique_chosentracestepsstepmultiplyleft + S (pfh_before_unique_chosentracestepsstep) = (p)) /\ (((exists pfa_gap_unique_chosentracestepsstepmultiplyright. pfa_gap_unique_chosentracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_unique_chosentracestepsstepmultiplyresultbound. pfa_gap_unique_chosentracestepsstepmultiplyresultbound + S (pfh_product_unique_chosentracestepsstep) = (p)) /\ ((exists pfa_offset_left_unique_chosentracestepsstepmultiplyresultcongruence pfa_offset_right_unique_chosentracestepsstepmultiplyresultcongruence. ((pfh_before_unique_chosentracestepsstep) * (t)) + (p) * pfa_offset_left_unique_chosentracestepsstepmultiplyresultcongruence = (pfh_product_unique_chosentracestepsstep) + (p) * pfa_offset_right_unique_chosentracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_unique_chosentracestepsstepaddleft. pfa_gap_unique_chosentracestepsstepaddleft + S (pfh_product_unique_chosentracestepsstep) = (p)) /\ (((exists pfa_gap_unique_chosentracestepsstepaddright. pfa_gap_unique_chosentracestepsstepaddright + S (pfh_coefficient_unique_chosentracestepsstep) = (p)) /\ ((((exists pfa_gap_unique_chosentracestepsstepaddresultbound. pfa_gap_unique_chosentracestepsstepaddresultbound + S (pfh_after_unique_chosentracestepsstep) = (p)) /\ ((exists pfa_offset_left_unique_chosentracestepsstepaddresultcongruence pfa_offset_right_unique_chosentracestepsstepaddresultcongruence. ((pfh_product_unique_chosentracestepsstep) + (pfh_coefficient_unique_chosentracestepsstep)) + (p) * pfa_offset_left_unique_chosentracestepsstepaddresultcongruence = (pfh_after_unique_chosentracestepsstep) + (p) * pfa_offset_right_unique_chosentracestepsstepaddresultcongruence))))))))))))))))))))))))))) /\ forall s. (exists pfh_trace_code_unique_other pfh_trace_scale_unique_other. (((exists pfa_gap_unique_othertracebase. pfa_gap_unique_othertracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_unique_othertraceinitial. ff_h_pfp_unique_othertraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_unique_other)) /\ exists ff_q_pfp_unique_othertraceinitial. pfh_trace_code_unique_other = ff_q_pfp_unique_othertraceinitial * S ((S (0)) * pfh_trace_scale_unique_other) + (0))) /\ (((((exists ff_h_pfp_unique_othertraceterminal. ff_h_pfp_unique_othertraceterminal + S (s) = S ((S (l)) * pfh_trace_scale_unique_other)) /\ exists ff_q_pfp_unique_othertraceterminal. pfh_trace_code_unique_other = ff_q_pfp_unique_othertraceterminal * S ((S (l)) * pfh_trace_scale_unique_other) + (s))) /\ ((forall pfh_index_unique_othertracesteps. (exists pfa_gap_unique_othertracestepsindex. pfa_gap_unique_othertracestepsindex + S (pfh_index_unique_othertracesteps) = (l)) -> (exists pfh_coefficient_unique_othertracestepsstep pfh_before_unique_othertracestepsstep pfh_after_unique_othertracestepsstep pfh_product_unique_othertracestepsstep. ((((exists ff_h_pfp_unique_othertracestepsstepcoefficient. ff_h_pfp_unique_othertracestepsstepcoefficient + S (pfh_coefficient_unique_othertracestepsstep) = S ((S (pfh_index_unique_othertracesteps)) * c)) /\ exists ff_q_pfp_unique_othertracestepsstepcoefficient. b = ff_q_pfp_unique_othertracestepsstepcoefficient * S ((S (pfh_index_unique_othertracesteps)) * c) + (pfh_coefficient_unique_othertracestepsstep))) /\ (((((exists ff_h_pfp_unique_othertracestepsstepbefore. ff_h_pfp_unique_othertracestepsstepbefore + S (pfh_before_unique_othertracestepsstep) = S ((S (pfh_index_unique_othertracesteps)) * pfh_trace_scale_unique_other)) /\ exists ff_q_pfp_unique_othertracestepsstepbefore. pfh_trace_code_unique_other = ff_q_pfp_unique_othertracestepsstepbefore * S ((S (pfh_index_unique_othertracesteps)) * pfh_trace_scale_unique_other) + (pfh_before_unique_othertracestepsstep))) /\ (((((exists ff_h_pfp_unique_othertracestepsstepafter. ff_h_pfp_unique_othertracestepsstepafter + S (pfh_after_unique_othertracestepsstep) = S ((S (S (pfh_index_unique_othertracesteps))) * pfh_trace_scale_unique_other)) /\ exists ff_q_pfp_unique_othertracestepsstepafter. pfh_trace_code_unique_other = ff_q_pfp_unique_othertracestepsstepafter * S ((S (S (pfh_index_unique_othertracesteps))) * pfh_trace_scale_unique_other) + (pfh_after_unique_othertracestepsstep))) /\ (((((exists pfa_gap_unique_othertracestepsstepmultiplyleft. pfa_gap_unique_othertracestepsstepmultiplyleft + S (pfh_before_unique_othertracestepsstep) = (p)) /\ (((exists pfa_gap_unique_othertracestepsstepmultiplyright. pfa_gap_unique_othertracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_unique_othertracestepsstepmultiplyresultbound. pfa_gap_unique_othertracestepsstepmultiplyresultbound + S (pfh_product_unique_othertracestepsstep) = (p)) /\ ((exists pfa_offset_left_unique_othertracestepsstepmultiplyresultcongruence pfa_offset_right_unique_othertracestepsstepmultiplyresultcongruence. ((pfh_before_unique_othertracestepsstep) * (t)) + (p) * pfa_offset_left_unique_othertracestepsstepmultiplyresultcongruence = (pfh_product_unique_othertracestepsstep) + (p) * pfa_offset_right_unique_othertracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_unique_othertracestepsstepaddleft. pfa_gap_unique_othertracestepsstepaddleft + S (pfh_product_unique_othertracestepsstep) = (p)) /\ (((exists pfa_gap_unique_othertracestepsstepaddright. pfa_gap_unique_othertracestepsstepaddright + S (pfh_coefficient_unique_othertracestepsstep) = (p)) /\ ((((exists pfa_gap_unique_othertracestepsstepaddresultbound. pfa_gap_unique_othertracestepsstepaddresultbound + S (pfh_after_unique_othertracestepsstep) = (p)) /\ ((exists pfa_offset_left_unique_othertracestepsstepaddresultcongruence pfa_offset_right_unique_othertracestepsstepaddresultcongruence. ((pfh_product_unique_othertracestepsstep) + (pfh_coefficient_unique_othertracestepsstep)) + (p) * pfa_offset_left_unique_othertracestepsstepaddresultcongruence = (pfh_after_unique_othertracestepsstep) + (p) * pfa_offset_right_unique_othertracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> s=r

forall p b c t. (~((p) = 1) /\ forall pfa_factor_left_empty_construct_prime pfa_factor_right_empty_construct_prime. (p) = pfa_factor_left_empty_construct_prime * pfa_factor_right_empty_construct_prime -> pfa_factor_left_empty_construct_prime = 1 \/ pfa_factor_right_empty_construct_prime = 1) -> (exists pfa_gap_empty_construct_base. pfa_gap_empty_construct_base + S (t) = (p)) -> (exists pfh_trace_code_empty_construct_execution pfh_trace_scale_empty_construct_execution. (((exists pfa_gap_empty_construct_executiontracebase. pfa_gap_empty_construct_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_empty_construct_executiontraceinitial. ff_h_pfp_empty_construct_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontraceinitial. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontraceinitial * S ((S (0)) * pfh_trace_scale_empty_construct_execution) + (0))) /\ (((((exists ff_h_pfp_empty_construct_executiontraceterminal. ff_h_pfp_empty_construct_executiontraceterminal + S (0) = S ((S (0)) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontraceterminal. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontraceterminal * S ((S (0)) * pfh_trace_scale_empty_construct_execution) + (0))) /\ ((forall pfh_index_empty_construct_executiontracesteps. (exists pfa_gap_empty_construct_executiontracestepsindex. pfa_gap_empty_construct_executiontracestepsindex + S (pfh_index_empty_construct_executiontracesteps) = (0)) -> (exists pfh_coefficient_empty_construct_executiontracestepsstep pfh_before_empty_construct_executiontracestepsstep pfh_after_empty_construct_executiontracestepsstep pfh_product_empty_construct_executiontracestepsstep. ((((exists ff_h_pfp_empty_construct_executiontracestepsstepcoefficient. ff_h_pfp_empty_construct_executiontracestepsstepcoefficient + S (pfh_coefficient_empty_construct_executiontracestepsstep) = S ((S (pfh_index_empty_construct_executiontracesteps)) * c)) /\ exists ff_q_pfp_empty_construct_executiontracestepsstepcoefficient. b = ff_q_pfp_empty_construct_executiontracestepsstepcoefficient * S ((S (pfh_index_empty_construct_executiontracesteps)) * c) + (pfh_coefficient_empty_construct_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_construct_executiontracestepsstepbefore. ff_h_pfp_empty_construct_executiontracestepsstepbefore + S (pfh_before_empty_construct_executiontracestepsstep) = S ((S (pfh_index_empty_construct_executiontracesteps)) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontracestepsstepbefore. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontracestepsstepbefore * S ((S (pfh_index_empty_construct_executiontracesteps)) * pfh_trace_scale_empty_construct_execution) + (pfh_before_empty_construct_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_construct_executiontracestepsstepafter. ff_h_pfp_empty_construct_executiontracestepsstepafter + S (pfh_after_empty_construct_executiontracestepsstep) = S ((S (S (pfh_index_empty_construct_executiontracesteps))) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontracestepsstepafter. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontracestepsstepafter * S ((S (S (pfh_index_empty_construct_executiontracesteps))) * pfh_trace_scale_empty_construct_execution) + (pfh_after_empty_construct_executiontracestepsstep))) /\ (((((exists pfa_gap_empty_construct_executiontracestepsstepmultiplyleft. pfa_gap_empty_construct_executiontracestepsstepmultiplyleft + S (pfh_before_empty_construct_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_construct_executiontracestepsstepmultiplyright. pfa_gap_empty_construct_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_empty_construct_executiontracestepsstepmultiplyresultbound. pfa_gap_empty_construct_executiontracestepsstepmultiplyresultbound + S (pfh_product_empty_construct_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_construct_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_empty_construct_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_empty_construct_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_empty_construct_executiontracestepsstepmultiplyresultcongruence = (pfh_product_empty_construct_executiontracestepsstep) + (p) * pfa_offset_right_empty_construct_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_empty_construct_executiontracestepsstepaddleft. pfa_gap_empty_construct_executiontracestepsstepaddleft + S (pfh_product_empty_construct_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_construct_executiontracestepsstepaddright. pfa_gap_empty_construct_executiontracestepsstepaddright + S (pfh_coefficient_empty_construct_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_empty_construct_executiontracestepsstepaddresultbound. pfa_gap_empty_construct_executiontracestepsstepaddresultbound + S (pfh_after_empty_construct_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_construct_executiontracestepsstepaddresultcongruence pfa_offset_right_empty_construct_executiontracestepsstepaddresultcongruence. ((pfh_product_empty_construct_executiontracestepsstep) + (pfh_coefficient_empty_construct_executiontracestepsstep)) + (p) * pfa_offset_left_empty_construct_executiontracestepsstepaddresultcongruence = (pfh_after_empty_construct_executiontracestepsstep) + (p) * pfa_offset_right_empty_construct_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

forall p b c t l a h k r. (~((p) = 1) /\ forall pfa_factor_left_successor_intro_prime pfa_factor_right_successor_intro_prime. (p) = pfa_factor_left_successor_intro_prime * pfa_factor_right_successor_intro_prime -> pfa_factor_left_successor_intro_prime = 1 \/ pfa_factor_right_successor_intro_prime = 1) -> (((exists ff_h_pfp_successor_intro_coefficient. ff_h_pfp_successor_intro_coefficient + S (a) = S ((S (l)) * c)) /\ exists ff_q_pfp_successor_intro_coefficient. b = ff_q_pfp_successor_intro_coefficient * S ((S (l)) * c) + (a))) -> (exists pfh_trace_code_successor_intro_prefix pfh_trace_scale_successor_intro_prefix. (((exists pfa_gap_successor_intro_prefixtracebase. pfa_gap_successor_intro_prefixtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_successor_intro_prefixtraceinitial. ff_h_pfp_successor_intro_prefixtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_successor_intro_prefix)) /\ exists ff_q_pfp_successor_intro_prefixtraceinitial. pfh_trace_code_successor_intro_prefix = ff_q_pfp_successor_intro_prefixtraceinitial * S ((S (0)) * pfh_trace_scale_successor_intro_prefix) + (0))) /\ (((((exists ff_h_pfp_successor_intro_prefixtraceterminal. ff_h_pfp_successor_intro_prefixtraceterminal + S (h) = S ((S (l)) * pfh_trace_scale_successor_intro_prefix)) /\ exists ff_q_pfp_successor_intro_prefixtraceterminal. pfh_trace_code_successor_intro_prefix = ff_q_pfp_successor_intro_prefixtraceterminal * S ((S (l)) * pfh_trace_scale_successor_intro_prefix) + (h))) /\ ((forall pfh_index_successor_intro_prefixtracesteps. (exists pfa_gap_successor_intro_prefixtracestepsindex. pfa_gap_successor_intro_prefixtracestepsindex + S (pfh_index_successor_intro_prefixtracesteps) = (l)) -> (exists pfh_coefficient_successor_intro_prefixtracestepsstep pfh_before_successor_intro_prefixtracestepsstep pfh_after_successor_intro_prefixtracestepsstep pfh_product_successor_intro_prefixtracestepsstep. ((((exists ff_h_pfp_successor_intro_prefixtracestepsstepcoefficient. ff_h_pfp_successor_intro_prefixtracestepsstepcoefficient + S (pfh_coefficient_successor_intro_prefixtracestepsstep) = S ((S (pfh_index_successor_intro_prefixtracesteps)) * c)) /\ exists ff_q_pfp_successor_intro_prefixtracestepsstepcoefficient. b = ff_q_pfp_successor_intro_prefixtracestepsstepcoefficient * S ((S (pfh_index_successor_intro_prefixtracesteps)) * c) + (pfh_coefficient_successor_intro_prefixtracestepsstep))) /\ (((((exists ff_h_pfp_successor_intro_prefixtracestepsstepbefore. ff_h_pfp_successor_intro_prefixtracestepsstepbefore + S (pfh_before_successor_intro_prefixtracestepsstep) = S ((S (pfh_index_successor_intro_prefixtracesteps)) * pfh_trace_scale_successor_intro_prefix)) /\ exists ff_q_pfp_successor_intro_prefixtracestepsstepbefore. pfh_trace_code_successor_intro_prefix = ff_q_pfp_successor_intro_prefixtracestepsstepbefore * S ((S (pfh_index_successor_intro_prefixtracesteps)) * pfh_trace_scale_successor_intro_prefix) + (pfh_before_successor_intro_prefixtracestepsstep))) /\ (((((exists ff_h_pfp_successor_intro_prefixtracestepsstepafter. ff_h_pfp_successor_intro_prefixtracestepsstepafter + S (pfh_after_successor_intro_prefixtracestepsstep) = S ((S (S (pfh_index_successor_intro_prefixtracesteps))) * pfh_trace_scale_successor_intro_prefix)) /\ exists ff_q_pfp_successor_intro_prefixtracestepsstepafter. pfh_trace_code_successor_intro_prefix = ff_q_pfp_successor_intro_prefixtracestepsstepafter * S ((S (S (pfh_index_successor_intro_prefixtracesteps))) * pfh_trace_scale_successor_intro_prefix) + (pfh_after_successor_intro_prefixtracestepsstep))) /\ (((((exists pfa_gap_successor_intro_prefixtracestepsstepmultiplyleft. pfa_gap_successor_intro_prefixtracestepsstepmultiplyleft + S (pfh_before_successor_intro_prefixtracestepsstep) = (p)) /\ (((exists pfa_gap_successor_intro_prefixtracestepsstepmultiplyright. pfa_gap_successor_intro_prefixtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_successor_intro_prefixtracestepsstepmultiplyresultbound. pfa_gap_successor_intro_prefixtracestepsstepmultiplyresultbound + S (pfh_product_successor_intro_prefixtracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_intro_prefixtracestepsstepmultiplyresultcongruence pfa_offset_right_successor_intro_prefixtracestepsstepmultiplyresultcongruence. ((pfh_before_successor_intro_prefixtracestepsstep) * (t)) + (p) * pfa_offset_left_successor_intro_prefixtracestepsstepmultiplyresultcongruence = (pfh_product_successor_intro_prefixtracestepsstep) + (p) * pfa_offset_right_successor_intro_prefixtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_successor_intro_prefixtracestepsstepaddleft. pfa_gap_successor_intro_prefixtracestepsstepaddleft + S (pfh_product_successor_intro_prefixtracestepsstep) = (p)) /\ (((exists pfa_gap_successor_intro_prefixtracestepsstepaddright. pfa_gap_successor_intro_prefixtracestepsstepaddright + S (pfh_coefficient_successor_intro_prefixtracestepsstep) = (p)) /\ ((((exists pfa_gap_successor_intro_prefixtracestepsstepaddresultbound. pfa_gap_successor_intro_prefixtracestepsstepaddresultbound + S (pfh_after_successor_intro_prefixtracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_intro_prefixtracestepsstepaddresultcongruence pfa_offset_right_successor_intro_prefixtracestepsstepaddresultcongruence. ((pfh_product_successor_intro_prefixtracestepsstep) + (pfh_coefficient_successor_intro_prefixtracestepsstep)) + (p) * pfa_offset_left_successor_intro_prefixtracestepsstepaddresultcongruence = (pfh_after_successor_intro_prefixtracestepsstep) + (p) * pfa_offset_right_successor_intro_prefixtracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (((exists pfa_gap_successor_intro_multiplyleft. pfa_gap_successor_intro_multiplyleft + S (h) = (p)) /\ (((exists pfa_gap_successor_intro_multiplyright. pfa_gap_successor_intro_multiplyright + S (t) = (p)) /\ ((((exists pfa_gap_successor_intro_multiplyresultbound. pfa_gap_successor_intro_multiplyresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_successor_intro_multiplyresultcongruence pfa_offset_right_successor_intro_multiplyresultcongruence. ((h) * (t)) + (p) * pfa_offset_left_successor_intro_multiplyresultcongruence = (k) + (p) * pfa_offset_right_successor_intro_multiplyresultcongruence))))))))) -> (((exists pfa_gap_successor_intro_addleft. pfa_gap_successor_intro_addleft + S (k) = (p)) /\ (((exists pfa_gap_successor_intro_addright. pfa_gap_successor_intro_addright + S (a) = (p)) /\ ((((exists pfa_gap_successor_intro_addresultbound. pfa_gap_successor_intro_addresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_successor_intro_addresultcongruence pfa_offset_right_successor_intro_addresultcongruence. ((k) + (a)) + (p) * pfa_offset_left_successor_intro_addresultcongruence = (r) + (p) * pfa_offset_right_successor_intro_addresultcongruence))))))))) -> (exists pfh_trace_code_successor_intro_execution pfh_trace_scale_successor_intro_execution. (((exists pfa_gap_successor_intro_executiontracebase. pfa_gap_successor_intro_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_successor_intro_executiontraceinitial. ff_h_pfp_successor_intro_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_successor_intro_execution)) /\ exists ff_q_pfp_successor_intro_executiontraceinitial. pfh_trace_code_successor_intro_execution = ff_q_pfp_successor_intro_executiontraceinitial * S ((S (0)) * pfh_trace_scale_successor_intro_execution) + (0))) /\ (((((exists ff_h_pfp_successor_intro_executiontraceterminal. ff_h_pfp_successor_intro_executiontraceterminal + S (r) = S ((S (S l)) * pfh_trace_scale_successor_intro_execution)) /\ exists ff_q_pfp_successor_intro_executiontraceterminal. pfh_trace_code_successor_intro_execution = ff_q_pfp_successor_intro_executiontraceterminal * S ((S (S l)) * pfh_trace_scale_successor_intro_execution) + (r))) /\ ((forall pfh_index_successor_intro_executiontracesteps. (exists pfa_gap_successor_intro_executiontracestepsindex. pfa_gap_successor_intro_executiontracestepsindex + S (pfh_index_successor_intro_executiontracesteps) = (S l)) -> (exists pfh_coefficient_successor_intro_executiontracestepsstep pfh_before_successor_intro_executiontracestepsstep pfh_after_successor_intro_executiontracestepsstep pfh_product_successor_intro_executiontracestepsstep. ((((exists ff_h_pfp_successor_intro_executiontracestepsstepcoefficient. ff_h_pfp_successor_intro_executiontracestepsstepcoefficient + S (pfh_coefficient_successor_intro_executiontracestepsstep) = S ((S (pfh_index_successor_intro_executiontracesteps)) * c)) /\ exists ff_q_pfp_successor_intro_executiontracestepsstepcoefficient. b = ff_q_pfp_successor_intro_executiontracestepsstepcoefficient * S ((S (pfh_index_successor_intro_executiontracesteps)) * c) + (pfh_coefficient_successor_intro_executiontracestepsstep))) /\ (((((exists ff_h_pfp_successor_intro_executiontracestepsstepbefore. ff_h_pfp_successor_intro_executiontracestepsstepbefore + S (pfh_before_successor_intro_executiontracestepsstep) = S ((S (pfh_index_successor_intro_executiontracesteps)) * pfh_trace_scale_successor_intro_execution)) /\ exists ff_q_pfp_successor_intro_executiontracestepsstepbefore. pfh_trace_code_successor_intro_execution = ff_q_pfp_successor_intro_executiontracestepsstepbefore * S ((S (pfh_index_successor_intro_executiontracesteps)) * pfh_trace_scale_successor_intro_execution) + (pfh_before_successor_intro_executiontracestepsstep))) /\ (((((exists ff_h_pfp_successor_intro_executiontracestepsstepafter. ff_h_pfp_successor_intro_executiontracestepsstepafter + S (pfh_after_successor_intro_executiontracestepsstep) = S ((S (S (pfh_index_successor_intro_executiontracesteps))) * pfh_trace_scale_successor_intro_execution)) /\ exists ff_q_pfp_successor_intro_executiontracestepsstepafter. pfh_trace_code_successor_intro_execution = ff_q_pfp_successor_intro_executiontracestepsstepafter * S ((S (S (pfh_index_successor_intro_executiontracesteps))) * pfh_trace_scale_successor_intro_execution) + (pfh_after_successor_intro_executiontracestepsstep))) /\ (((((exists pfa_gap_successor_intro_executiontracestepsstepmultiplyleft. pfa_gap_successor_intro_executiontracestepsstepmultiplyleft + S (pfh_before_successor_intro_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_successor_intro_executiontracestepsstepmultiplyright. pfa_gap_successor_intro_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_successor_intro_executiontracestepsstepmultiplyresultbound. pfa_gap_successor_intro_executiontracestepsstepmultiplyresultbound + S (pfh_product_successor_intro_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_intro_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_successor_intro_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_successor_intro_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_successor_intro_executiontracestepsstepmultiplyresultcongruence = (pfh_product_successor_intro_executiontracestepsstep) + (p) * pfa_offset_right_successor_intro_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_successor_intro_executiontracestepsstepaddleft. pfa_gap_successor_intro_executiontracestepsstepaddleft + S (pfh_product_successor_intro_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_successor_intro_executiontracestepsstepaddright. pfa_gap_successor_intro_executiontracestepsstepaddright + S (pfh_coefficient_successor_intro_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_successor_intro_executiontracestepsstepaddresultbound. pfa_gap_successor_intro_executiontracestepsstepaddresultbound + S (pfh_after_successor_intro_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_successor_intro_executiontracestepsstepaddresultcongruence pfa_offset_right_successor_intro_executiontracestepsstepaddresultcongruence. ((pfh_product_successor_intro_executiontracestepsstep) + (pfh_coefficient_successor_intro_executiontracestepsstep)) + (p) * pfa_offset_left_successor_intro_executiontracestepsstepaddresultcongruence = (pfh_after_successor_intro_executiontracestepsstep) + (p) * pfa_offset_right_successor_intro_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

forall p b c t a. (~((p) = 1) /\ forall pfa_factor_left_constant_prime pfa_factor_right_constant_prime. (p) = pfa_factor_left_constant_prime * pfa_factor_right_constant_prime -> pfa_factor_left_constant_prime = 1 \/ pfa_factor_right_constant_prime = 1) -> (exists pfa_gap_constant_base. pfa_gap_constant_base + S (t) = (p)) -> (exists pfa_gap_constant_value. pfa_gap_constant_value + S (a) = (p)) -> (((exists ff_h_pfp_constant_coefficient. ff_h_pfp_constant_coefficient + S (a) = S ((S (0)) * c)) /\ exists ff_q_pfp_constant_coefficient. b = ff_q_pfp_constant_coefficient * S ((S (0)) * c) + (a))) -> (exists pfh_trace_code_constant_execution pfh_trace_scale_constant_execution. (((exists pfa_gap_constant_executiontracebase. pfa_gap_constant_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_constant_executiontraceinitial. ff_h_pfp_constant_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontraceinitial. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontraceinitial * S ((S (0)) * pfh_trace_scale_constant_execution) + (0))) /\ (((((exists ff_h_pfp_constant_executiontraceterminal. ff_h_pfp_constant_executiontraceterminal + S (a) = S ((S (1)) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontraceterminal. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontraceterminal * S ((S (1)) * pfh_trace_scale_constant_execution) + (a))) /\ ((forall pfh_index_constant_executiontracesteps. (exists pfa_gap_constant_executiontracestepsindex. pfa_gap_constant_executiontracestepsindex + S (pfh_index_constant_executiontracesteps) = (1)) -> (exists pfh_coefficient_constant_executiontracestepsstep pfh_before_constant_executiontracestepsstep pfh_after_constant_executiontracestepsstep pfh_product_constant_executiontracestepsstep. ((((exists ff_h_pfp_constant_executiontracestepsstepcoefficient. ff_h_pfp_constant_executiontracestepsstepcoefficient + S (pfh_coefficient_constant_executiontracestepsstep) = S ((S (pfh_index_constant_executiontracesteps)) * c)) /\ exists ff_q_pfp_constant_executiontracestepsstepcoefficient. b = ff_q_pfp_constant_executiontracestepsstepcoefficient * S ((S (pfh_index_constant_executiontracesteps)) * c) + (pfh_coefficient_constant_executiontracestepsstep))) /\ (((((exists ff_h_pfp_constant_executiontracestepsstepbefore. ff_h_pfp_constant_executiontracestepsstepbefore + S (pfh_before_constant_executiontracestepsstep) = S ((S (pfh_index_constant_executiontracesteps)) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontracestepsstepbefore. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontracestepsstepbefore * S ((S (pfh_index_constant_executiontracesteps)) * pfh_trace_scale_constant_execution) + (pfh_before_constant_executiontracestepsstep))) /\ (((((exists ff_h_pfp_constant_executiontracestepsstepafter. ff_h_pfp_constant_executiontracestepsstepafter + S (pfh_after_constant_executiontracestepsstep) = S ((S (S (pfh_index_constant_executiontracesteps))) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontracestepsstepafter. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontracestepsstepafter * S ((S (S (pfh_index_constant_executiontracesteps))) * pfh_trace_scale_constant_execution) + (pfh_after_constant_executiontracestepsstep))) /\ (((((exists pfa_gap_constant_executiontracestepsstepmultiplyleft. pfa_gap_constant_executiontracestepsstepmultiplyleft + S (pfh_before_constant_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_constant_executiontracestepsstepmultiplyright. pfa_gap_constant_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_constant_executiontracestepsstepmultiplyresultbound. pfa_gap_constant_executiontracestepsstepmultiplyresultbound + S (pfh_product_constant_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_constant_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_constant_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_constant_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_constant_executiontracestepsstepmultiplyresultcongruence = (pfh_product_constant_executiontracestepsstep) + (p) * pfa_offset_right_constant_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_constant_executiontracestepsstepaddleft. pfa_gap_constant_executiontracestepsstepaddleft + S (pfh_product_constant_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_constant_executiontracestepsstepaddright. pfa_gap_constant_executiontracestepsstepaddright + S (pfh_coefficient_constant_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_constant_executiontracestepsstepaddresultbound. pfa_gap_constant_executiontracestepsstepaddresultbound + S (pfh_after_constant_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_constant_executiontracestepsstepaddresultcongruence pfa_offset_right_constant_executiontracestepsstepaddresultcongruence. ((pfh_product_constant_executiontracestepsstep) + (pfh_coefficient_constant_executiontracestepsstep)) + (p) * pfa_offset_left_constant_executiontracestepsstepaddresultcongruence = (pfh_after_constant_executiontracestepsstep) + (p) * pfa_offset_right_constant_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

forall p b c t l. (~((p) = 1) /\ forall pfa_factor_left_zero_prime pfa_factor_right_zero_prime. (p) = pfa_factor_left_zero_prime * pfa_factor_right_zero_prime -> pfa_factor_left_zero_prime = 1 \/ pfa_factor_right_zero_prime = 1) -> (exists pfa_gap_zero_base. pfa_gap_zero_base + S (t) = (p)) -> (forall pfp_repeat_index_zero_coefficients. (exists pfa_gap_zero_coefficientsindex. pfa_gap_zero_coefficientsindex + S (pfp_repeat_index_zero_coefficients) = (l)) -> (((exists ff_h_pfp_zero_coefficientsentry. ff_h_pfp_zero_coefficientsentry + S (0) = S ((S (pfp_repeat_index_zero_coefficients)) * c)) /\ exists ff_q_pfp_zero_coefficientsentry. b = ff_q_pfp_zero_coefficientsentry * S ((S (pfp_repeat_index_zero_coefficients)) * c) + (0)))) -> (exists pfh_trace_code_zero_execution pfh_trace_scale_zero_execution. (((exists pfa_gap_zero_executiontracebase. pfa_gap_zero_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_zero_executiontraceinitial. ff_h_pfp_zero_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_zero_execution)) /\ exists ff_q_pfp_zero_executiontraceinitial. pfh_trace_code_zero_execution = ff_q_pfp_zero_executiontraceinitial * S ((S (0)) * pfh_trace_scale_zero_execution) + (0))) /\ (((((exists ff_h_pfp_zero_executiontraceterminal. ff_h_pfp_zero_executiontraceterminal + S (0) = S ((S (l)) * pfh_trace_scale_zero_execution)) /\ exists ff_q_pfp_zero_executiontraceterminal. pfh_trace_code_zero_execution = ff_q_pfp_zero_executiontraceterminal * S ((S (l)) * pfh_trace_scale_zero_execution) + (0))) /\ ((forall pfh_index_zero_executiontracesteps. (exists pfa_gap_zero_executiontracestepsindex. pfa_gap_zero_executiontracestepsindex + S (pfh_index_zero_executiontracesteps) = (l)) -> (exists pfh_coefficient_zero_executiontracestepsstep pfh_before_zero_executiontracestepsstep pfh_after_zero_executiontracestepsstep pfh_product_zero_executiontracestepsstep. ((((exists ff_h_pfp_zero_executiontracestepsstepcoefficient. ff_h_pfp_zero_executiontracestepsstepcoefficient + S (pfh_coefficient_zero_executiontracestepsstep) = S ((S (pfh_index_zero_executiontracesteps)) * c)) /\ exists ff_q_pfp_zero_executiontracestepsstepcoefficient. b = ff_q_pfp_zero_executiontracestepsstepcoefficient * S ((S (pfh_index_zero_executiontracesteps)) * c) + (pfh_coefficient_zero_executiontracestepsstep))) /\ (((((exists ff_h_pfp_zero_executiontracestepsstepbefore. ff_h_pfp_zero_executiontracestepsstepbefore + S (pfh_before_zero_executiontracestepsstep) = S ((S (pfh_index_zero_executiontracesteps)) * pfh_trace_scale_zero_execution)) /\ exists ff_q_pfp_zero_executiontracestepsstepbefore. pfh_trace_code_zero_execution = ff_q_pfp_zero_executiontracestepsstepbefore * S ((S (pfh_index_zero_executiontracesteps)) * pfh_trace_scale_zero_execution) + (pfh_before_zero_executiontracestepsstep))) /\ (((((exists ff_h_pfp_zero_executiontracestepsstepafter. ff_h_pfp_zero_executiontracestepsstepafter + S (pfh_after_zero_executiontracestepsstep) = S ((S (S (pfh_index_zero_executiontracesteps))) * pfh_trace_scale_zero_execution)) /\ exists ff_q_pfp_zero_executiontracestepsstepafter. pfh_trace_code_zero_execution = ff_q_pfp_zero_executiontracestepsstepafter * S ((S (S (pfh_index_zero_executiontracesteps))) * pfh_trace_scale_zero_execution) + (pfh_after_zero_executiontracestepsstep))) /\ (((((exists pfa_gap_zero_executiontracestepsstepmultiplyleft. pfa_gap_zero_executiontracestepsstepmultiplyleft + S (pfh_before_zero_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_zero_executiontracestepsstepmultiplyright. pfa_gap_zero_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_zero_executiontracestepsstepmultiplyresultbound. pfa_gap_zero_executiontracestepsstepmultiplyresultbound + S (pfh_product_zero_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_zero_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_zero_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_zero_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_zero_executiontracestepsstepmultiplyresultcongruence = (pfh_product_zero_executiontracestepsstep) + (p) * pfa_offset_right_zero_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_zero_executiontracestepsstepaddleft. pfa_gap_zero_executiontracestepsstepaddleft + S (pfh_product_zero_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_zero_executiontracestepsstepaddright. pfa_gap_zero_executiontracestepsstepaddright + S (pfh_coefficient_zero_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_zero_executiontracestepsstepaddresultbound. pfa_gap_zero_executiontracestepsstepaddresultbound + S (pfh_after_zero_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_zero_executiontracestepsstepaddresultcongruence pfa_offset_right_zero_executiontracestepsstepaddresultcongruence. ((pfh_product_zero_executiontracestepsstep) + (pfh_coefficient_zero_executiontracestepsstep)) + (p) * pfa_offset_left_zero_executiontracestepsstepaddresultcongruence = (pfh_after_zero_executiontracestepsstep) + (p) * pfa_offset_right_zero_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

forall p b c d e t l n r. (~((p) = 1) /\ forall pfa_factor_left_iff_prime pfa_factor_right_iff_prime. (p) = pfa_factor_left_iff_prime * pfa_factor_right_iff_prime -> pfa_factor_left_iff_prime = 1 \/ pfa_factor_right_iff_prime = 1) -> (forall pfp_index_iff_reduction. (exists pfa_gap_iff_reductionindex. pfa_gap_iff_reductionindex + S (pfp_index_iff_reduction) = (l)) -> exists pfp_source_iff_reduction pfp_residue_iff_reduction. ((((exists ff_h_pfp_iff_reductionsource. ff_h_pfp_iff_reductionsource + S (pfp_source_iff_reduction) = S ((S (pfp_index_iff_reduction)) * c)) /\ exists ff_q_pfp_iff_reductionsource. b = ff_q_pfp_iff_reductionsource * S ((S (pfp_index_iff_reduction)) * c) + (pfp_source_iff_reduction))) /\ (((((exists ff_h_pfp_iff_reductiontarget. ff_h_pfp_iff_reductiontarget + S (pfp_residue_iff_reduction) = S ((S (pfp_index_iff_reduction)) * e)) /\ exists ff_q_pfp_iff_reductiontarget. d = ff_q_pfp_iff_reductiontarget * S ((S (pfp_index_iff_reduction)) * e) + (pfp_residue_iff_reduction))) /\ ((((exists pfa_gap_iff_reductionresiduebound. pfa_gap_iff_reductionresiduebound + S (pfp_residue_iff_reduction) = (p)) /\ ((exists pfa_offset_left_iff_reductionresiduecongruence pfa_offset_right_iff_reductionresiduecongruence. (pfp_source_iff_reduction) + (p) * pfa_offset_left_iff_reductionresiduecongruence = (pfp_residue_iff_reduction) + (p) * pfa_offset_right_iff_reductionresiduecongruence))))))))) -> (exists pfa_gap_iff_base. pfa_gap_iff_base + S (t) = (p)) -> (exists ff_u_ph_pfh_iff_natural ff_v_ph_pfh_iff_natural. ((((exists fs_h_ph_pfh_iff_natural_body_start. fs_h_ph_pfh_iff_natural_body_start + S (0) = S ((S (0)) * ff_v_ph_pfh_iff_natural)) /\ exists fs_q_ph_pfh_iff_natural_body_start. ff_u_ph_pfh_iff_natural = fs_q_ph_pfh_iff_natural_body_start * S ((S (0)) * ff_v_ph_pfh_iff_natural) + (0))) /\ ((((exists fs_h_ph_pfh_iff_natural_body_terminal. fs_h_ph_pfh_iff_natural_body_terminal + S (n) = S ((S (l)) * ff_v_ph_pfh_iff_natural)) /\ exists fs_q_ph_pfh_iff_natural_body_terminal. ff_u_ph_pfh_iff_natural = fs_q_ph_pfh_iff_natural_body_terminal * S ((S (l)) * ff_v_ph_pfh_iff_natural) + (n))) /\ forall ff_i_ph_pfh_iff_natural_body_steps. (exists ph_bound_pfh_iff_natural_body_steps. ph_bound_pfh_iff_natural_body_steps + S ff_i_ph_pfh_iff_natural_body_steps = l) -> exists ff_coefficient_ph_pfh_iff_natural_body_steps ff_previous_ph_pfh_iff_natural_body_steps ff_current_ph_pfh_iff_natural_body_steps. ((((exists fs_h_ph_pfh_iff_natural_body_steps_coefficient. fs_h_ph_pfh_iff_natural_body_steps_coefficient + S (ff_coefficient_ph_pfh_iff_natural_body_steps) = S ((S (ff_i_ph_pfh_iff_natural_body_steps)) * c)) /\ exists fs_q_ph_pfh_iff_natural_body_steps_coefficient. b = fs_q_ph_pfh_iff_natural_body_steps_coefficient * S ((S (ff_i_ph_pfh_iff_natural_body_steps)) * c) + (ff_coefficient_ph_pfh_iff_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_iff_natural_body_steps_before. fs_h_ph_pfh_iff_natural_body_steps_before + S (ff_previous_ph_pfh_iff_natural_body_steps) = S ((S (ff_i_ph_pfh_iff_natural_body_steps)) * ff_v_ph_pfh_iff_natural)) /\ exists fs_q_ph_pfh_iff_natural_body_steps_before. ff_u_ph_pfh_iff_natural = fs_q_ph_pfh_iff_natural_body_steps_before * S ((S (ff_i_ph_pfh_iff_natural_body_steps)) * ff_v_ph_pfh_iff_natural) + (ff_previous_ph_pfh_iff_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_iff_natural_body_steps_after. fs_h_ph_pfh_iff_natural_body_steps_after + S (ff_current_ph_pfh_iff_natural_body_steps) = S ((S (S ff_i_ph_pfh_iff_natural_body_steps)) * ff_v_ph_pfh_iff_natural)) /\ exists fs_q_ph_pfh_iff_natural_body_steps_after. ff_u_ph_pfh_iff_natural = fs_q_ph_pfh_iff_natural_body_steps_after * S ((S (S ff_i_ph_pfh_iff_natural_body_steps)) * ff_v_ph_pfh_iff_natural) + (ff_current_ph_pfh_iff_natural_body_steps))) /\ ff_current_ph_pfh_iff_natural_body_steps = ff_previous_ph_pfh_iff_natural_body_steps * t + ff_coefficient_ph_pfh_iff_natural_body_steps)))))) -> (((exists pfh_trace_code_iff_execution_forward pfh_trace_scale_iff_execution_forward. (((exists pfa_gap_iff_execution_forwardtracebase. pfa_gap_iff_execution_forwardtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_iff_execution_forwardtraceinitial. ff_h_pfp_iff_execution_forwardtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_iff_execution_forward)) /\ exists ff_q_pfp_iff_execution_forwardtraceinitial. pfh_trace_code_iff_execution_forward = ff_q_pfp_iff_execution_forwardtraceinitial * S ((S (0)) * pfh_trace_scale_iff_execution_forward) + (0))) /\ (((((exists ff_h_pfp_iff_execution_forwardtraceterminal. ff_h_pfp_iff_execution_forwardtraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_iff_execution_forward)) /\ exists ff_q_pfp_iff_execution_forwardtraceterminal. pfh_trace_code_iff_execution_forward = ff_q_pfp_iff_execution_forwardtraceterminal * S ((S (l)) * pfh_trace_scale_iff_execution_forward) + (r))) /\ ((forall pfh_index_iff_execution_forwardtracesteps. (exists pfa_gap_iff_execution_forwardtracestepsindex. pfa_gap_iff_execution_forwardtracestepsindex + S (pfh_index_iff_execution_forwardtracesteps) = (l)) -> (exists pfh_coefficient_iff_execution_forwardtracestepsstep pfh_before_iff_execution_forwardtracestepsstep pfh_after_iff_execution_forwardtracestepsstep pfh_product_iff_execution_forwardtracestepsstep. ((((exists ff_h_pfp_iff_execution_forwardtracestepsstepcoefficient. ff_h_pfp_iff_execution_forwardtracestepsstepcoefficient + S (pfh_coefficient_iff_execution_forwardtracestepsstep) = S ((S (pfh_index_iff_execution_forwardtracesteps)) * e)) /\ exists ff_q_pfp_iff_execution_forwardtracestepsstepcoefficient. d = ff_q_pfp_iff_execution_forwardtracestepsstepcoefficient * S ((S (pfh_index_iff_execution_forwardtracesteps)) * e) + (pfh_coefficient_iff_execution_forwardtracestepsstep))) /\ (((((exists ff_h_pfp_iff_execution_forwardtracestepsstepbefore. ff_h_pfp_iff_execution_forwardtracestepsstepbefore + S (pfh_before_iff_execution_forwardtracestepsstep) = S ((S (pfh_index_iff_execution_forwardtracesteps)) * pfh_trace_scale_iff_execution_forward)) /\ exists ff_q_pfp_iff_execution_forwardtracestepsstepbefore. pfh_trace_code_iff_execution_forward = ff_q_pfp_iff_execution_forwardtracestepsstepbefore * S ((S (pfh_index_iff_execution_forwardtracesteps)) * pfh_trace_scale_iff_execution_forward) + (pfh_before_iff_execution_forwardtracestepsstep))) /\ (((((exists ff_h_pfp_iff_execution_forwardtracestepsstepafter. ff_h_pfp_iff_execution_forwardtracestepsstepafter + S (pfh_after_iff_execution_forwardtracestepsstep) = S ((S (S (pfh_index_iff_execution_forwardtracesteps))) * pfh_trace_scale_iff_execution_forward)) /\ exists ff_q_pfp_iff_execution_forwardtracestepsstepafter. pfh_trace_code_iff_execution_forward = ff_q_pfp_iff_execution_forwardtracestepsstepafter * S ((S (S (pfh_index_iff_execution_forwardtracesteps))) * pfh_trace_scale_iff_execution_forward) + (pfh_after_iff_execution_forwardtracestepsstep))) /\ (((((exists pfa_gap_iff_execution_forwardtracestepsstepmultiplyleft. pfa_gap_iff_execution_forwardtracestepsstepmultiplyleft + S (pfh_before_iff_execution_forwardtracestepsstep) = (p)) /\ (((exists pfa_gap_iff_execution_forwardtracestepsstepmultiplyright. pfa_gap_iff_execution_forwardtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_iff_execution_forwardtracestepsstepmultiplyresultbound. pfa_gap_iff_execution_forwardtracestepsstepmultiplyresultbound + S (pfh_product_iff_execution_forwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_iff_execution_forwardtracestepsstepmultiplyresultcongruence pfa_offset_right_iff_execution_forwardtracestepsstepmultiplyresultcongruence. ((pfh_before_iff_execution_forwardtracestepsstep) * (t)) + (p) * pfa_offset_left_iff_execution_forwardtracestepsstepmultiplyresultcongruence = (pfh_product_iff_execution_forwardtracestepsstep) + (p) * pfa_offset_right_iff_execution_forwardtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_iff_execution_forwardtracestepsstepaddleft. pfa_gap_iff_execution_forwardtracestepsstepaddleft + S (pfh_product_iff_execution_forwardtracestepsstep) = (p)) /\ (((exists pfa_gap_iff_execution_forwardtracestepsstepaddright. pfa_gap_iff_execution_forwardtracestepsstepaddright + S (pfh_coefficient_iff_execution_forwardtracestepsstep) = (p)) /\ ((((exists pfa_gap_iff_execution_forwardtracestepsstepaddresultbound. pfa_gap_iff_execution_forwardtracestepsstepaddresultbound + S (pfh_after_iff_execution_forwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_iff_execution_forwardtracestepsstepaddresultcongruence pfa_offset_right_iff_execution_forwardtracestepsstepaddresultcongruence. ((pfh_product_iff_execution_forwardtracestepsstep) + (pfh_coefficient_iff_execution_forwardtracestepsstep)) + (p) * pfa_offset_left_iff_execution_forwardtracestepsstepaddresultcongruence = (pfh_after_iff_execution_forwardtracestepsstep) + (p) * pfa_offset_right_iff_execution_forwardtracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (((exists pfa_gap_iff_residue_forwardbound. pfa_gap_iff_residue_forwardbound + S (r) = (p)) /\ ((exists pfa_offset_left_iff_residue_forwardcongruence pfa_offset_right_iff_residue_forwardcongruence. (n) + (p) * pfa_offset_left_iff_residue_forwardcongruence = (r) + (p) * pfa_offset_right_iff_residue_forwardcongruence))))) /\ (((((exists pfa_gap_iff_residue_backwardbound. pfa_gap_iff_residue_backwardbound + S (r) = (p)) /\ ((exists pfa_offset_left_iff_residue_backwardcongruence pfa_offset_right_iff_residue_backwardcongruence. (n) + (p) * pfa_offset_left_iff_residue_backwardcongruence = (r) + (p) * pfa_offset_right_iff_residue_backwardcongruence)))) -> (exists pfh_trace_code_iff_execution_backward pfh_trace_scale_iff_execution_backward. (((exists pfa_gap_iff_execution_backwardtracebase. pfa_gap_iff_execution_backwardtracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_iff_execution_backwardtraceinitial. ff_h_pfp_iff_execution_backwardtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_iff_execution_backward)) /\ exists ff_q_pfp_iff_execution_backwardtraceinitial. pfh_trace_code_iff_execution_backward = ff_q_pfp_iff_execution_backwardtraceinitial * S ((S (0)) * pfh_trace_scale_iff_execution_backward) + (0))) /\ (((((exists ff_h_pfp_iff_execution_backwardtraceterminal. ff_h_pfp_iff_execution_backwardtraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_iff_execution_backward)) /\ exists ff_q_pfp_iff_execution_backwardtraceterminal. pfh_trace_code_iff_execution_backward = ff_q_pfp_iff_execution_backwardtraceterminal * S ((S (l)) * pfh_trace_scale_iff_execution_backward) + (r))) /\ ((forall pfh_index_iff_execution_backwardtracesteps. (exists pfa_gap_iff_execution_backwardtracestepsindex. pfa_gap_iff_execution_backwardtracestepsindex + S (pfh_index_iff_execution_backwardtracesteps) = (l)) -> (exists pfh_coefficient_iff_execution_backwardtracestepsstep pfh_before_iff_execution_backwardtracestepsstep pfh_after_iff_execution_backwardtracestepsstep pfh_product_iff_execution_backwardtracestepsstep. ((((exists ff_h_pfp_iff_execution_backwardtracestepsstepcoefficient. ff_h_pfp_iff_execution_backwardtracestepsstepcoefficient + S (pfh_coefficient_iff_execution_backwardtracestepsstep) = S ((S (pfh_index_iff_execution_backwardtracesteps)) * e)) /\ exists ff_q_pfp_iff_execution_backwardtracestepsstepcoefficient. d = ff_q_pfp_iff_execution_backwardtracestepsstepcoefficient * S ((S (pfh_index_iff_execution_backwardtracesteps)) * e) + (pfh_coefficient_iff_execution_backwardtracestepsstep))) /\ (((((exists ff_h_pfp_iff_execution_backwardtracestepsstepbefore. ff_h_pfp_iff_execution_backwardtracestepsstepbefore + S (pfh_before_iff_execution_backwardtracestepsstep) = S ((S (pfh_index_iff_execution_backwardtracesteps)) * pfh_trace_scale_iff_execution_backward)) /\ exists ff_q_pfp_iff_execution_backwardtracestepsstepbefore. pfh_trace_code_iff_execution_backward = ff_q_pfp_iff_execution_backwardtracestepsstepbefore * S ((S (pfh_index_iff_execution_backwardtracesteps)) * pfh_trace_scale_iff_execution_backward) + (pfh_before_iff_execution_backwardtracestepsstep))) /\ (((((exists ff_h_pfp_iff_execution_backwardtracestepsstepafter. ff_h_pfp_iff_execution_backwardtracestepsstepafter + S (pfh_after_iff_execution_backwardtracestepsstep) = S ((S (S (pfh_index_iff_execution_backwardtracesteps))) * pfh_trace_scale_iff_execution_backward)) /\ exists ff_q_pfp_iff_execution_backwardtracestepsstepafter. pfh_trace_code_iff_execution_backward = ff_q_pfp_iff_execution_backwardtracestepsstepafter * S ((S (S (pfh_index_iff_execution_backwardtracesteps))) * pfh_trace_scale_iff_execution_backward) + (pfh_after_iff_execution_backwardtracestepsstep))) /\ (((((exists pfa_gap_iff_execution_backwardtracestepsstepmultiplyleft. pfa_gap_iff_execution_backwardtracestepsstepmultiplyleft + S (pfh_before_iff_execution_backwardtracestepsstep) = (p)) /\ (((exists pfa_gap_iff_execution_backwardtracestepsstepmultiplyright. pfa_gap_iff_execution_backwardtracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_iff_execution_backwardtracestepsstepmultiplyresultbound. pfa_gap_iff_execution_backwardtracestepsstepmultiplyresultbound + S (pfh_product_iff_execution_backwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_iff_execution_backwardtracestepsstepmultiplyresultcongruence pfa_offset_right_iff_execution_backwardtracestepsstepmultiplyresultcongruence. ((pfh_before_iff_execution_backwardtracestepsstep) * (t)) + (p) * pfa_offset_left_iff_execution_backwardtracestepsstepmultiplyresultcongruence = (pfh_product_iff_execution_backwardtracestepsstep) + (p) * pfa_offset_right_iff_execution_backwardtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_iff_execution_backwardtracestepsstepaddleft. pfa_gap_iff_execution_backwardtracestepsstepaddleft + S (pfh_product_iff_execution_backwardtracestepsstep) = (p)) /\ (((exists pfa_gap_iff_execution_backwardtracestepsstepaddright. pfa_gap_iff_execution_backwardtracestepsstepaddright + S (pfh_coefficient_iff_execution_backwardtracestepsstep) = (p)) /\ ((((exists pfa_gap_iff_execution_backwardtracestepsstepaddresultbound. pfa_gap_iff_execution_backwardtracestepsstepaddresultbound + S (pfh_after_iff_execution_backwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_iff_execution_backwardtracestepsstepaddresultcongruence pfa_offset_right_iff_execution_backwardtracestepsstepaddresultcongruence. ((pfh_product_iff_execution_backwardtracestepsstep) + (pfh_coefficient_iff_execution_backwardtracestepsstep)) + (p) * pfa_offset_left_iff_execution_backwardtracestepsstepaddresultcongruence = (pfh_after_iff_execution_backwardtracestepsstep) + (p) * pfa_offset_right_iff_execution_backwardtracestepsstepaddresultcongruence))))))))))))))))))))))))))))))

forall p b c t l r. (~((p) = 1) /\ forall pfa_factor_left_result_bound_prime pfa_factor_right_result_bound_prime. (p) = pfa_factor_left_result_bound_prime * pfa_factor_right_result_bound_prime -> pfa_factor_left_result_bound_prime = 1 \/ pfa_factor_right_result_bound_prime = 1) -> (exists pfh_trace_code_result_bound_execution pfh_trace_scale_result_bound_execution. (((exists pfa_gap_result_bound_executiontracebase. pfa_gap_result_bound_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_result_bound_executiontraceinitial. ff_h_pfp_result_bound_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_result_bound_execution)) /\ exists ff_q_pfp_result_bound_executiontraceinitial. pfh_trace_code_result_bound_execution = ff_q_pfp_result_bound_executiontraceinitial * S ((S (0)) * pfh_trace_scale_result_bound_execution) + (0))) /\ (((((exists ff_h_pfp_result_bound_executiontraceterminal. ff_h_pfp_result_bound_executiontraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_result_bound_execution)) /\ exists ff_q_pfp_result_bound_executiontraceterminal. pfh_trace_code_result_bound_execution = ff_q_pfp_result_bound_executiontraceterminal * S ((S (l)) * pfh_trace_scale_result_bound_execution) + (r))) /\ ((forall pfh_index_result_bound_executiontracesteps. (exists pfa_gap_result_bound_executiontracestepsindex. pfa_gap_result_bound_executiontracestepsindex + S (pfh_index_result_bound_executiontracesteps) = (l)) -> (exists pfh_coefficient_result_bound_executiontracestepsstep pfh_before_result_bound_executiontracestepsstep pfh_after_result_bound_executiontracestepsstep pfh_product_result_bound_executiontracestepsstep. ((((exists ff_h_pfp_result_bound_executiontracestepsstepcoefficient. ff_h_pfp_result_bound_executiontracestepsstepcoefficient + S (pfh_coefficient_result_bound_executiontracestepsstep) = S ((S (pfh_index_result_bound_executiontracesteps)) * c)) /\ exists ff_q_pfp_result_bound_executiontracestepsstepcoefficient. b = ff_q_pfp_result_bound_executiontracestepsstepcoefficient * S ((S (pfh_index_result_bound_executiontracesteps)) * c) + (pfh_coefficient_result_bound_executiontracestepsstep))) /\ (((((exists ff_h_pfp_result_bound_executiontracestepsstepbefore. ff_h_pfp_result_bound_executiontracestepsstepbefore + S (pfh_before_result_bound_executiontracestepsstep) = S ((S (pfh_index_result_bound_executiontracesteps)) * pfh_trace_scale_result_bound_execution)) /\ exists ff_q_pfp_result_bound_executiontracestepsstepbefore. pfh_trace_code_result_bound_execution = ff_q_pfp_result_bound_executiontracestepsstepbefore * S ((S (pfh_index_result_bound_executiontracesteps)) * pfh_trace_scale_result_bound_execution) + (pfh_before_result_bound_executiontracestepsstep))) /\ (((((exists ff_h_pfp_result_bound_executiontracestepsstepafter. ff_h_pfp_result_bound_executiontracestepsstepafter + S (pfh_after_result_bound_executiontracestepsstep) = S ((S (S (pfh_index_result_bound_executiontracesteps))) * pfh_trace_scale_result_bound_execution)) /\ exists ff_q_pfp_result_bound_executiontracestepsstepafter. pfh_trace_code_result_bound_execution = ff_q_pfp_result_bound_executiontracestepsstepafter * S ((S (S (pfh_index_result_bound_executiontracesteps))) * pfh_trace_scale_result_bound_execution) + (pfh_after_result_bound_executiontracestepsstep))) /\ (((((exists pfa_gap_result_bound_executiontracestepsstepmultiplyleft. pfa_gap_result_bound_executiontracestepsstepmultiplyleft + S (pfh_before_result_bound_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_result_bound_executiontracestepsstepmultiplyright. pfa_gap_result_bound_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_result_bound_executiontracestepsstepmultiplyresultbound. pfa_gap_result_bound_executiontracestepsstepmultiplyresultbound + S (pfh_product_result_bound_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_result_bound_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_result_bound_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_result_bound_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_result_bound_executiontracestepsstepmultiplyresultcongruence = (pfh_product_result_bound_executiontracestepsstep) + (p) * pfa_offset_right_result_bound_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_result_bound_executiontracestepsstepaddleft. pfa_gap_result_bound_executiontracestepsstepaddleft + S (pfh_product_result_bound_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_result_bound_executiontracestepsstepaddright. pfa_gap_result_bound_executiontracestepsstepaddright + S (pfh_coefficient_result_bound_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_result_bound_executiontracestepsstepaddresultbound. pfa_gap_result_bound_executiontracestepsstepaddresultbound + S (pfh_after_result_bound_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_result_bound_executiontracestepsstepaddresultcongruence pfa_offset_right_result_bound_executiontracestepsstepaddresultcongruence. ((pfh_product_result_bound_executiontracestepsstep) + (pfh_coefficient_result_bound_executiontracestepsstep)) + (p) * pfa_offset_left_result_bound_executiontracestepsstepaddresultcongruence = (pfh_after_result_bound_executiontracestepsstep) + (p) * pfa_offset_right_result_bound_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (exists pfa_gap_result_bound. pfa_gap_result_bound + S (r) = (p))

forall p b c t l. (~((p) = 1) /\ forall pfa_factor_left_complete_prime pfa_factor_right_complete_prime. (p) = pfa_factor_left_complete_prime * pfa_factor_right_complete_prime -> pfa_factor_left_complete_prime = 1 \/ pfa_factor_right_complete_prime = 1) -> (exists pfa_gap_complete_base. pfa_gap_complete_base + S (t) = (p)) -> exists d e r. ((forall pfp_index_complete_reduction_result. (exists pfa_gap_complete_reduction_resultindex. pfa_gap_complete_reduction_resultindex + S (pfp_index_complete_reduction_result) = (l)) -> exists pfp_source_complete_reduction_result pfp_residue_complete_reduction_result. ((((exists ff_h_pfp_complete_reduction_resultsource. ff_h_pfp_complete_reduction_resultsource + S (pfp_source_complete_reduction_result) = S ((S (pfp_index_complete_reduction_result)) * c)) /\ exists ff_q_pfp_complete_reduction_resultsource. b = ff_q_pfp_complete_reduction_resultsource * S ((S (pfp_index_complete_reduction_result)) * c) + (pfp_source_complete_reduction_result))) /\ (((((exists ff_h_pfp_complete_reduction_resulttarget. ff_h_pfp_complete_reduction_resulttarget + S (pfp_residue_complete_reduction_result) = S ((S (pfp_index_complete_reduction_result)) * e)) /\ exists ff_q_pfp_complete_reduction_resulttarget. d = ff_q_pfp_complete_reduction_resulttarget * S ((S (pfp_index_complete_reduction_result)) * e) + (pfp_residue_complete_reduction_result))) /\ ((((exists pfa_gap_complete_reduction_resultresiduebound. pfa_gap_complete_reduction_resultresiduebound + S (pfp_residue_complete_reduction_result) = (p)) /\ ((exists pfa_offset_left_complete_reduction_resultresiduecongruence pfa_offset_right_complete_reduction_resultresiduecongruence. (pfp_source_complete_reduction_result) + (p) * pfa_offset_left_complete_reduction_resultresiduecongruence = (pfp_residue_complete_reduction_result) + (p) * pfa_offset_right_complete_reduction_resultresiduecongruence))))))))) /\ (((exists pfh_trace_code_complete_execution_result pfh_trace_scale_complete_execution_result. (((exists pfa_gap_complete_execution_resulttracebase. pfa_gap_complete_execution_resulttracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_complete_execution_resulttraceinitial. ff_h_pfp_complete_execution_resulttraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_complete_execution_result)) /\ exists ff_q_pfp_complete_execution_resulttraceinitial. pfh_trace_code_complete_execution_result = ff_q_pfp_complete_execution_resulttraceinitial * S ((S (0)) * pfh_trace_scale_complete_execution_result) + (0))) /\ (((((exists ff_h_pfp_complete_execution_resulttraceterminal. ff_h_pfp_complete_execution_resulttraceterminal + S (r) = S ((S (l)) * pfh_trace_scale_complete_execution_result)) /\ exists ff_q_pfp_complete_execution_resulttraceterminal. pfh_trace_code_complete_execution_result = ff_q_pfp_complete_execution_resulttraceterminal * S ((S (l)) * pfh_trace_scale_complete_execution_result) + (r))) /\ ((forall pfh_index_complete_execution_resulttracesteps. (exists pfa_gap_complete_execution_resulttracestepsindex. pfa_gap_complete_execution_resulttracestepsindex + S (pfh_index_complete_execution_resulttracesteps) = (l)) -> (exists pfh_coefficient_complete_execution_resulttracestepsstep pfh_before_complete_execution_resulttracestepsstep pfh_after_complete_execution_resulttracestepsstep pfh_product_complete_execution_resulttracestepsstep. ((((exists ff_h_pfp_complete_execution_resulttracestepsstepcoefficient. ff_h_pfp_complete_execution_resulttracestepsstepcoefficient + S (pfh_coefficient_complete_execution_resulttracestepsstep) = S ((S (pfh_index_complete_execution_resulttracesteps)) * e)) /\ exists ff_q_pfp_complete_execution_resulttracestepsstepcoefficient. d = ff_q_pfp_complete_execution_resulttracestepsstepcoefficient * S ((S (pfh_index_complete_execution_resulttracesteps)) * e) + (pfh_coefficient_complete_execution_resulttracestepsstep))) /\ (((((exists ff_h_pfp_complete_execution_resulttracestepsstepbefore. ff_h_pfp_complete_execution_resulttracestepsstepbefore + S (pfh_before_complete_execution_resulttracestepsstep) = S ((S (pfh_index_complete_execution_resulttracesteps)) * pfh_trace_scale_complete_execution_result)) /\ exists ff_q_pfp_complete_execution_resulttracestepsstepbefore. pfh_trace_code_complete_execution_result = ff_q_pfp_complete_execution_resulttracestepsstepbefore * S ((S (pfh_index_complete_execution_resulttracesteps)) * pfh_trace_scale_complete_execution_result) + (pfh_before_complete_execution_resulttracestepsstep))) /\ (((((exists ff_h_pfp_complete_execution_resulttracestepsstepafter. ff_h_pfp_complete_execution_resulttracestepsstepafter + S (pfh_after_complete_execution_resulttracestepsstep) = S ((S (S (pfh_index_complete_execution_resulttracesteps))) * pfh_trace_scale_complete_execution_result)) /\ exists ff_q_pfp_complete_execution_resulttracestepsstepafter. pfh_trace_code_complete_execution_result = ff_q_pfp_complete_execution_resulttracestepsstepafter * S ((S (S (pfh_index_complete_execution_resulttracesteps))) * pfh_trace_scale_complete_execution_result) + (pfh_after_complete_execution_resulttracestepsstep))) /\ (((((exists pfa_gap_complete_execution_resulttracestepsstepmultiplyleft. pfa_gap_complete_execution_resulttracestepsstepmultiplyleft + S (pfh_before_complete_execution_resulttracestepsstep) = (p)) /\ (((exists pfa_gap_complete_execution_resulttracestepsstepmultiplyright. pfa_gap_complete_execution_resulttracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_complete_execution_resulttracestepsstepmultiplyresultbound. pfa_gap_complete_execution_resulttracestepsstepmultiplyresultbound + S (pfh_product_complete_execution_resulttracestepsstep) = (p)) /\ ((exists pfa_offset_left_complete_execution_resulttracestepsstepmultiplyresultcongruence pfa_offset_right_complete_execution_resulttracestepsstepmultiplyresultcongruence. ((pfh_before_complete_execution_resulttracestepsstep) * (t)) + (p) * pfa_offset_left_complete_execution_resulttracestepsstepmultiplyresultcongruence = (pfh_product_complete_execution_resulttracestepsstep) + (p) * pfa_offset_right_complete_execution_resulttracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_complete_execution_resulttracestepsstepaddleft. pfa_gap_complete_execution_resulttracestepsstepaddleft + S (pfh_product_complete_execution_resulttracestepsstep) = (p)) /\ (((exists pfa_gap_complete_execution_resulttracestepsstepaddright. pfa_gap_complete_execution_resulttracestepsstepaddright + S (pfh_coefficient_complete_execution_resulttracestepsstep) = (p)) /\ ((((exists pfa_gap_complete_execution_resulttracestepsstepaddresultbound. pfa_gap_complete_execution_resulttracestepsstepaddresultbound + S (pfh_after_complete_execution_resulttracestepsstep) = (p)) /\ ((exists pfa_offset_left_complete_execution_resulttracestepsstepaddresultcongruence pfa_offset_right_complete_execution_resulttracestepsstepaddresultcongruence. ((pfh_product_complete_execution_resulttracestepsstep) + (pfh_coefficient_complete_execution_resulttracestepsstep)) + (p) * pfa_offset_left_complete_execution_resulttracestepsstepaddresultcongruence = (pfh_after_complete_execution_resulttracestepsstep) + (p) * pfa_offset_right_complete_execution_resulttracestepsstepaddresultcongruence))))))))))))))))))))))))))) /\ ((forall n. (exists ff_u_ph_pfh_complete_natural ff_v_ph_pfh_complete_natural. ((((exists fs_h_ph_pfh_complete_natural_body_start. fs_h_ph_pfh_complete_natural_body_start + S (0) = S ((S (0)) * ff_v_ph_pfh_complete_natural)) /\ exists fs_q_ph_pfh_complete_natural_body_start. ff_u_ph_pfh_complete_natural = fs_q_ph_pfh_complete_natural_body_start * S ((S (0)) * ff_v_ph_pfh_complete_natural) + (0))) /\ ((((exists fs_h_ph_pfh_complete_natural_body_terminal. fs_h_ph_pfh_complete_natural_body_terminal + S (n) = S ((S (l)) * ff_v_ph_pfh_complete_natural)) /\ exists fs_q_ph_pfh_complete_natural_body_terminal. ff_u_ph_pfh_complete_natural = fs_q_ph_pfh_complete_natural_body_terminal * S ((S (l)) * ff_v_ph_pfh_complete_natural) + (n))) /\ forall ff_i_ph_pfh_complete_natural_body_steps. (exists ph_bound_pfh_complete_natural_body_steps. ph_bound_pfh_complete_natural_body_steps + S ff_i_ph_pfh_complete_natural_body_steps = l) -> exists ff_coefficient_ph_pfh_complete_natural_body_steps ff_previous_ph_pfh_complete_natural_body_steps ff_current_ph_pfh_complete_natural_body_steps. ((((exists fs_h_ph_pfh_complete_natural_body_steps_coefficient. fs_h_ph_pfh_complete_natural_body_steps_coefficient + S (ff_coefficient_ph_pfh_complete_natural_body_steps) = S ((S (ff_i_ph_pfh_complete_natural_body_steps)) * c)) /\ exists fs_q_ph_pfh_complete_natural_body_steps_coefficient. b = fs_q_ph_pfh_complete_natural_body_steps_coefficient * S ((S (ff_i_ph_pfh_complete_natural_body_steps)) * c) + (ff_coefficient_ph_pfh_complete_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_complete_natural_body_steps_before. fs_h_ph_pfh_complete_natural_body_steps_before + S (ff_previous_ph_pfh_complete_natural_body_steps) = S ((S (ff_i_ph_pfh_complete_natural_body_steps)) * ff_v_ph_pfh_complete_natural)) /\ exists fs_q_ph_pfh_complete_natural_body_steps_before. ff_u_ph_pfh_complete_natural = fs_q_ph_pfh_complete_natural_body_steps_before * S ((S (ff_i_ph_pfh_complete_natural_body_steps)) * ff_v_ph_pfh_complete_natural) + (ff_previous_ph_pfh_complete_natural_body_steps))) /\ ((((exists fs_h_ph_pfh_complete_natural_body_steps_after. fs_h_ph_pfh_complete_natural_body_steps_after + S (ff_current_ph_pfh_complete_natural_body_steps) = S ((S (S ff_i_ph_pfh_complete_natural_body_steps)) * ff_v_ph_pfh_complete_natural)) /\ exists fs_q_ph_pfh_complete_natural_body_steps_after. ff_u_ph_pfh_complete_natural = fs_q_ph_pfh_complete_natural_body_steps_after * S ((S (S ff_i_ph_pfh_complete_natural_body_steps)) * ff_v_ph_pfh_complete_natural) + (ff_current_ph_pfh_complete_natural_body_steps))) /\ ff_current_ph_pfh_complete_natural_body_steps = ff_previous_ph_pfh_complete_natural_body_steps * t + ff_coefficient_ph_pfh_complete_natural_body_steps)))))) -> (((exists pfa_gap_complete_residuebound. pfa_gap_complete_residuebound + S (r) = (p)) /\ ((exists pfa_offset_left_complete_residuecongruence pfa_offset_right_complete_residuecongruence. (n) + (p) * pfa_offset_left_complete_residuecongruence = (r) + (p) * pfa_offset_right_complete_residuecongruence)))))))))

forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

forall b c i x y. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ((exists h. h + S y = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + y) -> x = y

forall p b c l. ~(p = 0) -> exists qb qc rb rc. (forall fdp_index_exists_result. (exists gsp_lt_gap_exists_result_index_bound. gsp_lt_gap_exists_result_index_bound + S fdp_index_exists_result = l) -> exists fdp_value_exists_result fdp_quotient_exists_result fdp_remainder_exists_result. (((exists ff_h_fdp_exists_result_source. ff_h_fdp_exists_result_source + S (fdp_value_exists_result) = S ((S (fdp_index_exists_result)) * c)) /\ exists ff_q_fdp_exists_result_source. b = ff_q_fdp_exists_result_source * S ((S (fdp_index_exists_result)) * c) + (fdp_value_exists_result))) /\ ((((exists ff_h_fdp_exists_result_quotient_entry. ff_h_fdp_exists_result_quotient_entry + S (fdp_quotient_exists_result) = S ((S (fdp_index_exists_result)) * qc)) /\ exists ff_q_fdp_exists_result_quotient_entry. qb = ff_q_fdp_exists_result_quotient_entry * S ((S (fdp_index_exists_result)) * qc) + (fdp_quotient_exists_result))) /\ ((((exists ff_h_fdp_exists_result_remainder_entry. ff_h_fdp_exists_result_remainder_entry + S (fdp_remainder_exists_result) = S ((S (fdp_index_exists_result)) * rc)) /\ exists ff_q_fdp_exists_result_remainder_entry. rb = ff_q_fdp_exists_result_remainder_entry * S ((S (fdp_index_exists_result)) * rc) + (fdp_remainder_exists_result))) /\ (fdp_value_exists_result = p * fdp_quotient_exists_result + fdp_remainder_exists_result /\ (exists gsp_lt_gap_exists_result_remainder_bound. gsp_lt_gap_exists_result_remainder_bound + S fdp_remainder_exists_result = p)))))

forall b c t n. (exists ff_u_ph_zero ff_v_ph_zero. ((((exists fs_h_ph_zero_body_start. fs_h_ph_zero_body_start + S (0) = S ((S (0)) * ff_v_ph_zero)) /\ exists fs_q_ph_zero_body_start. ff_u_ph_zero = fs_q_ph_zero_body_start * S ((S (0)) * ff_v_ph_zero) + (0))) /\ ((((exists fs_h_ph_zero_body_terminal. fs_h_ph_zero_body_terminal + S (n) = S ((S (0)) * ff_v_ph_zero)) /\ exists fs_q_ph_zero_body_terminal. ff_u_ph_zero = fs_q_ph_zero_body_terminal * S ((S (0)) * ff_v_ph_zero) + (n))) /\ forall ff_i_ph_zero_body_steps. (exists ph_bound_zero_body_steps. ph_bound_zero_body_steps + S ff_i_ph_zero_body_steps = 0) -> exists ff_coefficient_ph_zero_body_steps ff_previous_ph_zero_body_steps ff_current_ph_zero_body_steps. ((((exists fs_h_ph_zero_body_steps_coefficient. fs_h_ph_zero_body_steps_coefficient + S (ff_coefficient_ph_zero_body_steps) = S ((S (ff_i_ph_zero_body_steps)) * c)) /\ exists fs_q_ph_zero_body_steps_coefficient. b = fs_q_ph_zero_body_steps_coefficient * S ((S (ff_i_ph_zero_body_steps)) * c) + (ff_coefficient_ph_zero_body_steps))) /\ ((((exists fs_h_ph_zero_body_steps_before. fs_h_ph_zero_body_steps_before + S (ff_previous_ph_zero_body_steps) = S ((S (ff_i_ph_zero_body_steps)) * ff_v_ph_zero)) /\ exists fs_q_ph_zero_body_steps_before. ff_u_ph_zero = fs_q_ph_zero_body_steps_before * S ((S (ff_i_ph_zero_body_steps)) * ff_v_ph_zero) + (ff_previous_ph_zero_body_steps))) /\ ((((exists fs_h_ph_zero_body_steps_after. fs_h_ph_zero_body_steps_after + S (ff_current_ph_zero_body_steps) = S ((S (S ff_i_ph_zero_body_steps)) * ff_v_ph_zero)) /\ exists fs_q_ph_zero_body_steps_after. ff_u_ph_zero = fs_q_ph_zero_body_steps_after * S ((S (S ff_i_ph_zero_body_steps)) * ff_v_ph_zero) + (ff_current_ph_zero_body_steps))) /\ ff_current_ph_zero_body_steps = ff_previous_ph_zero_body_steps * t + ff_coefficient_ph_zero_body_steps)))))) -> n = 0

forall b c t l. exists n. (exists ff_u_ph_root_value ff_v_ph_root_value. ((((exists fs_h_ph_root_value_body_start. fs_h_ph_root_value_body_start + S (0) = S ((S (0)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_start. ff_u_ph_root_value = fs_q_ph_root_value_body_start * S ((S (0)) * ff_v_ph_root_value) + (0))) /\ ((((exists fs_h_ph_root_value_body_terminal. fs_h_ph_root_value_body_terminal + S (n) = S ((S (l)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_terminal. ff_u_ph_root_value = fs_q_ph_root_value_body_terminal * S ((S (l)) * ff_v_ph_root_value) + (n))) /\ forall ff_i_ph_root_value_body_steps. (exists ph_bound_root_value_body_steps. ph_bound_root_value_body_steps + S ff_i_ph_root_value_body_steps = l) -> exists ff_coefficient_ph_root_value_body_steps ff_previous_ph_root_value_body_steps ff_current_ph_root_value_body_steps. ((((exists fs_h_ph_root_value_body_steps_coefficient. fs_h_ph_root_value_body_steps_coefficient + S (ff_coefficient_ph_root_value_body_steps) = S ((S (ff_i_ph_root_value_body_steps)) * c)) /\ exists fs_q_ph_root_value_body_steps_coefficient. b = fs_q_ph_root_value_body_steps_coefficient * S ((S (ff_i_ph_root_value_body_steps)) * c) + (ff_coefficient_ph_root_value_body_steps))) /\ ((((exists fs_h_ph_root_value_body_steps_before. fs_h_ph_root_value_body_steps_before + S (ff_previous_ph_root_value_body_steps) = S ((S (ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_steps_before. ff_u_ph_root_value = fs_q_ph_root_value_body_steps_before * S ((S (ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value) + (ff_previous_ph_root_value_body_steps))) /\ ((((exists fs_h_ph_root_value_body_steps_after. fs_h_ph_root_value_body_steps_after + S (ff_current_ph_root_value_body_steps) = S ((S (S ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_steps_after. ff_u_ph_root_value = fs_q_ph_root_value_body_steps_after * S ((S (S ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value) + (ff_current_ph_root_value_body_steps))) /\ ff_current_ph_root_value_body_steps = ff_previous_ph_root_value_body_steps * t + ff_coefficient_ph_root_value_body_steps))))))

forall b c t l n. (exists ff_u_ph_successor ff_v_ph_successor. ((((exists fs_h_ph_successor_body_start. fs_h_ph_successor_body_start + S (0) = S ((S (0)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_start. ff_u_ph_successor = fs_q_ph_successor_body_start * S ((S (0)) * ff_v_ph_successor) + (0))) /\ ((((exists fs_h_ph_successor_body_terminal. fs_h_ph_successor_body_terminal + S (n) = S ((S (S l)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_terminal. ff_u_ph_successor = fs_q_ph_successor_body_terminal * S ((S (S l)) * ff_v_ph_successor) + (n))) /\ forall ff_i_ph_successor_body_steps. (exists ph_bound_successor_body_steps. ph_bound_successor_body_steps + S ff_i_ph_successor_body_steps = S l) -> exists ff_coefficient_ph_successor_body_steps ff_previous_ph_successor_body_steps ff_current_ph_successor_body_steps. ((((exists fs_h_ph_successor_body_steps_coefficient. fs_h_ph_successor_body_steps_coefficient + S (ff_coefficient_ph_successor_body_steps) = S ((S (ff_i_ph_successor_body_steps)) * c)) /\ exists fs_q_ph_successor_body_steps_coefficient. b = fs_q_ph_successor_body_steps_coefficient * S ((S (ff_i_ph_successor_body_steps)) * c) + (ff_coefficient_ph_successor_body_steps))) /\ ((((exists fs_h_ph_successor_body_steps_before. fs_h_ph_successor_body_steps_before + S (ff_previous_ph_successor_body_steps) = S ((S (ff_i_ph_successor_body_steps)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_steps_before. ff_u_ph_successor = fs_q_ph_successor_body_steps_before * S ((S (ff_i_ph_successor_body_steps)) * ff_v_ph_successor) + (ff_previous_ph_successor_body_steps))) /\ ((((exists fs_h_ph_successor_body_steps_after. fs_h_ph_successor_body_steps_after + S (ff_current_ph_successor_body_steps) = S ((S (S ff_i_ph_successor_body_steps)) * ff_v_ph_successor)) /\ exists fs_q_ph_successor_body_steps_after. ff_u_ph_successor = fs_q_ph_successor_body_steps_after * S ((S (S ff_i_ph_successor_body_steps)) * ff_v_ph_successor) + (ff_current_ph_successor_body_steps))) /\ ff_current_ph_successor_body_steps = ff_previous_ph_successor_body_steps * t + ff_coefficient_ph_successor_body_steps)))))) -> exists a r. ((((exists fs_h_ph_final_coefficient. fs_h_ph_final_coefficient + S (a) = S ((S (l)) * c)) /\ exists fs_q_ph_final_coefficient. b = fs_q_ph_final_coefficient * S ((S (l)) * c) + (a))) /\ ((exists ff_u_ph_prefix ff_v_ph_prefix. ((((exists fs_h_ph_prefix_body_start. fs_h_ph_prefix_body_start + S (0) = S ((S (0)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_start. ff_u_ph_prefix = fs_q_ph_prefix_body_start * S ((S (0)) * ff_v_ph_prefix) + (0))) /\ ((((exists fs_h_ph_prefix_body_terminal. fs_h_ph_prefix_body_terminal + S (r) = S ((S (l)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_terminal. ff_u_ph_prefix = fs_q_ph_prefix_body_terminal * S ((S (l)) * ff_v_ph_prefix) + (r))) /\ forall ff_i_ph_prefix_body_steps. (exists ph_bound_prefix_body_steps. ph_bound_prefix_body_steps + S ff_i_ph_prefix_body_steps = l) -> exists ff_coefficient_ph_prefix_body_steps ff_previous_ph_prefix_body_steps ff_current_ph_prefix_body_steps. ((((exists fs_h_ph_prefix_body_steps_coefficient. fs_h_ph_prefix_body_steps_coefficient + S (ff_coefficient_ph_prefix_body_steps) = S ((S (ff_i_ph_prefix_body_steps)) * c)) /\ exists fs_q_ph_prefix_body_steps_coefficient. b = fs_q_ph_prefix_body_steps_coefficient * S ((S (ff_i_ph_prefix_body_steps)) * c) + (ff_coefficient_ph_prefix_body_steps))) /\ ((((exists fs_h_ph_prefix_body_steps_before. fs_h_ph_prefix_body_steps_before + S (ff_previous_ph_prefix_body_steps) = S ((S (ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_steps_before. ff_u_ph_prefix = fs_q_ph_prefix_body_steps_before * S ((S (ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix) + (ff_previous_ph_prefix_body_steps))) /\ ((((exists fs_h_ph_prefix_body_steps_after. fs_h_ph_prefix_body_steps_after + S (ff_current_ph_prefix_body_steps) = S ((S (S ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix)) /\ exists fs_q_ph_prefix_body_steps_after. ff_u_ph_prefix = fs_q_ph_prefix_body_steps_after * S ((S (S ff_i_ph_prefix_body_steps)) * ff_v_ph_prefix) + (ff_current_ph_prefix_body_steps))) /\ ff_current_ph_prefix_body_steps = ff_previous_ph_prefix_body_steps * t + ff_coefficient_ph_prefix_body_steps)))))) /\ n = r * t + a))

forall mb mc sb sc l. exists tb tc. (forall ff_index_mcp_add_add_exists ff_left_mcp_add_add_exists ff_right_mcp_add_add_exists ff_target_mcp_add_add_exists. (exists mcp_gap_add_exists_bound. mcp_gap_add_exists_bound + S (ff_index_mcp_add_add_exists) = (l)) -> (((exists fs_h_mcp_add_exists_left. fs_h_mcp_add_exists_left + S (ff_left_mcp_add_add_exists) = S ((S (ff_index_mcp_add_add_exists)) * mc)) /\ exists fs_q_mcp_add_exists_left. mb = fs_q_mcp_add_exists_left * S ((S (ff_index_mcp_add_add_exists)) * mc) + (ff_left_mcp_add_add_exists))) -> (((exists fs_h_mcp_add_exists_right. fs_h_mcp_add_exists_right + S (ff_right_mcp_add_add_exists) = S ((S (ff_index_mcp_add_add_exists)) * sc)) /\ exists fs_q_mcp_add_exists_right. sb = fs_q_mcp_add_exists_right * S ((S (ff_index_mcp_add_add_exists)) * sc) + (ff_right_mcp_add_add_exists))) -> (((exists fs_h_mcp_add_exists_target. fs_h_mcp_add_exists_target + S (ff_target_mcp_add_add_exists) = S ((S (ff_index_mcp_add_add_exists)) * tc)) /\ exists fs_q_mcp_add_exists_target. tb = fs_q_mcp_add_exists_target * S ((S (ff_index_mcp_add_add_exists)) * tc) + (ff_target_mcp_add_add_exists))) -> ff_target_mcp_add_add_exists = ff_left_mcp_add_add_exists + ff_right_mcp_add_add_exists)

forall mb mc sb sc l. exists tb tc. (forall fpmp_index_recode_exists_result fpmp_left_recode_exists_result fpmp_right_recode_exists_result fpmp_target_recode_exists_result. (exists fpmp_gap_recode_exists_result. fpmp_gap_recode_exists_result + S fpmp_index_recode_exists_result = l) -> (((exists ff_h_fpmp_recode_exists_result_left. ff_h_fpmp_recode_exists_result_left + S (fpmp_left_recode_exists_result) = S ((S (fpmp_index_recode_exists_result)) * mc)) /\ exists ff_q_fpmp_recode_exists_result_left. mb = ff_q_fpmp_recode_exists_result_left * S ((S (fpmp_index_recode_exists_result)) * mc) + (fpmp_left_recode_exists_result))) -> (((exists ff_h_fpmp_recode_exists_result_right. ff_h_fpmp_recode_exists_result_right + S (fpmp_right_recode_exists_result) = S ((S (fpmp_index_recode_exists_result)) * sc)) /\ exists ff_q_fpmp_recode_exists_result_right. sb = ff_q_fpmp_recode_exists_result_right * S ((S (fpmp_index_recode_exists_result)) * sc) + (fpmp_right_recode_exists_result))) -> (((exists ff_h_fpmp_recode_exists_result_target. ff_h_fpmp_recode_exists_result_target + S (fpmp_target_recode_exists_result) = S ((S (fpmp_index_recode_exists_result)) * tc)) /\ exists ff_q_fpmp_recode_exists_result_target. tb = ff_q_fpmp_recode_exists_result_target * S ((S (fpmp_index_recode_exists_result)) * tc) + (fpmp_target_recode_exists_result))) -> fpmp_target_recode_exists_result = fpmp_left_recode_exists_result * fpmp_right_recode_exists_result)

forall a l. exists b c. (forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> (((exists ff_h_r_decoded. ff_h_r_decoded + S (a) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_decoded. b = ff_q_r_decoded * S ((S (ff_i_r)) * c) + (a))))

forall m n r s. (((exists ff_gap_binary_value. ff_gap_binary_value + S (r) = m) /\ (exists ff_left_binary_value_congruence ff_right_binary_value_congruence. (n) + m * ff_left_binary_value_congruence = (r) + m * ff_right_binary_value_congruence))) -> (((exists ff_gap_binary_other. ff_gap_binary_other + S (s) = m) /\ (exists ff_left_binary_other_congruence ff_right_binary_other_congruence. (n) + m * ff_left_binary_other_congruence = (s) + m * ff_right_binary_other_congruence))) -> r = s

forall a b. (exists k. k + a = b) -> exists r. r + a = S b

forall b c B. (forall fom_index_mrf_empty_bounded. (exists fom_gap_mrf_empty_bounded_index_bound. fom_gap_mrf_empty_bounded_index_bound + S (fom_index_mrf_empty_bounded) = 0) -> exists fom_value_mrf_empty_bounded. ((((exists fom_beta_height_mrf_empty_bounded_entry. fom_beta_height_mrf_empty_bounded_entry + S (fom_value_mrf_empty_bounded) = S ((S (fom_index_mrf_empty_bounded)) * c)) /\ exists fom_beta_quotient_mrf_empty_bounded_entry. b = fom_beta_quotient_mrf_empty_bounded_entry * S ((S (fom_index_mrf_empty_bounded)) * c) + (fom_value_mrf_empty_bounded))) /\ (exists fom_gap_mrf_empty_bounded_value_bound. fom_gap_mrf_empty_bounded_value_bound + S (fom_value_mrf_empty_bounded) = B)))

forall b c l B a. (forall fom_index_mrf_extend_source. (exists fom_gap_mrf_extend_source_index_bound. fom_gap_mrf_extend_source_index_bound + S (fom_index_mrf_extend_source) = l) -> exists fom_value_mrf_extend_source. ((((exists fom_beta_height_mrf_extend_source_entry. fom_beta_height_mrf_extend_source_entry + S (fom_value_mrf_extend_source) = S ((S (fom_index_mrf_extend_source)) * c)) /\ exists fom_beta_quotient_mrf_extend_source_entry. b = fom_beta_quotient_mrf_extend_source_entry * S ((S (fom_index_mrf_extend_source)) * c) + (fom_value_mrf_extend_source))) /\ (exists fom_gap_mrf_extend_source_value_bound. fom_gap_mrf_extend_source_value_bound + S (fom_value_mrf_extend_source) = B))) -> (((exists ff_h_mdr_extend_entry. ff_h_mdr_extend_entry + S (a) = S ((S (l)) * c)) /\ exists ff_q_mdr_extend_entry. b = ff_q_mdr_extend_entry * S ((S (l)) * c) + (a))) -> (exists mdr_gap_extend_bound. mdr_gap_extend_bound + S (a) = (B)) -> (forall fom_index_mrf_extend_result. (exists fom_gap_mrf_extend_result_index_bound. fom_gap_mrf_extend_result_index_bound + S (fom_index_mrf_extend_result) = S l) -> exists fom_value_mrf_extend_result. ((((exists fom_beta_height_mrf_extend_result_entry. fom_beta_height_mrf_extend_result_entry + S (fom_value_mrf_extend_result) = S ((S (fom_index_mrf_extend_result)) * c)) /\ exists fom_beta_quotient_mrf_extend_result_entry. b = fom_beta_quotient_mrf_extend_result_entry * S ((S (fom_index_mrf_extend_result)) * c) + (fom_value_mrf_extend_result))) /\ (exists fom_gap_mrf_extend_result_value_bound. fom_gap_mrf_extend_result_value_bound + S (fom_value_mrf_extend_result) = B)))

forall b c l B i a. (forall fom_index_mrf_value_source. (exists fom_gap_mrf_value_source_index_bound. fom_gap_mrf_value_source_index_bound + S (fom_index_mrf_value_source) = l) -> exists fom_value_mrf_value_source. ((((exists fom_beta_height_mrf_value_source_entry. fom_beta_height_mrf_value_source_entry + S (fom_value_mrf_value_source) = S ((S (fom_index_mrf_value_source)) * c)) /\ exists fom_beta_quotient_mrf_value_source_entry. b = fom_beta_quotient_mrf_value_source_entry * S ((S (fom_index_mrf_value_source)) * c) + (fom_value_mrf_value_source))) /\ (exists fom_gap_mrf_value_source_value_bound. fom_gap_mrf_value_source_value_bound + S (fom_value_mrf_value_source) = B))) -> (exists mdr_gap_value_index. mdr_gap_value_index + S (i) = (l)) -> (((exists ff_h_mdr_value_entry. ff_h_mdr_value_entry + S (a) = S ((S (i)) * c)) /\ exists ff_q_mdr_value_entry. b = ff_q_mdr_value_entry * S ((S (i)) * c) + (a))) -> (exists mdr_gap_value_bound. mdr_gap_value_bound + S (a) = (B))

forall n m. n * m = m * n

forall p a b c x y u v. (((exists pfa_gap_addassoc_firstleft. pfa_gap_addassoc_firstleft + S (a) = (p)) /\ (((exists pfa_gap_addassoc_firstright. pfa_gap_addassoc_firstright + S (b) = (p)) /\ ((((exists pfa_gap_addassoc_firstresultbound. pfa_gap_addassoc_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_addassoc_firstresultcongruence pfa_offset_right_addassoc_firstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addassoc_firstresultcongruence = (x) + (p) * pfa_offset_right_addassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_addassoc_leftleft. pfa_gap_addassoc_leftleft + S (x) = (p)) /\ (((exists pfa_gap_addassoc_leftright. pfa_gap_addassoc_leftright + S (c) = (p)) /\ ((((exists pfa_gap_addassoc_leftresultbound. pfa_gap_addassoc_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_addassoc_leftresultcongruence pfa_offset_right_addassoc_leftresultcongruence. ((x) + (c)) + (p) * pfa_offset_left_addassoc_leftresultcongruence = (u) + (p) * pfa_offset_right_addassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_addassoc_secondleft. pfa_gap_addassoc_secondleft + S (b) = (p)) /\ (((exists pfa_gap_addassoc_secondright. pfa_gap_addassoc_secondright + S (c) = (p)) /\ ((((exists pfa_gap_addassoc_secondresultbound. pfa_gap_addassoc_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_addassoc_secondresultcongruence pfa_offset_right_addassoc_secondresultcongruence. ((b) + (c)) + (p) * pfa_offset_left_addassoc_secondresultcongruence = (y) + (p) * pfa_offset_right_addassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_addassoc_rightleft. pfa_gap_addassoc_rightleft + S (a) = (p)) /\ (((exists pfa_gap_addassoc_rightright. pfa_gap_addassoc_rightright + S (y) = (p)) /\ ((((exists pfa_gap_addassoc_rightresultbound. pfa_gap_addassoc_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addassoc_rightresultcongruence pfa_offset_right_addassoc_rightresultcongruence. ((a) + (y)) + (p) * pfa_offset_left_addassoc_rightresultcongruence = (v) + (p) * pfa_offset_right_addassoc_rightresultcongruence))))))))) -> u = v

forall p a b c. (((exists pfa_gap_addcomm_sourceleft. pfa_gap_addcomm_sourceleft + S (a) = (p)) /\ (((exists pfa_gap_addcomm_sourceright. pfa_gap_addcomm_sourceright + S (b) = (p)) /\ ((((exists pfa_gap_addcomm_sourceresultbound. pfa_gap_addcomm_sourceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_sourceresultcongruence pfa_offset_right_addcomm_sourceresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addcomm_sourceresultcongruence = (c) + (p) * pfa_offset_right_addcomm_sourceresultcongruence))))))))) -> (((exists pfa_gap_addcomm_targetleft. pfa_gap_addcomm_targetleft + S (b) = (p)) /\ (((exists pfa_gap_addcomm_targetright. pfa_gap_addcomm_targetright + S (a) = (p)) /\ ((((exists pfa_gap_addcomm_targetresultbound. pfa_gap_addcomm_targetresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addcomm_targetresultcongruence pfa_offset_right_addcomm_targetresultcongruence. ((b) + (a)) + (p) * pfa_offset_left_addcomm_targetresultcongruence = (c) + (p) * pfa_offset_right_addcomm_targetresultcongruence)))))))))

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_adddomain pfa_factor_right_adddomain. (p) = pfa_factor_left_adddomain * pfa_factor_right_adddomain -> pfa_factor_left_adddomain = 1 \/ pfa_factor_right_adddomain = 1) -> (exists pfa_gap_addleft. pfa_gap_addleft + S (a) = (p)) -> (exists pfa_gap_addright. pfa_gap_addright + S (b) = (p)) -> exists c. (((exists pfa_gap_addexistsleft. pfa_gap_addexistsleft + S (a) = (p)) /\ (((exists pfa_gap_addexistsright. pfa_gap_addexistsright + S (b) = (p)) /\ ((((exists pfa_gap_addexistsresultbound. pfa_gap_addexistsresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addexistsresultcongruence pfa_offset_right_addexistsresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addexistsresultcongruence = (c) + (p) * pfa_offset_right_addexistsresultcongruence)))))))))

forall p a b c d. (((exists pfa_gap_addfirstleft. pfa_gap_addfirstleft + S (a) = (p)) /\ (((exists pfa_gap_addfirstright. pfa_gap_addfirstright + S (b) = (p)) /\ ((((exists pfa_gap_addfirstresultbound. pfa_gap_addfirstresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addfirstresultcongruence pfa_offset_right_addfirstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addfirstresultcongruence = (c) + (p) * pfa_offset_right_addfirstresultcongruence))))))))) -> (((exists pfa_gap_addsecondleft. pfa_gap_addsecondleft + S (a) = (p)) /\ (((exists pfa_gap_addsecondright. pfa_gap_addsecondright + S (b) = (p)) /\ ((((exists pfa_gap_addsecondresultbound. pfa_gap_addsecondresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_addsecondresultcongruence pfa_offset_right_addsecondresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addsecondresultcongruence = (d) + (p) * pfa_offset_right_addsecondresultcongruence))))))))) -> c = d

forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_add_domain pfa_factor_right_zero_add_domain. (p) = pfa_factor_left_zero_add_domain * pfa_factor_right_zero_add_domain -> pfa_factor_left_zero_add_domain = 1 \/ pfa_factor_right_zero_add_domain = 1) -> (exists pfa_gap_zero_add_bound. pfa_gap_zero_add_bound + S (a) = (p)) -> (((exists pfa_gap_zero_addleft. pfa_gap_zero_addleft + S (0) = (p)) /\ (((exists pfa_gap_zero_addright. pfa_gap_zero_addright + S (a) = (p)) /\ ((((exists pfa_gap_zero_addresultbound. pfa_gap_zero_addresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_zero_addresultcongruence pfa_offset_right_zero_addresultcongruence. ((0) + (a)) + (p) * pfa_offset_left_zero_addresultcongruence = (a) + (p) * pfa_offset_right_zero_addresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_add_zero_domain pfa_factor_right_add_zero_domain. (p) = pfa_factor_left_add_zero_domain * pfa_factor_right_add_zero_domain -> pfa_factor_left_add_zero_domain = 1 \/ pfa_factor_right_add_zero_domain = 1) -> (exists pfa_gap_add_zero_bound. pfa_gap_add_zero_bound + S (a) = (p)) -> (((exists pfa_gap_add_zeroleft. pfa_gap_add_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_add_zeroright. pfa_gap_add_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_add_zeroresultbound. pfa_gap_add_zeroresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_add_zeroresultcongruence pfa_offset_right_add_zeroresultcongruence. ((a) + (0)) + (p) * pfa_offset_left_add_zeroresultcongruence = (a) + (p) * pfa_offset_right_add_zeroresultcongruence)))))))))

forall p a b c s x y u v. (((exists pfa_gap_leftdistribution_sumleft. pfa_gap_leftdistribution_sumleft + S (b) = (p)) /\ (((exists pfa_gap_leftdistribution_sumright. pfa_gap_leftdistribution_sumright + S (c) = (p)) /\ ((((exists pfa_gap_leftdistribution_sumresultbound. pfa_gap_leftdistribution_sumresultbound + S (s) = (p)) /\ ((exists pfa_offset_left_leftdistribution_sumresultcongruence pfa_offset_right_leftdistribution_sumresultcongruence. ((b) + (c)) + (p) * pfa_offset_left_leftdistribution_sumresultcongruence = (s) + (p) * pfa_offset_right_leftdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_leftleft. pfa_gap_leftdistribution_leftleft + S (a) = (p)) /\ (((exists pfa_gap_leftdistribution_leftright. pfa_gap_leftdistribution_leftright + S (s) = (p)) /\ ((((exists pfa_gap_leftdistribution_leftresultbound. pfa_gap_leftdistribution_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_leftdistribution_leftresultcongruence pfa_offset_right_leftdistribution_leftresultcongruence. ((a) * (s)) + (p) * pfa_offset_left_leftdistribution_leftresultcongruence = (u) + (p) * pfa_offset_right_leftdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_firstleft. pfa_gap_leftdistribution_firstleft + S (a) = (p)) /\ (((exists pfa_gap_leftdistribution_firstright. pfa_gap_leftdistribution_firstright + S (b) = (p)) /\ ((((exists pfa_gap_leftdistribution_firstresultbound. pfa_gap_leftdistribution_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_leftdistribution_firstresultcongruence pfa_offset_right_leftdistribution_firstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_leftdistribution_firstresultcongruence = (x) + (p) * pfa_offset_right_leftdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_secondleft. pfa_gap_leftdistribution_secondleft + S (a) = (p)) /\ (((exists pfa_gap_leftdistribution_secondright. pfa_gap_leftdistribution_secondright + S (c) = (p)) /\ ((((exists pfa_gap_leftdistribution_secondresultbound. pfa_gap_leftdistribution_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_leftdistribution_secondresultcongruence pfa_offset_right_leftdistribution_secondresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_leftdistribution_secondresultcongruence = (y) + (p) * pfa_offset_right_leftdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_leftdistribution_rightleft. pfa_gap_leftdistribution_rightleft + S (x) = (p)) /\ (((exists pfa_gap_leftdistribution_rightright. pfa_gap_leftdistribution_rightright + S (y) = (p)) /\ ((((exists pfa_gap_leftdistribution_rightresultbound. pfa_gap_leftdistribution_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_leftdistribution_rightresultcongruence pfa_offset_right_leftdistribution_rightresultcongruence. ((x) + (y)) + (p) * pfa_offset_left_leftdistribution_rightresultcongruence = (v) + (p) * pfa_offset_right_leftdistribution_rightresultcongruence))))))))) -> u = v

forall p a b c x y u v. (((exists pfa_gap_multiplyassoc_firstleft. pfa_gap_multiplyassoc_firstleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyassoc_firstright. pfa_gap_multiplyassoc_firstright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyassoc_firstresultbound. pfa_gap_multiplyassoc_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_firstresultcongruence pfa_offset_right_multiplyassoc_firstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyassoc_firstresultcongruence = (x) + (p) * pfa_offset_right_multiplyassoc_firstresultcongruence))))))))) -> (((exists pfa_gap_multiplyassoc_leftleft. pfa_gap_multiplyassoc_leftleft + S (x) = (p)) /\ (((exists pfa_gap_multiplyassoc_leftright. pfa_gap_multiplyassoc_leftright + S (c) = (p)) /\ ((((exists pfa_gap_multiplyassoc_leftresultbound. pfa_gap_multiplyassoc_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_leftresultcongruence pfa_offset_right_multiplyassoc_leftresultcongruence. ((x) * (c)) + (p) * pfa_offset_left_multiplyassoc_leftresultcongruence = (u) + (p) * pfa_offset_right_multiplyassoc_leftresultcongruence))))))))) -> (((exists pfa_gap_multiplyassoc_secondleft. pfa_gap_multiplyassoc_secondleft + S (b) = (p)) /\ (((exists pfa_gap_multiplyassoc_secondright. pfa_gap_multiplyassoc_secondright + S (c) = (p)) /\ ((((exists pfa_gap_multiplyassoc_secondresultbound. pfa_gap_multiplyassoc_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_secondresultcongruence pfa_offset_right_multiplyassoc_secondresultcongruence. ((b) * (c)) + (p) * pfa_offset_left_multiplyassoc_secondresultcongruence = (y) + (p) * pfa_offset_right_multiplyassoc_secondresultcongruence))))))))) -> (((exists pfa_gap_multiplyassoc_rightleft. pfa_gap_multiplyassoc_rightleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyassoc_rightright. pfa_gap_multiplyassoc_rightright + S (y) = (p)) /\ ((((exists pfa_gap_multiplyassoc_rightresultbound. pfa_gap_multiplyassoc_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplyassoc_rightresultcongruence pfa_offset_right_multiplyassoc_rightresultcongruence. ((a) * (y)) + (p) * pfa_offset_left_multiplyassoc_rightresultcongruence = (v) + (p) * pfa_offset_right_multiplyassoc_rightresultcongruence))))))))) -> u = v

forall p a b. (~((p) = 1) /\ forall pfa_factor_left_multiplydomain pfa_factor_right_multiplydomain. (p) = pfa_factor_left_multiplydomain * pfa_factor_right_multiplydomain -> pfa_factor_left_multiplydomain = 1 \/ pfa_factor_right_multiplydomain = 1) -> (exists pfa_gap_multiplyleft. pfa_gap_multiplyleft + S (a) = (p)) -> (exists pfa_gap_multiplyright. pfa_gap_multiplyright + S (b) = (p)) -> exists c. (((exists pfa_gap_multiplyexistsleft. pfa_gap_multiplyexistsleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyexistsright. pfa_gap_multiplyexistsright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyexistsresultbound. pfa_gap_multiplyexistsresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyexistsresultcongruence pfa_offset_right_multiplyexistsresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyexistsresultcongruence = (c) + (p) * pfa_offset_right_multiplyexistsresultcongruence)))))))))

forall p a b c d. (((exists pfa_gap_multiplyfirstleft. pfa_gap_multiplyfirstleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyfirstright. pfa_gap_multiplyfirstright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyfirstresultbound. pfa_gap_multiplyfirstresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyfirstresultcongruence pfa_offset_right_multiplyfirstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyfirstresultcongruence = (c) + (p) * pfa_offset_right_multiplyfirstresultcongruence))))))))) -> (((exists pfa_gap_multiplysecondleft. pfa_gap_multiplysecondleft + S (a) = (p)) /\ (((exists pfa_gap_multiplysecondright. pfa_gap_multiplysecondright + S (b) = (p)) /\ ((((exists pfa_gap_multiplysecondresultbound. pfa_gap_multiplysecondresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_multiplysecondresultcongruence pfa_offset_right_multiplysecondresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplysecondresultcongruence = (d) + (p) * pfa_offset_right_multiplysecondresultcongruence))))))))) -> c = d

forall p a. (~((p) = 1) /\ forall pfa_factor_left_one_multiply_domain pfa_factor_right_one_multiply_domain. (p) = pfa_factor_left_one_multiply_domain * pfa_factor_right_one_multiply_domain -> pfa_factor_left_one_multiply_domain = 1 \/ pfa_factor_right_one_multiply_domain = 1) -> (exists pfa_gap_one_multiply_bound. pfa_gap_one_multiply_bound + S (a) = (p)) -> (((exists pfa_gap_one_multiplyleft. pfa_gap_one_multiplyleft + S (1) = (p)) /\ (((exists pfa_gap_one_multiplyright. pfa_gap_one_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_one_multiplyresultbound. pfa_gap_one_multiplyresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_one_multiplyresultcongruence pfa_offset_right_one_multiplyresultcongruence. ((1) * (a)) + (p) * pfa_offset_left_one_multiplyresultcongruence = (a) + (p) * pfa_offset_right_one_multiplyresultcongruence)))))))))

forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_multiply_domain pfa_factor_right_zero_multiply_domain. (p) = pfa_factor_left_zero_multiply_domain * pfa_factor_right_zero_multiply_domain -> pfa_factor_left_zero_multiply_domain = 1 \/ pfa_factor_right_zero_multiply_domain = 1) -> (exists pfa_gap_zero_multiply_bound. pfa_gap_zero_multiply_bound + S (a) = (p)) -> (((exists pfa_gap_zero_multiplyleft. pfa_gap_zero_multiplyleft + S (0) = (p)) /\ (((exists pfa_gap_zero_multiplyright. pfa_gap_zero_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_zero_multiplyresultbound. pfa_gap_zero_multiplyresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_zero_multiplyresultcongruence pfa_offset_right_zero_multiplyresultcongruence. ((0) * (a)) + (p) * pfa_offset_left_zero_multiplyresultcongruence = (0) + (p) * pfa_offset_right_zero_multiplyresultcongruence)))))))))

forall p a b x y z. (((exists pfa_gap_addnatural_leftbound. pfa_gap_addnatural_leftbound + S (x) = (p)) /\ ((exists pfa_offset_left_addnatural_leftcongruence pfa_offset_right_addnatural_leftcongruence. (a) + (p) * pfa_offset_left_addnatural_leftcongruence = (x) + (p) * pfa_offset_right_addnatural_leftcongruence)))) -> (((exists pfa_gap_addnatural_rightbound. pfa_gap_addnatural_rightbound + S (y) = (p)) /\ ((exists pfa_offset_left_addnatural_rightcongruence pfa_offset_right_addnatural_rightcongruence. (b) + (p) * pfa_offset_left_addnatural_rightcongruence = (y) + (p) * pfa_offset_right_addnatural_rightcongruence)))) -> (((exists pfa_gap_addnatural_resultleft. pfa_gap_addnatural_resultleft + S (x) = (p)) /\ (((exists pfa_gap_addnatural_resultright. pfa_gap_addnatural_resultright + S (y) = (p)) /\ ((((exists pfa_gap_addnatural_resultresultbound. pfa_gap_addnatural_resultresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_addnatural_resultresultcongruence pfa_offset_right_addnatural_resultresultcongruence. ((x) + (y)) + (p) * pfa_offset_left_addnatural_resultresultcongruence = (z) + (p) * pfa_offset_right_addnatural_resultresultcongruence))))))))) -> (((exists pfa_gap_addnatural_homomorphismbound. pfa_gap_addnatural_homomorphismbound + S (z) = (p)) /\ ((exists pfa_offset_left_addnatural_homomorphismcongruence pfa_offset_right_addnatural_homomorphismcongruence. (a + b) + (p) * pfa_offset_left_addnatural_homomorphismcongruence = (z) + (p) * pfa_offset_right_addnatural_homomorphismcongruence))))

forall p a r. (exists pfa_gap_bounded_value. pfa_gap_bounded_value + S (a) = (p)) -> (((exists pfa_gap_bounded_resultbound. pfa_gap_bounded_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_bounded_resultcongruence pfa_offset_right_bounded_resultcongruence. (a) + (p) * pfa_offset_left_bounded_resultcongruence = (r) + (p) * pfa_offset_right_bounded_resultcongruence)))) -> r = a

forall p n m r. n = m -> (((exists pfa_gap_input_equal_sourcebound. pfa_gap_input_equal_sourcebound + S (r) = (p)) /\ ((exists pfa_offset_left_input_equal_sourcecongruence pfa_offset_right_input_equal_sourcecongruence. (m) + (p) * pfa_offset_left_input_equal_sourcecongruence = (r) + (p) * pfa_offset_right_input_equal_sourcecongruence)))) -> (((exists pfa_gap_input_equal_targetbound. pfa_gap_input_equal_targetbound + S (r) = (p)) /\ ((exists pfa_offset_left_input_equal_targetcongruence pfa_offset_right_input_equal_targetcongruence. (n) + (p) * pfa_offset_left_input_equal_targetcongruence = (r) + (p) * pfa_offset_right_input_equal_targetcongruence))))

forall p a b x y z. (((exists pfa_gap_multiplynatural_leftbound. pfa_gap_multiplynatural_leftbound + S (x) = (p)) /\ ((exists pfa_offset_left_multiplynatural_leftcongruence pfa_offset_right_multiplynatural_leftcongruence. (a) + (p) * pfa_offset_left_multiplynatural_leftcongruence = (x) + (p) * pfa_offset_right_multiplynatural_leftcongruence)))) -> (((exists pfa_gap_multiplynatural_rightbound. pfa_gap_multiplynatural_rightbound + S (y) = (p)) /\ ((exists pfa_offset_left_multiplynatural_rightcongruence pfa_offset_right_multiplynatural_rightcongruence. (b) + (p) * pfa_offset_left_multiplynatural_rightcongruence = (y) + (p) * pfa_offset_right_multiplynatural_rightcongruence)))) -> (((exists pfa_gap_multiplynatural_resultleft. pfa_gap_multiplynatural_resultleft + S (x) = (p)) /\ (((exists pfa_gap_multiplynatural_resultright. pfa_gap_multiplynatural_resultright + S (y) = (p)) /\ ((((exists pfa_gap_multiplynatural_resultresultbound. pfa_gap_multiplynatural_resultresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_resultresultcongruence pfa_offset_right_multiplynatural_resultresultcongruence. ((x) * (y)) + (p) * pfa_offset_left_multiplynatural_resultresultcongruence = (z) + (p) * pfa_offset_right_multiplynatural_resultresultcongruence))))))))) -> (((exists pfa_gap_multiplynatural_homomorphismbound. pfa_gap_multiplynatural_homomorphismbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_homomorphismcongruence pfa_offset_right_multiplynatural_homomorphismcongruence. (a * b) + (p) * pfa_offset_left_multiplynatural_homomorphismcongruence = (z) + (p) * pfa_offset_right_multiplynatural_homomorphismcongruence))))

forall p a. (exists pfa_gap_reflexive_bound. pfa_gap_reflexive_bound + S (a) = (p)) -> (((exists pfa_gap_reflexivebound. pfa_gap_reflexivebound + S (a) = (p)) /\ ((exists pfa_offset_left_reflexivecongruence pfa_offset_right_reflexivecongruence. (a) + (p) * pfa_offset_left_reflexivecongruence = (a) + (p) * pfa_offset_right_reflexivecongruence))))

forall p a b c s x y u v. (((exists pfa_gap_rightdistribution_sumleft. pfa_gap_rightdistribution_sumleft + S (b) = (p)) /\ (((exists pfa_gap_rightdistribution_sumright. pfa_gap_rightdistribution_sumright + S (c) = (p)) /\ ((((exists pfa_gap_rightdistribution_sumresultbound. pfa_gap_rightdistribution_sumresultbound + S (s) = (p)) /\ ((exists pfa_offset_left_rightdistribution_sumresultcongruence pfa_offset_right_rightdistribution_sumresultcongruence. ((b) + (c)) + (p) * pfa_offset_left_rightdistribution_sumresultcongruence = (s) + (p) * pfa_offset_right_rightdistribution_sumresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_leftleft. pfa_gap_rightdistribution_leftleft + S (s) = (p)) /\ (((exists pfa_gap_rightdistribution_leftright. pfa_gap_rightdistribution_leftright + S (a) = (p)) /\ ((((exists pfa_gap_rightdistribution_leftresultbound. pfa_gap_rightdistribution_leftresultbound + S (u) = (p)) /\ ((exists pfa_offset_left_rightdistribution_leftresultcongruence pfa_offset_right_rightdistribution_leftresultcongruence. ((s) * (a)) + (p) * pfa_offset_left_rightdistribution_leftresultcongruence = (u) + (p) * pfa_offset_right_rightdistribution_leftresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_firstleft. pfa_gap_rightdistribution_firstleft + S (b) = (p)) /\ (((exists pfa_gap_rightdistribution_firstright. pfa_gap_rightdistribution_firstright + S (a) = (p)) /\ ((((exists pfa_gap_rightdistribution_firstresultbound. pfa_gap_rightdistribution_firstresultbound + S (x) = (p)) /\ ((exists pfa_offset_left_rightdistribution_firstresultcongruence pfa_offset_right_rightdistribution_firstresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_rightdistribution_firstresultcongruence = (x) + (p) * pfa_offset_right_rightdistribution_firstresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_secondleft. pfa_gap_rightdistribution_secondleft + S (c) = (p)) /\ (((exists pfa_gap_rightdistribution_secondright. pfa_gap_rightdistribution_secondright + S (a) = (p)) /\ ((((exists pfa_gap_rightdistribution_secondresultbound. pfa_gap_rightdistribution_secondresultbound + S (y) = (p)) /\ ((exists pfa_offset_left_rightdistribution_secondresultcongruence pfa_offset_right_rightdistribution_secondresultcongruence. ((c) * (a)) + (p) * pfa_offset_left_rightdistribution_secondresultcongruence = (y) + (p) * pfa_offset_right_rightdistribution_secondresultcongruence))))))))) -> (((exists pfa_gap_rightdistribution_rightleft. pfa_gap_rightdistribution_rightleft + S (x) = (p)) /\ (((exists pfa_gap_rightdistribution_rightright. pfa_gap_rightdistribution_rightright + S (y) = (p)) /\ ((((exists pfa_gap_rightdistribution_rightresultbound. pfa_gap_rightdistribution_rightresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_rightdistribution_rightresultcongruence pfa_offset_right_rightdistribution_rightresultcongruence. ((x) + (y)) + (p) * pfa_offset_left_rightdistribution_rightresultcongruence = (v) + (p) * pfa_offset_right_rightdistribution_rightresultcongruence))))))))) -> u = v

forall p. (~((p) = 1) /\ forall pfa_factor_left_zero_domain pfa_factor_right_zero_domain. (p) = pfa_factor_left_zero_domain * pfa_factor_right_zero_domain -> pfa_factor_left_zero_domain = 1 \/ pfa_factor_right_zero_domain = 1) -> (exists pfa_gap_zero_bound. pfa_gap_zero_bound + S (0) = (p))

forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 0)

forall p. ((~(p = 1) /\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \/ frm_prime_right_bpvl_prime = 1)) -> (exists bpvl_gap_prime_two. bpvl_gap_prime_two + (2) = (p))

forall m b q x. b = q * m + x -> exists u v. b + m * u = x + m * v

forall a b. (exists k. k + a = b) -> exists r. r + S a = S b

forall n. 0 + n = n