Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
293 checked bundle nodes; 740 dependency edges.
Literal checked proof bundle · SHA-256 fec8cf768ef2b94430d58d947daa0affada315bbc5160a03991dc4d2550dd0e9
Full G009 is admitted. Full G091 remains open; polynomial prerequisites do not prove arbitrary prime-power fields.
Exact theorem nodes and inherited prerequisites
prime_field_subtract_existsExact checked statement
forall p a b. (~((p) = 1) /\ forall pfa_factor_left_scalar_prime pfa_factor_right_scalar_prime. (p) = pfa_factor_left_scalar_prime * pfa_factor_right_scalar_prime -> pfa_factor_left_scalar_prime = 1 \/ pfa_factor_right_scalar_prime = 1) -> (exists pfa_gap_scalar_left. pfa_gap_scalar_left + S (a) = (p)) -> (exists pfa_gap_scalar_right. pfa_gap_scalar_right + S (b) = (p)) -> exists r. (((exists pfa_gap_scalar_subtractleft. pfa_gap_scalar_subtractleft + S (b) = (p)) /\ (((exists pfa_gap_scalar_subtractright. pfa_gap_scalar_subtractright + S (r) = (p)) /\ ((((exists pfa_gap_scalar_subtractresultbound. pfa_gap_scalar_subtractresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_scalar_subtractresultcongruence pfa_offset_right_scalar_subtractresultcongruence. ((b) + (r)) + (p) * pfa_offset_left_scalar_subtractresultcongruence = (a) + (p) * pfa_offset_right_scalar_subtractresultcongruence)))))))))prime_field_subtract_equal_zeroExact checked statement
forall p a r. (~((p) = 1) /\ forall pfa_factor_left_scalar_equal_prime pfa_factor_right_scalar_equal_prime. (p) = pfa_factor_left_scalar_equal_prime * pfa_factor_right_scalar_equal_prime -> pfa_factor_left_scalar_equal_prime = 1 \/ pfa_factor_right_scalar_equal_prime = 1) -> (((exists pfa_gap_scalar_equal_sumleft. pfa_gap_scalar_equal_sumleft + S (a) = (p)) /\ (((exists pfa_gap_scalar_equal_sumright. pfa_gap_scalar_equal_sumright + S (r) = (p)) /\ ((((exists pfa_gap_scalar_equal_sumresultbound. pfa_gap_scalar_equal_sumresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_scalar_equal_sumresultcongruence pfa_offset_right_scalar_equal_sumresultcongruence. ((a) + (r)) + (p) * pfa_offset_left_scalar_equal_sumresultcongruence = (a) + (p) * pfa_offset_right_scalar_equal_sumresultcongruence))))))))) -> r=0prime_field_polynomial_negate_emptyExact checked statement
forall p ab ac rb rc. forall pfs_index_negate_empty. (exists pfa_gap_negate_emptyindex. pfa_gap_negate_emptyindex + S (pfs_index_negate_empty) = (0)) -> exists pfs_source_negate_empty pfs_result_negate_empty. ((((exists ff_h_pfp_negate_emptysource. ff_h_pfp_negate_emptysource + S (pfs_source_negate_empty) = S ((S (pfs_index_negate_empty)) * ac)) /\ exists ff_q_pfp_negate_emptysource. ab = ff_q_pfp_negate_emptysource * S ((S (pfs_index_negate_empty)) * ac) + (pfs_source_negate_empty))) /\ (((((exists ff_h_pfp_negate_emptyresult. ff_h_pfp_negate_emptyresult + S (pfs_result_negate_empty) = S ((S (pfs_index_negate_empty)) * rc)) /\ exists ff_q_pfp_negate_emptyresult. rb = ff_q_pfp_negate_emptyresult * S ((S (pfs_index_negate_empty)) * rc) + (pfs_result_negate_empty))) /\ ((((exists pfa_gap_negate_emptyoperationadditionleft. pfa_gap_negate_emptyoperationadditionleft + S (pfs_source_negate_empty) = (p)) /\ (((exists pfa_gap_negate_emptyoperationadditionright. pfa_gap_negate_emptyoperationadditionright + S (pfs_result_negate_empty) = (p)) /\ ((((exists pfa_gap_negate_emptyoperationadditionresultbound. pfa_gap_negate_emptyoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_emptyoperationadditionresultcongruence pfa_offset_right_negate_emptyoperationadditionresultcongruence. ((pfs_source_negate_empty) + (pfs_result_negate_empty)) + (p) * pfa_offset_left_negate_emptyoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_emptyoperationadditionresultcongruence)))))))))))))prime_field_polynomial_negate_existsExact checked statement
forall p ab ac l. (~((p) = 1) /\ forall pfa_factor_left_negate_exists_prime pfa_factor_right_negate_exists_prime. (p) = pfa_factor_left_negate_exists_prime * pfa_factor_right_negate_exists_prime -> pfa_factor_left_negate_exists_prime = 1 \/ pfa_factor_right_negate_exists_prime = 1) -> (forall fom_index_pfp_negate_exists_ab. (exists fom_gap_pfp_negate_exists_ab_index_bound. fom_gap_pfp_negate_exists_ab_index_bound + S (fom_index_pfp_negate_exists_ab) = l) -> exists fom_value_pfp_negate_exists_ab. ((((exists fom_beta_height_pfp_negate_exists_ab_entry. fom_beta_height_pfp_negate_exists_ab_entry + S (fom_value_pfp_negate_exists_ab) = S ((S (fom_index_pfp_negate_exists_ab)) * ac)) /\ exists fom_beta_quotient_pfp_negate_exists_ab_entry. ab = fom_beta_quotient_pfp_negate_exists_ab_entry * S ((S (fom_index_pfp_negate_exists_ab)) * ac) + (fom_value_pfp_negate_exists_ab))) /\ (exists fom_gap_pfp_negate_exists_ab_value_bound. fom_gap_pfp_negate_exists_ab_value_bound + S (fom_value_pfp_negate_exists_ab) = p))) -> exists rb rc. (forall pfs_index_negate_exists_result. (exists pfa_gap_negate_exists_resultindex. pfa_gap_negate_exists_resultindex + S (pfs_index_negate_exists_result) = (l)) -> exists pfs_source_negate_exists_result pfs_result_negate_exists_result. ((((exists ff_h_pfp_negate_exists_resultsource. ff_h_pfp_negate_exists_resultsource + S (pfs_source_negate_exists_result) = S ((S (pfs_index_negate_exists_result)) * ac)) /\ exists ff_q_pfp_negate_exists_resultsource. ab = ff_q_pfp_negate_exists_resultsource * S ((S (pfs_index_negate_exists_result)) * ac) + (pfs_source_negate_exists_result))) /\ (((((exists ff_h_pfp_negate_exists_resultresult. ff_h_pfp_negate_exists_resultresult + S (pfs_result_negate_exists_result) = S ((S (pfs_index_negate_exists_result)) * rc)) /\ exists ff_q_pfp_negate_exists_resultresult. rb = ff_q_pfp_negate_exists_resultresult * S ((S (pfs_index_negate_exists_result)) * rc) + (pfs_result_negate_exists_result))) /\ ((((exists pfa_gap_negate_exists_resultoperationadditionleft. pfa_gap_negate_exists_resultoperationadditionleft + S (pfs_source_negate_exists_result) = (p)) /\ (((exists pfa_gap_negate_exists_resultoperationadditionright. pfa_gap_negate_exists_resultoperationadditionright + S (pfs_result_negate_exists_result) = (p)) /\ ((((exists pfa_gap_negate_exists_resultoperationadditionresultbound. pfa_gap_negate_exists_resultoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_exists_resultoperationadditionresultcongruence pfa_offset_right_negate_exists_resultoperationadditionresultcongruence. ((pfs_source_negate_exists_result) + (pfs_result_negate_exists_result)) + (p) * pfa_offset_left_negate_exists_resultoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_exists_resultoperationadditionresultcongruence))))))))))))))prime_field_polynomial_negate_entryExact checked statement
forall p ab ac rb rc l i a r. (forall pfs_index_negate_entry_graph. (exists pfa_gap_negate_entry_graphindex. pfa_gap_negate_entry_graphindex + S (pfs_index_negate_entry_graph) = (l)) -> exists pfs_source_negate_entry_graph pfs_result_negate_entry_graph. ((((exists ff_h_pfp_negate_entry_graphsource. ff_h_pfp_negate_entry_graphsource + S (pfs_source_negate_entry_graph) = S ((S (pfs_index_negate_entry_graph)) * ac)) /\ exists ff_q_pfp_negate_entry_graphsource. ab = ff_q_pfp_negate_entry_graphsource * S ((S (pfs_index_negate_entry_graph)) * ac) + (pfs_source_negate_entry_graph))) /\ (((((exists ff_h_pfp_negate_entry_graphresult. ff_h_pfp_negate_entry_graphresult + S (pfs_result_negate_entry_graph) = S ((S (pfs_index_negate_entry_graph)) * rc)) /\ exists ff_q_pfp_negate_entry_graphresult. rb = ff_q_pfp_negate_entry_graphresult * S ((S (pfs_index_negate_entry_graph)) * rc) + (pfs_result_negate_entry_graph))) /\ ((((exists pfa_gap_negate_entry_graphoperationadditionleft. pfa_gap_negate_entry_graphoperationadditionleft + S (pfs_source_negate_entry_graph) = (p)) /\ (((exists pfa_gap_negate_entry_graphoperationadditionright. pfa_gap_negate_entry_graphoperationadditionright + S (pfs_result_negate_entry_graph) = (p)) /\ ((((exists pfa_gap_negate_entry_graphoperationadditionresultbound. pfa_gap_negate_entry_graphoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_entry_graphoperationadditionresultcongruence pfa_offset_right_negate_entry_graphoperationadditionresultcongruence. ((pfs_source_negate_entry_graph) + (pfs_result_negate_entry_graph)) + (p) * pfa_offset_left_negate_entry_graphoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_entry_graphoperationadditionresultcongruence)))))))))))))) -> (exists pfa_gap_negate_entry_index. pfa_gap_negate_entry_index + S (i) = (l)) -> (((exists ff_h_pfp_negate_entry_a. ff_h_pfp_negate_entry_a + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_negate_entry_a. ab = ff_q_pfp_negate_entry_a * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_negate_entry_r. ff_h_pfp_negate_entry_r + S (r) = S ((S (i)) * rc)) /\ exists ff_q_pfp_negate_entry_r. rb = ff_q_pfp_negate_entry_r * S ((S (i)) * rc) + (r))) -> (((exists pfa_gap_negate_entry_resultadditionleft. pfa_gap_negate_entry_resultadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_entry_resultadditionright. pfa_gap_negate_entry_resultadditionright + S (r) = (p)) /\ ((((exists pfa_gap_negate_entry_resultadditionresultbound. pfa_gap_negate_entry_resultadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_entry_resultadditionresultcongruence pfa_offset_right_negate_entry_resultadditionresultcongruence. ((a) + (r)) + (p) * pfa_offset_left_negate_entry_resultadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_entry_resultadditionresultcongruence)))))))))prime_field_polynomial_negate_boundedExact checked statement
forall p ab ac rb rc l. (forall pfs_index_negate_bounded_graph. (exists pfa_gap_negate_bounded_graphindex. pfa_gap_negate_bounded_graphindex + S (pfs_index_negate_bounded_graph) = (l)) -> exists pfs_source_negate_bounded_graph pfs_result_negate_bounded_graph. ((((exists ff_h_pfp_negate_bounded_graphsource. ff_h_pfp_negate_bounded_graphsource + S (pfs_source_negate_bounded_graph) = S ((S (pfs_index_negate_bounded_graph)) * ac)) /\ exists ff_q_pfp_negate_bounded_graphsource. ab = ff_q_pfp_negate_bounded_graphsource * S ((S (pfs_index_negate_bounded_graph)) * ac) + (pfs_source_negate_bounded_graph))) /\ (((((exists ff_h_pfp_negate_bounded_graphresult. ff_h_pfp_negate_bounded_graphresult + S (pfs_result_negate_bounded_graph) = S ((S (pfs_index_negate_bounded_graph)) * rc)) /\ exists ff_q_pfp_negate_bounded_graphresult. rb = ff_q_pfp_negate_bounded_graphresult * S ((S (pfs_index_negate_bounded_graph)) * rc) + (pfs_result_negate_bounded_graph))) /\ ((((exists pfa_gap_negate_bounded_graphoperationadditionleft. pfa_gap_negate_bounded_graphoperationadditionleft + S (pfs_source_negate_bounded_graph) = (p)) /\ (((exists pfa_gap_negate_bounded_graphoperationadditionright. pfa_gap_negate_bounded_graphoperationadditionright + S (pfs_result_negate_bounded_graph) = (p)) /\ ((((exists pfa_gap_negate_bounded_graphoperationadditionresultbound. pfa_gap_negate_bounded_graphoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_bounded_graphoperationadditionresultcongruence pfa_offset_right_negate_bounded_graphoperationadditionresultcongruence. ((pfs_source_negate_bounded_graph) + (pfs_result_negate_bounded_graph)) + (p) * pfa_offset_left_negate_bounded_graphoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_bounded_graphoperationadditionresultcongruence)))))))))))))) -> ((forall fom_index_pfp_negate_bounded_ab. (exists fom_gap_pfp_negate_bounded_ab_index_bound. fom_gap_pfp_negate_bounded_ab_index_bound + S (fom_index_pfp_negate_bounded_ab) = l) -> exists fom_value_pfp_negate_bounded_ab. ((((exists fom_beta_height_pfp_negate_bounded_ab_entry. fom_beta_height_pfp_negate_bounded_ab_entry + S (fom_value_pfp_negate_bounded_ab) = S ((S (fom_index_pfp_negate_bounded_ab)) * ac)) /\ exists fom_beta_quotient_pfp_negate_bounded_ab_entry. ab = fom_beta_quotient_pfp_negate_bounded_ab_entry * S ((S (fom_index_pfp_negate_bounded_ab)) * ac) + (fom_value_pfp_negate_bounded_ab))) /\ (exists fom_gap_pfp_negate_bounded_ab_value_bound. fom_gap_pfp_negate_bounded_ab_value_bound + S (fom_value_pfp_negate_bounded_ab) = p))) /\ ((forall fom_index_pfp_negate_bounded_rb. (exists fom_gap_pfp_negate_bounded_rb_index_bound. fom_gap_pfp_negate_bounded_rb_index_bound + S (fom_index_pfp_negate_bounded_rb) = l) -> exists fom_value_pfp_negate_bounded_rb. ((((exists fom_beta_height_pfp_negate_bounded_rb_entry. fom_beta_height_pfp_negate_bounded_rb_entry + S (fom_value_pfp_negate_bounded_rb) = S ((S (fom_index_pfp_negate_bounded_rb)) * rc)) /\ exists fom_beta_quotient_pfp_negate_bounded_rb_entry. rb = fom_beta_quotient_pfp_negate_bounded_rb_entry * S ((S (fom_index_pfp_negate_bounded_rb)) * rc) + (fom_value_pfp_negate_bounded_rb))) /\ (exists fom_gap_pfp_negate_bounded_rb_value_bound. fom_gap_pfp_negate_bounded_rb_value_bound + S (fom_value_pfp_negate_bounded_rb) = p)))))prime_field_polynomial_negate_functionalExact checked statement
forall p ab ac rb rc db dc l. (forall pfs_index_negate_functional_first. (exists pfa_gap_negate_functional_firstindex. pfa_gap_negate_functional_firstindex + S (pfs_index_negate_functional_first) = (l)) -> exists pfs_source_negate_functional_first pfs_result_negate_functional_first. ((((exists ff_h_pfp_negate_functional_firstsource. ff_h_pfp_negate_functional_firstsource + S (pfs_source_negate_functional_first) = S ((S (pfs_index_negate_functional_first)) * ac)) /\ exists ff_q_pfp_negate_functional_firstsource. ab = ff_q_pfp_negate_functional_firstsource * S ((S (pfs_index_negate_functional_first)) * ac) + (pfs_source_negate_functional_first))) /\ (((((exists ff_h_pfp_negate_functional_firstresult. ff_h_pfp_negate_functional_firstresult + S (pfs_result_negate_functional_first) = S ((S (pfs_index_negate_functional_first)) * rc)) /\ exists ff_q_pfp_negate_functional_firstresult. rb = ff_q_pfp_negate_functional_firstresult * S ((S (pfs_index_negate_functional_first)) * rc) + (pfs_result_negate_functional_first))) /\ ((((exists pfa_gap_negate_functional_firstoperationadditionleft. pfa_gap_negate_functional_firstoperationadditionleft + S (pfs_source_negate_functional_first) = (p)) /\ (((exists pfa_gap_negate_functional_firstoperationadditionright. pfa_gap_negate_functional_firstoperationadditionright + S (pfs_result_negate_functional_first) = (p)) /\ ((((exists pfa_gap_negate_functional_firstoperationadditionresultbound. pfa_gap_negate_functional_firstoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_functional_firstoperationadditionresultcongruence pfa_offset_right_negate_functional_firstoperationadditionresultcongruence. ((pfs_source_negate_functional_first) + (pfs_result_negate_functional_first)) + (p) * pfa_offset_left_negate_functional_firstoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_functional_firstoperationadditionresultcongruence)))))))))))))) -> (forall pfs_index_negate_functional_second. (exists pfa_gap_negate_functional_secondindex. pfa_gap_negate_functional_secondindex + S (pfs_index_negate_functional_second) = (l)) -> exists pfs_source_negate_functional_second pfs_result_negate_functional_second. ((((exists ff_h_pfp_negate_functional_secondsource. ff_h_pfp_negate_functional_secondsource + S (pfs_source_negate_functional_second) = S ((S (pfs_index_negate_functional_second)) * ac)) /\ exists ff_q_pfp_negate_functional_secondsource. ab = ff_q_pfp_negate_functional_secondsource * S ((S (pfs_index_negate_functional_second)) * ac) + (pfs_source_negate_functional_second))) /\ (((((exists ff_h_pfp_negate_functional_secondresult. ff_h_pfp_negate_functional_secondresult + S (pfs_result_negate_functional_second) = S ((S (pfs_index_negate_functional_second)) * dc)) /\ exists ff_q_pfp_negate_functional_secondresult. db = ff_q_pfp_negate_functional_secondresult * S ((S (pfs_index_negate_functional_second)) * dc) + (pfs_result_negate_functional_second))) /\ ((((exists pfa_gap_negate_functional_secondoperationadditionleft. pfa_gap_negate_functional_secondoperationadditionleft + S (pfs_source_negate_functional_second) = (p)) /\ (((exists pfa_gap_negate_functional_secondoperationadditionright. pfa_gap_negate_functional_secondoperationadditionright + S (pfs_result_negate_functional_second) = (p)) /\ ((((exists pfa_gap_negate_functional_secondoperationadditionresultbound. pfa_gap_negate_functional_secondoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_functional_secondoperationadditionresultcongruence pfa_offset_right_negate_functional_secondoperationadditionresultcongruence. ((pfs_source_negate_functional_second) + (pfs_result_negate_functional_second)) + (p) * pfa_offset_left_negate_functional_secondoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_functional_secondoperationadditionresultcongruence)))))))))))))) -> (forall mdr_i_pfp_negate_functional_result mdr_a_pfp_negate_functional_result. (exists mdr_gap_pfp_negate_functional_resultb. mdr_gap_pfp_negate_functional_resultb + S (mdr_i_pfp_negate_functional_result) = (l)) -> (((exists ff_h_mdr_pfp_negate_functional_resulto. ff_h_mdr_pfp_negate_functional_resulto + S (mdr_a_pfp_negate_functional_result) = S ((S (mdr_i_pfp_negate_functional_result)) * rc)) /\ exists ff_q_mdr_pfp_negate_functional_resulto. rb = ff_q_mdr_pfp_negate_functional_resulto * S ((S (mdr_i_pfp_negate_functional_result)) * rc) + (mdr_a_pfp_negate_functional_result))) -> (((exists ff_h_mdr_pfp_negate_functional_resultn. ff_h_mdr_pfp_negate_functional_resultn + S (mdr_a_pfp_negate_functional_result) = S ((S (mdr_i_pfp_negate_functional_result)) * dc)) /\ exists ff_q_mdr_pfp_negate_functional_resultn. db = ff_q_mdr_pfp_negate_functional_resultn * S ((S (mdr_i_pfp_negate_functional_result)) * dc) + (mdr_a_pfp_negate_functional_result))))prime_field_polynomial_negate_transportExact checked statement
forall p ab ac rb rc AB AC RB RC l. (forall mdr_i_pfp_negate_transport_ab mdr_a_pfp_negate_transport_ab. (exists mdr_gap_pfp_negate_transport_abb. mdr_gap_pfp_negate_transport_abb + S (mdr_i_pfp_negate_transport_ab) = (l)) -> (((exists ff_h_mdr_pfp_negate_transport_abo. ff_h_mdr_pfp_negate_transport_abo + S (mdr_a_pfp_negate_transport_ab) = S ((S (mdr_i_pfp_negate_transport_ab)) * ac)) /\ exists ff_q_mdr_pfp_negate_transport_abo. ab = ff_q_mdr_pfp_negate_transport_abo * S ((S (mdr_i_pfp_negate_transport_ab)) * ac) + (mdr_a_pfp_negate_transport_ab))) -> (((exists ff_h_mdr_pfp_negate_transport_abn. ff_h_mdr_pfp_negate_transport_abn + S (mdr_a_pfp_negate_transport_ab) = S ((S (mdr_i_pfp_negate_transport_ab)) * AC)) /\ exists ff_q_mdr_pfp_negate_transport_abn. AB = ff_q_mdr_pfp_negate_transport_abn * S ((S (mdr_i_pfp_negate_transport_ab)) * AC) + (mdr_a_pfp_negate_transport_ab)))) -> (forall mdr_i_pfp_negate_transport_rb mdr_a_pfp_negate_transport_rb. (exists mdr_gap_pfp_negate_transport_rbb. mdr_gap_pfp_negate_transport_rbb + S (mdr_i_pfp_negate_transport_rb) = (l)) -> (((exists ff_h_mdr_pfp_negate_transport_rbo. ff_h_mdr_pfp_negate_transport_rbo + S (mdr_a_pfp_negate_transport_rb) = S ((S (mdr_i_pfp_negate_transport_rb)) * rc)) /\ exists ff_q_mdr_pfp_negate_transport_rbo. rb = ff_q_mdr_pfp_negate_transport_rbo * S ((S (mdr_i_pfp_negate_transport_rb)) * rc) + (mdr_a_pfp_negate_transport_rb))) -> (((exists ff_h_mdr_pfp_negate_transport_rbn. ff_h_mdr_pfp_negate_transport_rbn + S (mdr_a_pfp_negate_transport_rb) = S ((S (mdr_i_pfp_negate_transport_rb)) * RC)) /\ exists ff_q_mdr_pfp_negate_transport_rbn. RB = ff_q_mdr_pfp_negate_transport_rbn * S ((S (mdr_i_pfp_negate_transport_rb)) * RC) + (mdr_a_pfp_negate_transport_rb)))) -> (forall pfs_index_negate_transport_old. (exists pfa_gap_negate_transport_oldindex. pfa_gap_negate_transport_oldindex + S (pfs_index_negate_transport_old) = (l)) -> exists pfs_source_negate_transport_old pfs_result_negate_transport_old. ((((exists ff_h_pfp_negate_transport_oldsource. ff_h_pfp_negate_transport_oldsource + S (pfs_source_negate_transport_old) = S ((S (pfs_index_negate_transport_old)) * ac)) /\ exists ff_q_pfp_negate_transport_oldsource. ab = ff_q_pfp_negate_transport_oldsource * S ((S (pfs_index_negate_transport_old)) * ac) + (pfs_source_negate_transport_old))) /\ (((((exists ff_h_pfp_negate_transport_oldresult. ff_h_pfp_negate_transport_oldresult + S (pfs_result_negate_transport_old) = S ((S (pfs_index_negate_transport_old)) * rc)) /\ exists ff_q_pfp_negate_transport_oldresult. rb = ff_q_pfp_negate_transport_oldresult * S ((S (pfs_index_negate_transport_old)) * rc) + (pfs_result_negate_transport_old))) /\ ((((exists pfa_gap_negate_transport_oldoperationadditionleft. pfa_gap_negate_transport_oldoperationadditionleft + S (pfs_source_negate_transport_old) = (p)) /\ (((exists pfa_gap_negate_transport_oldoperationadditionright. pfa_gap_negate_transport_oldoperationadditionright + S (pfs_result_negate_transport_old) = (p)) /\ ((((exists pfa_gap_negate_transport_oldoperationadditionresultbound. pfa_gap_negate_transport_oldoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_transport_oldoperationadditionresultcongruence pfa_offset_right_negate_transport_oldoperationadditionresultcongruence. ((pfs_source_negate_transport_old) + (pfs_result_negate_transport_old)) + (p) * pfa_offset_left_negate_transport_oldoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_transport_oldoperationadditionresultcongruence)))))))))))))) -> (forall pfs_index_negate_transport_new. (exists pfa_gap_negate_transport_newindex. pfa_gap_negate_transport_newindex + S (pfs_index_negate_transport_new) = (l)) -> exists pfs_source_negate_transport_new pfs_result_negate_transport_new. ((((exists ff_h_pfp_negate_transport_newsource. ff_h_pfp_negate_transport_newsource + S (pfs_source_negate_transport_new) = S ((S (pfs_index_negate_transport_new)) * AC)) /\ exists ff_q_pfp_negate_transport_newsource. AB = ff_q_pfp_negate_transport_newsource * S ((S (pfs_index_negate_transport_new)) * AC) + (pfs_source_negate_transport_new))) /\ (((((exists ff_h_pfp_negate_transport_newresult. ff_h_pfp_negate_transport_newresult + S (pfs_result_negate_transport_new) = S ((S (pfs_index_negate_transport_new)) * RC)) /\ exists ff_q_pfp_negate_transport_newresult. RB = ff_q_pfp_negate_transport_newresult * S ((S (pfs_index_negate_transport_new)) * RC) + (pfs_result_negate_transport_new))) /\ ((((exists pfa_gap_negate_transport_newoperationadditionleft. pfa_gap_negate_transport_newoperationadditionleft + S (pfs_source_negate_transport_new) = (p)) /\ (((exists pfa_gap_negate_transport_newoperationadditionright. pfa_gap_negate_transport_newoperationadditionright + S (pfs_result_negate_transport_new) = (p)) /\ ((((exists pfa_gap_negate_transport_newoperationadditionresultbound. pfa_gap_negate_transport_newoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_transport_newoperationadditionresultcongruence pfa_offset_right_negate_transport_newoperationadditionresultcongruence. ((pfs_source_negate_transport_new) + (pfs_result_negate_transport_new)) + (p) * pfa_offset_left_negate_transport_newoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_transport_newoperationadditionresultcongruence))))))))))))))prime_field_polynomial_negate_involutiveExact checked statement
forall p ab ac rb rc l. (forall pfs_index_neg_inv_old. (exists pfa_gap_neg_inv_oldindex. pfa_gap_neg_inv_oldindex + S (pfs_index_neg_inv_old) = (l)) -> exists pfs_source_neg_inv_old pfs_result_neg_inv_old. ((((exists ff_h_pfp_neg_inv_oldsource. ff_h_pfp_neg_inv_oldsource + S (pfs_source_neg_inv_old) = S ((S (pfs_index_neg_inv_old)) * ac)) /\ exists ff_q_pfp_neg_inv_oldsource. ab = ff_q_pfp_neg_inv_oldsource * S ((S (pfs_index_neg_inv_old)) * ac) + (pfs_source_neg_inv_old))) /\ (((((exists ff_h_pfp_neg_inv_oldresult. ff_h_pfp_neg_inv_oldresult + S (pfs_result_neg_inv_old) = S ((S (pfs_index_neg_inv_old)) * rc)) /\ exists ff_q_pfp_neg_inv_oldresult. rb = ff_q_pfp_neg_inv_oldresult * S ((S (pfs_index_neg_inv_old)) * rc) + (pfs_result_neg_inv_old))) /\ ((((exists pfa_gap_neg_inv_oldoperationadditionleft. pfa_gap_neg_inv_oldoperationadditionleft + S (pfs_source_neg_inv_old) = (p)) /\ (((exists pfa_gap_neg_inv_oldoperationadditionright. pfa_gap_neg_inv_oldoperationadditionright + S (pfs_result_neg_inv_old) = (p)) /\ ((((exists pfa_gap_neg_inv_oldoperationadditionresultbound. pfa_gap_neg_inv_oldoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_neg_inv_oldoperationadditionresultcongruence pfa_offset_right_neg_inv_oldoperationadditionresultcongruence. ((pfs_source_neg_inv_old) + (pfs_result_neg_inv_old)) + (p) * pfa_offset_left_neg_inv_oldoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_neg_inv_oldoperationadditionresultcongruence)))))))))))))) -> (forall pfs_index_neg_inv_new. (exists pfa_gap_neg_inv_newindex. pfa_gap_neg_inv_newindex + S (pfs_index_neg_inv_new) = (l)) -> exists pfs_source_neg_inv_new pfs_result_neg_inv_new. ((((exists ff_h_pfp_neg_inv_newsource. ff_h_pfp_neg_inv_newsource + S (pfs_source_neg_inv_new) = S ((S (pfs_index_neg_inv_new)) * rc)) /\ exists ff_q_pfp_neg_inv_newsource. rb = ff_q_pfp_neg_inv_newsource * S ((S (pfs_index_neg_inv_new)) * rc) + (pfs_source_neg_inv_new))) /\ (((((exists ff_h_pfp_neg_inv_newresult. ff_h_pfp_neg_inv_newresult + S (pfs_result_neg_inv_new) = S ((S (pfs_index_neg_inv_new)) * ac)) /\ exists ff_q_pfp_neg_inv_newresult. ab = ff_q_pfp_neg_inv_newresult * S ((S (pfs_index_neg_inv_new)) * ac) + (pfs_result_neg_inv_new))) /\ ((((exists pfa_gap_neg_inv_newoperationadditionleft. pfa_gap_neg_inv_newoperationadditionleft + S (pfs_source_neg_inv_new) = (p)) /\ (((exists pfa_gap_neg_inv_newoperationadditionright. pfa_gap_neg_inv_newoperationadditionright + S (pfs_result_neg_inv_new) = (p)) /\ ((((exists pfa_gap_neg_inv_newoperationadditionresultbound. pfa_gap_neg_inv_newoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_neg_inv_newoperationadditionresultcongruence pfa_offset_right_neg_inv_newoperationadditionresultcongruence. ((pfs_source_neg_inv_new) + (pfs_result_neg_inv_new)) + (p) * pfa_offset_left_neg_inv_newoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_neg_inv_newoperationadditionresultcongruence))))))))))))))prime_field_polynomial_negate_zeroExact checked statement
forall p b c l. (~((p) = 1) /\ forall pfa_factor_left_neg_zero_prime pfa_factor_right_neg_zero_prime. (p) = pfa_factor_left_neg_zero_prime * pfa_factor_right_neg_zero_prime -> pfa_factor_left_neg_zero_prime = 1 \/ pfa_factor_right_neg_zero_prime = 1) -> (forall pfp_repeat_index_neg_zero_prefix. (exists pfa_gap_neg_zero_prefixindex. pfa_gap_neg_zero_prefixindex + S (pfp_repeat_index_neg_zero_prefix) = (l)) -> (((exists ff_h_pfp_neg_zero_prefixentry. ff_h_pfp_neg_zero_prefixentry + S (0) = S ((S (pfp_repeat_index_neg_zero_prefix)) * c)) /\ exists ff_q_pfp_neg_zero_prefixentry. b = ff_q_pfp_neg_zero_prefixentry * S ((S (pfp_repeat_index_neg_zero_prefix)) * c) + (0)))) -> (forall pfs_index_neg_zero_result. (exists pfa_gap_neg_zero_resultindex. pfa_gap_neg_zero_resultindex + S (pfs_index_neg_zero_result) = (l)) -> exists pfs_source_neg_zero_result pfs_result_neg_zero_result. ((((exists ff_h_pfp_neg_zero_resultsource. ff_h_pfp_neg_zero_resultsource + S (pfs_source_neg_zero_result) = S ((S (pfs_index_neg_zero_result)) * c)) /\ exists ff_q_pfp_neg_zero_resultsource. b = ff_q_pfp_neg_zero_resultsource * S ((S (pfs_index_neg_zero_result)) * c) + (pfs_source_neg_zero_result))) /\ (((((exists ff_h_pfp_neg_zero_resultresult. ff_h_pfp_neg_zero_resultresult + S (pfs_result_neg_zero_result) = S ((S (pfs_index_neg_zero_result)) * c)) /\ exists ff_q_pfp_neg_zero_resultresult. b = ff_q_pfp_neg_zero_resultresult * S ((S (pfs_index_neg_zero_result)) * c) + (pfs_result_neg_zero_result))) /\ ((((exists pfa_gap_neg_zero_resultoperationadditionleft. pfa_gap_neg_zero_resultoperationadditionleft + S (pfs_source_neg_zero_result) = (p)) /\ (((exists pfa_gap_neg_zero_resultoperationadditionright. pfa_gap_neg_zero_resultoperationadditionright + S (pfs_result_neg_zero_result) = (p)) /\ ((((exists pfa_gap_neg_zero_resultoperationadditionresultbound. pfa_gap_neg_zero_resultoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_neg_zero_resultoperationadditionresultcongruence pfa_offset_right_neg_zero_resultoperationadditionresultcongruence. ((pfs_source_neg_zero_result) + (pfs_result_neg_zero_result)) + (p) * pfa_offset_left_neg_zero_resultoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_neg_zero_resultoperationadditionresultcongruence))))))))))))))prime_field_polynomial_negate_add_zeroExact checked statement
forall p ab ac rb rc zb zc l. (forall pfs_index_neg_add_graph. (exists pfa_gap_neg_add_graphindex. pfa_gap_neg_add_graphindex + S (pfs_index_neg_add_graph) = (l)) -> exists pfs_source_neg_add_graph pfs_result_neg_add_graph. ((((exists ff_h_pfp_neg_add_graphsource. ff_h_pfp_neg_add_graphsource + S (pfs_source_neg_add_graph) = S ((S (pfs_index_neg_add_graph)) * ac)) /\ exists ff_q_pfp_neg_add_graphsource. ab = ff_q_pfp_neg_add_graphsource * S ((S (pfs_index_neg_add_graph)) * ac) + (pfs_source_neg_add_graph))) /\ (((((exists ff_h_pfp_neg_add_graphresult. ff_h_pfp_neg_add_graphresult + S (pfs_result_neg_add_graph) = S ((S (pfs_index_neg_add_graph)) * rc)) /\ exists ff_q_pfp_neg_add_graphresult. rb = ff_q_pfp_neg_add_graphresult * S ((S (pfs_index_neg_add_graph)) * rc) + (pfs_result_neg_add_graph))) /\ ((((exists pfa_gap_neg_add_graphoperationadditionleft. pfa_gap_neg_add_graphoperationadditionleft + S (pfs_source_neg_add_graph) = (p)) /\ (((exists pfa_gap_neg_add_graphoperationadditionright. pfa_gap_neg_add_graphoperationadditionright + S (pfs_result_neg_add_graph) = (p)) /\ ((((exists pfa_gap_neg_add_graphoperationadditionresultbound. pfa_gap_neg_add_graphoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_neg_add_graphoperationadditionresultcongruence pfa_offset_right_neg_add_graphoperationadditionresultcongruence. ((pfs_source_neg_add_graph) + (pfs_result_neg_add_graph)) + (p) * pfa_offset_left_neg_add_graphoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_neg_add_graphoperationadditionresultcongruence)))))))))))))) -> (forall pfp_repeat_index_neg_add_zero. (exists pfa_gap_neg_add_zeroindex. pfa_gap_neg_add_zeroindex + S (pfp_repeat_index_neg_add_zero) = (l)) -> (((exists ff_h_pfp_neg_add_zeroentry. ff_h_pfp_neg_add_zeroentry + S (0) = S ((S (pfp_repeat_index_neg_add_zero)) * zc)) /\ exists ff_q_pfp_neg_add_zeroentry. zb = ff_q_pfp_neg_add_zeroentry * S ((S (pfp_repeat_index_neg_add_zero)) * zc) + (0)))) -> (forall pfp_index_neg_add_result. (exists pfa_gap_neg_add_resultindex. pfa_gap_neg_add_resultindex + S (pfp_index_neg_add_result) = (l)) -> exists pfp_left_neg_add_result pfp_right_neg_add_result pfp_value_neg_add_result. ((((exists ff_h_pfp_neg_add_resultleft. ff_h_pfp_neg_add_resultleft + S (pfp_left_neg_add_result) = S ((S (pfp_index_neg_add_result)) * ac)) /\ exists ff_q_pfp_neg_add_resultleft. ab = ff_q_pfp_neg_add_resultleft * S ((S (pfp_index_neg_add_result)) * ac) + (pfp_left_neg_add_result))) /\ (((((exists ff_h_pfp_neg_add_resultright. ff_h_pfp_neg_add_resultright + S (pfp_right_neg_add_result) = S ((S (pfp_index_neg_add_result)) * rc)) /\ exists ff_q_pfp_neg_add_resultright. rb = ff_q_pfp_neg_add_resultright * S ((S (pfp_index_neg_add_result)) * rc) + (pfp_right_neg_add_result))) /\ (((((exists ff_h_pfp_neg_add_resulttarget. ff_h_pfp_neg_add_resulttarget + S (pfp_value_neg_add_result) = S ((S (pfp_index_neg_add_result)) * zc)) /\ exists ff_q_pfp_neg_add_resulttarget. zb = ff_q_pfp_neg_add_resulttarget * S ((S (pfp_index_neg_add_result)) * zc) + (pfp_value_neg_add_result))) /\ ((((exists pfa_gap_neg_add_resultoperationleft. pfa_gap_neg_add_resultoperationleft + S (pfp_left_neg_add_result) = (p)) /\ (((exists pfa_gap_neg_add_resultoperationright. pfa_gap_neg_add_resultoperationright + S (pfp_right_neg_add_result) = (p)) /\ ((((exists pfa_gap_neg_add_resultoperationresultbound. pfa_gap_neg_add_resultoperationresultbound + S (pfp_value_neg_add_result) = (p)) /\ ((exists pfa_offset_left_neg_add_resultoperationresultcongruence pfa_offset_right_neg_add_resultoperationresultcongruence. ((pfp_left_neg_add_result) + (pfp_right_neg_add_result)) + (p) * pfa_offset_left_neg_add_resultoperationresultcongruence = (pfp_value_neg_add_result) + (p) * pfa_offset_right_neg_add_resultoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_emptyExact checked statement
forall p ab ac bb bc rb rc. forall pfs_index_subtract_empty. (exists pfa_gap_subtract_emptyindex. pfa_gap_subtract_emptyindex + S (pfs_index_subtract_empty) = (0)) -> exists pfs_left_subtract_empty pfs_right_subtract_empty pfs_result_subtract_empty. ((((exists ff_h_pfp_subtract_emptyleft. ff_h_pfp_subtract_emptyleft + S (pfs_left_subtract_empty) = S ((S (pfs_index_subtract_empty)) * ac)) /\ exists ff_q_pfp_subtract_emptyleft. ab = ff_q_pfp_subtract_emptyleft * S ((S (pfs_index_subtract_empty)) * ac) + (pfs_left_subtract_empty))) /\ (((((exists ff_h_pfp_subtract_emptyright. ff_h_pfp_subtract_emptyright + S (pfs_right_subtract_empty) = S ((S (pfs_index_subtract_empty)) * bc)) /\ exists ff_q_pfp_subtract_emptyright. bb = ff_q_pfp_subtract_emptyright * S ((S (pfs_index_subtract_empty)) * bc) + (pfs_right_subtract_empty))) /\ (((((exists ff_h_pfp_subtract_emptyresult. ff_h_pfp_subtract_emptyresult + S (pfs_result_subtract_empty) = S ((S (pfs_index_subtract_empty)) * rc)) /\ exists ff_q_pfp_subtract_emptyresult. rb = ff_q_pfp_subtract_emptyresult * S ((S (pfs_index_subtract_empty)) * rc) + (pfs_result_subtract_empty))) /\ ((((exists pfa_gap_subtract_emptyoperationleft. pfa_gap_subtract_emptyoperationleft + S (pfs_right_subtract_empty) = (p)) /\ (((exists pfa_gap_subtract_emptyoperationright. pfa_gap_subtract_emptyoperationright + S (pfs_result_subtract_empty) = (p)) /\ ((((exists pfa_gap_subtract_emptyoperationresultbound. pfa_gap_subtract_emptyoperationresultbound + S (pfs_left_subtract_empty) = (p)) /\ ((exists pfa_offset_left_subtract_emptyoperationresultcongruence pfa_offset_right_subtract_emptyoperationresultcongruence. ((pfs_right_subtract_empty) + (pfs_result_subtract_empty)) + (p) * pfa_offset_left_subtract_emptyoperationresultcongruence = (pfs_left_subtract_empty) + (p) * pfa_offset_right_subtract_emptyoperationresultcongruence)))))))))))))))prime_field_polynomial_subtract_existsExact checked statement
forall p ab ac bb bc l. (~((p) = 1) /\ forall pfa_factor_left_subtract_exists_prime pfa_factor_right_subtract_exists_prime. (p) = pfa_factor_left_subtract_exists_prime * pfa_factor_right_subtract_exists_prime -> pfa_factor_left_subtract_exists_prime = 1 \/ pfa_factor_right_subtract_exists_prime = 1) -> (forall fom_index_pfp_subtract_exists_ab. (exists fom_gap_pfp_subtract_exists_ab_index_bound. fom_gap_pfp_subtract_exists_ab_index_bound + S (fom_index_pfp_subtract_exists_ab) = l) -> exists fom_value_pfp_subtract_exists_ab. ((((exists fom_beta_height_pfp_subtract_exists_ab_entry. fom_beta_height_pfp_subtract_exists_ab_entry + S (fom_value_pfp_subtract_exists_ab) = S ((S (fom_index_pfp_subtract_exists_ab)) * ac)) /\ exists fom_beta_quotient_pfp_subtract_exists_ab_entry. ab = fom_beta_quotient_pfp_subtract_exists_ab_entry * S ((S (fom_index_pfp_subtract_exists_ab)) * ac) + (fom_value_pfp_subtract_exists_ab))) /\ (exists fom_gap_pfp_subtract_exists_ab_value_bound. fom_gap_pfp_subtract_exists_ab_value_bound + S (fom_value_pfp_subtract_exists_ab) = p))) -> (forall fom_index_pfp_subtract_exists_bb. (exists fom_gap_pfp_subtract_exists_bb_index_bound. fom_gap_pfp_subtract_exists_bb_index_bound + S (fom_index_pfp_subtract_exists_bb) = l) -> exists fom_value_pfp_subtract_exists_bb. ((((exists fom_beta_height_pfp_subtract_exists_bb_entry. fom_beta_height_pfp_subtract_exists_bb_entry + S (fom_value_pfp_subtract_exists_bb) = S ((S (fom_index_pfp_subtract_exists_bb)) * bc)) /\ exists fom_beta_quotient_pfp_subtract_exists_bb_entry. bb = fom_beta_quotient_pfp_subtract_exists_bb_entry * S ((S (fom_index_pfp_subtract_exists_bb)) * bc) + (fom_value_pfp_subtract_exists_bb))) /\ (exists fom_gap_pfp_subtract_exists_bb_value_bound. fom_gap_pfp_subtract_exists_bb_value_bound + S (fom_value_pfp_subtract_exists_bb) = p))) -> exists rb rc. (forall pfs_index_subtract_exists_result. (exists pfa_gap_subtract_exists_resultindex. pfa_gap_subtract_exists_resultindex + S (pfs_index_subtract_exists_result) = (l)) -> exists pfs_left_subtract_exists_result pfs_right_subtract_exists_result pfs_result_subtract_exists_result. ((((exists ff_h_pfp_subtract_exists_resultleft. ff_h_pfp_subtract_exists_resultleft + S (pfs_left_subtract_exists_result) = S ((S (pfs_index_subtract_exists_result)) * ac)) /\ exists ff_q_pfp_subtract_exists_resultleft. ab = ff_q_pfp_subtract_exists_resultleft * S ((S (pfs_index_subtract_exists_result)) * ac) + (pfs_left_subtract_exists_result))) /\ (((((exists ff_h_pfp_subtract_exists_resultright. ff_h_pfp_subtract_exists_resultright + S (pfs_right_subtract_exists_result) = S ((S (pfs_index_subtract_exists_result)) * bc)) /\ exists ff_q_pfp_subtract_exists_resultright. bb = ff_q_pfp_subtract_exists_resultright * S ((S (pfs_index_subtract_exists_result)) * bc) + (pfs_right_subtract_exists_result))) /\ (((((exists ff_h_pfp_subtract_exists_resultresult. ff_h_pfp_subtract_exists_resultresult + S (pfs_result_subtract_exists_result) = S ((S (pfs_index_subtract_exists_result)) * rc)) /\ exists ff_q_pfp_subtract_exists_resultresult. rb = ff_q_pfp_subtract_exists_resultresult * S ((S (pfs_index_subtract_exists_result)) * rc) + (pfs_result_subtract_exists_result))) /\ ((((exists pfa_gap_subtract_exists_resultoperationleft. pfa_gap_subtract_exists_resultoperationleft + S (pfs_right_subtract_exists_result) = (p)) /\ (((exists pfa_gap_subtract_exists_resultoperationright. pfa_gap_subtract_exists_resultoperationright + S (pfs_result_subtract_exists_result) = (p)) /\ ((((exists pfa_gap_subtract_exists_resultoperationresultbound. pfa_gap_subtract_exists_resultoperationresultbound + S (pfs_left_subtract_exists_result) = (p)) /\ ((exists pfa_offset_left_subtract_exists_resultoperationresultcongruence pfa_offset_right_subtract_exists_resultoperationresultcongruence. ((pfs_right_subtract_exists_result) + (pfs_result_subtract_exists_result)) + (p) * pfa_offset_left_subtract_exists_resultoperationresultcongruence = (pfs_left_subtract_exists_result) + (p) * pfa_offset_right_subtract_exists_resultoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_entryExact checked statement
forall p ab ac bb bc rb rc l i a b r. (forall pfs_index_subtract_entry_graph. (exists pfa_gap_subtract_entry_graphindex. pfa_gap_subtract_entry_graphindex + S (pfs_index_subtract_entry_graph) = (l)) -> exists pfs_left_subtract_entry_graph pfs_right_subtract_entry_graph pfs_result_subtract_entry_graph. ((((exists ff_h_pfp_subtract_entry_graphleft. ff_h_pfp_subtract_entry_graphleft + S (pfs_left_subtract_entry_graph) = S ((S (pfs_index_subtract_entry_graph)) * ac)) /\ exists ff_q_pfp_subtract_entry_graphleft. ab = ff_q_pfp_subtract_entry_graphleft * S ((S (pfs_index_subtract_entry_graph)) * ac) + (pfs_left_subtract_entry_graph))) /\ (((((exists ff_h_pfp_subtract_entry_graphright. ff_h_pfp_subtract_entry_graphright + S (pfs_right_subtract_entry_graph) = S ((S (pfs_index_subtract_entry_graph)) * bc)) /\ exists ff_q_pfp_subtract_entry_graphright. bb = ff_q_pfp_subtract_entry_graphright * S ((S (pfs_index_subtract_entry_graph)) * bc) + (pfs_right_subtract_entry_graph))) /\ (((((exists ff_h_pfp_subtract_entry_graphresult. ff_h_pfp_subtract_entry_graphresult + S (pfs_result_subtract_entry_graph) = S ((S (pfs_index_subtract_entry_graph)) * rc)) /\ exists ff_q_pfp_subtract_entry_graphresult. rb = ff_q_pfp_subtract_entry_graphresult * S ((S (pfs_index_subtract_entry_graph)) * rc) + (pfs_result_subtract_entry_graph))) /\ ((((exists pfa_gap_subtract_entry_graphoperationleft. pfa_gap_subtract_entry_graphoperationleft + S (pfs_right_subtract_entry_graph) = (p)) /\ (((exists pfa_gap_subtract_entry_graphoperationright. pfa_gap_subtract_entry_graphoperationright + S (pfs_result_subtract_entry_graph) = (p)) /\ ((((exists pfa_gap_subtract_entry_graphoperationresultbound. pfa_gap_subtract_entry_graphoperationresultbound + S (pfs_left_subtract_entry_graph) = (p)) /\ ((exists pfa_offset_left_subtract_entry_graphoperationresultcongruence pfa_offset_right_subtract_entry_graphoperationresultcongruence. ((pfs_right_subtract_entry_graph) + (pfs_result_subtract_entry_graph)) + (p) * pfa_offset_left_subtract_entry_graphoperationresultcongruence = (pfs_left_subtract_entry_graph) + (p) * pfa_offset_right_subtract_entry_graphoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_subtract_entry_index. pfa_gap_subtract_entry_index + S (i) = (l)) -> (((exists ff_h_pfp_subtract_entry_a. ff_h_pfp_subtract_entry_a + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_subtract_entry_a. ab = ff_q_pfp_subtract_entry_a * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_subtract_entry_b. ff_h_pfp_subtract_entry_b + S (b) = S ((S (i)) * bc)) /\ exists ff_q_pfp_subtract_entry_b. bb = ff_q_pfp_subtract_entry_b * S ((S (i)) * bc) + (b))) -> (((exists ff_h_pfp_subtract_entry_r. ff_h_pfp_subtract_entry_r + S (r) = S ((S (i)) * rc)) /\ exists ff_q_pfp_subtract_entry_r. rb = ff_q_pfp_subtract_entry_r * S ((S (i)) * rc) + (r))) -> (((exists pfa_gap_subtract_entry_resultleft. pfa_gap_subtract_entry_resultleft + S (b) = (p)) /\ (((exists pfa_gap_subtract_entry_resultright. pfa_gap_subtract_entry_resultright + S (r) = (p)) /\ ((((exists pfa_gap_subtract_entry_resultresultbound. pfa_gap_subtract_entry_resultresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_subtract_entry_resultresultcongruence pfa_offset_right_subtract_entry_resultresultcongruence. ((b) + (r)) + (p) * pfa_offset_left_subtract_entry_resultresultcongruence = (a) + (p) * pfa_offset_right_subtract_entry_resultresultcongruence)))))))))prime_field_polynomial_subtract_boundedExact checked statement
forall p ab ac bb bc rb rc l. (forall pfs_index_subtract_bounded_graph. (exists pfa_gap_subtract_bounded_graphindex. pfa_gap_subtract_bounded_graphindex + S (pfs_index_subtract_bounded_graph) = (l)) -> exists pfs_left_subtract_bounded_graph pfs_right_subtract_bounded_graph pfs_result_subtract_bounded_graph. ((((exists ff_h_pfp_subtract_bounded_graphleft. ff_h_pfp_subtract_bounded_graphleft + S (pfs_left_subtract_bounded_graph) = S ((S (pfs_index_subtract_bounded_graph)) * ac)) /\ exists ff_q_pfp_subtract_bounded_graphleft. ab = ff_q_pfp_subtract_bounded_graphleft * S ((S (pfs_index_subtract_bounded_graph)) * ac) + (pfs_left_subtract_bounded_graph))) /\ (((((exists ff_h_pfp_subtract_bounded_graphright. ff_h_pfp_subtract_bounded_graphright + S (pfs_right_subtract_bounded_graph) = S ((S (pfs_index_subtract_bounded_graph)) * bc)) /\ exists ff_q_pfp_subtract_bounded_graphright. bb = ff_q_pfp_subtract_bounded_graphright * S ((S (pfs_index_subtract_bounded_graph)) * bc) + (pfs_right_subtract_bounded_graph))) /\ (((((exists ff_h_pfp_subtract_bounded_graphresult. ff_h_pfp_subtract_bounded_graphresult + S (pfs_result_subtract_bounded_graph) = S ((S (pfs_index_subtract_bounded_graph)) * rc)) /\ exists ff_q_pfp_subtract_bounded_graphresult. rb = ff_q_pfp_subtract_bounded_graphresult * S ((S (pfs_index_subtract_bounded_graph)) * rc) + (pfs_result_subtract_bounded_graph))) /\ ((((exists pfa_gap_subtract_bounded_graphoperationleft. pfa_gap_subtract_bounded_graphoperationleft + S (pfs_right_subtract_bounded_graph) = (p)) /\ (((exists pfa_gap_subtract_bounded_graphoperationright. pfa_gap_subtract_bounded_graphoperationright + S (pfs_result_subtract_bounded_graph) = (p)) /\ ((((exists pfa_gap_subtract_bounded_graphoperationresultbound. pfa_gap_subtract_bounded_graphoperationresultbound + S (pfs_left_subtract_bounded_graph) = (p)) /\ ((exists pfa_offset_left_subtract_bounded_graphoperationresultcongruence pfa_offset_right_subtract_bounded_graphoperationresultcongruence. ((pfs_right_subtract_bounded_graph) + (pfs_result_subtract_bounded_graph)) + (p) * pfa_offset_left_subtract_bounded_graphoperationresultcongruence = (pfs_left_subtract_bounded_graph) + (p) * pfa_offset_right_subtract_bounded_graphoperationresultcongruence)))))))))))))))) -> ((forall fom_index_pfp_subtract_bounded_ab. (exists fom_gap_pfp_subtract_bounded_ab_index_bound. fom_gap_pfp_subtract_bounded_ab_index_bound + S (fom_index_pfp_subtract_bounded_ab) = l) -> exists fom_value_pfp_subtract_bounded_ab. ((((exists fom_beta_height_pfp_subtract_bounded_ab_entry. fom_beta_height_pfp_subtract_bounded_ab_entry + S (fom_value_pfp_subtract_bounded_ab) = S ((S (fom_index_pfp_subtract_bounded_ab)) * ac)) /\ exists fom_beta_quotient_pfp_subtract_bounded_ab_entry. ab = fom_beta_quotient_pfp_subtract_bounded_ab_entry * S ((S (fom_index_pfp_subtract_bounded_ab)) * ac) + (fom_value_pfp_subtract_bounded_ab))) /\ (exists fom_gap_pfp_subtract_bounded_ab_value_bound. fom_gap_pfp_subtract_bounded_ab_value_bound + S (fom_value_pfp_subtract_bounded_ab) = p))) /\ (((forall fom_index_pfp_subtract_bounded_bb. (exists fom_gap_pfp_subtract_bounded_bb_index_bound. fom_gap_pfp_subtract_bounded_bb_index_bound + S (fom_index_pfp_subtract_bounded_bb) = l) -> exists fom_value_pfp_subtract_bounded_bb. ((((exists fom_beta_height_pfp_subtract_bounded_bb_entry. fom_beta_height_pfp_subtract_bounded_bb_entry + S (fom_value_pfp_subtract_bounded_bb) = S ((S (fom_index_pfp_subtract_bounded_bb)) * bc)) /\ exists fom_beta_quotient_pfp_subtract_bounded_bb_entry. bb = fom_beta_quotient_pfp_subtract_bounded_bb_entry * S ((S (fom_index_pfp_subtract_bounded_bb)) * bc) + (fom_value_pfp_subtract_bounded_bb))) /\ (exists fom_gap_pfp_subtract_bounded_bb_value_bound. fom_gap_pfp_subtract_bounded_bb_value_bound + S (fom_value_pfp_subtract_bounded_bb) = p))) /\ ((forall fom_index_pfp_subtract_bounded_rb. (exists fom_gap_pfp_subtract_bounded_rb_index_bound. fom_gap_pfp_subtract_bounded_rb_index_bound + S (fom_index_pfp_subtract_bounded_rb) = l) -> exists fom_value_pfp_subtract_bounded_rb. ((((exists fom_beta_height_pfp_subtract_bounded_rb_entry. fom_beta_height_pfp_subtract_bounded_rb_entry + S (fom_value_pfp_subtract_bounded_rb) = S ((S (fom_index_pfp_subtract_bounded_rb)) * rc)) /\ exists fom_beta_quotient_pfp_subtract_bounded_rb_entry. rb = fom_beta_quotient_pfp_subtract_bounded_rb_entry * S ((S (fom_index_pfp_subtract_bounded_rb)) * rc) + (fom_value_pfp_subtract_bounded_rb))) /\ (exists fom_gap_pfp_subtract_bounded_rb_value_bound. fom_gap_pfp_subtract_bounded_rb_value_bound + S (fom_value_pfp_subtract_bounded_rb) = p)))))))prime_field_polynomial_subtract_functionalExact checked statement
forall p ab ac bb bc rb rc db dc l. (forall pfs_index_subtract_functional_first. (exists pfa_gap_subtract_functional_firstindex. pfa_gap_subtract_functional_firstindex + S (pfs_index_subtract_functional_first) = (l)) -> exists pfs_left_subtract_functional_first pfs_right_subtract_functional_first pfs_result_subtract_functional_first. ((((exists ff_h_pfp_subtract_functional_firstleft. ff_h_pfp_subtract_functional_firstleft + S (pfs_left_subtract_functional_first) = S ((S (pfs_index_subtract_functional_first)) * ac)) /\ exists ff_q_pfp_subtract_functional_firstleft. ab = ff_q_pfp_subtract_functional_firstleft * S ((S (pfs_index_subtract_functional_first)) * ac) + (pfs_left_subtract_functional_first))) /\ (((((exists ff_h_pfp_subtract_functional_firstright. ff_h_pfp_subtract_functional_firstright + S (pfs_right_subtract_functional_first) = S ((S (pfs_index_subtract_functional_first)) * bc)) /\ exists ff_q_pfp_subtract_functional_firstright. bb = ff_q_pfp_subtract_functional_firstright * S ((S (pfs_index_subtract_functional_first)) * bc) + (pfs_right_subtract_functional_first))) /\ (((((exists ff_h_pfp_subtract_functional_firstresult. ff_h_pfp_subtract_functional_firstresult + S (pfs_result_subtract_functional_first) = S ((S (pfs_index_subtract_functional_first)) * rc)) /\ exists ff_q_pfp_subtract_functional_firstresult. rb = ff_q_pfp_subtract_functional_firstresult * S ((S (pfs_index_subtract_functional_first)) * rc) + (pfs_result_subtract_functional_first))) /\ ((((exists pfa_gap_subtract_functional_firstoperationleft. pfa_gap_subtract_functional_firstoperationleft + S (pfs_right_subtract_functional_first) = (p)) /\ (((exists pfa_gap_subtract_functional_firstoperationright. pfa_gap_subtract_functional_firstoperationright + S (pfs_result_subtract_functional_first) = (p)) /\ ((((exists pfa_gap_subtract_functional_firstoperationresultbound. pfa_gap_subtract_functional_firstoperationresultbound + S (pfs_left_subtract_functional_first) = (p)) /\ ((exists pfa_offset_left_subtract_functional_firstoperationresultcongruence pfa_offset_right_subtract_functional_firstoperationresultcongruence. ((pfs_right_subtract_functional_first) + (pfs_result_subtract_functional_first)) + (p) * pfa_offset_left_subtract_functional_firstoperationresultcongruence = (pfs_left_subtract_functional_first) + (p) * pfa_offset_right_subtract_functional_firstoperationresultcongruence)))))))))))))))) -> (forall pfs_index_subtract_functional_second. (exists pfa_gap_subtract_functional_secondindex. pfa_gap_subtract_functional_secondindex + S (pfs_index_subtract_functional_second) = (l)) -> exists pfs_left_subtract_functional_second pfs_right_subtract_functional_second pfs_result_subtract_functional_second. ((((exists ff_h_pfp_subtract_functional_secondleft. ff_h_pfp_subtract_functional_secondleft + S (pfs_left_subtract_functional_second) = S ((S (pfs_index_subtract_functional_second)) * ac)) /\ exists ff_q_pfp_subtract_functional_secondleft. ab = ff_q_pfp_subtract_functional_secondleft * S ((S (pfs_index_subtract_functional_second)) * ac) + (pfs_left_subtract_functional_second))) /\ (((((exists ff_h_pfp_subtract_functional_secondright. ff_h_pfp_subtract_functional_secondright + S (pfs_right_subtract_functional_second) = S ((S (pfs_index_subtract_functional_second)) * bc)) /\ exists ff_q_pfp_subtract_functional_secondright. bb = ff_q_pfp_subtract_functional_secondright * S ((S (pfs_index_subtract_functional_second)) * bc) + (pfs_right_subtract_functional_second))) /\ (((((exists ff_h_pfp_subtract_functional_secondresult. ff_h_pfp_subtract_functional_secondresult + S (pfs_result_subtract_functional_second) = S ((S (pfs_index_subtract_functional_second)) * dc)) /\ exists ff_q_pfp_subtract_functional_secondresult. db = ff_q_pfp_subtract_functional_secondresult * S ((S (pfs_index_subtract_functional_second)) * dc) + (pfs_result_subtract_functional_second))) /\ ((((exists pfa_gap_subtract_functional_secondoperationleft. pfa_gap_subtract_functional_secondoperationleft + S (pfs_right_subtract_functional_second) = (p)) /\ (((exists pfa_gap_subtract_functional_secondoperationright. pfa_gap_subtract_functional_secondoperationright + S (pfs_result_subtract_functional_second) = (p)) /\ ((((exists pfa_gap_subtract_functional_secondoperationresultbound. pfa_gap_subtract_functional_secondoperationresultbound + S (pfs_left_subtract_functional_second) = (p)) /\ ((exists pfa_offset_left_subtract_functional_secondoperationresultcongruence pfa_offset_right_subtract_functional_secondoperationresultcongruence. ((pfs_right_subtract_functional_second) + (pfs_result_subtract_functional_second)) + (p) * pfa_offset_left_subtract_functional_secondoperationresultcongruence = (pfs_left_subtract_functional_second) + (p) * pfa_offset_right_subtract_functional_secondoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_subtract_functional_result mdr_a_pfp_subtract_functional_result. (exists mdr_gap_pfp_subtract_functional_resultb. mdr_gap_pfp_subtract_functional_resultb + S (mdr_i_pfp_subtract_functional_result) = (l)) -> (((exists ff_h_mdr_pfp_subtract_functional_resulto. ff_h_mdr_pfp_subtract_functional_resulto + S (mdr_a_pfp_subtract_functional_result) = S ((S (mdr_i_pfp_subtract_functional_result)) * rc)) /\ exists ff_q_mdr_pfp_subtract_functional_resulto. rb = ff_q_mdr_pfp_subtract_functional_resulto * S ((S (mdr_i_pfp_subtract_functional_result)) * rc) + (mdr_a_pfp_subtract_functional_result))) -> (((exists ff_h_mdr_pfp_subtract_functional_resultn. ff_h_mdr_pfp_subtract_functional_resultn + S (mdr_a_pfp_subtract_functional_result) = S ((S (mdr_i_pfp_subtract_functional_result)) * dc)) /\ exists ff_q_mdr_pfp_subtract_functional_resultn. db = ff_q_mdr_pfp_subtract_functional_resultn * S ((S (mdr_i_pfp_subtract_functional_result)) * dc) + (mdr_a_pfp_subtract_functional_result))))prime_field_polynomial_subtract_transportExact checked statement
forall p ab ac bb bc rb rc AB AC BB BC RB RC l. (forall mdr_i_pfp_subtract_transport_ab mdr_a_pfp_subtract_transport_ab. (exists mdr_gap_pfp_subtract_transport_abb. mdr_gap_pfp_subtract_transport_abb + S (mdr_i_pfp_subtract_transport_ab) = (l)) -> (((exists ff_h_mdr_pfp_subtract_transport_abo. ff_h_mdr_pfp_subtract_transport_abo + S (mdr_a_pfp_subtract_transport_ab) = S ((S (mdr_i_pfp_subtract_transport_ab)) * ac)) /\ exists ff_q_mdr_pfp_subtract_transport_abo. ab = ff_q_mdr_pfp_subtract_transport_abo * S ((S (mdr_i_pfp_subtract_transport_ab)) * ac) + (mdr_a_pfp_subtract_transport_ab))) -> (((exists ff_h_mdr_pfp_subtract_transport_abn. ff_h_mdr_pfp_subtract_transport_abn + S (mdr_a_pfp_subtract_transport_ab) = S ((S (mdr_i_pfp_subtract_transport_ab)) * AC)) /\ exists ff_q_mdr_pfp_subtract_transport_abn. AB = ff_q_mdr_pfp_subtract_transport_abn * S ((S (mdr_i_pfp_subtract_transport_ab)) * AC) + (mdr_a_pfp_subtract_transport_ab)))) -> (forall mdr_i_pfp_subtract_transport_bb mdr_a_pfp_subtract_transport_bb. (exists mdr_gap_pfp_subtract_transport_bbb. mdr_gap_pfp_subtract_transport_bbb + S (mdr_i_pfp_subtract_transport_bb) = (l)) -> (((exists ff_h_mdr_pfp_subtract_transport_bbo. ff_h_mdr_pfp_subtract_transport_bbo + S (mdr_a_pfp_subtract_transport_bb) = S ((S (mdr_i_pfp_subtract_transport_bb)) * bc)) /\ exists ff_q_mdr_pfp_subtract_transport_bbo. bb = ff_q_mdr_pfp_subtract_transport_bbo * S ((S (mdr_i_pfp_subtract_transport_bb)) * bc) + (mdr_a_pfp_subtract_transport_bb))) -> (((exists ff_h_mdr_pfp_subtract_transport_bbn. ff_h_mdr_pfp_subtract_transport_bbn + S (mdr_a_pfp_subtract_transport_bb) = S ((S (mdr_i_pfp_subtract_transport_bb)) * BC)) /\ exists ff_q_mdr_pfp_subtract_transport_bbn. BB = ff_q_mdr_pfp_subtract_transport_bbn * S ((S (mdr_i_pfp_subtract_transport_bb)) * BC) + (mdr_a_pfp_subtract_transport_bb)))) -> (forall mdr_i_pfp_subtract_transport_rb mdr_a_pfp_subtract_transport_rb. (exists mdr_gap_pfp_subtract_transport_rbb. mdr_gap_pfp_subtract_transport_rbb + S (mdr_i_pfp_subtract_transport_rb) = (l)) -> (((exists ff_h_mdr_pfp_subtract_transport_rbo. ff_h_mdr_pfp_subtract_transport_rbo + S (mdr_a_pfp_subtract_transport_rb) = S ((S (mdr_i_pfp_subtract_transport_rb)) * rc)) /\ exists ff_q_mdr_pfp_subtract_transport_rbo. rb = ff_q_mdr_pfp_subtract_transport_rbo * S ((S (mdr_i_pfp_subtract_transport_rb)) * rc) + (mdr_a_pfp_subtract_transport_rb))) -> (((exists ff_h_mdr_pfp_subtract_transport_rbn. ff_h_mdr_pfp_subtract_transport_rbn + S (mdr_a_pfp_subtract_transport_rb) = S ((S (mdr_i_pfp_subtract_transport_rb)) * RC)) /\ exists ff_q_mdr_pfp_subtract_transport_rbn. RB = ff_q_mdr_pfp_subtract_transport_rbn * S ((S (mdr_i_pfp_subtract_transport_rb)) * RC) + (mdr_a_pfp_subtract_transport_rb)))) -> (forall pfs_index_subtract_transport_old. (exists pfa_gap_subtract_transport_oldindex. pfa_gap_subtract_transport_oldindex + S (pfs_index_subtract_transport_old) = (l)) -> exists pfs_left_subtract_transport_old pfs_right_subtract_transport_old pfs_result_subtract_transport_old. ((((exists ff_h_pfp_subtract_transport_oldleft. ff_h_pfp_subtract_transport_oldleft + S (pfs_left_subtract_transport_old) = S ((S (pfs_index_subtract_transport_old)) * ac)) /\ exists ff_q_pfp_subtract_transport_oldleft. ab = ff_q_pfp_subtract_transport_oldleft * S ((S (pfs_index_subtract_transport_old)) * ac) + (pfs_left_subtract_transport_old))) /\ (((((exists ff_h_pfp_subtract_transport_oldright. ff_h_pfp_subtract_transport_oldright + S (pfs_right_subtract_transport_old) = S ((S (pfs_index_subtract_transport_old)) * bc)) /\ exists ff_q_pfp_subtract_transport_oldright. bb = ff_q_pfp_subtract_transport_oldright * S ((S (pfs_index_subtract_transport_old)) * bc) + (pfs_right_subtract_transport_old))) /\ (((((exists ff_h_pfp_subtract_transport_oldresult. ff_h_pfp_subtract_transport_oldresult + S (pfs_result_subtract_transport_old) = S ((S (pfs_index_subtract_transport_old)) * rc)) /\ exists ff_q_pfp_subtract_transport_oldresult. rb = ff_q_pfp_subtract_transport_oldresult * S ((S (pfs_index_subtract_transport_old)) * rc) + (pfs_result_subtract_transport_old))) /\ ((((exists pfa_gap_subtract_transport_oldoperationleft. pfa_gap_subtract_transport_oldoperationleft + S (pfs_right_subtract_transport_old) = (p)) /\ (((exists pfa_gap_subtract_transport_oldoperationright. pfa_gap_subtract_transport_oldoperationright + S (pfs_result_subtract_transport_old) = (p)) /\ ((((exists pfa_gap_subtract_transport_oldoperationresultbound. pfa_gap_subtract_transport_oldoperationresultbound + S (pfs_left_subtract_transport_old) = (p)) /\ ((exists pfa_offset_left_subtract_transport_oldoperationresultcongruence pfa_offset_right_subtract_transport_oldoperationresultcongruence. ((pfs_right_subtract_transport_old) + (pfs_result_subtract_transport_old)) + (p) * pfa_offset_left_subtract_transport_oldoperationresultcongruence = (pfs_left_subtract_transport_old) + (p) * pfa_offset_right_subtract_transport_oldoperationresultcongruence)))))))))))))))) -> (forall pfs_index_subtract_transport_new. (exists pfa_gap_subtract_transport_newindex. pfa_gap_subtract_transport_newindex + S (pfs_index_subtract_transport_new) = (l)) -> exists pfs_left_subtract_transport_new pfs_right_subtract_transport_new pfs_result_subtract_transport_new. ((((exists ff_h_pfp_subtract_transport_newleft. ff_h_pfp_subtract_transport_newleft + S (pfs_left_subtract_transport_new) = S ((S (pfs_index_subtract_transport_new)) * AC)) /\ exists ff_q_pfp_subtract_transport_newleft. AB = ff_q_pfp_subtract_transport_newleft * S ((S (pfs_index_subtract_transport_new)) * AC) + (pfs_left_subtract_transport_new))) /\ (((((exists ff_h_pfp_subtract_transport_newright. ff_h_pfp_subtract_transport_newright + S (pfs_right_subtract_transport_new) = S ((S (pfs_index_subtract_transport_new)) * BC)) /\ exists ff_q_pfp_subtract_transport_newright. BB = ff_q_pfp_subtract_transport_newright * S ((S (pfs_index_subtract_transport_new)) * BC) + (pfs_right_subtract_transport_new))) /\ (((((exists ff_h_pfp_subtract_transport_newresult. ff_h_pfp_subtract_transport_newresult + S (pfs_result_subtract_transport_new) = S ((S (pfs_index_subtract_transport_new)) * RC)) /\ exists ff_q_pfp_subtract_transport_newresult. RB = ff_q_pfp_subtract_transport_newresult * S ((S (pfs_index_subtract_transport_new)) * RC) + (pfs_result_subtract_transport_new))) /\ ((((exists pfa_gap_subtract_transport_newoperationleft. pfa_gap_subtract_transport_newoperationleft + S (pfs_right_subtract_transport_new) = (p)) /\ (((exists pfa_gap_subtract_transport_newoperationright. pfa_gap_subtract_transport_newoperationright + S (pfs_result_subtract_transport_new) = (p)) /\ ((((exists pfa_gap_subtract_transport_newoperationresultbound. pfa_gap_subtract_transport_newoperationresultbound + S (pfs_left_subtract_transport_new) = (p)) /\ ((exists pfa_offset_left_subtract_transport_newoperationresultcongruence pfa_offset_right_subtract_transport_newoperationresultcongruence. ((pfs_right_subtract_transport_new) + (pfs_result_subtract_transport_new)) + (p) * pfa_offset_left_subtract_transport_newoperationresultcongruence = (pfs_left_subtract_transport_new) + (p) * pfa_offset_right_subtract_transport_newoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_recover_addExact checked statement
forall p ab ac bb bc rb rc l. (forall pfs_index_sub_bridge_old_recover_add. (exists pfa_gap_sub_bridge_old_recover_addindex. pfa_gap_sub_bridge_old_recover_addindex + S (pfs_index_sub_bridge_old_recover_add) = (l)) -> exists pfs_left_sub_bridge_old_recover_add pfs_right_sub_bridge_old_recover_add pfs_result_sub_bridge_old_recover_add. ((((exists ff_h_pfp_sub_bridge_old_recover_addleft. ff_h_pfp_sub_bridge_old_recover_addleft + S (pfs_left_sub_bridge_old_recover_add) = S ((S (pfs_index_sub_bridge_old_recover_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_old_recover_addleft. ab = ff_q_pfp_sub_bridge_old_recover_addleft * S ((S (pfs_index_sub_bridge_old_recover_add)) * ac) + (pfs_left_sub_bridge_old_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_recover_addright. ff_h_pfp_sub_bridge_old_recover_addright + S (pfs_right_sub_bridge_old_recover_add) = S ((S (pfs_index_sub_bridge_old_recover_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_old_recover_addright. bb = ff_q_pfp_sub_bridge_old_recover_addright * S ((S (pfs_index_sub_bridge_old_recover_add)) * bc) + (pfs_right_sub_bridge_old_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_recover_addresult. ff_h_pfp_sub_bridge_old_recover_addresult + S (pfs_result_sub_bridge_old_recover_add) = S ((S (pfs_index_sub_bridge_old_recover_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_old_recover_addresult. rb = ff_q_pfp_sub_bridge_old_recover_addresult * S ((S (pfs_index_sub_bridge_old_recover_add)) * rc) + (pfs_result_sub_bridge_old_recover_add))) /\ ((((exists pfa_gap_sub_bridge_old_recover_addoperationleft. pfa_gap_sub_bridge_old_recover_addoperationleft + S (pfs_right_sub_bridge_old_recover_add) = (p)) /\ (((exists pfa_gap_sub_bridge_old_recover_addoperationright. pfa_gap_sub_bridge_old_recover_addoperationright + S (pfs_result_sub_bridge_old_recover_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_old_recover_addoperationresultbound. pfa_gap_sub_bridge_old_recover_addoperationresultbound + S (pfs_left_sub_bridge_old_recover_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_old_recover_addoperationresultcongruence pfa_offset_right_sub_bridge_old_recover_addoperationresultcongruence. ((pfs_right_sub_bridge_old_recover_add) + (pfs_result_sub_bridge_old_recover_add)) + (p) * pfa_offset_left_sub_bridge_old_recover_addoperationresultcongruence = (pfs_left_sub_bridge_old_recover_add) + (p) * pfa_offset_right_sub_bridge_old_recover_addoperationresultcongruence)))))))))))))))) -> (forall pfp_index_sub_bridge_new_recover_add. (exists pfa_gap_sub_bridge_new_recover_addindex. pfa_gap_sub_bridge_new_recover_addindex + S (pfp_index_sub_bridge_new_recover_add) = (l)) -> exists pfp_left_sub_bridge_new_recover_add pfp_right_sub_bridge_new_recover_add pfp_value_sub_bridge_new_recover_add. ((((exists ff_h_pfp_sub_bridge_new_recover_addleft. ff_h_pfp_sub_bridge_new_recover_addleft + S (pfp_left_sub_bridge_new_recover_add) = S ((S (pfp_index_sub_bridge_new_recover_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_new_recover_addleft. bb = ff_q_pfp_sub_bridge_new_recover_addleft * S ((S (pfp_index_sub_bridge_new_recover_add)) * bc) + (pfp_left_sub_bridge_new_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_recover_addright. ff_h_pfp_sub_bridge_new_recover_addright + S (pfp_right_sub_bridge_new_recover_add) = S ((S (pfp_index_sub_bridge_new_recover_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_new_recover_addright. rb = ff_q_pfp_sub_bridge_new_recover_addright * S ((S (pfp_index_sub_bridge_new_recover_add)) * rc) + (pfp_right_sub_bridge_new_recover_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_recover_addtarget. ff_h_pfp_sub_bridge_new_recover_addtarget + S (pfp_value_sub_bridge_new_recover_add) = S ((S (pfp_index_sub_bridge_new_recover_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_new_recover_addtarget. ab = ff_q_pfp_sub_bridge_new_recover_addtarget * S ((S (pfp_index_sub_bridge_new_recover_add)) * ac) + (pfp_value_sub_bridge_new_recover_add))) /\ ((((exists pfa_gap_sub_bridge_new_recover_addoperationleft. pfa_gap_sub_bridge_new_recover_addoperationleft + S (pfp_left_sub_bridge_new_recover_add) = (p)) /\ (((exists pfa_gap_sub_bridge_new_recover_addoperationright. pfa_gap_sub_bridge_new_recover_addoperationright + S (pfp_right_sub_bridge_new_recover_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_new_recover_addoperationresultbound. pfa_gap_sub_bridge_new_recover_addoperationresultbound + S (pfp_value_sub_bridge_new_recover_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_new_recover_addoperationresultcongruence pfa_offset_right_sub_bridge_new_recover_addoperationresultcongruence. ((pfp_left_sub_bridge_new_recover_add) + (pfp_right_sub_bridge_new_recover_add)) + (p) * pfa_offset_left_sub_bridge_new_recover_addoperationresultcongruence = (pfp_value_sub_bridge_new_recover_add) + (p) * pfa_offset_right_sub_bridge_new_recover_addoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_from_addExact checked statement
forall p ab ac bb bc rb rc l. (forall pfp_index_sub_bridge_old_from_add. (exists pfa_gap_sub_bridge_old_from_addindex. pfa_gap_sub_bridge_old_from_addindex + S (pfp_index_sub_bridge_old_from_add) = (l)) -> exists pfp_left_sub_bridge_old_from_add pfp_right_sub_bridge_old_from_add pfp_value_sub_bridge_old_from_add. ((((exists ff_h_pfp_sub_bridge_old_from_addleft. ff_h_pfp_sub_bridge_old_from_addleft + S (pfp_left_sub_bridge_old_from_add) = S ((S (pfp_index_sub_bridge_old_from_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_old_from_addleft. bb = ff_q_pfp_sub_bridge_old_from_addleft * S ((S (pfp_index_sub_bridge_old_from_add)) * bc) + (pfp_left_sub_bridge_old_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_from_addright. ff_h_pfp_sub_bridge_old_from_addright + S (pfp_right_sub_bridge_old_from_add) = S ((S (pfp_index_sub_bridge_old_from_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_old_from_addright. rb = ff_q_pfp_sub_bridge_old_from_addright * S ((S (pfp_index_sub_bridge_old_from_add)) * rc) + (pfp_right_sub_bridge_old_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_old_from_addtarget. ff_h_pfp_sub_bridge_old_from_addtarget + S (pfp_value_sub_bridge_old_from_add) = S ((S (pfp_index_sub_bridge_old_from_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_old_from_addtarget. ab = ff_q_pfp_sub_bridge_old_from_addtarget * S ((S (pfp_index_sub_bridge_old_from_add)) * ac) + (pfp_value_sub_bridge_old_from_add))) /\ ((((exists pfa_gap_sub_bridge_old_from_addoperationleft. pfa_gap_sub_bridge_old_from_addoperationleft + S (pfp_left_sub_bridge_old_from_add) = (p)) /\ (((exists pfa_gap_sub_bridge_old_from_addoperationright. pfa_gap_sub_bridge_old_from_addoperationright + S (pfp_right_sub_bridge_old_from_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_old_from_addoperationresultbound. pfa_gap_sub_bridge_old_from_addoperationresultbound + S (pfp_value_sub_bridge_old_from_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_old_from_addoperationresultcongruence pfa_offset_right_sub_bridge_old_from_addoperationresultcongruence. ((pfp_left_sub_bridge_old_from_add) + (pfp_right_sub_bridge_old_from_add)) + (p) * pfa_offset_left_sub_bridge_old_from_addoperationresultcongruence = (pfp_value_sub_bridge_old_from_add) + (p) * pfa_offset_right_sub_bridge_old_from_addoperationresultcongruence)))))))))))))))) -> (forall pfs_index_sub_bridge_new_from_add. (exists pfa_gap_sub_bridge_new_from_addindex. pfa_gap_sub_bridge_new_from_addindex + S (pfs_index_sub_bridge_new_from_add) = (l)) -> exists pfs_left_sub_bridge_new_from_add pfs_right_sub_bridge_new_from_add pfs_result_sub_bridge_new_from_add. ((((exists ff_h_pfp_sub_bridge_new_from_addleft. ff_h_pfp_sub_bridge_new_from_addleft + S (pfs_left_sub_bridge_new_from_add) = S ((S (pfs_index_sub_bridge_new_from_add)) * ac)) /\ exists ff_q_pfp_sub_bridge_new_from_addleft. ab = ff_q_pfp_sub_bridge_new_from_addleft * S ((S (pfs_index_sub_bridge_new_from_add)) * ac) + (pfs_left_sub_bridge_new_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_from_addright. ff_h_pfp_sub_bridge_new_from_addright + S (pfs_right_sub_bridge_new_from_add) = S ((S (pfs_index_sub_bridge_new_from_add)) * bc)) /\ exists ff_q_pfp_sub_bridge_new_from_addright. bb = ff_q_pfp_sub_bridge_new_from_addright * S ((S (pfs_index_sub_bridge_new_from_add)) * bc) + (pfs_right_sub_bridge_new_from_add))) /\ (((((exists ff_h_pfp_sub_bridge_new_from_addresult. ff_h_pfp_sub_bridge_new_from_addresult + S (pfs_result_sub_bridge_new_from_add) = S ((S (pfs_index_sub_bridge_new_from_add)) * rc)) /\ exists ff_q_pfp_sub_bridge_new_from_addresult. rb = ff_q_pfp_sub_bridge_new_from_addresult * S ((S (pfs_index_sub_bridge_new_from_add)) * rc) + (pfs_result_sub_bridge_new_from_add))) /\ ((((exists pfa_gap_sub_bridge_new_from_addoperationleft. pfa_gap_sub_bridge_new_from_addoperationleft + S (pfs_right_sub_bridge_new_from_add) = (p)) /\ (((exists pfa_gap_sub_bridge_new_from_addoperationright. pfa_gap_sub_bridge_new_from_addoperationright + S (pfs_result_sub_bridge_new_from_add) = (p)) /\ ((((exists pfa_gap_sub_bridge_new_from_addoperationresultbound. pfa_gap_sub_bridge_new_from_addoperationresultbound + S (pfs_left_sub_bridge_new_from_add) = (p)) /\ ((exists pfa_offset_left_sub_bridge_new_from_addoperationresultcongruence pfa_offset_right_sub_bridge_new_from_addoperationresultcongruence. ((pfs_right_sub_bridge_new_from_add) + (pfs_result_sub_bridge_new_from_add)) + (p) * pfa_offset_left_sub_bridge_new_from_addoperationresultcongruence = (pfs_left_sub_bridge_new_from_add) + (p) * pfa_offset_right_sub_bridge_new_from_addoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_self_zeroExact checked statement
forall p ab ac zb zc l. (~((p) = 1) /\ forall pfa_factor_left_sub_self_prime pfa_factor_right_sub_self_prime. (p) = pfa_factor_left_sub_self_prime * pfa_factor_right_sub_self_prime -> pfa_factor_left_sub_self_prime = 1 \/ pfa_factor_right_sub_self_prime = 1) -> (forall fom_index_pfp_sub_self_coeff. (exists fom_gap_pfp_sub_self_coeff_index_bound. fom_gap_pfp_sub_self_coeff_index_bound + S (fom_index_pfp_sub_self_coeff) = l) -> exists fom_value_pfp_sub_self_coeff. ((((exists fom_beta_height_pfp_sub_self_coeff_entry. fom_beta_height_pfp_sub_self_coeff_entry + S (fom_value_pfp_sub_self_coeff) = S ((S (fom_index_pfp_sub_self_coeff)) * ac)) /\ exists fom_beta_quotient_pfp_sub_self_coeff_entry. ab = fom_beta_quotient_pfp_sub_self_coeff_entry * S ((S (fom_index_pfp_sub_self_coeff)) * ac) + (fom_value_pfp_sub_self_coeff))) /\ (exists fom_gap_pfp_sub_self_coeff_value_bound. fom_gap_pfp_sub_self_coeff_value_bound + S (fom_value_pfp_sub_self_coeff) = p))) -> (forall pfp_repeat_index_sub_self_zero. (exists pfa_gap_sub_self_zeroindex. pfa_gap_sub_self_zeroindex + S (pfp_repeat_index_sub_self_zero) = (l)) -> (((exists ff_h_pfp_sub_self_zeroentry. ff_h_pfp_sub_self_zeroentry + S (0) = S ((S (pfp_repeat_index_sub_self_zero)) * zc)) /\ exists ff_q_pfp_sub_self_zeroentry. zb = ff_q_pfp_sub_self_zeroentry * S ((S (pfp_repeat_index_sub_self_zero)) * zc) + (0)))) -> (forall pfs_index_sub_self_result. (exists pfa_gap_sub_self_resultindex. pfa_gap_sub_self_resultindex + S (pfs_index_sub_self_result) = (l)) -> exists pfs_left_sub_self_result pfs_right_sub_self_result pfs_result_sub_self_result. ((((exists ff_h_pfp_sub_self_resultleft. ff_h_pfp_sub_self_resultleft + S (pfs_left_sub_self_result) = S ((S (pfs_index_sub_self_result)) * ac)) /\ exists ff_q_pfp_sub_self_resultleft. ab = ff_q_pfp_sub_self_resultleft * S ((S (pfs_index_sub_self_result)) * ac) + (pfs_left_sub_self_result))) /\ (((((exists ff_h_pfp_sub_self_resultright. ff_h_pfp_sub_self_resultright + S (pfs_right_sub_self_result) = S ((S (pfs_index_sub_self_result)) * ac)) /\ exists ff_q_pfp_sub_self_resultright. ab = ff_q_pfp_sub_self_resultright * S ((S (pfs_index_sub_self_result)) * ac) + (pfs_right_sub_self_result))) /\ (((((exists ff_h_pfp_sub_self_resultresult. ff_h_pfp_sub_self_resultresult + S (pfs_result_sub_self_result) = S ((S (pfs_index_sub_self_result)) * zc)) /\ exists ff_q_pfp_sub_self_resultresult. zb = ff_q_pfp_sub_self_resultresult * S ((S (pfs_index_sub_self_result)) * zc) + (pfs_result_sub_self_result))) /\ ((((exists pfa_gap_sub_self_resultoperationleft. pfa_gap_sub_self_resultoperationleft + S (pfs_right_sub_self_result) = (p)) /\ (((exists pfa_gap_sub_self_resultoperationright. pfa_gap_sub_self_resultoperationright + S (pfs_result_sub_self_result) = (p)) /\ ((((exists pfa_gap_sub_self_resultoperationresultbound. pfa_gap_sub_self_resultoperationresultbound + S (pfs_left_sub_self_result) = (p)) /\ ((exists pfa_offset_left_sub_self_resultoperationresultcongruence pfa_offset_right_sub_self_resultoperationresultcongruence. ((pfs_right_sub_self_result) + (pfs_result_sub_self_result)) + (p) * pfa_offset_left_sub_self_resultoperationresultcongruence = (pfs_left_sub_self_result) + (p) * pfa_offset_right_sub_self_resultoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_zero_rightExact checked statement
forall p ab ac zb zc l. (~((p) = 1) /\ forall pfa_factor_left_sub_zero_prime pfa_factor_right_sub_zero_prime. (p) = pfa_factor_left_sub_zero_prime * pfa_factor_right_sub_zero_prime -> pfa_factor_left_sub_zero_prime = 1 \/ pfa_factor_right_sub_zero_prime = 1) -> (forall fom_index_pfp_sub_zero_coeff. (exists fom_gap_pfp_sub_zero_coeff_index_bound. fom_gap_pfp_sub_zero_coeff_index_bound + S (fom_index_pfp_sub_zero_coeff) = l) -> exists fom_value_pfp_sub_zero_coeff. ((((exists fom_beta_height_pfp_sub_zero_coeff_entry. fom_beta_height_pfp_sub_zero_coeff_entry + S (fom_value_pfp_sub_zero_coeff) = S ((S (fom_index_pfp_sub_zero_coeff)) * ac)) /\ exists fom_beta_quotient_pfp_sub_zero_coeff_entry. ab = fom_beta_quotient_pfp_sub_zero_coeff_entry * S ((S (fom_index_pfp_sub_zero_coeff)) * ac) + (fom_value_pfp_sub_zero_coeff))) /\ (exists fom_gap_pfp_sub_zero_coeff_value_bound. fom_gap_pfp_sub_zero_coeff_value_bound + S (fom_value_pfp_sub_zero_coeff) = p))) -> (forall pfp_repeat_index_sub_zero_prefix. (exists pfa_gap_sub_zero_prefixindex. pfa_gap_sub_zero_prefixindex + S (pfp_repeat_index_sub_zero_prefix) = (l)) -> (((exists ff_h_pfp_sub_zero_prefixentry. ff_h_pfp_sub_zero_prefixentry + S (0) = S ((S (pfp_repeat_index_sub_zero_prefix)) * zc)) /\ exists ff_q_pfp_sub_zero_prefixentry. zb = ff_q_pfp_sub_zero_prefixentry * S ((S (pfp_repeat_index_sub_zero_prefix)) * zc) + (0)))) -> (forall pfs_index_sub_zero_result. (exists pfa_gap_sub_zero_resultindex. pfa_gap_sub_zero_resultindex + S (pfs_index_sub_zero_result) = (l)) -> exists pfs_left_sub_zero_result pfs_right_sub_zero_result pfs_result_sub_zero_result. ((((exists ff_h_pfp_sub_zero_resultleft. ff_h_pfp_sub_zero_resultleft + S (pfs_left_sub_zero_result) = S ((S (pfs_index_sub_zero_result)) * ac)) /\ exists ff_q_pfp_sub_zero_resultleft. ab = ff_q_pfp_sub_zero_resultleft * S ((S (pfs_index_sub_zero_result)) * ac) + (pfs_left_sub_zero_result))) /\ (((((exists ff_h_pfp_sub_zero_resultright. ff_h_pfp_sub_zero_resultright + S (pfs_right_sub_zero_result) = S ((S (pfs_index_sub_zero_result)) * zc)) /\ exists ff_q_pfp_sub_zero_resultright. zb = ff_q_pfp_sub_zero_resultright * S ((S (pfs_index_sub_zero_result)) * zc) + (pfs_right_sub_zero_result))) /\ (((((exists ff_h_pfp_sub_zero_resultresult. ff_h_pfp_sub_zero_resultresult + S (pfs_result_sub_zero_result) = S ((S (pfs_index_sub_zero_result)) * ac)) /\ exists ff_q_pfp_sub_zero_resultresult. ab = ff_q_pfp_sub_zero_resultresult * S ((S (pfs_index_sub_zero_result)) * ac) + (pfs_result_sub_zero_result))) /\ ((((exists pfa_gap_sub_zero_resultoperationleft. pfa_gap_sub_zero_resultoperationleft + S (pfs_right_sub_zero_result) = (p)) /\ (((exists pfa_gap_sub_zero_resultoperationright. pfa_gap_sub_zero_resultoperationright + S (pfs_result_sub_zero_result) = (p)) /\ ((((exists pfa_gap_sub_zero_resultoperationresultbound. pfa_gap_sub_zero_resultoperationresultbound + S (pfs_left_sub_zero_result) = (p)) /\ ((exists pfa_offset_left_sub_zero_resultoperationresultcongruence pfa_offset_right_sub_zero_resultoperationresultcongruence. ((pfs_right_sub_zero_result) + (pfs_result_sub_zero_result)) + (p) * pfa_offset_left_sub_zero_resultoperationresultcongruence = (pfs_left_sub_zero_result) + (p) * pfa_offset_right_sub_zero_resultoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_zero_leftExact checked statement
forall p bb bc rb rc zb zc l. (forall pfs_index_sub_zero_left_neg. (exists pfa_gap_sub_zero_left_negindex. pfa_gap_sub_zero_left_negindex + S (pfs_index_sub_zero_left_neg) = (l)) -> exists pfs_source_sub_zero_left_neg pfs_result_sub_zero_left_neg. ((((exists ff_h_pfp_sub_zero_left_negsource. ff_h_pfp_sub_zero_left_negsource + S (pfs_source_sub_zero_left_neg) = S ((S (pfs_index_sub_zero_left_neg)) * bc)) /\ exists ff_q_pfp_sub_zero_left_negsource. bb = ff_q_pfp_sub_zero_left_negsource * S ((S (pfs_index_sub_zero_left_neg)) * bc) + (pfs_source_sub_zero_left_neg))) /\ (((((exists ff_h_pfp_sub_zero_left_negresult. ff_h_pfp_sub_zero_left_negresult + S (pfs_result_sub_zero_left_neg) = S ((S (pfs_index_sub_zero_left_neg)) * rc)) /\ exists ff_q_pfp_sub_zero_left_negresult. rb = ff_q_pfp_sub_zero_left_negresult * S ((S (pfs_index_sub_zero_left_neg)) * rc) + (pfs_result_sub_zero_left_neg))) /\ ((((exists pfa_gap_sub_zero_left_negoperationadditionleft. pfa_gap_sub_zero_left_negoperationadditionleft + S (pfs_source_sub_zero_left_neg) = (p)) /\ (((exists pfa_gap_sub_zero_left_negoperationadditionright. pfa_gap_sub_zero_left_negoperationadditionright + S (pfs_result_sub_zero_left_neg) = (p)) /\ ((((exists pfa_gap_sub_zero_left_negoperationadditionresultbound. pfa_gap_sub_zero_left_negoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_sub_zero_left_negoperationadditionresultcongruence pfa_offset_right_sub_zero_left_negoperationadditionresultcongruence. ((pfs_source_sub_zero_left_neg) + (pfs_result_sub_zero_left_neg)) + (p) * pfa_offset_left_sub_zero_left_negoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_sub_zero_left_negoperationadditionresultcongruence)))))))))))))) -> (forall pfp_repeat_index_sub_zero_left_zero. (exists pfa_gap_sub_zero_left_zeroindex. pfa_gap_sub_zero_left_zeroindex + S (pfp_repeat_index_sub_zero_left_zero) = (l)) -> (((exists ff_h_pfp_sub_zero_left_zeroentry. ff_h_pfp_sub_zero_left_zeroentry + S (0) = S ((S (pfp_repeat_index_sub_zero_left_zero)) * zc)) /\ exists ff_q_pfp_sub_zero_left_zeroentry. zb = ff_q_pfp_sub_zero_left_zeroentry * S ((S (pfp_repeat_index_sub_zero_left_zero)) * zc) + (0)))) -> (forall pfs_index_sub_zero_left_result. (exists pfa_gap_sub_zero_left_resultindex. pfa_gap_sub_zero_left_resultindex + S (pfs_index_sub_zero_left_result) = (l)) -> exists pfs_left_sub_zero_left_result pfs_right_sub_zero_left_result pfs_result_sub_zero_left_result. ((((exists ff_h_pfp_sub_zero_left_resultleft. ff_h_pfp_sub_zero_left_resultleft + S (pfs_left_sub_zero_left_result) = S ((S (pfs_index_sub_zero_left_result)) * zc)) /\ exists ff_q_pfp_sub_zero_left_resultleft. zb = ff_q_pfp_sub_zero_left_resultleft * S ((S (pfs_index_sub_zero_left_result)) * zc) + (pfs_left_sub_zero_left_result))) /\ (((((exists ff_h_pfp_sub_zero_left_resultright. ff_h_pfp_sub_zero_left_resultright + S (pfs_right_sub_zero_left_result) = S ((S (pfs_index_sub_zero_left_result)) * bc)) /\ exists ff_q_pfp_sub_zero_left_resultright. bb = ff_q_pfp_sub_zero_left_resultright * S ((S (pfs_index_sub_zero_left_result)) * bc) + (pfs_right_sub_zero_left_result))) /\ (((((exists ff_h_pfp_sub_zero_left_resultresult. ff_h_pfp_sub_zero_left_resultresult + S (pfs_result_sub_zero_left_result) = S ((S (pfs_index_sub_zero_left_result)) * rc)) /\ exists ff_q_pfp_sub_zero_left_resultresult. rb = ff_q_pfp_sub_zero_left_resultresult * S ((S (pfs_index_sub_zero_left_result)) * rc) + (pfs_result_sub_zero_left_result))) /\ ((((exists pfa_gap_sub_zero_left_resultoperationleft. pfa_gap_sub_zero_left_resultoperationleft + S (pfs_right_sub_zero_left_result) = (p)) /\ (((exists pfa_gap_sub_zero_left_resultoperationright. pfa_gap_sub_zero_left_resultoperationright + S (pfs_result_sub_zero_left_result) = (p)) /\ ((((exists pfa_gap_sub_zero_left_resultoperationresultbound. pfa_gap_sub_zero_left_resultoperationresultbound + S (pfs_left_sub_zero_left_result) = (p)) /\ ((exists pfa_offset_left_sub_zero_left_resultoperationresultcongruence pfa_offset_right_sub_zero_left_resultoperationresultcongruence. ((pfs_right_sub_zero_left_result) + (pfs_result_sub_zero_left_result)) + (p) * pfa_offset_left_sub_zero_left_resultoperationresultcongruence = (pfs_left_sub_zero_left_result) + (p) * pfa_offset_right_sub_zero_left_resultoperationresultcongruence))))))))))))))))prime_field_polynomial_subtract_equal_entry_zeroExact checked statement
forall p ab ac bb bc rb rc l i a r. (~((p) = 1) /\ forall pfa_factor_left_sub_entry_zero_prime pfa_factor_right_sub_entry_zero_prime. (p) = pfa_factor_left_sub_entry_zero_prime * pfa_factor_right_sub_entry_zero_prime -> pfa_factor_left_sub_entry_zero_prime = 1 \/ pfa_factor_right_sub_entry_zero_prime = 1) -> (forall pfs_index_sub_entry_zero_graph. (exists pfa_gap_sub_entry_zero_graphindex. pfa_gap_sub_entry_zero_graphindex + S (pfs_index_sub_entry_zero_graph) = (l)) -> exists pfs_left_sub_entry_zero_graph pfs_right_sub_entry_zero_graph pfs_result_sub_entry_zero_graph. ((((exists ff_h_pfp_sub_entry_zero_graphleft. ff_h_pfp_sub_entry_zero_graphleft + S (pfs_left_sub_entry_zero_graph) = S ((S (pfs_index_sub_entry_zero_graph)) * ac)) /\ exists ff_q_pfp_sub_entry_zero_graphleft. ab = ff_q_pfp_sub_entry_zero_graphleft * S ((S (pfs_index_sub_entry_zero_graph)) * ac) + (pfs_left_sub_entry_zero_graph))) /\ (((((exists ff_h_pfp_sub_entry_zero_graphright. ff_h_pfp_sub_entry_zero_graphright + S (pfs_right_sub_entry_zero_graph) = S ((S (pfs_index_sub_entry_zero_graph)) * bc)) /\ exists ff_q_pfp_sub_entry_zero_graphright. bb = ff_q_pfp_sub_entry_zero_graphright * S ((S (pfs_index_sub_entry_zero_graph)) * bc) + (pfs_right_sub_entry_zero_graph))) /\ (((((exists ff_h_pfp_sub_entry_zero_graphresult. ff_h_pfp_sub_entry_zero_graphresult + S (pfs_result_sub_entry_zero_graph) = S ((S (pfs_index_sub_entry_zero_graph)) * rc)) /\ exists ff_q_pfp_sub_entry_zero_graphresult. rb = ff_q_pfp_sub_entry_zero_graphresult * S ((S (pfs_index_sub_entry_zero_graph)) * rc) + (pfs_result_sub_entry_zero_graph))) /\ ((((exists pfa_gap_sub_entry_zero_graphoperationleft. pfa_gap_sub_entry_zero_graphoperationleft + S (pfs_right_sub_entry_zero_graph) = (p)) /\ (((exists pfa_gap_sub_entry_zero_graphoperationright. pfa_gap_sub_entry_zero_graphoperationright + S (pfs_result_sub_entry_zero_graph) = (p)) /\ ((((exists pfa_gap_sub_entry_zero_graphoperationresultbound. pfa_gap_sub_entry_zero_graphoperationresultbound + S (pfs_left_sub_entry_zero_graph) = (p)) /\ ((exists pfa_offset_left_sub_entry_zero_graphoperationresultcongruence pfa_offset_right_sub_entry_zero_graphoperationresultcongruence. ((pfs_right_sub_entry_zero_graph) + (pfs_result_sub_entry_zero_graph)) + (p) * pfa_offset_left_sub_entry_zero_graphoperationresultcongruence = (pfs_left_sub_entry_zero_graph) + (p) * pfa_offset_right_sub_entry_zero_graphoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_sub_entry_zero_index. pfa_gap_sub_entry_zero_index + S (i) = (l)) -> (((exists ff_h_pfp_sub_entry_zero_a. ff_h_pfp_sub_entry_zero_a + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_sub_entry_zero_a. ab = ff_q_pfp_sub_entry_zero_a * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_sub_entry_zero_b. ff_h_pfp_sub_entry_zero_b + S (a) = S ((S (i)) * bc)) /\ exists ff_q_pfp_sub_entry_zero_b. bb = ff_q_pfp_sub_entry_zero_b * S ((S (i)) * bc) + (a))) -> (((exists ff_h_pfp_sub_entry_zero_r. ff_h_pfp_sub_entry_zero_r + S (r) = S ((S (i)) * rc)) /\ exists ff_q_pfp_sub_entry_zero_r. rb = ff_q_pfp_sub_entry_zero_r * S ((S (i)) * rc) + (r))) -> r=0prime_field_polynomial_subtract_equal_zeroExact checked statement
forall p ab ac bb bc rb rc l. (~((p) = 1) /\ forall pfa_factor_left_sub_equal_prime pfa_factor_right_sub_equal_prime. (p) = pfa_factor_left_sub_equal_prime * pfa_factor_right_sub_equal_prime -> pfa_factor_left_sub_equal_prime = 1 \/ pfa_factor_right_sub_equal_prime = 1) -> (forall mdr_i_pfp_sub_equal_inputs mdr_a_pfp_sub_equal_inputs. (exists mdr_gap_pfp_sub_equal_inputsb. mdr_gap_pfp_sub_equal_inputsb + S (mdr_i_pfp_sub_equal_inputs) = (l)) -> (((exists ff_h_mdr_pfp_sub_equal_inputso. ff_h_mdr_pfp_sub_equal_inputso + S (mdr_a_pfp_sub_equal_inputs) = S ((S (mdr_i_pfp_sub_equal_inputs)) * ac)) /\ exists ff_q_mdr_pfp_sub_equal_inputso. ab = ff_q_mdr_pfp_sub_equal_inputso * S ((S (mdr_i_pfp_sub_equal_inputs)) * ac) + (mdr_a_pfp_sub_equal_inputs))) -> (((exists ff_h_mdr_pfp_sub_equal_inputsn. ff_h_mdr_pfp_sub_equal_inputsn + S (mdr_a_pfp_sub_equal_inputs) = S ((S (mdr_i_pfp_sub_equal_inputs)) * bc)) /\ exists ff_q_mdr_pfp_sub_equal_inputsn. bb = ff_q_mdr_pfp_sub_equal_inputsn * S ((S (mdr_i_pfp_sub_equal_inputs)) * bc) + (mdr_a_pfp_sub_equal_inputs)))) -> (forall pfs_index_sub_equal_graph. (exists pfa_gap_sub_equal_graphindex. pfa_gap_sub_equal_graphindex + S (pfs_index_sub_equal_graph) = (l)) -> exists pfs_left_sub_equal_graph pfs_right_sub_equal_graph pfs_result_sub_equal_graph. ((((exists ff_h_pfp_sub_equal_graphleft. ff_h_pfp_sub_equal_graphleft + S (pfs_left_sub_equal_graph) = S ((S (pfs_index_sub_equal_graph)) * ac)) /\ exists ff_q_pfp_sub_equal_graphleft. ab = ff_q_pfp_sub_equal_graphleft * S ((S (pfs_index_sub_equal_graph)) * ac) + (pfs_left_sub_equal_graph))) /\ (((((exists ff_h_pfp_sub_equal_graphright. ff_h_pfp_sub_equal_graphright + S (pfs_right_sub_equal_graph) = S ((S (pfs_index_sub_equal_graph)) * bc)) /\ exists ff_q_pfp_sub_equal_graphright. bb = ff_q_pfp_sub_equal_graphright * S ((S (pfs_index_sub_equal_graph)) * bc) + (pfs_right_sub_equal_graph))) /\ (((((exists ff_h_pfp_sub_equal_graphresult. ff_h_pfp_sub_equal_graphresult + S (pfs_result_sub_equal_graph) = S ((S (pfs_index_sub_equal_graph)) * rc)) /\ exists ff_q_pfp_sub_equal_graphresult. rb = ff_q_pfp_sub_equal_graphresult * S ((S (pfs_index_sub_equal_graph)) * rc) + (pfs_result_sub_equal_graph))) /\ ((((exists pfa_gap_sub_equal_graphoperationleft. pfa_gap_sub_equal_graphoperationleft + S (pfs_right_sub_equal_graph) = (p)) /\ (((exists pfa_gap_sub_equal_graphoperationright. pfa_gap_sub_equal_graphoperationright + S (pfs_result_sub_equal_graph) = (p)) /\ ((((exists pfa_gap_sub_equal_graphoperationresultbound. pfa_gap_sub_equal_graphoperationresultbound + S (pfs_left_sub_equal_graph) = (p)) /\ ((exists pfa_offset_left_sub_equal_graphoperationresultcongruence pfa_offset_right_sub_equal_graphoperationresultcongruence. ((pfs_right_sub_equal_graph) + (pfs_result_sub_equal_graph)) + (p) * pfa_offset_left_sub_equal_graphoperationresultcongruence = (pfs_left_sub_equal_graph) + (p) * pfa_offset_right_sub_equal_graphoperationresultcongruence)))))))))))))))) -> (forall pfp_repeat_index_sub_equal_result. (exists pfa_gap_sub_equal_resultindex. pfa_gap_sub_equal_resultindex + S (pfp_repeat_index_sub_equal_result) = (l)) -> (((exists ff_h_pfp_sub_equal_resultentry. ff_h_pfp_sub_equal_resultentry + S (0) = S ((S (pfp_repeat_index_sub_equal_result)) * rc)) /\ exists ff_q_pfp_sub_equal_resultentry. rb = ff_q_pfp_sub_equal_resultentry * S ((S (pfp_repeat_index_sub_equal_result)) * rc) + (0))))prime_field_polynomial_subtract_add_cancelExact checked statement
forall p ab ac bb bc cb cc rb rc l. (forall pfp_index_sub_cancel_sum. (exists pfa_gap_sub_cancel_sumindex. pfa_gap_sub_cancel_sumindex + S (pfp_index_sub_cancel_sum) = (l)) -> exists pfp_left_sub_cancel_sum pfp_right_sub_cancel_sum pfp_value_sub_cancel_sum. ((((exists ff_h_pfp_sub_cancel_sumleft. ff_h_pfp_sub_cancel_sumleft + S (pfp_left_sub_cancel_sum) = S ((S (pfp_index_sub_cancel_sum)) * ac)) /\ exists ff_q_pfp_sub_cancel_sumleft. ab = ff_q_pfp_sub_cancel_sumleft * S ((S (pfp_index_sub_cancel_sum)) * ac) + (pfp_left_sub_cancel_sum))) /\ (((((exists ff_h_pfp_sub_cancel_sumright. ff_h_pfp_sub_cancel_sumright + S (pfp_right_sub_cancel_sum) = S ((S (pfp_index_sub_cancel_sum)) * bc)) /\ exists ff_q_pfp_sub_cancel_sumright. bb = ff_q_pfp_sub_cancel_sumright * S ((S (pfp_index_sub_cancel_sum)) * bc) + (pfp_right_sub_cancel_sum))) /\ (((((exists ff_h_pfp_sub_cancel_sumtarget. ff_h_pfp_sub_cancel_sumtarget + S (pfp_value_sub_cancel_sum) = S ((S (pfp_index_sub_cancel_sum)) * cc)) /\ exists ff_q_pfp_sub_cancel_sumtarget. cb = ff_q_pfp_sub_cancel_sumtarget * S ((S (pfp_index_sub_cancel_sum)) * cc) + (pfp_value_sub_cancel_sum))) /\ ((((exists pfa_gap_sub_cancel_sumoperationleft. pfa_gap_sub_cancel_sumoperationleft + S (pfp_left_sub_cancel_sum) = (p)) /\ (((exists pfa_gap_sub_cancel_sumoperationright. pfa_gap_sub_cancel_sumoperationright + S (pfp_right_sub_cancel_sum) = (p)) /\ ((((exists pfa_gap_sub_cancel_sumoperationresultbound. pfa_gap_sub_cancel_sumoperationresultbound + S (pfp_value_sub_cancel_sum) = (p)) /\ ((exists pfa_offset_left_sub_cancel_sumoperationresultcongruence pfa_offset_right_sub_cancel_sumoperationresultcongruence. ((pfp_left_sub_cancel_sum) + (pfp_right_sub_cancel_sum)) + (p) * pfa_offset_left_sub_cancel_sumoperationresultcongruence = (pfp_value_sub_cancel_sum) + (p) * pfa_offset_right_sub_cancel_sumoperationresultcongruence)))))))))))))))) -> (forall pfs_index_sub_cancel_difference. (exists pfa_gap_sub_cancel_differenceindex. pfa_gap_sub_cancel_differenceindex + S (pfs_index_sub_cancel_difference) = (l)) -> exists pfs_left_sub_cancel_difference pfs_right_sub_cancel_difference pfs_result_sub_cancel_difference. ((((exists ff_h_pfp_sub_cancel_differenceleft. ff_h_pfp_sub_cancel_differenceleft + S (pfs_left_sub_cancel_difference) = S ((S (pfs_index_sub_cancel_difference)) * cc)) /\ exists ff_q_pfp_sub_cancel_differenceleft. cb = ff_q_pfp_sub_cancel_differenceleft * S ((S (pfs_index_sub_cancel_difference)) * cc) + (pfs_left_sub_cancel_difference))) /\ (((((exists ff_h_pfp_sub_cancel_differenceright. ff_h_pfp_sub_cancel_differenceright + S (pfs_right_sub_cancel_difference) = S ((S (pfs_index_sub_cancel_difference)) * ac)) /\ exists ff_q_pfp_sub_cancel_differenceright. ab = ff_q_pfp_sub_cancel_differenceright * S ((S (pfs_index_sub_cancel_difference)) * ac) + (pfs_right_sub_cancel_difference))) /\ (((((exists ff_h_pfp_sub_cancel_differenceresult. ff_h_pfp_sub_cancel_differenceresult + S (pfs_result_sub_cancel_difference) = S ((S (pfs_index_sub_cancel_difference)) * rc)) /\ exists ff_q_pfp_sub_cancel_differenceresult. rb = ff_q_pfp_sub_cancel_differenceresult * S ((S (pfs_index_sub_cancel_difference)) * rc) + (pfs_result_sub_cancel_difference))) /\ ((((exists pfa_gap_sub_cancel_differenceoperationleft. pfa_gap_sub_cancel_differenceoperationleft + S (pfs_right_sub_cancel_difference) = (p)) /\ (((exists pfa_gap_sub_cancel_differenceoperationright. pfa_gap_sub_cancel_differenceoperationright + S (pfs_result_sub_cancel_difference) = (p)) /\ ((((exists pfa_gap_sub_cancel_differenceoperationresultbound. pfa_gap_sub_cancel_differenceoperationresultbound + S (pfs_left_sub_cancel_difference) = (p)) /\ ((exists pfa_offset_left_sub_cancel_differenceoperationresultcongruence pfa_offset_right_sub_cancel_differenceoperationresultcongruence. ((pfs_right_sub_cancel_difference) + (pfs_result_sub_cancel_difference)) + (p) * pfa_offset_left_sub_cancel_differenceoperationresultcongruence = (pfs_left_sub_cancel_difference) + (p) * pfa_offset_right_sub_cancel_differenceoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_sub_cancel_result mdr_a_pfp_sub_cancel_result. (exists mdr_gap_pfp_sub_cancel_resultb. mdr_gap_pfp_sub_cancel_resultb + S (mdr_i_pfp_sub_cancel_result) = (l)) -> (((exists ff_h_mdr_pfp_sub_cancel_resulto. ff_h_mdr_pfp_sub_cancel_resulto + S (mdr_a_pfp_sub_cancel_result) = S ((S (mdr_i_pfp_sub_cancel_result)) * bc)) /\ exists ff_q_mdr_pfp_sub_cancel_resulto. bb = ff_q_mdr_pfp_sub_cancel_resulto * S ((S (mdr_i_pfp_sub_cancel_result)) * bc) + (mdr_a_pfp_sub_cancel_result))) -> (((exists ff_h_mdr_pfp_sub_cancel_resultn. ff_h_mdr_pfp_sub_cancel_resultn + S (mdr_a_pfp_sub_cancel_result) = S ((S (mdr_i_pfp_sub_cancel_result)) * rc)) /\ exists ff_q_mdr_pfp_sub_cancel_resultn. rb = ff_q_mdr_pfp_sub_cancel_resultn * S ((S (mdr_i_pfp_sub_cancel_result)) * rc) + (mdr_a_pfp_sub_cancel_result))))prime_field_polynomial_subtract_common_right_cancelExact checked statement
forall p ab ac bb bc cb cc rb rc l. (forall pfs_index_sub_cancel_right_first. (exists pfa_gap_sub_cancel_right_firstindex. pfa_gap_sub_cancel_right_firstindex + S (pfs_index_sub_cancel_right_first) = (l)) -> exists pfs_left_sub_cancel_right_first pfs_right_sub_cancel_right_first pfs_result_sub_cancel_right_first. ((((exists ff_h_pfp_sub_cancel_right_firstleft. ff_h_pfp_sub_cancel_right_firstleft + S (pfs_left_sub_cancel_right_first) = S ((S (pfs_index_sub_cancel_right_first)) * ac)) /\ exists ff_q_pfp_sub_cancel_right_firstleft. ab = ff_q_pfp_sub_cancel_right_firstleft * S ((S (pfs_index_sub_cancel_right_first)) * ac) + (pfs_left_sub_cancel_right_first))) /\ (((((exists ff_h_pfp_sub_cancel_right_firstright. ff_h_pfp_sub_cancel_right_firstright + S (pfs_right_sub_cancel_right_first) = S ((S (pfs_index_sub_cancel_right_first)) * cc)) /\ exists ff_q_pfp_sub_cancel_right_firstright. cb = ff_q_pfp_sub_cancel_right_firstright * S ((S (pfs_index_sub_cancel_right_first)) * cc) + (pfs_right_sub_cancel_right_first))) /\ (((((exists ff_h_pfp_sub_cancel_right_firstresult. ff_h_pfp_sub_cancel_right_firstresult + S (pfs_result_sub_cancel_right_first) = S ((S (pfs_index_sub_cancel_right_first)) * rc)) /\ exists ff_q_pfp_sub_cancel_right_firstresult. rb = ff_q_pfp_sub_cancel_right_firstresult * S ((S (pfs_index_sub_cancel_right_first)) * rc) + (pfs_result_sub_cancel_right_first))) /\ ((((exists pfa_gap_sub_cancel_right_firstoperationleft. pfa_gap_sub_cancel_right_firstoperationleft + S (pfs_right_sub_cancel_right_first) = (p)) /\ (((exists pfa_gap_sub_cancel_right_firstoperationright. pfa_gap_sub_cancel_right_firstoperationright + S (pfs_result_sub_cancel_right_first) = (p)) /\ ((((exists pfa_gap_sub_cancel_right_firstoperationresultbound. pfa_gap_sub_cancel_right_firstoperationresultbound + S (pfs_left_sub_cancel_right_first) = (p)) /\ ((exists pfa_offset_left_sub_cancel_right_firstoperationresultcongruence pfa_offset_right_sub_cancel_right_firstoperationresultcongruence. ((pfs_right_sub_cancel_right_first) + (pfs_result_sub_cancel_right_first)) + (p) * pfa_offset_left_sub_cancel_right_firstoperationresultcongruence = (pfs_left_sub_cancel_right_first) + (p) * pfa_offset_right_sub_cancel_right_firstoperationresultcongruence)))))))))))))))) -> (forall pfs_index_sub_cancel_right_second. (exists pfa_gap_sub_cancel_right_secondindex. pfa_gap_sub_cancel_right_secondindex + S (pfs_index_sub_cancel_right_second) = (l)) -> exists pfs_left_sub_cancel_right_second pfs_right_sub_cancel_right_second pfs_result_sub_cancel_right_second. ((((exists ff_h_pfp_sub_cancel_right_secondleft. ff_h_pfp_sub_cancel_right_secondleft + S (pfs_left_sub_cancel_right_second) = S ((S (pfs_index_sub_cancel_right_second)) * bc)) /\ exists ff_q_pfp_sub_cancel_right_secondleft. bb = ff_q_pfp_sub_cancel_right_secondleft * S ((S (pfs_index_sub_cancel_right_second)) * bc) + (pfs_left_sub_cancel_right_second))) /\ (((((exists ff_h_pfp_sub_cancel_right_secondright. ff_h_pfp_sub_cancel_right_secondright + S (pfs_right_sub_cancel_right_second) = S ((S (pfs_index_sub_cancel_right_second)) * cc)) /\ exists ff_q_pfp_sub_cancel_right_secondright. cb = ff_q_pfp_sub_cancel_right_secondright * S ((S (pfs_index_sub_cancel_right_second)) * cc) + (pfs_right_sub_cancel_right_second))) /\ (((((exists ff_h_pfp_sub_cancel_right_secondresult. ff_h_pfp_sub_cancel_right_secondresult + S (pfs_result_sub_cancel_right_second) = S ((S (pfs_index_sub_cancel_right_second)) * rc)) /\ exists ff_q_pfp_sub_cancel_right_secondresult. rb = ff_q_pfp_sub_cancel_right_secondresult * S ((S (pfs_index_sub_cancel_right_second)) * rc) + (pfs_result_sub_cancel_right_second))) /\ ((((exists pfa_gap_sub_cancel_right_secondoperationleft. pfa_gap_sub_cancel_right_secondoperationleft + S (pfs_right_sub_cancel_right_second) = (p)) /\ (((exists pfa_gap_sub_cancel_right_secondoperationright. pfa_gap_sub_cancel_right_secondoperationright + S (pfs_result_sub_cancel_right_second) = (p)) /\ ((((exists pfa_gap_sub_cancel_right_secondoperationresultbound. pfa_gap_sub_cancel_right_secondoperationresultbound + S (pfs_left_sub_cancel_right_second) = (p)) /\ ((exists pfa_offset_left_sub_cancel_right_secondoperationresultcongruence pfa_offset_right_sub_cancel_right_secondoperationresultcongruence. ((pfs_right_sub_cancel_right_second) + (pfs_result_sub_cancel_right_second)) + (p) * pfa_offset_left_sub_cancel_right_secondoperationresultcongruence = (pfs_left_sub_cancel_right_second) + (p) * pfa_offset_right_sub_cancel_right_secondoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_sub_cancel_right_result mdr_a_pfp_sub_cancel_right_result. (exists mdr_gap_pfp_sub_cancel_right_resultb. mdr_gap_pfp_sub_cancel_right_resultb + S (mdr_i_pfp_sub_cancel_right_result) = (l)) -> (((exists ff_h_mdr_pfp_sub_cancel_right_resulto. ff_h_mdr_pfp_sub_cancel_right_resulto + S (mdr_a_pfp_sub_cancel_right_result) = S ((S (mdr_i_pfp_sub_cancel_right_result)) * ac)) /\ exists ff_q_mdr_pfp_sub_cancel_right_resulto. ab = ff_q_mdr_pfp_sub_cancel_right_resulto * S ((S (mdr_i_pfp_sub_cancel_right_result)) * ac) + (mdr_a_pfp_sub_cancel_right_result))) -> (((exists ff_h_mdr_pfp_sub_cancel_right_resultn. ff_h_mdr_pfp_sub_cancel_right_resultn + S (mdr_a_pfp_sub_cancel_right_result) = S ((S (mdr_i_pfp_sub_cancel_right_result)) * bc)) /\ exists ff_q_mdr_pfp_sub_cancel_right_resultn. bb = ff_q_mdr_pfp_sub_cancel_right_resultn * S ((S (mdr_i_pfp_sub_cancel_right_result)) * bc) + (mdr_a_pfp_sub_cancel_right_result))))prime_field_polynomial_suffix_existsExact checked statement
forall b c t M. exists d e. (forall pftrim_index_suffix_exists pftrim_value_suffix_exists. (exists pfa_gap_suffix_existsbound. pfa_gap_suffix_existsbound + S (pftrim_index_suffix_exists) = (M)) -> (((exists ff_h_pfp_suffix_existssource. ff_h_pfp_suffix_existssource + S (pftrim_value_suffix_exists) = S ((S ((t)+pftrim_index_suffix_exists)) * c)) /\ exists ff_q_pfp_suffix_existssource. b = ff_q_pfp_suffix_existssource * S ((S ((t)+pftrim_index_suffix_exists)) * c) + (pftrim_value_suffix_exists))) -> (((exists ff_h_pfp_suffix_existsoutput. ff_h_pfp_suffix_existsoutput + S (pftrim_value_suffix_exists) = S ((S (pftrim_index_suffix_exists)) * e)) /\ exists ff_q_pfp_suffix_existsoutput. d = ff_q_pfp_suffix_existsoutput * S ((S (pftrim_index_suffix_exists)) * e) + (pftrim_value_suffix_exists))))prime_field_polynomial_suffix_entryExact checked statement
forall b c t d e M i a r. (forall pftrim_index_suffix_entry pftrim_value_suffix_entry. (exists pfa_gap_suffix_entrybound. pfa_gap_suffix_entrybound + S (pftrim_index_suffix_entry) = (M)) -> (((exists ff_h_pfp_suffix_entrysource. ff_h_pfp_suffix_entrysource + S (pftrim_value_suffix_entry) = S ((S ((t)+pftrim_index_suffix_entry)) * c)) /\ exists ff_q_pfp_suffix_entrysource. b = ff_q_pfp_suffix_entrysource * S ((S ((t)+pftrim_index_suffix_entry)) * c) + (pftrim_value_suffix_entry))) -> (((exists ff_h_pfp_suffix_entryoutput. ff_h_pfp_suffix_entryoutput + S (pftrim_value_suffix_entry) = S ((S (pftrim_index_suffix_entry)) * e)) /\ exists ff_q_pfp_suffix_entryoutput. d = ff_q_pfp_suffix_entryoutput * S ((S (pftrim_index_suffix_entry)) * e) + (pftrim_value_suffix_entry)))) -> (exists pfa_gap_suffix_entry_bound. pfa_gap_suffix_entry_bound + S (i) = (M)) -> (((exists ff_h_pfp_suffix_entry_source. ff_h_pfp_suffix_entry_source + S (a) = S ((S (t+i)) * c)) /\ exists ff_q_pfp_suffix_entry_source. b = ff_q_pfp_suffix_entry_source * S ((S (t+i)) * c) + (a))) -> (((exists ff_h_pfp_suffix_entry_output. ff_h_pfp_suffix_entry_output + S (r) = S ((S (i)) * e)) /\ exists ff_q_pfp_suffix_entry_output. d = ff_q_pfp_suffix_entry_output * S ((S (i)) * e) + (r))) -> r=aprime_field_polynomial_suffix_boundedExact checked statement
forall p b c L t d e M. L=t+M -> (forall fom_index_pfp_suffix_input_coefficients. (exists fom_gap_pfp_suffix_input_coefficients_index_bound. fom_gap_pfp_suffix_input_coefficients_index_bound + S (fom_index_pfp_suffix_input_coefficients) = L) -> exists fom_value_pfp_suffix_input_coefficients. ((((exists fom_beta_height_pfp_suffix_input_coefficients_entry. fom_beta_height_pfp_suffix_input_coefficients_entry + S (fom_value_pfp_suffix_input_coefficients) = S ((S (fom_index_pfp_suffix_input_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_suffix_input_coefficients_entry. b = fom_beta_quotient_pfp_suffix_input_coefficients_entry * S ((S (fom_index_pfp_suffix_input_coefficients)) * c) + (fom_value_pfp_suffix_input_coefficients))) /\ (exists fom_gap_pfp_suffix_input_coefficients_value_bound. fom_gap_pfp_suffix_input_coefficients_value_bound + S (fom_value_pfp_suffix_input_coefficients) = p))) -> (forall pftrim_index_suffix_bounded pftrim_value_suffix_bounded. (exists pfa_gap_suffix_boundedbound. pfa_gap_suffix_boundedbound + S (pftrim_index_suffix_bounded) = (M)) -> (((exists ff_h_pfp_suffix_boundedsource. ff_h_pfp_suffix_boundedsource + S (pftrim_value_suffix_bounded) = S ((S ((t)+pftrim_index_suffix_bounded)) * c)) /\ exists ff_q_pfp_suffix_boundedsource. b = ff_q_pfp_suffix_boundedsource * S ((S ((t)+pftrim_index_suffix_bounded)) * c) + (pftrim_value_suffix_bounded))) -> (((exists ff_h_pfp_suffix_boundedoutput. ff_h_pfp_suffix_boundedoutput + S (pftrim_value_suffix_bounded) = S ((S (pftrim_index_suffix_bounded)) * e)) /\ exists ff_q_pfp_suffix_boundedoutput. d = ff_q_pfp_suffix_boundedoutput * S ((S (pftrim_index_suffix_bounded)) * e) + (pftrim_value_suffix_bounded)))) -> (forall fom_index_pfp_suffix_output_coefficients. (exists fom_gap_pfp_suffix_output_coefficients_index_bound. fom_gap_pfp_suffix_output_coefficients_index_bound + S (fom_index_pfp_suffix_output_coefficients) = M) -> exists fom_value_pfp_suffix_output_coefficients. ((((exists fom_beta_height_pfp_suffix_output_coefficients_entry. fom_beta_height_pfp_suffix_output_coefficients_entry + S (fom_value_pfp_suffix_output_coefficients) = S ((S (fom_index_pfp_suffix_output_coefficients)) * e)) /\ exists fom_beta_quotient_pfp_suffix_output_coefficients_entry. d = fom_beta_quotient_pfp_suffix_output_coefficients_entry * S ((S (fom_index_pfp_suffix_output_coefficients)) * e) + (fom_value_pfp_suffix_output_coefficients))) /\ (exists fom_gap_pfp_suffix_output_coefficients_value_bound. fom_gap_pfp_suffix_output_coefficients_value_bound + S (fom_value_pfp_suffix_output_coefficients) = p)))prime_field_polynomial_suffix_equalExact checked statement
forall b c t d e f g M. (forall pftrim_index_suffix_equal_left pftrim_value_suffix_equal_left. (exists pfa_gap_suffix_equal_leftbound. pfa_gap_suffix_equal_leftbound + S (pftrim_index_suffix_equal_left) = (M)) -> (((exists ff_h_pfp_suffix_equal_leftsource. ff_h_pfp_suffix_equal_leftsource + S (pftrim_value_suffix_equal_left) = S ((S ((t)+pftrim_index_suffix_equal_left)) * c)) /\ exists ff_q_pfp_suffix_equal_leftsource. b = ff_q_pfp_suffix_equal_leftsource * S ((S ((t)+pftrim_index_suffix_equal_left)) * c) + (pftrim_value_suffix_equal_left))) -> (((exists ff_h_pfp_suffix_equal_leftoutput. ff_h_pfp_suffix_equal_leftoutput + S (pftrim_value_suffix_equal_left) = S ((S (pftrim_index_suffix_equal_left)) * e)) /\ exists ff_q_pfp_suffix_equal_leftoutput. d = ff_q_pfp_suffix_equal_leftoutput * S ((S (pftrim_index_suffix_equal_left)) * e) + (pftrim_value_suffix_equal_left)))) -> (forall pftrim_index_suffix_equal_right pftrim_value_suffix_equal_right. (exists pfa_gap_suffix_equal_rightbound. pfa_gap_suffix_equal_rightbound + S (pftrim_index_suffix_equal_right) = (M)) -> (((exists ff_h_pfp_suffix_equal_rightsource. ff_h_pfp_suffix_equal_rightsource + S (pftrim_value_suffix_equal_right) = S ((S ((t)+pftrim_index_suffix_equal_right)) * c)) /\ exists ff_q_pfp_suffix_equal_rightsource. b = ff_q_pfp_suffix_equal_rightsource * S ((S ((t)+pftrim_index_suffix_equal_right)) * c) + (pftrim_value_suffix_equal_right))) -> (((exists ff_h_pfp_suffix_equal_rightoutput. ff_h_pfp_suffix_equal_rightoutput + S (pftrim_value_suffix_equal_right) = S ((S (pftrim_index_suffix_equal_right)) * g)) /\ exists ff_q_pfp_suffix_equal_rightoutput. f = ff_q_pfp_suffix_equal_rightoutput * S ((S (pftrim_index_suffix_equal_right)) * g) + (pftrim_value_suffix_equal_right)))) -> (forall mdr_i_pfp_suffix_equal_result mdr_a_pfp_suffix_equal_result. (exists mdr_gap_pfp_suffix_equal_resultb. mdr_gap_pfp_suffix_equal_resultb + S (mdr_i_pfp_suffix_equal_result) = (M)) -> (((exists ff_h_mdr_pfp_suffix_equal_resulto. ff_h_mdr_pfp_suffix_equal_resulto + S (mdr_a_pfp_suffix_equal_result) = S ((S (mdr_i_pfp_suffix_equal_result)) * e)) /\ exists ff_q_mdr_pfp_suffix_equal_resulto. d = ff_q_mdr_pfp_suffix_equal_resulto * S ((S (mdr_i_pfp_suffix_equal_result)) * e) + (mdr_a_pfp_suffix_equal_result))) -> (((exists ff_h_mdr_pfp_suffix_equal_resultn. ff_h_mdr_pfp_suffix_equal_resultn + S (mdr_a_pfp_suffix_equal_result) = S ((S (mdr_i_pfp_suffix_equal_result)) * g)) /\ exists ff_q_mdr_pfp_suffix_equal_resultn. f = ff_q_mdr_pfp_suffix_equal_resultn * S ((S (mdr_i_pfp_suffix_equal_result)) * g) + (mdr_a_pfp_suffix_equal_result))))prime_field_polynomial_leading_zero_cut_existsExact checked statement
forall b c L. exists t M. ((((L)=(t)+(M)) /\ (((forall pfp_repeat_index_cut_existszero. (exists pfa_gap_cut_existszeroindex. pfa_gap_cut_existszeroindex + S (pfp_repeat_index_cut_existszero) = (t)) -> (((exists ff_h_pfp_cut_existszeroentry. ff_h_pfp_cut_existszeroentry + S (0) = S ((S (pfp_repeat_index_cut_existszero)) * c)) /\ exists ff_q_pfp_cut_existszeroentry. b = ff_q_pfp_cut_existszeroentry * S ((S (pfp_repeat_index_cut_existszero)) * c) + (0)))) /\ (((M)=0 \/ (((~((M)=0)) /\ ((exists pftrim_leading_cut_existshead. ((((exists ff_h_pfp_cut_existsheadentry. ff_h_pfp_cut_existsheadentry + S (pftrim_leading_cut_existshead) = S ((S (t)) * c)) /\ exists ff_q_pfp_cut_existsheadentry. b = ff_q_pfp_cut_existsheadentry * S ((S (t)) * c) + (pftrim_leading_cut_existshead))) /\ ((~(pftrim_leading_cut_existshead=0))))))))))))))prime_field_polynomial_trim_from_cutExact checked statement
forall p b c L t d e M. (forall fom_index_pfp_cut_to_trim_coefficients. (exists fom_gap_pfp_cut_to_trim_coefficients_index_bound. fom_gap_pfp_cut_to_trim_coefficients_index_bound + S (fom_index_pfp_cut_to_trim_coefficients) = L) -> exists fom_value_pfp_cut_to_trim_coefficients. ((((exists fom_beta_height_pfp_cut_to_trim_coefficients_entry. fom_beta_height_pfp_cut_to_trim_coefficients_entry + S (fom_value_pfp_cut_to_trim_coefficients) = S ((S (fom_index_pfp_cut_to_trim_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_cut_to_trim_coefficients_entry. b = fom_beta_quotient_pfp_cut_to_trim_coefficients_entry * S ((S (fom_index_pfp_cut_to_trim_coefficients)) * c) + (fom_value_pfp_cut_to_trim_coefficients))) /\ (exists fom_gap_pfp_cut_to_trim_coefficients_value_bound. fom_gap_pfp_cut_to_trim_coefficients_value_bound + S (fom_value_pfp_cut_to_trim_coefficients) = p))) -> ((((L)=(t)+(M)) /\ (((forall pfp_repeat_index_cut_to_trim_sourcezero. (exists pfa_gap_cut_to_trim_sourcezeroindex. pfa_gap_cut_to_trim_sourcezeroindex + S (pfp_repeat_index_cut_to_trim_sourcezero) = (t)) -> (((exists ff_h_pfp_cut_to_trim_sourcezeroentry. ff_h_pfp_cut_to_trim_sourcezeroentry + S (0) = S ((S (pfp_repeat_index_cut_to_trim_sourcezero)) * c)) /\ exists ff_q_pfp_cut_to_trim_sourcezeroentry. b = ff_q_pfp_cut_to_trim_sourcezeroentry * S ((S (pfp_repeat_index_cut_to_trim_sourcezero)) * c) + (0)))) /\ (((M)=0 \/ (((~((M)=0)) /\ ((exists pftrim_leading_cut_to_trim_sourcehead. ((((exists ff_h_pfp_cut_to_trim_sourceheadentry. ff_h_pfp_cut_to_trim_sourceheadentry + S (pftrim_leading_cut_to_trim_sourcehead) = S ((S (t)) * c)) /\ exists ff_q_pfp_cut_to_trim_sourceheadentry. b = ff_q_pfp_cut_to_trim_sourceheadentry * S ((S (t)) * c) + (pftrim_leading_cut_to_trim_sourcehead))) /\ ((~(pftrim_leading_cut_to_trim_sourcehead=0)))))))))))))) -> (forall pftrim_index_cut_to_trim_suffix pftrim_value_cut_to_trim_suffix. (exists pfa_gap_cut_to_trim_suffixbound. pfa_gap_cut_to_trim_suffixbound + S (pftrim_index_cut_to_trim_suffix) = (M)) -> (((exists ff_h_pfp_cut_to_trim_suffixsource. ff_h_pfp_cut_to_trim_suffixsource + S (pftrim_value_cut_to_trim_suffix) = S ((S ((t)+pftrim_index_cut_to_trim_suffix)) * c)) /\ exists ff_q_pfp_cut_to_trim_suffixsource. b = ff_q_pfp_cut_to_trim_suffixsource * S ((S ((t)+pftrim_index_cut_to_trim_suffix)) * c) + (pftrim_value_cut_to_trim_suffix))) -> (((exists ff_h_pfp_cut_to_trim_suffixoutput. ff_h_pfp_cut_to_trim_suffixoutput + S (pftrim_value_cut_to_trim_suffix) = S ((S (pftrim_index_cut_to_trim_suffix)) * e)) /\ exists ff_q_pfp_cut_to_trim_suffixoutput. d = ff_q_pfp_cut_to_trim_suffixoutput * S ((S (pftrim_index_cut_to_trim_suffix)) * e) + (pftrim_value_cut_to_trim_suffix)))) -> ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_cut_to_trim_resultinput. (exists fom_gap_pfp_cut_to_trim_resultinput_index_bound. fom_gap_pfp_cut_to_trim_resultinput_index_bound + S (fom_index_pfp_cut_to_trim_resultinput) = L) -> exists fom_value_pfp_cut_to_trim_resultinput. ((((exists fom_beta_height_pfp_cut_to_trim_resultinput_entry. fom_beta_height_pfp_cut_to_trim_resultinput_entry + S (fom_value_pfp_cut_to_trim_resultinput) = S ((S (fom_index_pfp_cut_to_trim_resultinput)) * c)) /\ exists fom_beta_quotient_pfp_cut_to_trim_resultinput_entry. b = fom_beta_quotient_pfp_cut_to_trim_resultinput_entry * S ((S (fom_index_pfp_cut_to_trim_resultinput)) * c) + (fom_value_pfp_cut_to_trim_resultinput))) /\ (exists fom_gap_pfp_cut_to_trim_resultinput_value_bound. fom_gap_pfp_cut_to_trim_resultinput_value_bound + S (fom_value_pfp_cut_to_trim_resultinput) = p))) /\ (((forall pfp_repeat_index_cut_to_trim_resultremoved. (exists pfa_gap_cut_to_trim_resultremovedindex. pfa_gap_cut_to_trim_resultremovedindex + S (pfp_repeat_index_cut_to_trim_resultremoved) = (t)) -> (((exists ff_h_pfp_cut_to_trim_resultremovedentry. ff_h_pfp_cut_to_trim_resultremovedentry + S (0) = S ((S (pfp_repeat_index_cut_to_trim_resultremoved)) * c)) /\ exists ff_q_pfp_cut_to_trim_resultremovedentry. b = ff_q_pfp_cut_to_trim_resultremovedentry * S ((S (pfp_repeat_index_cut_to_trim_resultremoved)) * c) + (0)))) /\ (((forall pftrim_index_cut_to_trim_resultsuffix pftrim_value_cut_to_trim_resultsuffix. (exists pfa_gap_cut_to_trim_resultsuffixbound. pfa_gap_cut_to_trim_resultsuffixbound + S (pftrim_index_cut_to_trim_resultsuffix) = (M)) -> (((exists ff_h_pfp_cut_to_trim_resultsuffixsource. ff_h_pfp_cut_to_trim_resultsuffixsource + S (pftrim_value_cut_to_trim_resultsuffix) = S ((S ((t)+pftrim_index_cut_to_trim_resultsuffix)) * c)) /\ exists ff_q_pfp_cut_to_trim_resultsuffixsource. b = ff_q_pfp_cut_to_trim_resultsuffixsource * S ((S ((t)+pftrim_index_cut_to_trim_resultsuffix)) * c) + (pftrim_value_cut_to_trim_resultsuffix))) -> (((exists ff_h_pfp_cut_to_trim_resultsuffixoutput. ff_h_pfp_cut_to_trim_resultsuffixoutput + S (pftrim_value_cut_to_trim_resultsuffix) = S ((S (pftrim_index_cut_to_trim_resultsuffix)) * e)) /\ exists ff_q_pfp_cut_to_trim_resultsuffixoutput. d = ff_q_pfp_cut_to_trim_resultsuffixoutput * S ((S (pftrim_index_cut_to_trim_resultsuffix)) * e) + (pftrim_value_cut_to_trim_resultsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_cut_to_trim_resultnormal. ((((exists ff_h_pfp_cut_to_trim_resultnormalentry. ff_h_pfp_cut_to_trim_resultnormalentry + S (pftrim_leading_cut_to_trim_resultnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_cut_to_trim_resultnormalentry. d = ff_q_pfp_cut_to_trim_resultnormalentry * S ((S (0)) * e) + (pftrim_leading_cut_to_trim_resultnormal))) /\ ((~(pftrim_leading_cut_to_trim_resultnormal=0)))))))))))))))prime_field_polynomial_trim_existsExact checked statement
forall p b c L. (forall fom_index_pfp_trim_exists_source. (exists fom_gap_pfp_trim_exists_source_index_bound. fom_gap_pfp_trim_exists_source_index_bound + S (fom_index_pfp_trim_exists_source) = L) -> exists fom_value_pfp_trim_exists_source. ((((exists fom_beta_height_pfp_trim_exists_source_entry. fom_beta_height_pfp_trim_exists_source_entry + S (fom_value_pfp_trim_exists_source) = S ((S (fom_index_pfp_trim_exists_source)) * c)) /\ exists fom_beta_quotient_pfp_trim_exists_source_entry. b = fom_beta_quotient_pfp_trim_exists_source_entry * S ((S (fom_index_pfp_trim_exists_source)) * c) + (fom_value_pfp_trim_exists_source))) /\ (exists fom_gap_pfp_trim_exists_source_value_bound. fom_gap_pfp_trim_exists_source_value_bound + S (fom_value_pfp_trim_exists_source) = p))) -> exists t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_exists_resultinput. (exists fom_gap_pfp_trim_exists_resultinput_index_bound. fom_gap_pfp_trim_exists_resultinput_index_bound + S (fom_index_pfp_trim_exists_resultinput) = L) -> exists fom_value_pfp_trim_exists_resultinput. ((((exists fom_beta_height_pfp_trim_exists_resultinput_entry. fom_beta_height_pfp_trim_exists_resultinput_entry + S (fom_value_pfp_trim_exists_resultinput) = S ((S (fom_index_pfp_trim_exists_resultinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_exists_resultinput_entry. b = fom_beta_quotient_pfp_trim_exists_resultinput_entry * S ((S (fom_index_pfp_trim_exists_resultinput)) * c) + (fom_value_pfp_trim_exists_resultinput))) /\ (exists fom_gap_pfp_trim_exists_resultinput_value_bound. fom_gap_pfp_trim_exists_resultinput_value_bound + S (fom_value_pfp_trim_exists_resultinput) = p))) /\ (((forall pfp_repeat_index_trim_exists_resultremoved. (exists pfa_gap_trim_exists_resultremovedindex. pfa_gap_trim_exists_resultremovedindex + S (pfp_repeat_index_trim_exists_resultremoved) = (t)) -> (((exists ff_h_pfp_trim_exists_resultremovedentry. ff_h_pfp_trim_exists_resultremovedentry + S (0) = S ((S (pfp_repeat_index_trim_exists_resultremoved)) * c)) /\ exists ff_q_pfp_trim_exists_resultremovedentry. b = ff_q_pfp_trim_exists_resultremovedentry * S ((S (pfp_repeat_index_trim_exists_resultremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_exists_resultsuffix pftrim_value_trim_exists_resultsuffix. (exists pfa_gap_trim_exists_resultsuffixbound. pfa_gap_trim_exists_resultsuffixbound + S (pftrim_index_trim_exists_resultsuffix) = (M)) -> (((exists ff_h_pfp_trim_exists_resultsuffixsource. ff_h_pfp_trim_exists_resultsuffixsource + S (pftrim_value_trim_exists_resultsuffix) = S ((S ((t)+pftrim_index_trim_exists_resultsuffix)) * c)) /\ exists ff_q_pfp_trim_exists_resultsuffixsource. b = ff_q_pfp_trim_exists_resultsuffixsource * S ((S ((t)+pftrim_index_trim_exists_resultsuffix)) * c) + (pftrim_value_trim_exists_resultsuffix))) -> (((exists ff_h_pfp_trim_exists_resultsuffixoutput. ff_h_pfp_trim_exists_resultsuffixoutput + S (pftrim_value_trim_exists_resultsuffix) = S ((S (pftrim_index_trim_exists_resultsuffix)) * e)) /\ exists ff_q_pfp_trim_exists_resultsuffixoutput. d = ff_q_pfp_trim_exists_resultsuffixoutput * S ((S (pftrim_index_trim_exists_resultsuffix)) * e) + (pftrim_value_trim_exists_resultsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_exists_resultnormal. ((((exists ff_h_pfp_trim_exists_resultnormalentry. ff_h_pfp_trim_exists_resultnormalentry + S (pftrim_leading_trim_exists_resultnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_exists_resultnormalentry. d = ff_q_pfp_trim_exists_resultnormalentry * S ((S (0)) * e) + (pftrim_leading_trim_exists_resultnormal))) /\ ((~(pftrim_leading_trim_exists_resultnormal=0)))))))))))))))prime_field_polynomial_trim_empty_inputExact checked statement
forall p b c d e. ((((0)=(0)+(0)) /\ (((forall fom_index_pfp_empty_triminput. (exists fom_gap_pfp_empty_triminput_index_bound. fom_gap_pfp_empty_triminput_index_bound + S (fom_index_pfp_empty_triminput) = 0) -> exists fom_value_pfp_empty_triminput. ((((exists fom_beta_height_pfp_empty_triminput_entry. fom_beta_height_pfp_empty_triminput_entry + S (fom_value_pfp_empty_triminput) = S ((S (fom_index_pfp_empty_triminput)) * c)) /\ exists fom_beta_quotient_pfp_empty_triminput_entry. b = fom_beta_quotient_pfp_empty_triminput_entry * S ((S (fom_index_pfp_empty_triminput)) * c) + (fom_value_pfp_empty_triminput))) /\ (exists fom_gap_pfp_empty_triminput_value_bound. fom_gap_pfp_empty_triminput_value_bound + S (fom_value_pfp_empty_triminput) = p))) /\ (((forall pfp_repeat_index_empty_trimremoved. (exists pfa_gap_empty_trimremovedindex. pfa_gap_empty_trimremovedindex + S (pfp_repeat_index_empty_trimremoved) = (0)) -> (((exists ff_h_pfp_empty_trimremovedentry. ff_h_pfp_empty_trimremovedentry + S (0) = S ((S (pfp_repeat_index_empty_trimremoved)) * c)) /\ exists ff_q_pfp_empty_trimremovedentry. b = ff_q_pfp_empty_trimremovedentry * S ((S (pfp_repeat_index_empty_trimremoved)) * c) + (0)))) /\ (((forall pftrim_index_empty_trimsuffix pftrim_value_empty_trimsuffix. (exists pfa_gap_empty_trimsuffixbound. pfa_gap_empty_trimsuffixbound + S (pftrim_index_empty_trimsuffix) = (0)) -> (((exists ff_h_pfp_empty_trimsuffixsource. ff_h_pfp_empty_trimsuffixsource + S (pftrim_value_empty_trimsuffix) = S ((S ((0)+pftrim_index_empty_trimsuffix)) * c)) /\ exists ff_q_pfp_empty_trimsuffixsource. b = ff_q_pfp_empty_trimsuffixsource * S ((S ((0)+pftrim_index_empty_trimsuffix)) * c) + (pftrim_value_empty_trimsuffix))) -> (((exists ff_h_pfp_empty_trimsuffixoutput. ff_h_pfp_empty_trimsuffixoutput + S (pftrim_value_empty_trimsuffix) = S ((S (pftrim_index_empty_trimsuffix)) * e)) /\ exists ff_q_pfp_empty_trimsuffixoutput. d = ff_q_pfp_empty_trimsuffixoutput * S ((S (pftrim_index_empty_trimsuffix)) * e) + (pftrim_value_empty_trimsuffix)))) /\ (((0)=0 \/ (exists pftrim_leading_empty_trimnormal. ((((exists ff_h_pfp_empty_trimnormalentry. ff_h_pfp_empty_trimnormalentry + S (pftrim_leading_empty_trimnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_empty_trimnormalentry. d = ff_q_pfp_empty_trimnormalentry * S ((S (0)) * e) + (pftrim_leading_empty_trimnormal))) /\ ((~(pftrim_leading_empty_trimnormal=0)))))))))))))))prime_field_polynomial_trim_output_coefficientsExact checked statement
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_bounded_sourceinput. (exists fom_gap_pfp_trim_bounded_sourceinput_index_bound. fom_gap_pfp_trim_bounded_sourceinput_index_bound + S (fom_index_pfp_trim_bounded_sourceinput) = L) -> exists fom_value_pfp_trim_bounded_sourceinput. ((((exists fom_beta_height_pfp_trim_bounded_sourceinput_entry. fom_beta_height_pfp_trim_bounded_sourceinput_entry + S (fom_value_pfp_trim_bounded_sourceinput) = S ((S (fom_index_pfp_trim_bounded_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_bounded_sourceinput_entry. b = fom_beta_quotient_pfp_trim_bounded_sourceinput_entry * S ((S (fom_index_pfp_trim_bounded_sourceinput)) * c) + (fom_value_pfp_trim_bounded_sourceinput))) /\ (exists fom_gap_pfp_trim_bounded_sourceinput_value_bound. fom_gap_pfp_trim_bounded_sourceinput_value_bound + S (fom_value_pfp_trim_bounded_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_bounded_sourceremoved. (exists pfa_gap_trim_bounded_sourceremovedindex. pfa_gap_trim_bounded_sourceremovedindex + S (pfp_repeat_index_trim_bounded_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_bounded_sourceremovedentry. ff_h_pfp_trim_bounded_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_bounded_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_bounded_sourceremovedentry. b = ff_q_pfp_trim_bounded_sourceremovedentry * S ((S (pfp_repeat_index_trim_bounded_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_bounded_sourcesuffix pftrim_value_trim_bounded_sourcesuffix. (exists pfa_gap_trim_bounded_sourcesuffixbound. pfa_gap_trim_bounded_sourcesuffixbound + S (pftrim_index_trim_bounded_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_bounded_sourcesuffixsource. ff_h_pfp_trim_bounded_sourcesuffixsource + S (pftrim_value_trim_bounded_sourcesuffix) = S ((S ((t)+pftrim_index_trim_bounded_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_bounded_sourcesuffixsource. b = ff_q_pfp_trim_bounded_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_bounded_sourcesuffix)) * c) + (pftrim_value_trim_bounded_sourcesuffix))) -> (((exists ff_h_pfp_trim_bounded_sourcesuffixoutput. ff_h_pfp_trim_bounded_sourcesuffixoutput + S (pftrim_value_trim_bounded_sourcesuffix) = S ((S (pftrim_index_trim_bounded_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_bounded_sourcesuffixoutput. d = ff_q_pfp_trim_bounded_sourcesuffixoutput * S ((S (pftrim_index_trim_bounded_sourcesuffix)) * e) + (pftrim_value_trim_bounded_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_bounded_sourcenormal. ((((exists ff_h_pfp_trim_bounded_sourcenormalentry. ff_h_pfp_trim_bounded_sourcenormalentry + S (pftrim_leading_trim_bounded_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_bounded_sourcenormalentry. d = ff_q_pfp_trim_bounded_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_bounded_sourcenormal))) /\ ((~(pftrim_leading_trim_bounded_sourcenormal=0))))))))))))))) -> (forall fom_index_pfp_trim_bounded_result. (exists fom_gap_pfp_trim_bounded_result_index_bound. fom_gap_pfp_trim_bounded_result_index_bound + S (fom_index_pfp_trim_bounded_result) = M) -> exists fom_value_pfp_trim_bounded_result. ((((exists fom_beta_height_pfp_trim_bounded_result_entry. fom_beta_height_pfp_trim_bounded_result_entry + S (fom_value_pfp_trim_bounded_result) = S ((S (fom_index_pfp_trim_bounded_result)) * e)) /\ exists fom_beta_quotient_pfp_trim_bounded_result_entry. d = fom_beta_quotient_pfp_trim_bounded_result_entry * S ((S (fom_index_pfp_trim_bounded_result)) * e) + (fom_value_pfp_trim_bounded_result))) /\ (exists fom_gap_pfp_trim_bounded_result_value_bound. fom_gap_pfp_trim_bounded_result_value_bound + S (fom_value_pfp_trim_bounded_result) = p)))prime_field_polynomial_trim_length_boundsExact checked statement
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_length_sourceinput. (exists fom_gap_pfp_trim_length_sourceinput_index_bound. fom_gap_pfp_trim_length_sourceinput_index_bound + S (fom_index_pfp_trim_length_sourceinput) = L) -> exists fom_value_pfp_trim_length_sourceinput. ((((exists fom_beta_height_pfp_trim_length_sourceinput_entry. fom_beta_height_pfp_trim_length_sourceinput_entry + S (fom_value_pfp_trim_length_sourceinput) = S ((S (fom_index_pfp_trim_length_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_length_sourceinput_entry. b = fom_beta_quotient_pfp_trim_length_sourceinput_entry * S ((S (fom_index_pfp_trim_length_sourceinput)) * c) + (fom_value_pfp_trim_length_sourceinput))) /\ (exists fom_gap_pfp_trim_length_sourceinput_value_bound. fom_gap_pfp_trim_length_sourceinput_value_bound + S (fom_value_pfp_trim_length_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_length_sourceremoved. (exists pfa_gap_trim_length_sourceremovedindex. pfa_gap_trim_length_sourceremovedindex + S (pfp_repeat_index_trim_length_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_length_sourceremovedentry. ff_h_pfp_trim_length_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_length_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_length_sourceremovedentry. b = ff_q_pfp_trim_length_sourceremovedentry * S ((S (pfp_repeat_index_trim_length_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_length_sourcesuffix pftrim_value_trim_length_sourcesuffix. (exists pfa_gap_trim_length_sourcesuffixbound. pfa_gap_trim_length_sourcesuffixbound + S (pftrim_index_trim_length_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_length_sourcesuffixsource. ff_h_pfp_trim_length_sourcesuffixsource + S (pftrim_value_trim_length_sourcesuffix) = S ((S ((t)+pftrim_index_trim_length_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_length_sourcesuffixsource. b = ff_q_pfp_trim_length_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_length_sourcesuffix)) * c) + (pftrim_value_trim_length_sourcesuffix))) -> (((exists ff_h_pfp_trim_length_sourcesuffixoutput. ff_h_pfp_trim_length_sourcesuffixoutput + S (pftrim_value_trim_length_sourcesuffix) = S ((S (pftrim_index_trim_length_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_length_sourcesuffixoutput. d = ff_q_pfp_trim_length_sourcesuffixoutput * S ((S (pftrim_index_trim_length_sourcesuffix)) * e) + (pftrim_value_trim_length_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_length_sourcenormal. ((((exists ff_h_pfp_trim_length_sourcenormalentry. ff_h_pfp_trim_length_sourcenormalentry + S (pftrim_leading_trim_length_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_length_sourcenormalentry. d = ff_q_pfp_trim_length_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_length_sourcenormal))) /\ ((~(pftrim_leading_trim_length_sourcenormal=0))))))))))))))) -> ((exists pftrim_gap_removed_bound. pftrim_gap_removed_bound+(t)=(L)) /\ ((exists pftrim_gap_retained_bound. pftrim_gap_retained_bound+(M)=(L))))prime_field_polynomial_trim_leading_source_nonzeroExact checked statement
forall p b c L t d e M a. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_nonzero_sourceinput. (exists fom_gap_pfp_trim_nonzero_sourceinput_index_bound. fom_gap_pfp_trim_nonzero_sourceinput_index_bound + S (fom_index_pfp_trim_nonzero_sourceinput) = L) -> exists fom_value_pfp_trim_nonzero_sourceinput. ((((exists fom_beta_height_pfp_trim_nonzero_sourceinput_entry. fom_beta_height_pfp_trim_nonzero_sourceinput_entry + S (fom_value_pfp_trim_nonzero_sourceinput) = S ((S (fom_index_pfp_trim_nonzero_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_nonzero_sourceinput_entry. b = fom_beta_quotient_pfp_trim_nonzero_sourceinput_entry * S ((S (fom_index_pfp_trim_nonzero_sourceinput)) * c) + (fom_value_pfp_trim_nonzero_sourceinput))) /\ (exists fom_gap_pfp_trim_nonzero_sourceinput_value_bound. fom_gap_pfp_trim_nonzero_sourceinput_value_bound + S (fom_value_pfp_trim_nonzero_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_nonzero_sourceremoved. (exists pfa_gap_trim_nonzero_sourceremovedindex. pfa_gap_trim_nonzero_sourceremovedindex + S (pfp_repeat_index_trim_nonzero_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_nonzero_sourceremovedentry. ff_h_pfp_trim_nonzero_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_nonzero_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_nonzero_sourceremovedentry. b = ff_q_pfp_trim_nonzero_sourceremovedentry * S ((S (pfp_repeat_index_trim_nonzero_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_nonzero_sourcesuffix pftrim_value_trim_nonzero_sourcesuffix. (exists pfa_gap_trim_nonzero_sourcesuffixbound. pfa_gap_trim_nonzero_sourcesuffixbound + S (pftrim_index_trim_nonzero_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_nonzero_sourcesuffixsource. ff_h_pfp_trim_nonzero_sourcesuffixsource + S (pftrim_value_trim_nonzero_sourcesuffix) = S ((S ((t)+pftrim_index_trim_nonzero_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_nonzero_sourcesuffixsource. b = ff_q_pfp_trim_nonzero_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_nonzero_sourcesuffix)) * c) + (pftrim_value_trim_nonzero_sourcesuffix))) -> (((exists ff_h_pfp_trim_nonzero_sourcesuffixoutput. ff_h_pfp_trim_nonzero_sourcesuffixoutput + S (pftrim_value_trim_nonzero_sourcesuffix) = S ((S (pftrim_index_trim_nonzero_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_nonzero_sourcesuffixoutput. d = ff_q_pfp_trim_nonzero_sourcesuffixoutput * S ((S (pftrim_index_trim_nonzero_sourcesuffix)) * e) + (pftrim_value_trim_nonzero_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_nonzero_sourcenormal. ((((exists ff_h_pfp_trim_nonzero_sourcenormalentry. ff_h_pfp_trim_nonzero_sourcenormalentry + S (pftrim_leading_trim_nonzero_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_nonzero_sourcenormalentry. d = ff_q_pfp_trim_nonzero_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_nonzero_sourcenormal))) /\ ((~(pftrim_leading_trim_nonzero_sourcenormal=0))))))))))))))) -> ~(M=0) -> (((exists ff_h_pfp_trim_leading_input. ff_h_pfp_trim_leading_input + S (a) = S ((S (t)) * c)) /\ exists ff_q_pfp_trim_leading_input. b = ff_q_pfp_trim_leading_input * S ((S (t)) * c) + (a))) -> ~(a=0)prime_field_polynomial_trim_zero_of_emptyExact checked statement
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_empty_sourceinput. (exists fom_gap_pfp_trim_empty_sourceinput_index_bound. fom_gap_pfp_trim_empty_sourceinput_index_bound + S (fom_index_pfp_trim_empty_sourceinput) = L) -> exists fom_value_pfp_trim_empty_sourceinput. ((((exists fom_beta_height_pfp_trim_empty_sourceinput_entry. fom_beta_height_pfp_trim_empty_sourceinput_entry + S (fom_value_pfp_trim_empty_sourceinput) = S ((S (fom_index_pfp_trim_empty_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_empty_sourceinput_entry. b = fom_beta_quotient_pfp_trim_empty_sourceinput_entry * S ((S (fom_index_pfp_trim_empty_sourceinput)) * c) + (fom_value_pfp_trim_empty_sourceinput))) /\ (exists fom_gap_pfp_trim_empty_sourceinput_value_bound. fom_gap_pfp_trim_empty_sourceinput_value_bound + S (fom_value_pfp_trim_empty_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_empty_sourceremoved. (exists pfa_gap_trim_empty_sourceremovedindex. pfa_gap_trim_empty_sourceremovedindex + S (pfp_repeat_index_trim_empty_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_empty_sourceremovedentry. ff_h_pfp_trim_empty_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_empty_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_empty_sourceremovedentry. b = ff_q_pfp_trim_empty_sourceremovedentry * S ((S (pfp_repeat_index_trim_empty_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_empty_sourcesuffix pftrim_value_trim_empty_sourcesuffix. (exists pfa_gap_trim_empty_sourcesuffixbound. pfa_gap_trim_empty_sourcesuffixbound + S (pftrim_index_trim_empty_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_empty_sourcesuffixsource. ff_h_pfp_trim_empty_sourcesuffixsource + S (pftrim_value_trim_empty_sourcesuffix) = S ((S ((t)+pftrim_index_trim_empty_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_empty_sourcesuffixsource. b = ff_q_pfp_trim_empty_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_empty_sourcesuffix)) * c) + (pftrim_value_trim_empty_sourcesuffix))) -> (((exists ff_h_pfp_trim_empty_sourcesuffixoutput. ff_h_pfp_trim_empty_sourcesuffixoutput + S (pftrim_value_trim_empty_sourcesuffix) = S ((S (pftrim_index_trim_empty_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_empty_sourcesuffixoutput. d = ff_q_pfp_trim_empty_sourcesuffixoutput * S ((S (pftrim_index_trim_empty_sourcesuffix)) * e) + (pftrim_value_trim_empty_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_empty_sourcenormal. ((((exists ff_h_pfp_trim_empty_sourcenormalentry. ff_h_pfp_trim_empty_sourcenormalentry + S (pftrim_leading_trim_empty_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_empty_sourcenormalentry. d = ff_q_pfp_trim_empty_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_empty_sourcenormal))) /\ ((~(pftrim_leading_trim_empty_sourcenormal=0))))))))))))))) -> M=0 -> (forall pfp_repeat_index_trim_zero_input. (exists pfa_gap_trim_zero_inputindex. pfa_gap_trim_zero_inputindex + S (pfp_repeat_index_trim_zero_input) = (L)) -> (((exists ff_h_pfp_trim_zero_inputentry. ff_h_pfp_trim_zero_inputentry + S (0) = S ((S (pfp_repeat_index_trim_zero_input)) * c)) /\ exists ff_q_pfp_trim_zero_inputentry. b = ff_q_pfp_trim_zero_inputentry * S ((S (pfp_repeat_index_trim_zero_input)) * c) + (0))))prime_field_polynomial_trim_empty_of_zeroExact checked statement
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_all_zeroinput. (exists fom_gap_pfp_trim_all_zeroinput_index_bound. fom_gap_pfp_trim_all_zeroinput_index_bound + S (fom_index_pfp_trim_all_zeroinput) = L) -> exists fom_value_pfp_trim_all_zeroinput. ((((exists fom_beta_height_pfp_trim_all_zeroinput_entry. fom_beta_height_pfp_trim_all_zeroinput_entry + S (fom_value_pfp_trim_all_zeroinput) = S ((S (fom_index_pfp_trim_all_zeroinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_all_zeroinput_entry. b = fom_beta_quotient_pfp_trim_all_zeroinput_entry * S ((S (fom_index_pfp_trim_all_zeroinput)) * c) + (fom_value_pfp_trim_all_zeroinput))) /\ (exists fom_gap_pfp_trim_all_zeroinput_value_bound. fom_gap_pfp_trim_all_zeroinput_value_bound + S (fom_value_pfp_trim_all_zeroinput) = p))) /\ (((forall pfp_repeat_index_trim_all_zeroremoved. (exists pfa_gap_trim_all_zeroremovedindex. pfa_gap_trim_all_zeroremovedindex + S (pfp_repeat_index_trim_all_zeroremoved) = (t)) -> (((exists ff_h_pfp_trim_all_zeroremovedentry. ff_h_pfp_trim_all_zeroremovedentry + S (0) = S ((S (pfp_repeat_index_trim_all_zeroremoved)) * c)) /\ exists ff_q_pfp_trim_all_zeroremovedentry. b = ff_q_pfp_trim_all_zeroremovedentry * S ((S (pfp_repeat_index_trim_all_zeroremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_all_zerosuffix pftrim_value_trim_all_zerosuffix. (exists pfa_gap_trim_all_zerosuffixbound. pfa_gap_trim_all_zerosuffixbound + S (pftrim_index_trim_all_zerosuffix) = (M)) -> (((exists ff_h_pfp_trim_all_zerosuffixsource. ff_h_pfp_trim_all_zerosuffixsource + S (pftrim_value_trim_all_zerosuffix) = S ((S ((t)+pftrim_index_trim_all_zerosuffix)) * c)) /\ exists ff_q_pfp_trim_all_zerosuffixsource. b = ff_q_pfp_trim_all_zerosuffixsource * S ((S ((t)+pftrim_index_trim_all_zerosuffix)) * c) + (pftrim_value_trim_all_zerosuffix))) -> (((exists ff_h_pfp_trim_all_zerosuffixoutput. ff_h_pfp_trim_all_zerosuffixoutput + S (pftrim_value_trim_all_zerosuffix) = S ((S (pftrim_index_trim_all_zerosuffix)) * e)) /\ exists ff_q_pfp_trim_all_zerosuffixoutput. d = ff_q_pfp_trim_all_zerosuffixoutput * S ((S (pftrim_index_trim_all_zerosuffix)) * e) + (pftrim_value_trim_all_zerosuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_all_zeronormal. ((((exists ff_h_pfp_trim_all_zeronormalentry. ff_h_pfp_trim_all_zeronormalentry + S (pftrim_leading_trim_all_zeronormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_all_zeronormalentry. d = ff_q_pfp_trim_all_zeronormalentry * S ((S (0)) * e) + (pftrim_leading_trim_all_zeronormal))) /\ ((~(pftrim_leading_trim_all_zeronormal=0))))))))))))))) -> (forall pfp_repeat_index_trim_all_zero_input. (exists pfa_gap_trim_all_zero_inputindex. pfa_gap_trim_all_zero_inputindex + S (pfp_repeat_index_trim_all_zero_input) = (L)) -> (((exists ff_h_pfp_trim_all_zero_inputentry. ff_h_pfp_trim_all_zero_inputentry + S (0) = S ((S (pfp_repeat_index_trim_all_zero_input)) * c)) /\ exists ff_q_pfp_trim_all_zero_inputentry. b = ff_q_pfp_trim_all_zero_inputentry * S ((S (pfp_repeat_index_trim_all_zero_input)) * c) + (0)))) -> M=0prime_field_polynomial_trim_zero_iffExact checked statement
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_zero_equivalenceinput. (exists fom_gap_pfp_trim_zero_equivalenceinput_index_bound. fom_gap_pfp_trim_zero_equivalenceinput_index_bound + S (fom_index_pfp_trim_zero_equivalenceinput) = L) -> exists fom_value_pfp_trim_zero_equivalenceinput. ((((exists fom_beta_height_pfp_trim_zero_equivalenceinput_entry. fom_beta_height_pfp_trim_zero_equivalenceinput_entry + S (fom_value_pfp_trim_zero_equivalenceinput) = S ((S (fom_index_pfp_trim_zero_equivalenceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_zero_equivalenceinput_entry. b = fom_beta_quotient_pfp_trim_zero_equivalenceinput_entry * S ((S (fom_index_pfp_trim_zero_equivalenceinput)) * c) + (fom_value_pfp_trim_zero_equivalenceinput))) /\ (exists fom_gap_pfp_trim_zero_equivalenceinput_value_bound. fom_gap_pfp_trim_zero_equivalenceinput_value_bound + S (fom_value_pfp_trim_zero_equivalenceinput) = p))) /\ (((forall pfp_repeat_index_trim_zero_equivalenceremoved. (exists pfa_gap_trim_zero_equivalenceremovedindex. pfa_gap_trim_zero_equivalenceremovedindex + S (pfp_repeat_index_trim_zero_equivalenceremoved) = (t)) -> (((exists ff_h_pfp_trim_zero_equivalenceremovedentry. ff_h_pfp_trim_zero_equivalenceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_zero_equivalenceremoved)) * c)) /\ exists ff_q_pfp_trim_zero_equivalenceremovedentry. b = ff_q_pfp_trim_zero_equivalenceremovedentry * S ((S (pfp_repeat_index_trim_zero_equivalenceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_zero_equivalencesuffix pftrim_value_trim_zero_equivalencesuffix. (exists pfa_gap_trim_zero_equivalencesuffixbound. pfa_gap_trim_zero_equivalencesuffixbound + S (pftrim_index_trim_zero_equivalencesuffix) = (M)) -> (((exists ff_h_pfp_trim_zero_equivalencesuffixsource. ff_h_pfp_trim_zero_equivalencesuffixsource + S (pftrim_value_trim_zero_equivalencesuffix) = S ((S ((t)+pftrim_index_trim_zero_equivalencesuffix)) * c)) /\ exists ff_q_pfp_trim_zero_equivalencesuffixsource. b = ff_q_pfp_trim_zero_equivalencesuffixsource * S ((S ((t)+pftrim_index_trim_zero_equivalencesuffix)) * c) + (pftrim_value_trim_zero_equivalencesuffix))) -> (((exists ff_h_pfp_trim_zero_equivalencesuffixoutput. ff_h_pfp_trim_zero_equivalencesuffixoutput + S (pftrim_value_trim_zero_equivalencesuffix) = S ((S (pftrim_index_trim_zero_equivalencesuffix)) * e)) /\ exists ff_q_pfp_trim_zero_equivalencesuffixoutput. d = ff_q_pfp_trim_zero_equivalencesuffixoutput * S ((S (pftrim_index_trim_zero_equivalencesuffix)) * e) + (pftrim_value_trim_zero_equivalencesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_zero_equivalencenormal. ((((exists ff_h_pfp_trim_zero_equivalencenormalentry. ff_h_pfp_trim_zero_equivalencenormalentry + S (pftrim_leading_trim_zero_equivalencenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_zero_equivalencenormalentry. d = ff_q_pfp_trim_zero_equivalencenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_zero_equivalencenormal))) /\ ((~(pftrim_leading_trim_zero_equivalencenormal=0))))))))))))))) -> ((M=0 -> (forall pfp_repeat_index_trim_zero_forward. (exists pfa_gap_trim_zero_forwardindex. pfa_gap_trim_zero_forwardindex + S (pfp_repeat_index_trim_zero_forward) = (L)) -> (((exists ff_h_pfp_trim_zero_forwardentry. ff_h_pfp_trim_zero_forwardentry + S (0) = S ((S (pfp_repeat_index_trim_zero_forward)) * c)) /\ exists ff_q_pfp_trim_zero_forwardentry. b = ff_q_pfp_trim_zero_forwardentry * S ((S (pfp_repeat_index_trim_zero_forward)) * c) + (0))))) /\ (((forall pfp_repeat_index_trim_zero_backward. (exists pfa_gap_trim_zero_backwardindex. pfa_gap_trim_zero_backwardindex + S (pfp_repeat_index_trim_zero_backward) = (L)) -> (((exists ff_h_pfp_trim_zero_backwardentry. ff_h_pfp_trim_zero_backwardentry + S (0) = S ((S (pfp_repeat_index_trim_zero_backward)) * c)) /\ exists ff_q_pfp_trim_zero_backwardentry. b = ff_q_pfp_trim_zero_backwardentry * S ((S (pfp_repeat_index_trim_zero_backward)) * c) + (0)))) -> M=0)))prime_field_polynomial_trim_removed_leExact checked statement
forall p b c L t d e M u f g N. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_unique_firstinput. (exists fom_gap_pfp_unique_firstinput_index_bound. fom_gap_pfp_unique_firstinput_index_bound + S (fom_index_pfp_unique_firstinput) = L) -> exists fom_value_pfp_unique_firstinput. ((((exists fom_beta_height_pfp_unique_firstinput_entry. fom_beta_height_pfp_unique_firstinput_entry + S (fom_value_pfp_unique_firstinput) = S ((S (fom_index_pfp_unique_firstinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_firstinput_entry. b = fom_beta_quotient_pfp_unique_firstinput_entry * S ((S (fom_index_pfp_unique_firstinput)) * c) + (fom_value_pfp_unique_firstinput))) /\ (exists fom_gap_pfp_unique_firstinput_value_bound. fom_gap_pfp_unique_firstinput_value_bound + S (fom_value_pfp_unique_firstinput) = p))) /\ (((forall pfp_repeat_index_unique_firstremoved. (exists pfa_gap_unique_firstremovedindex. pfa_gap_unique_firstremovedindex + S (pfp_repeat_index_unique_firstremoved) = (t)) -> (((exists ff_h_pfp_unique_firstremovedentry. ff_h_pfp_unique_firstremovedentry + S (0) = S ((S (pfp_repeat_index_unique_firstremoved)) * c)) /\ exists ff_q_pfp_unique_firstremovedentry. b = ff_q_pfp_unique_firstremovedentry * S ((S (pfp_repeat_index_unique_firstremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_firstsuffix pftrim_value_unique_firstsuffix. (exists pfa_gap_unique_firstsuffixbound. pfa_gap_unique_firstsuffixbound + S (pftrim_index_unique_firstsuffix) = (M)) -> (((exists ff_h_pfp_unique_firstsuffixsource. ff_h_pfp_unique_firstsuffixsource + S (pftrim_value_unique_firstsuffix) = S ((S ((t)+pftrim_index_unique_firstsuffix)) * c)) /\ exists ff_q_pfp_unique_firstsuffixsource. b = ff_q_pfp_unique_firstsuffixsource * S ((S ((t)+pftrim_index_unique_firstsuffix)) * c) + (pftrim_value_unique_firstsuffix))) -> (((exists ff_h_pfp_unique_firstsuffixoutput. ff_h_pfp_unique_firstsuffixoutput + S (pftrim_value_unique_firstsuffix) = S ((S (pftrim_index_unique_firstsuffix)) * e)) /\ exists ff_q_pfp_unique_firstsuffixoutput. d = ff_q_pfp_unique_firstsuffixoutput * S ((S (pftrim_index_unique_firstsuffix)) * e) + (pftrim_value_unique_firstsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_unique_firstnormal. ((((exists ff_h_pfp_unique_firstnormalentry. ff_h_pfp_unique_firstnormalentry + S (pftrim_leading_unique_firstnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_unique_firstnormalentry. d = ff_q_pfp_unique_firstnormalentry * S ((S (0)) * e) + (pftrim_leading_unique_firstnormal))) /\ ((~(pftrim_leading_unique_firstnormal=0))))))))))))))) -> ((((L)=(u)+(N)) /\ (((forall fom_index_pfp_unique_secondinput. (exists fom_gap_pfp_unique_secondinput_index_bound. fom_gap_pfp_unique_secondinput_index_bound + S (fom_index_pfp_unique_secondinput) = L) -> exists fom_value_pfp_unique_secondinput. ((((exists fom_beta_height_pfp_unique_secondinput_entry. fom_beta_height_pfp_unique_secondinput_entry + S (fom_value_pfp_unique_secondinput) = S ((S (fom_index_pfp_unique_secondinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_secondinput_entry. b = fom_beta_quotient_pfp_unique_secondinput_entry * S ((S (fom_index_pfp_unique_secondinput)) * c) + (fom_value_pfp_unique_secondinput))) /\ (exists fom_gap_pfp_unique_secondinput_value_bound. fom_gap_pfp_unique_secondinput_value_bound + S (fom_value_pfp_unique_secondinput) = p))) /\ (((forall pfp_repeat_index_unique_secondremoved. (exists pfa_gap_unique_secondremovedindex. pfa_gap_unique_secondremovedindex + S (pfp_repeat_index_unique_secondremoved) = (u)) -> (((exists ff_h_pfp_unique_secondremovedentry. ff_h_pfp_unique_secondremovedentry + S (0) = S ((S (pfp_repeat_index_unique_secondremoved)) * c)) /\ exists ff_q_pfp_unique_secondremovedentry. b = ff_q_pfp_unique_secondremovedentry * S ((S (pfp_repeat_index_unique_secondremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_secondsuffix pftrim_value_unique_secondsuffix. (exists pfa_gap_unique_secondsuffixbound. pfa_gap_unique_secondsuffixbound + S (pftrim_index_unique_secondsuffix) = (N)) -> (((exists ff_h_pfp_unique_secondsuffixsource. ff_h_pfp_unique_secondsuffixsource + S (pftrim_value_unique_secondsuffix) = S ((S ((u)+pftrim_index_unique_secondsuffix)) * c)) /\ exists ff_q_pfp_unique_secondsuffixsource. b = ff_q_pfp_unique_secondsuffixsource * S ((S ((u)+pftrim_index_unique_secondsuffix)) * c) + (pftrim_value_unique_secondsuffix))) -> (((exists ff_h_pfp_unique_secondsuffixoutput. ff_h_pfp_unique_secondsuffixoutput + S (pftrim_value_unique_secondsuffix) = S ((S (pftrim_index_unique_secondsuffix)) * g)) /\ exists ff_q_pfp_unique_secondsuffixoutput. f = ff_q_pfp_unique_secondsuffixoutput * S ((S (pftrim_index_unique_secondsuffix)) * g) + (pftrim_value_unique_secondsuffix)))) /\ (((N)=0 \/ (exists pftrim_leading_unique_secondnormal. ((((exists ff_h_pfp_unique_secondnormalentry. ff_h_pfp_unique_secondnormalentry + S (pftrim_leading_unique_secondnormal) = S ((S (0)) * g)) /\ exists ff_q_pfp_unique_secondnormalentry. f = ff_q_pfp_unique_secondnormalentry * S ((S (0)) * g) + (pftrim_leading_unique_secondnormal))) /\ ((~(pftrim_leading_unique_secondnormal=0))))))))))))))) -> (exists pftrim_gap_unique_removed_le. pftrim_gap_unique_removed_le+(t)=(u))prime_field_polynomial_trim_removed_count_uniqueExact checked statement
forall p b c L t d e M u f g N. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_unique_firstinput. (exists fom_gap_pfp_unique_firstinput_index_bound. fom_gap_pfp_unique_firstinput_index_bound + S (fom_index_pfp_unique_firstinput) = L) -> exists fom_value_pfp_unique_firstinput. ((((exists fom_beta_height_pfp_unique_firstinput_entry. fom_beta_height_pfp_unique_firstinput_entry + S (fom_value_pfp_unique_firstinput) = S ((S (fom_index_pfp_unique_firstinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_firstinput_entry. b = fom_beta_quotient_pfp_unique_firstinput_entry * S ((S (fom_index_pfp_unique_firstinput)) * c) + (fom_value_pfp_unique_firstinput))) /\ (exists fom_gap_pfp_unique_firstinput_value_bound. fom_gap_pfp_unique_firstinput_value_bound + S (fom_value_pfp_unique_firstinput) = p))) /\ (((forall pfp_repeat_index_unique_firstremoved. (exists pfa_gap_unique_firstremovedindex. pfa_gap_unique_firstremovedindex + S (pfp_repeat_index_unique_firstremoved) = (t)) -> (((exists ff_h_pfp_unique_firstremovedentry. ff_h_pfp_unique_firstremovedentry + S (0) = S ((S (pfp_repeat_index_unique_firstremoved)) * c)) /\ exists ff_q_pfp_unique_firstremovedentry. b = ff_q_pfp_unique_firstremovedentry * S ((S (pfp_repeat_index_unique_firstremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_firstsuffix pftrim_value_unique_firstsuffix. (exists pfa_gap_unique_firstsuffixbound. pfa_gap_unique_firstsuffixbound + S (pftrim_index_unique_firstsuffix) = (M)) -> (((exists ff_h_pfp_unique_firstsuffixsource. ff_h_pfp_unique_firstsuffixsource + S (pftrim_value_unique_firstsuffix) = S ((S ((t)+pftrim_index_unique_firstsuffix)) * c)) /\ exists ff_q_pfp_unique_firstsuffixsource. b = ff_q_pfp_unique_firstsuffixsource * S ((S ((t)+pftrim_index_unique_firstsuffix)) * c) + (pftrim_value_unique_firstsuffix))) -> (((exists ff_h_pfp_unique_firstsuffixoutput. ff_h_pfp_unique_firstsuffixoutput + S (pftrim_value_unique_firstsuffix) = S ((S (pftrim_index_unique_firstsuffix)) * e)) /\ exists ff_q_pfp_unique_firstsuffixoutput. d = ff_q_pfp_unique_firstsuffixoutput * S ((S (pftrim_index_unique_firstsuffix)) * e) + (pftrim_value_unique_firstsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_unique_firstnormal. ((((exists ff_h_pfp_unique_firstnormalentry. ff_h_pfp_unique_firstnormalentry + S (pftrim_leading_unique_firstnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_unique_firstnormalentry. d = ff_q_pfp_unique_firstnormalentry * S ((S (0)) * e) + (pftrim_leading_unique_firstnormal))) /\ ((~(pftrim_leading_unique_firstnormal=0))))))))))))))) -> ((((L)=(u)+(N)) /\ (((forall fom_index_pfp_unique_secondinput. (exists fom_gap_pfp_unique_secondinput_index_bound. fom_gap_pfp_unique_secondinput_index_bound + S (fom_index_pfp_unique_secondinput) = L) -> exists fom_value_pfp_unique_secondinput. ((((exists fom_beta_height_pfp_unique_secondinput_entry. fom_beta_height_pfp_unique_secondinput_entry + S (fom_value_pfp_unique_secondinput) = S ((S (fom_index_pfp_unique_secondinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_secondinput_entry. b = fom_beta_quotient_pfp_unique_secondinput_entry * S ((S (fom_index_pfp_unique_secondinput)) * c) + (fom_value_pfp_unique_secondinput))) /\ (exists fom_gap_pfp_unique_secondinput_value_bound. fom_gap_pfp_unique_secondinput_value_bound + S (fom_value_pfp_unique_secondinput) = p))) /\ (((forall pfp_repeat_index_unique_secondremoved. (exists pfa_gap_unique_secondremovedindex. pfa_gap_unique_secondremovedindex + S (pfp_repeat_index_unique_secondremoved) = (u)) -> (((exists ff_h_pfp_unique_secondremovedentry. ff_h_pfp_unique_secondremovedentry + S (0) = S ((S (pfp_repeat_index_unique_secondremoved)) * c)) /\ exists ff_q_pfp_unique_secondremovedentry. b = ff_q_pfp_unique_secondremovedentry * S ((S (pfp_repeat_index_unique_secondremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_secondsuffix pftrim_value_unique_secondsuffix. (exists pfa_gap_unique_secondsuffixbound. pfa_gap_unique_secondsuffixbound + S (pftrim_index_unique_secondsuffix) = (N)) -> (((exists ff_h_pfp_unique_secondsuffixsource. ff_h_pfp_unique_secondsuffixsource + S (pftrim_value_unique_secondsuffix) = S ((S ((u)+pftrim_index_unique_secondsuffix)) * c)) /\ exists ff_q_pfp_unique_secondsuffixsource. b = ff_q_pfp_unique_secondsuffixsource * S ((S ((u)+pftrim_index_unique_secondsuffix)) * c) + (pftrim_value_unique_secondsuffix))) -> (((exists ff_h_pfp_unique_secondsuffixoutput. ff_h_pfp_unique_secondsuffixoutput + S (pftrim_value_unique_secondsuffix) = S ((S (pftrim_index_unique_secondsuffix)) * g)) /\ exists ff_q_pfp_unique_secondsuffixoutput. f = ff_q_pfp_unique_secondsuffixoutput * S ((S (pftrim_index_unique_secondsuffix)) * g) + (pftrim_value_unique_secondsuffix)))) /\ (((N)=0 \/ (exists pftrim_leading_unique_secondnormal. ((((exists ff_h_pfp_unique_secondnormalentry. ff_h_pfp_unique_secondnormalentry + S (pftrim_leading_unique_secondnormal) = S ((S (0)) * g)) /\ exists ff_q_pfp_unique_secondnormalentry. f = ff_q_pfp_unique_secondnormalentry * S ((S (0)) * g) + (pftrim_leading_unique_secondnormal))) /\ ((~(pftrim_leading_unique_secondnormal=0))))))))))))))) -> t=uprime_field_polynomial_trim_retained_length_uniqueExact checked statement
forall p b c L t d e M u f g N. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_unique_firstinput. (exists fom_gap_pfp_unique_firstinput_index_bound. fom_gap_pfp_unique_firstinput_index_bound + S (fom_index_pfp_unique_firstinput) = L) -> exists fom_value_pfp_unique_firstinput. ((((exists fom_beta_height_pfp_unique_firstinput_entry. fom_beta_height_pfp_unique_firstinput_entry + S (fom_value_pfp_unique_firstinput) = S ((S (fom_index_pfp_unique_firstinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_firstinput_entry. b = fom_beta_quotient_pfp_unique_firstinput_entry * S ((S (fom_index_pfp_unique_firstinput)) * c) + (fom_value_pfp_unique_firstinput))) /\ (exists fom_gap_pfp_unique_firstinput_value_bound. fom_gap_pfp_unique_firstinput_value_bound + S (fom_value_pfp_unique_firstinput) = p))) /\ (((forall pfp_repeat_index_unique_firstremoved. (exists pfa_gap_unique_firstremovedindex. pfa_gap_unique_firstremovedindex + S (pfp_repeat_index_unique_firstremoved) = (t)) -> (((exists ff_h_pfp_unique_firstremovedentry. ff_h_pfp_unique_firstremovedentry + S (0) = S ((S (pfp_repeat_index_unique_firstremoved)) * c)) /\ exists ff_q_pfp_unique_firstremovedentry. b = ff_q_pfp_unique_firstremovedentry * S ((S (pfp_repeat_index_unique_firstremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_firstsuffix pftrim_value_unique_firstsuffix. (exists pfa_gap_unique_firstsuffixbound. pfa_gap_unique_firstsuffixbound + S (pftrim_index_unique_firstsuffix) = (M)) -> (((exists ff_h_pfp_unique_firstsuffixsource. ff_h_pfp_unique_firstsuffixsource + S (pftrim_value_unique_firstsuffix) = S ((S ((t)+pftrim_index_unique_firstsuffix)) * c)) /\ exists ff_q_pfp_unique_firstsuffixsource. b = ff_q_pfp_unique_firstsuffixsource * S ((S ((t)+pftrim_index_unique_firstsuffix)) * c) + (pftrim_value_unique_firstsuffix))) -> (((exists ff_h_pfp_unique_firstsuffixoutput. ff_h_pfp_unique_firstsuffixoutput + S (pftrim_value_unique_firstsuffix) = S ((S (pftrim_index_unique_firstsuffix)) * e)) /\ exists ff_q_pfp_unique_firstsuffixoutput. d = ff_q_pfp_unique_firstsuffixoutput * S ((S (pftrim_index_unique_firstsuffix)) * e) + (pftrim_value_unique_firstsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_unique_firstnormal. ((((exists ff_h_pfp_unique_firstnormalentry. ff_h_pfp_unique_firstnormalentry + S (pftrim_leading_unique_firstnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_unique_firstnormalentry. d = ff_q_pfp_unique_firstnormalentry * S ((S (0)) * e) + (pftrim_leading_unique_firstnormal))) /\ ((~(pftrim_leading_unique_firstnormal=0))))))))))))))) -> ((((L)=(u)+(N)) /\ (((forall fom_index_pfp_unique_secondinput. (exists fom_gap_pfp_unique_secondinput_index_bound. fom_gap_pfp_unique_secondinput_index_bound + S (fom_index_pfp_unique_secondinput) = L) -> exists fom_value_pfp_unique_secondinput. ((((exists fom_beta_height_pfp_unique_secondinput_entry. fom_beta_height_pfp_unique_secondinput_entry + S (fom_value_pfp_unique_secondinput) = S ((S (fom_index_pfp_unique_secondinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_secondinput_entry. b = fom_beta_quotient_pfp_unique_secondinput_entry * S ((S (fom_index_pfp_unique_secondinput)) * c) + (fom_value_pfp_unique_secondinput))) /\ (exists fom_gap_pfp_unique_secondinput_value_bound. fom_gap_pfp_unique_secondinput_value_bound + S (fom_value_pfp_unique_secondinput) = p))) /\ (((forall pfp_repeat_index_unique_secondremoved. (exists pfa_gap_unique_secondremovedindex. pfa_gap_unique_secondremovedindex + S (pfp_repeat_index_unique_secondremoved) = (u)) -> (((exists ff_h_pfp_unique_secondremovedentry. ff_h_pfp_unique_secondremovedentry + S (0) = S ((S (pfp_repeat_index_unique_secondremoved)) * c)) /\ exists ff_q_pfp_unique_secondremovedentry. b = ff_q_pfp_unique_secondremovedentry * S ((S (pfp_repeat_index_unique_secondremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_secondsuffix pftrim_value_unique_secondsuffix. (exists pfa_gap_unique_secondsuffixbound. pfa_gap_unique_secondsuffixbound + S (pftrim_index_unique_secondsuffix) = (N)) -> (((exists ff_h_pfp_unique_secondsuffixsource. ff_h_pfp_unique_secondsuffixsource + S (pftrim_value_unique_secondsuffix) = S ((S ((u)+pftrim_index_unique_secondsuffix)) * c)) /\ exists ff_q_pfp_unique_secondsuffixsource. b = ff_q_pfp_unique_secondsuffixsource * S ((S ((u)+pftrim_index_unique_secondsuffix)) * c) + (pftrim_value_unique_secondsuffix))) -> (((exists ff_h_pfp_unique_secondsuffixoutput. ff_h_pfp_unique_secondsuffixoutput + S (pftrim_value_unique_secondsuffix) = S ((S (pftrim_index_unique_secondsuffix)) * g)) /\ exists ff_q_pfp_unique_secondsuffixoutput. f = ff_q_pfp_unique_secondsuffixoutput * S ((S (pftrim_index_unique_secondsuffix)) * g) + (pftrim_value_unique_secondsuffix)))) /\ (((N)=0 \/ (exists pftrim_leading_unique_secondnormal. ((((exists ff_h_pfp_unique_secondnormalentry. ff_h_pfp_unique_secondnormalentry + S (pftrim_leading_unique_secondnormal) = S ((S (0)) * g)) /\ exists ff_q_pfp_unique_secondnormalentry. f = ff_q_pfp_unique_secondnormalentry * S ((S (0)) * g) + (pftrim_leading_unique_secondnormal))) /\ ((~(pftrim_leading_unique_secondnormal=0))))))))))))))) -> M=Nprime_field_polynomial_trim_output_equalExact checked statement
forall p b c L t d e M u f g N. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_unique_firstinput. (exists fom_gap_pfp_unique_firstinput_index_bound. fom_gap_pfp_unique_firstinput_index_bound + S (fom_index_pfp_unique_firstinput) = L) -> exists fom_value_pfp_unique_firstinput. ((((exists fom_beta_height_pfp_unique_firstinput_entry. fom_beta_height_pfp_unique_firstinput_entry + S (fom_value_pfp_unique_firstinput) = S ((S (fom_index_pfp_unique_firstinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_firstinput_entry. b = fom_beta_quotient_pfp_unique_firstinput_entry * S ((S (fom_index_pfp_unique_firstinput)) * c) + (fom_value_pfp_unique_firstinput))) /\ (exists fom_gap_pfp_unique_firstinput_value_bound. fom_gap_pfp_unique_firstinput_value_bound + S (fom_value_pfp_unique_firstinput) = p))) /\ (((forall pfp_repeat_index_unique_firstremoved. (exists pfa_gap_unique_firstremovedindex. pfa_gap_unique_firstremovedindex + S (pfp_repeat_index_unique_firstremoved) = (t)) -> (((exists ff_h_pfp_unique_firstremovedentry. ff_h_pfp_unique_firstremovedentry + S (0) = S ((S (pfp_repeat_index_unique_firstremoved)) * c)) /\ exists ff_q_pfp_unique_firstremovedentry. b = ff_q_pfp_unique_firstremovedentry * S ((S (pfp_repeat_index_unique_firstremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_firstsuffix pftrim_value_unique_firstsuffix. (exists pfa_gap_unique_firstsuffixbound. pfa_gap_unique_firstsuffixbound + S (pftrim_index_unique_firstsuffix) = (M)) -> (((exists ff_h_pfp_unique_firstsuffixsource. ff_h_pfp_unique_firstsuffixsource + S (pftrim_value_unique_firstsuffix) = S ((S ((t)+pftrim_index_unique_firstsuffix)) * c)) /\ exists ff_q_pfp_unique_firstsuffixsource. b = ff_q_pfp_unique_firstsuffixsource * S ((S ((t)+pftrim_index_unique_firstsuffix)) * c) + (pftrim_value_unique_firstsuffix))) -> (((exists ff_h_pfp_unique_firstsuffixoutput. ff_h_pfp_unique_firstsuffixoutput + S (pftrim_value_unique_firstsuffix) = S ((S (pftrim_index_unique_firstsuffix)) * e)) /\ exists ff_q_pfp_unique_firstsuffixoutput. d = ff_q_pfp_unique_firstsuffixoutput * S ((S (pftrim_index_unique_firstsuffix)) * e) + (pftrim_value_unique_firstsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_unique_firstnormal. ((((exists ff_h_pfp_unique_firstnormalentry. ff_h_pfp_unique_firstnormalentry + S (pftrim_leading_unique_firstnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_unique_firstnormalentry. d = ff_q_pfp_unique_firstnormalentry * S ((S (0)) * e) + (pftrim_leading_unique_firstnormal))) /\ ((~(pftrim_leading_unique_firstnormal=0))))))))))))))) -> ((((L)=(u)+(N)) /\ (((forall fom_index_pfp_unique_secondinput. (exists fom_gap_pfp_unique_secondinput_index_bound. fom_gap_pfp_unique_secondinput_index_bound + S (fom_index_pfp_unique_secondinput) = L) -> exists fom_value_pfp_unique_secondinput. ((((exists fom_beta_height_pfp_unique_secondinput_entry. fom_beta_height_pfp_unique_secondinput_entry + S (fom_value_pfp_unique_secondinput) = S ((S (fom_index_pfp_unique_secondinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_secondinput_entry. b = fom_beta_quotient_pfp_unique_secondinput_entry * S ((S (fom_index_pfp_unique_secondinput)) * c) + (fom_value_pfp_unique_secondinput))) /\ (exists fom_gap_pfp_unique_secondinput_value_bound. fom_gap_pfp_unique_secondinput_value_bound + S (fom_value_pfp_unique_secondinput) = p))) /\ (((forall pfp_repeat_index_unique_secondremoved. (exists pfa_gap_unique_secondremovedindex. pfa_gap_unique_secondremovedindex + S (pfp_repeat_index_unique_secondremoved) = (u)) -> (((exists ff_h_pfp_unique_secondremovedentry. ff_h_pfp_unique_secondremovedentry + S (0) = S ((S (pfp_repeat_index_unique_secondremoved)) * c)) /\ exists ff_q_pfp_unique_secondremovedentry. b = ff_q_pfp_unique_secondremovedentry * S ((S (pfp_repeat_index_unique_secondremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_secondsuffix pftrim_value_unique_secondsuffix. (exists pfa_gap_unique_secondsuffixbound. pfa_gap_unique_secondsuffixbound + S (pftrim_index_unique_secondsuffix) = (N)) -> (((exists ff_h_pfp_unique_secondsuffixsource. ff_h_pfp_unique_secondsuffixsource + S (pftrim_value_unique_secondsuffix) = S ((S ((u)+pftrim_index_unique_secondsuffix)) * c)) /\ exists ff_q_pfp_unique_secondsuffixsource. b = ff_q_pfp_unique_secondsuffixsource * S ((S ((u)+pftrim_index_unique_secondsuffix)) * c) + (pftrim_value_unique_secondsuffix))) -> (((exists ff_h_pfp_unique_secondsuffixoutput. ff_h_pfp_unique_secondsuffixoutput + S (pftrim_value_unique_secondsuffix) = S ((S (pftrim_index_unique_secondsuffix)) * g)) /\ exists ff_q_pfp_unique_secondsuffixoutput. f = ff_q_pfp_unique_secondsuffixoutput * S ((S (pftrim_index_unique_secondsuffix)) * g) + (pftrim_value_unique_secondsuffix)))) /\ (((N)=0 \/ (exists pftrim_leading_unique_secondnormal. ((((exists ff_h_pfp_unique_secondnormalentry. ff_h_pfp_unique_secondnormalentry + S (pftrim_leading_unique_secondnormal) = S ((S (0)) * g)) /\ exists ff_q_pfp_unique_secondnormalentry. f = ff_q_pfp_unique_secondnormalentry * S ((S (0)) * g) + (pftrim_leading_unique_secondnormal))) /\ ((~(pftrim_leading_unique_secondnormal=0))))))))))))))) -> (forall mdr_i_pfp_unique_coefficients mdr_a_pfp_unique_coefficients. (exists mdr_gap_pfp_unique_coefficientsb. mdr_gap_pfp_unique_coefficientsb + S (mdr_i_pfp_unique_coefficients) = (M)) -> (((exists ff_h_mdr_pfp_unique_coefficientso. ff_h_mdr_pfp_unique_coefficientso + S (mdr_a_pfp_unique_coefficients) = S ((S (mdr_i_pfp_unique_coefficients)) * e)) /\ exists ff_q_mdr_pfp_unique_coefficientso. d = ff_q_mdr_pfp_unique_coefficientso * S ((S (mdr_i_pfp_unique_coefficients)) * e) + (mdr_a_pfp_unique_coefficients))) -> (((exists ff_h_mdr_pfp_unique_coefficientsn. ff_h_mdr_pfp_unique_coefficientsn + S (mdr_a_pfp_unique_coefficients) = S ((S (mdr_i_pfp_unique_coefficients)) * g)) /\ exists ff_q_mdr_pfp_unique_coefficientsn. f = ff_q_mdr_pfp_unique_coefficientsn * S ((S (mdr_i_pfp_unique_coefficients)) * g) + (mdr_a_pfp_unique_coefficients))))prime_field_polynomial_trim_exists_uniqueExact checked statement
forall p b c L. (forall fom_index_pfp_unique_input. (exists fom_gap_pfp_unique_input_index_bound. fom_gap_pfp_unique_input_index_bound + S (fom_index_pfp_unique_input) = L) -> exists fom_value_pfp_unique_input. ((((exists fom_beta_height_pfp_unique_input_entry. fom_beta_height_pfp_unique_input_entry + S (fom_value_pfp_unique_input) = S ((S (fom_index_pfp_unique_input)) * c)) /\ exists fom_beta_quotient_pfp_unique_input_entry. b = fom_beta_quotient_pfp_unique_input_entry * S ((S (fom_index_pfp_unique_input)) * c) + (fom_value_pfp_unique_input))) /\ (exists fom_gap_pfp_unique_input_value_bound. fom_gap_pfp_unique_input_value_bound + S (fom_value_pfp_unique_input) = p))) -> exists t d e M. (((((L)=(t)+(M)) /\ (((forall fom_index_pfp_unique_constructedinput. (exists fom_gap_pfp_unique_constructedinput_index_bound. fom_gap_pfp_unique_constructedinput_index_bound + S (fom_index_pfp_unique_constructedinput) = L) -> exists fom_value_pfp_unique_constructedinput. ((((exists fom_beta_height_pfp_unique_constructedinput_entry. fom_beta_height_pfp_unique_constructedinput_entry + S (fom_value_pfp_unique_constructedinput) = S ((S (fom_index_pfp_unique_constructedinput)) * c)) /\ exists fom_beta_quotient_pfp_unique_constructedinput_entry. b = fom_beta_quotient_pfp_unique_constructedinput_entry * S ((S (fom_index_pfp_unique_constructedinput)) * c) + (fom_value_pfp_unique_constructedinput))) /\ (exists fom_gap_pfp_unique_constructedinput_value_bound. fom_gap_pfp_unique_constructedinput_value_bound + S (fom_value_pfp_unique_constructedinput) = p))) /\ (((forall pfp_repeat_index_unique_constructedremoved. (exists pfa_gap_unique_constructedremovedindex. pfa_gap_unique_constructedremovedindex + S (pfp_repeat_index_unique_constructedremoved) = (t)) -> (((exists ff_h_pfp_unique_constructedremovedentry. ff_h_pfp_unique_constructedremovedentry + S (0) = S ((S (pfp_repeat_index_unique_constructedremoved)) * c)) /\ exists ff_q_pfp_unique_constructedremovedentry. b = ff_q_pfp_unique_constructedremovedentry * S ((S (pfp_repeat_index_unique_constructedremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_constructedsuffix pftrim_value_unique_constructedsuffix. (exists pfa_gap_unique_constructedsuffixbound. pfa_gap_unique_constructedsuffixbound + S (pftrim_index_unique_constructedsuffix) = (M)) -> (((exists ff_h_pfp_unique_constructedsuffixsource. ff_h_pfp_unique_constructedsuffixsource + S (pftrim_value_unique_constructedsuffix) = S ((S ((t)+pftrim_index_unique_constructedsuffix)) * c)) /\ exists ff_q_pfp_unique_constructedsuffixsource. b = ff_q_pfp_unique_constructedsuffixsource * S ((S ((t)+pftrim_index_unique_constructedsuffix)) * c) + (pftrim_value_unique_constructedsuffix))) -> (((exists ff_h_pfp_unique_constructedsuffixoutput. ff_h_pfp_unique_constructedsuffixoutput + S (pftrim_value_unique_constructedsuffix) = S ((S (pftrim_index_unique_constructedsuffix)) * e)) /\ exists ff_q_pfp_unique_constructedsuffixoutput. d = ff_q_pfp_unique_constructedsuffixoutput * S ((S (pftrim_index_unique_constructedsuffix)) * e) + (pftrim_value_unique_constructedsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_unique_constructednormal. ((((exists ff_h_pfp_unique_constructednormalentry. ff_h_pfp_unique_constructednormalentry + S (pftrim_leading_unique_constructednormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_unique_constructednormalentry. d = ff_q_pfp_unique_constructednormalentry * S ((S (0)) * e) + (pftrim_leading_unique_constructednormal))) /\ ((~(pftrim_leading_unique_constructednormal=0))))))))))))))) /\ ((forall u f g N. ((((L)=(u)+(N)) /\ (((forall fom_index_pfp_unique_comparisoninput. (exists fom_gap_pfp_unique_comparisoninput_index_bound. fom_gap_pfp_unique_comparisoninput_index_bound + S (fom_index_pfp_unique_comparisoninput) = L) -> exists fom_value_pfp_unique_comparisoninput. ((((exists fom_beta_height_pfp_unique_comparisoninput_entry. fom_beta_height_pfp_unique_comparisoninput_entry + S (fom_value_pfp_unique_comparisoninput) = S ((S (fom_index_pfp_unique_comparisoninput)) * c)) /\ exists fom_beta_quotient_pfp_unique_comparisoninput_entry. b = fom_beta_quotient_pfp_unique_comparisoninput_entry * S ((S (fom_index_pfp_unique_comparisoninput)) * c) + (fom_value_pfp_unique_comparisoninput))) /\ (exists fom_gap_pfp_unique_comparisoninput_value_bound. fom_gap_pfp_unique_comparisoninput_value_bound + S (fom_value_pfp_unique_comparisoninput) = p))) /\ (((forall pfp_repeat_index_unique_comparisonremoved. (exists pfa_gap_unique_comparisonremovedindex. pfa_gap_unique_comparisonremovedindex + S (pfp_repeat_index_unique_comparisonremoved) = (u)) -> (((exists ff_h_pfp_unique_comparisonremovedentry. ff_h_pfp_unique_comparisonremovedentry + S (0) = S ((S (pfp_repeat_index_unique_comparisonremoved)) * c)) /\ exists ff_q_pfp_unique_comparisonremovedentry. b = ff_q_pfp_unique_comparisonremovedentry * S ((S (pfp_repeat_index_unique_comparisonremoved)) * c) + (0)))) /\ (((forall pftrim_index_unique_comparisonsuffix pftrim_value_unique_comparisonsuffix. (exists pfa_gap_unique_comparisonsuffixbound. pfa_gap_unique_comparisonsuffixbound + S (pftrim_index_unique_comparisonsuffix) = (N)) -> (((exists ff_h_pfp_unique_comparisonsuffixsource. ff_h_pfp_unique_comparisonsuffixsource + S (pftrim_value_unique_comparisonsuffix) = S ((S ((u)+pftrim_index_unique_comparisonsuffix)) * c)) /\ exists ff_q_pfp_unique_comparisonsuffixsource. b = ff_q_pfp_unique_comparisonsuffixsource * S ((S ((u)+pftrim_index_unique_comparisonsuffix)) * c) + (pftrim_value_unique_comparisonsuffix))) -> (((exists ff_h_pfp_unique_comparisonsuffixoutput. ff_h_pfp_unique_comparisonsuffixoutput + S (pftrim_value_unique_comparisonsuffix) = S ((S (pftrim_index_unique_comparisonsuffix)) * g)) /\ exists ff_q_pfp_unique_comparisonsuffixoutput. f = ff_q_pfp_unique_comparisonsuffixoutput * S ((S (pftrim_index_unique_comparisonsuffix)) * g) + (pftrim_value_unique_comparisonsuffix)))) /\ (((N)=0 \/ (exists pftrim_leading_unique_comparisonnormal. ((((exists ff_h_pfp_unique_comparisonnormalentry. ff_h_pfp_unique_comparisonnormalentry + S (pftrim_leading_unique_comparisonnormal) = S ((S (0)) * g)) /\ exists ff_q_pfp_unique_comparisonnormalentry. f = ff_q_pfp_unique_comparisonnormalentry * S ((S (0)) * g) + (pftrim_leading_unique_comparisonnormal))) /\ ((~(pftrim_leading_unique_comparisonnormal=0))))))))))))))) -> (((t=u) /\ (((M=N) /\ ((forall mdr_i_pfp_exists_unique_values mdr_a_pfp_exists_unique_values. (exists mdr_gap_pfp_exists_unique_valuesb. mdr_gap_pfp_exists_unique_valuesb + S (mdr_i_pfp_exists_unique_values) = (M)) -> (((exists ff_h_mdr_pfp_exists_unique_valueso. ff_h_mdr_pfp_exists_unique_valueso + S (mdr_a_pfp_exists_unique_values) = S ((S (mdr_i_pfp_exists_unique_values)) * e)) /\ exists ff_q_mdr_pfp_exists_unique_valueso. d = ff_q_mdr_pfp_exists_unique_valueso * S ((S (mdr_i_pfp_exists_unique_values)) * e) + (mdr_a_pfp_exists_unique_values))) -> (((exists ff_h_mdr_pfp_exists_unique_valuesn. ff_h_mdr_pfp_exists_unique_valuesn + S (mdr_a_pfp_exists_unique_values) = S ((S (mdr_i_pfp_exists_unique_values)) * g)) /\ exists ff_q_mdr_pfp_exists_unique_valuesn. f = ff_q_mdr_pfp_exists_unique_valuesn * S ((S (mdr_i_pfp_exists_unique_values)) * g) + (mdr_a_pfp_exists_unique_values))))))))))))prime_field_polynomial_trim_represented_degreeExact checked statement
forall p b c L t d e M q. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_degree_inputinput. (exists fom_gap_pfp_trim_degree_inputinput_index_bound. fom_gap_pfp_trim_degree_inputinput_index_bound + S (fom_index_pfp_trim_degree_inputinput) = L) -> exists fom_value_pfp_trim_degree_inputinput. ((((exists fom_beta_height_pfp_trim_degree_inputinput_entry. fom_beta_height_pfp_trim_degree_inputinput_entry + S (fom_value_pfp_trim_degree_inputinput) = S ((S (fom_index_pfp_trim_degree_inputinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_degree_inputinput_entry. b = fom_beta_quotient_pfp_trim_degree_inputinput_entry * S ((S (fom_index_pfp_trim_degree_inputinput)) * c) + (fom_value_pfp_trim_degree_inputinput))) /\ (exists fom_gap_pfp_trim_degree_inputinput_value_bound. fom_gap_pfp_trim_degree_inputinput_value_bound + S (fom_value_pfp_trim_degree_inputinput) = p))) /\ (((forall pfp_repeat_index_trim_degree_inputremoved. (exists pfa_gap_trim_degree_inputremovedindex. pfa_gap_trim_degree_inputremovedindex + S (pfp_repeat_index_trim_degree_inputremoved) = (t)) -> (((exists ff_h_pfp_trim_degree_inputremovedentry. ff_h_pfp_trim_degree_inputremovedentry + S (0) = S ((S (pfp_repeat_index_trim_degree_inputremoved)) * c)) /\ exists ff_q_pfp_trim_degree_inputremovedentry. b = ff_q_pfp_trim_degree_inputremovedentry * S ((S (pfp_repeat_index_trim_degree_inputremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_degree_inputsuffix pftrim_value_trim_degree_inputsuffix. (exists pfa_gap_trim_degree_inputsuffixbound. pfa_gap_trim_degree_inputsuffixbound + S (pftrim_index_trim_degree_inputsuffix) = (M)) -> (((exists ff_h_pfp_trim_degree_inputsuffixsource. ff_h_pfp_trim_degree_inputsuffixsource + S (pftrim_value_trim_degree_inputsuffix) = S ((S ((t)+pftrim_index_trim_degree_inputsuffix)) * c)) /\ exists ff_q_pfp_trim_degree_inputsuffixsource. b = ff_q_pfp_trim_degree_inputsuffixsource * S ((S ((t)+pftrim_index_trim_degree_inputsuffix)) * c) + (pftrim_value_trim_degree_inputsuffix))) -> (((exists ff_h_pfp_trim_degree_inputsuffixoutput. ff_h_pfp_trim_degree_inputsuffixoutput + S (pftrim_value_trim_degree_inputsuffix) = S ((S (pftrim_index_trim_degree_inputsuffix)) * e)) /\ exists ff_q_pfp_trim_degree_inputsuffixoutput. d = ff_q_pfp_trim_degree_inputsuffixoutput * S ((S (pftrim_index_trim_degree_inputsuffix)) * e) + (pftrim_value_trim_degree_inputsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_degree_inputnormal. ((((exists ff_h_pfp_trim_degree_inputnormalentry. ff_h_pfp_trim_degree_inputnormalentry + S (pftrim_leading_trim_degree_inputnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_degree_inputnormalentry. d = ff_q_pfp_trim_degree_inputnormalentry * S ((S (0)) * e) + (pftrim_leading_trim_degree_inputnormal))) /\ ((~(pftrim_leading_trim_degree_inputnormal=0))))))))))))))) -> M=S q -> ((((M)=S (q)) /\ (((forall fom_index_pfp_trim_degree_resultcoefficients. (exists fom_gap_pfp_trim_degree_resultcoefficients_index_bound. fom_gap_pfp_trim_degree_resultcoefficients_index_bound + S (fom_index_pfp_trim_degree_resultcoefficients) = M) -> exists fom_value_pfp_trim_degree_resultcoefficients. ((((exists fom_beta_height_pfp_trim_degree_resultcoefficients_entry. fom_beta_height_pfp_trim_degree_resultcoefficients_entry + S (fom_value_pfp_trim_degree_resultcoefficients) = S ((S (fom_index_pfp_trim_degree_resultcoefficients)) * e)) /\ exists fom_beta_quotient_pfp_trim_degree_resultcoefficients_entry. d = fom_beta_quotient_pfp_trim_degree_resultcoefficients_entry * S ((S (fom_index_pfp_trim_degree_resultcoefficients)) * e) + (fom_value_pfp_trim_degree_resultcoefficients))) /\ (exists fom_gap_pfp_trim_degree_resultcoefficients_value_bound. fom_gap_pfp_trim_degree_resultcoefficients_value_bound + S (fom_value_pfp_trim_degree_resultcoefficients) = p))) /\ ((exists pfd_leading_trim_degree_result. ((((exists ff_h_pfp_trim_degree_resultentry. ff_h_pfp_trim_degree_resultentry + S (pfd_leading_trim_degree_result) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_degree_resultentry. d = ff_q_pfp_trim_degree_resultentry * S ((S (0)) * e) + (pfd_leading_trim_degree_result))) /\ ((~(pfd_leading_trim_degree_result=0))))))))))prime_field_polynomial_trim_nonempty_degree_existsExact checked statement
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_nonempty_inputinput. (exists fom_gap_pfp_trim_nonempty_inputinput_index_bound. fom_gap_pfp_trim_nonempty_inputinput_index_bound + S (fom_index_pfp_trim_nonempty_inputinput) = L) -> exists fom_value_pfp_trim_nonempty_inputinput. ((((exists fom_beta_height_pfp_trim_nonempty_inputinput_entry. fom_beta_height_pfp_trim_nonempty_inputinput_entry + S (fom_value_pfp_trim_nonempty_inputinput) = S ((S (fom_index_pfp_trim_nonempty_inputinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_nonempty_inputinput_entry. b = fom_beta_quotient_pfp_trim_nonempty_inputinput_entry * S ((S (fom_index_pfp_trim_nonempty_inputinput)) * c) + (fom_value_pfp_trim_nonempty_inputinput))) /\ (exists fom_gap_pfp_trim_nonempty_inputinput_value_bound. fom_gap_pfp_trim_nonempty_inputinput_value_bound + S (fom_value_pfp_trim_nonempty_inputinput) = p))) /\ (((forall pfp_repeat_index_trim_nonempty_inputremoved. (exists pfa_gap_trim_nonempty_inputremovedindex. pfa_gap_trim_nonempty_inputremovedindex + S (pfp_repeat_index_trim_nonempty_inputremoved) = (t)) -> (((exists ff_h_pfp_trim_nonempty_inputremovedentry. ff_h_pfp_trim_nonempty_inputremovedentry + S (0) = S ((S (pfp_repeat_index_trim_nonempty_inputremoved)) * c)) /\ exists ff_q_pfp_trim_nonempty_inputremovedentry. b = ff_q_pfp_trim_nonempty_inputremovedentry * S ((S (pfp_repeat_index_trim_nonempty_inputremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_nonempty_inputsuffix pftrim_value_trim_nonempty_inputsuffix. (exists pfa_gap_trim_nonempty_inputsuffixbound. pfa_gap_trim_nonempty_inputsuffixbound + S (pftrim_index_trim_nonempty_inputsuffix) = (M)) -> (((exists ff_h_pfp_trim_nonempty_inputsuffixsource. ff_h_pfp_trim_nonempty_inputsuffixsource + S (pftrim_value_trim_nonempty_inputsuffix) = S ((S ((t)+pftrim_index_trim_nonempty_inputsuffix)) * c)) /\ exists ff_q_pfp_trim_nonempty_inputsuffixsource. b = ff_q_pfp_trim_nonempty_inputsuffixsource * S ((S ((t)+pftrim_index_trim_nonempty_inputsuffix)) * c) + (pftrim_value_trim_nonempty_inputsuffix))) -> (((exists ff_h_pfp_trim_nonempty_inputsuffixoutput. ff_h_pfp_trim_nonempty_inputsuffixoutput + S (pftrim_value_trim_nonempty_inputsuffix) = S ((S (pftrim_index_trim_nonempty_inputsuffix)) * e)) /\ exists ff_q_pfp_trim_nonempty_inputsuffixoutput. d = ff_q_pfp_trim_nonempty_inputsuffixoutput * S ((S (pftrim_index_trim_nonempty_inputsuffix)) * e) + (pftrim_value_trim_nonempty_inputsuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_nonempty_inputnormal. ((((exists ff_h_pfp_trim_nonempty_inputnormalentry. ff_h_pfp_trim_nonempty_inputnormalentry + S (pftrim_leading_trim_nonempty_inputnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_nonempty_inputnormalentry. d = ff_q_pfp_trim_nonempty_inputnormalentry * S ((S (0)) * e) + (pftrim_leading_trim_nonempty_inputnormal))) /\ ((~(pftrim_leading_trim_nonempty_inputnormal=0))))))))))))))) -> ~(M=0) -> exists q. ((((M)=S (q)) /\ (((forall fom_index_pfp_trim_nonempty_degreecoefficients. (exists fom_gap_pfp_trim_nonempty_degreecoefficients_index_bound. fom_gap_pfp_trim_nonempty_degreecoefficients_index_bound + S (fom_index_pfp_trim_nonempty_degreecoefficients) = M) -> exists fom_value_pfp_trim_nonempty_degreecoefficients. ((((exists fom_beta_height_pfp_trim_nonempty_degreecoefficients_entry. fom_beta_height_pfp_trim_nonempty_degreecoefficients_entry + S (fom_value_pfp_trim_nonempty_degreecoefficients) = S ((S (fom_index_pfp_trim_nonempty_degreecoefficients)) * e)) /\ exists fom_beta_quotient_pfp_trim_nonempty_degreecoefficients_entry. d = fom_beta_quotient_pfp_trim_nonempty_degreecoefficients_entry * S ((S (fom_index_pfp_trim_nonempty_degreecoefficients)) * e) + (fom_value_pfp_trim_nonempty_degreecoefficients))) /\ (exists fom_gap_pfp_trim_nonempty_degreecoefficients_value_bound. fom_gap_pfp_trim_nonempty_degreecoefficients_value_bound + S (fom_value_pfp_trim_nonempty_degreecoefficients) = p))) /\ ((exists pfd_leading_trim_nonempty_degree. ((((exists ff_h_pfp_trim_nonempty_degreeentry. ff_h_pfp_trim_nonempty_degreeentry + S (pfd_leading_trim_nonempty_degree) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_nonempty_degreeentry. d = ff_q_pfp_trim_nonempty_degreeentry * S ((S (0)) * e) + (pfd_leading_trim_nonempty_degree))) /\ ((~(pfd_leading_trim_nonempty_degree=0))))))))))prime_field_polynomial_trim_represented_identityExact checked statement
forall p b c L q. ((((L)=S (q)) /\ (((forall fom_index_pfp_trim_already_normalcoefficients. (exists fom_gap_pfp_trim_already_normalcoefficients_index_bound. fom_gap_pfp_trim_already_normalcoefficients_index_bound + S (fom_index_pfp_trim_already_normalcoefficients) = L) -> exists fom_value_pfp_trim_already_normalcoefficients. ((((exists fom_beta_height_pfp_trim_already_normalcoefficients_entry. fom_beta_height_pfp_trim_already_normalcoefficients_entry + S (fom_value_pfp_trim_already_normalcoefficients) = S ((S (fom_index_pfp_trim_already_normalcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_trim_already_normalcoefficients_entry. b = fom_beta_quotient_pfp_trim_already_normalcoefficients_entry * S ((S (fom_index_pfp_trim_already_normalcoefficients)) * c) + (fom_value_pfp_trim_already_normalcoefficients))) /\ (exists fom_gap_pfp_trim_already_normalcoefficients_value_bound. fom_gap_pfp_trim_already_normalcoefficients_value_bound + S (fom_value_pfp_trim_already_normalcoefficients) = p))) /\ ((exists pfd_leading_trim_already_normal. ((((exists ff_h_pfp_trim_already_normalentry. ff_h_pfp_trim_already_normalentry + S (pfd_leading_trim_already_normal) = S ((S (0)) * c)) /\ exists ff_q_pfp_trim_already_normalentry. b = ff_q_pfp_trim_already_normalentry * S ((S (0)) * c) + (pfd_leading_trim_already_normal))) /\ ((~(pfd_leading_trim_already_normal=0)))))))))) -> ((((L)=(0)+(L)) /\ (((forall fom_index_pfp_trim_identityinput. (exists fom_gap_pfp_trim_identityinput_index_bound. fom_gap_pfp_trim_identityinput_index_bound + S (fom_index_pfp_trim_identityinput) = L) -> exists fom_value_pfp_trim_identityinput. ((((exists fom_beta_height_pfp_trim_identityinput_entry. fom_beta_height_pfp_trim_identityinput_entry + S (fom_value_pfp_trim_identityinput) = S ((S (fom_index_pfp_trim_identityinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_identityinput_entry. b = fom_beta_quotient_pfp_trim_identityinput_entry * S ((S (fom_index_pfp_trim_identityinput)) * c) + (fom_value_pfp_trim_identityinput))) /\ (exists fom_gap_pfp_trim_identityinput_value_bound. fom_gap_pfp_trim_identityinput_value_bound + S (fom_value_pfp_trim_identityinput) = p))) /\ (((forall pfp_repeat_index_trim_identityremoved. (exists pfa_gap_trim_identityremovedindex. pfa_gap_trim_identityremovedindex + S (pfp_repeat_index_trim_identityremoved) = (0)) -> (((exists ff_h_pfp_trim_identityremovedentry. ff_h_pfp_trim_identityremovedentry + S (0) = S ((S (pfp_repeat_index_trim_identityremoved)) * c)) /\ exists ff_q_pfp_trim_identityremovedentry. b = ff_q_pfp_trim_identityremovedentry * S ((S (pfp_repeat_index_trim_identityremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_identitysuffix pftrim_value_trim_identitysuffix. (exists pfa_gap_trim_identitysuffixbound. pfa_gap_trim_identitysuffixbound + S (pftrim_index_trim_identitysuffix) = (L)) -> (((exists ff_h_pfp_trim_identitysuffixsource. ff_h_pfp_trim_identitysuffixsource + S (pftrim_value_trim_identitysuffix) = S ((S ((0)+pftrim_index_trim_identitysuffix)) * c)) /\ exists ff_q_pfp_trim_identitysuffixsource. b = ff_q_pfp_trim_identitysuffixsource * S ((S ((0)+pftrim_index_trim_identitysuffix)) * c) + (pftrim_value_trim_identitysuffix))) -> (((exists ff_h_pfp_trim_identitysuffixoutput. ff_h_pfp_trim_identitysuffixoutput + S (pftrim_value_trim_identitysuffix) = S ((S (pftrim_index_trim_identitysuffix)) * c)) /\ exists ff_q_pfp_trim_identitysuffixoutput. b = ff_q_pfp_trim_identitysuffixoutput * S ((S (pftrim_index_trim_identitysuffix)) * c) + (pftrim_value_trim_identitysuffix)))) /\ (((L)=0 \/ (exists pftrim_leading_trim_identitynormal. ((((exists ff_h_pfp_trim_identitynormalentry. ff_h_pfp_trim_identitynormalentry + S (pftrim_leading_trim_identitynormal) = S ((S (0)) * c)) /\ exists ff_q_pfp_trim_identitynormalentry. b = ff_q_pfp_trim_identitynormalentry * S ((S (0)) * c) + (pftrim_leading_trim_identitynormal))) /\ ((~(pftrim_leading_trim_identitynormal=0)))))))))))))))prime_field_polynomial_monic_leading_valueExact checked statement
forall p b c L a. (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_value_sourcecoefficients. (exists fom_gap_pfp_monic_value_sourcecoefficients_index_bound. fom_gap_pfp_monic_value_sourcecoefficients_index_bound + S (fom_index_pfp_monic_value_sourcecoefficients) = L) -> exists fom_value_pfp_monic_value_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_value_sourcecoefficients_entry. fom_beta_height_pfp_monic_value_sourcecoefficients_entry + S (fom_value_pfp_monic_value_sourcecoefficients) = S ((S (fom_index_pfp_monic_value_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_value_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_value_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_value_sourcecoefficients)) * c) + (fom_value_pfp_monic_value_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_value_sourcecoefficients_value_bound. fom_gap_pfp_monic_value_sourcecoefficients_value_bound + S (fom_value_pfp_monic_value_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_value_sourceleading. ff_h_pfp_monic_value_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_value_sourceleading. b = ff_q_pfp_monic_value_sourceleading * S ((S (0)) * c) + (1)))))))) -> (((exists ff_h_pfp_monic_value_entry. ff_h_pfp_monic_value_entry + S (a) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_value_entry. b = ff_q_pfp_monic_value_entry * S ((S (0)) * c) + (a))) -> a=1prime_field_polynomial_monic_represented_degreeExact checked statement
forall p b c L d. (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_degree_sourcecoefficients. (exists fom_gap_pfp_monic_degree_sourcecoefficients_index_bound. fom_gap_pfp_monic_degree_sourcecoefficients_index_bound + S (fom_index_pfp_monic_degree_sourcecoefficients) = L) -> exists fom_value_pfp_monic_degree_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_degree_sourcecoefficients_entry. fom_beta_height_pfp_monic_degree_sourcecoefficients_entry + S (fom_value_pfp_monic_degree_sourcecoefficients) = S ((S (fom_index_pfp_monic_degree_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_degree_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_degree_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_degree_sourcecoefficients)) * c) + (fom_value_pfp_monic_degree_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_degree_sourcecoefficients_value_bound. fom_gap_pfp_monic_degree_sourcecoefficients_value_bound + S (fom_value_pfp_monic_degree_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_degree_sourceleading. ff_h_pfp_monic_degree_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_degree_sourceleading. b = ff_q_pfp_monic_degree_sourceleading * S ((S (0)) * c) + (1)))))))) -> L=S d -> ((((L)=S (d)) /\ (((forall fom_index_pfp_monic_degree_resultcoefficients. (exists fom_gap_pfp_monic_degree_resultcoefficients_index_bound. fom_gap_pfp_monic_degree_resultcoefficients_index_bound + S (fom_index_pfp_monic_degree_resultcoefficients) = L) -> exists fom_value_pfp_monic_degree_resultcoefficients. ((((exists fom_beta_height_pfp_monic_degree_resultcoefficients_entry. fom_beta_height_pfp_monic_degree_resultcoefficients_entry + S (fom_value_pfp_monic_degree_resultcoefficients) = S ((S (fom_index_pfp_monic_degree_resultcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_degree_resultcoefficients_entry. b = fom_beta_quotient_pfp_monic_degree_resultcoefficients_entry * S ((S (fom_index_pfp_monic_degree_resultcoefficients)) * c) + (fom_value_pfp_monic_degree_resultcoefficients))) /\ (exists fom_gap_pfp_monic_degree_resultcoefficients_value_bound. fom_gap_pfp_monic_degree_resultcoefficients_value_bound + S (fom_value_pfp_monic_degree_resultcoefficients) = p))) /\ ((exists pfd_leading_monic_degree_result. ((((exists ff_h_pfp_monic_degree_resultentry. ff_h_pfp_monic_degree_resultentry + S (pfd_leading_monic_degree_result) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_degree_resultentry. b = ff_q_pfp_monic_degree_resultentry * S ((S (0)) * c) + (pfd_leading_monic_degree_result))) /\ ((~(pfd_leading_monic_degree_result=0))))))))))prime_field_polynomial_monic_transportExact checked statement
forall p b c B C L. (forall mdr_i_pfp_monic_recode mdr_a_pfp_monic_recode. (exists mdr_gap_pfp_monic_recodeb. mdr_gap_pfp_monic_recodeb + S (mdr_i_pfp_monic_recode) = (L)) -> (((exists ff_h_mdr_pfp_monic_recodeo. ff_h_mdr_pfp_monic_recodeo + S (mdr_a_pfp_monic_recode) = S ((S (mdr_i_pfp_monic_recode)) * c)) /\ exists ff_q_mdr_pfp_monic_recodeo. b = ff_q_mdr_pfp_monic_recodeo * S ((S (mdr_i_pfp_monic_recode)) * c) + (mdr_a_pfp_monic_recode))) -> (((exists ff_h_mdr_pfp_monic_recoden. ff_h_mdr_pfp_monic_recoden + S (mdr_a_pfp_monic_recode) = S ((S (mdr_i_pfp_monic_recode)) * C)) /\ exists ff_q_mdr_pfp_monic_recoden. B = ff_q_mdr_pfp_monic_recoden * S ((S (mdr_i_pfp_monic_recode)) * C) + (mdr_a_pfp_monic_recode)))) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_oldcoefficients. (exists fom_gap_pfp_monic_oldcoefficients_index_bound. fom_gap_pfp_monic_oldcoefficients_index_bound + S (fom_index_pfp_monic_oldcoefficients) = L) -> exists fom_value_pfp_monic_oldcoefficients. ((((exists fom_beta_height_pfp_monic_oldcoefficients_entry. fom_beta_height_pfp_monic_oldcoefficients_entry + S (fom_value_pfp_monic_oldcoefficients) = S ((S (fom_index_pfp_monic_oldcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_oldcoefficients_entry. b = fom_beta_quotient_pfp_monic_oldcoefficients_entry * S ((S (fom_index_pfp_monic_oldcoefficients)) * c) + (fom_value_pfp_monic_oldcoefficients))) /\ (exists fom_gap_pfp_monic_oldcoefficients_value_bound. fom_gap_pfp_monic_oldcoefficients_value_bound + S (fom_value_pfp_monic_oldcoefficients) = p))) /\ ((((exists ff_h_pfp_monic_oldleading. ff_h_pfp_monic_oldleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_oldleading. b = ff_q_pfp_monic_oldleading * S ((S (0)) * c) + (1)))))))) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_newcoefficients. (exists fom_gap_pfp_monic_newcoefficients_index_bound. fom_gap_pfp_monic_newcoefficients_index_bound + S (fom_index_pfp_monic_newcoefficients) = L) -> exists fom_value_pfp_monic_newcoefficients. ((((exists fom_beta_height_pfp_monic_newcoefficients_entry. fom_beta_height_pfp_monic_newcoefficients_entry + S (fom_value_pfp_monic_newcoefficients) = S ((S (fom_index_pfp_monic_newcoefficients)) * C)) /\ exists fom_beta_quotient_pfp_monic_newcoefficients_entry. B = fom_beta_quotient_pfp_monic_newcoefficients_entry * S ((S (fom_index_pfp_monic_newcoefficients)) * C) + (fom_value_pfp_monic_newcoefficients))) /\ (exists fom_gap_pfp_monic_newcoefficients_value_bound. fom_gap_pfp_monic_newcoefficients_value_bound + S (fom_value_pfp_monic_newcoefficients) = p))) /\ ((((exists ff_h_pfp_monic_newleading. ff_h_pfp_monic_newleading + S (1) = S ((S (0)) * C)) /\ exists ff_q_pfp_monic_newleading. B = ff_q_pfp_monic_newleading * S ((S (0)) * C) + (1))))))))prime_field_polynomial_monic_constantExact checked statement
forall p b c. (((~((1) = 0)) /\ (((forall fom_index_pfp_monic_constant_sourcecoefficients. (exists fom_gap_pfp_monic_constant_sourcecoefficients_index_bound. fom_gap_pfp_monic_constant_sourcecoefficients_index_bound + S (fom_index_pfp_monic_constant_sourcecoefficients) = 1) -> exists fom_value_pfp_monic_constant_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_constant_sourcecoefficients_entry. fom_beta_height_pfp_monic_constant_sourcecoefficients_entry + S (fom_value_pfp_monic_constant_sourcecoefficients) = S ((S (fom_index_pfp_monic_constant_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_constant_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_constant_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_constant_sourcecoefficients)) * c) + (fom_value_pfp_monic_constant_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_constant_sourcecoefficients_value_bound. fom_gap_pfp_monic_constant_sourcecoefficients_value_bound + S (fom_value_pfp_monic_constant_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_constant_sourceleading. ff_h_pfp_monic_constant_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_constant_sourceleading. b = ff_q_pfp_monic_constant_sourceleading * S ((S (0)) * c) + (1)))))))) -> (forall pfp_repeat_index_monic_constant_value. (exists pfa_gap_monic_constant_valueindex. pfa_gap_monic_constant_valueindex + S (pfp_repeat_index_monic_constant_value) = (1)) -> (((exists ff_h_pfp_monic_constant_valueentry. ff_h_pfp_monic_constant_valueentry + S (1) = S ((S (pfp_repeat_index_monic_constant_value)) * c)) /\ exists ff_q_pfp_monic_constant_valueentry. b = ff_q_pfp_monic_constant_valueentry * S ((S (pfp_repeat_index_monic_constant_value)) * c) + (1))))prime_field_polynomial_monic_normalization_inverseExact checked statement
forall p k ab ac bb bc L a. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_inverse_source. ((((exists ff_h_pfp_normalization_inverse_sourcesource. ff_h_pfp_normalization_inverse_sourcesource + S (pfm_leading_normalization_inverse_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_inverse_sourcesource. ab = ff_q_pfp_normalization_inverse_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_inverse_source))) /\ ((((~((pfm_leading_normalization_inverse_source) = 0)) /\ ((((exists pfa_gap_normalization_inverse_sourceinversemultiplicationleft. pfa_gap_normalization_inverse_sourceinversemultiplicationleft + S (pfm_leading_normalization_inverse_source) = (p)) /\ (((exists pfa_gap_normalization_inverse_sourceinversemultiplicationright. pfa_gap_normalization_inverse_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_inverse_sourceinversemultiplicationresultbound. pfa_gap_normalization_inverse_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_inverse_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_inverse_source) * (k)) + (p) * pfa_offset_left_normalization_inverse_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_inverse_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_inverse_sourcescalescalar. pfa_gap_normalization_inverse_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_inverse_sourcescale. (exists pfa_gap_normalization_inverse_sourcescaleindex. pfa_gap_normalization_inverse_sourcescaleindex + S (pfp_index_normalization_inverse_sourcescale) = (L)) -> exists pfp_source_normalization_inverse_sourcescale pfp_value_normalization_inverse_sourcescale. ((((exists ff_h_pfp_normalization_inverse_sourcescalesource. ff_h_pfp_normalization_inverse_sourcescalesource + S (pfp_source_normalization_inverse_sourcescale) = S ((S (pfp_index_normalization_inverse_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_inverse_sourcescalesource. ab = ff_q_pfp_normalization_inverse_sourcescalesource * S ((S (pfp_index_normalization_inverse_sourcescale)) * ac) + (pfp_source_normalization_inverse_sourcescale))) /\ (((((exists ff_h_pfp_normalization_inverse_sourcescaletarget. ff_h_pfp_normalization_inverse_sourcescaletarget + S (pfp_value_normalization_inverse_sourcescale) = S ((S (pfp_index_normalization_inverse_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_inverse_sourcescaletarget. bb = ff_q_pfp_normalization_inverse_sourcescaletarget * S ((S (pfp_index_normalization_inverse_sourcescale)) * bc) + (pfp_value_normalization_inverse_sourcescale))) /\ ((((exists pfa_gap_normalization_inverse_sourcescaleoperationleft. pfa_gap_normalization_inverse_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_inverse_sourcescaleoperationright. pfa_gap_normalization_inverse_sourcescaleoperationright + S (pfp_source_normalization_inverse_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_inverse_sourcescaleoperationresultbound. pfa_gap_normalization_inverse_sourcescaleoperationresultbound + S (pfp_value_normalization_inverse_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_sourcescaleoperationresultcongruence pfa_offset_right_normalization_inverse_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_inverse_sourcescale)) + (p) * pfa_offset_left_normalization_inverse_sourcescaleoperationresultcongruence = (pfp_value_normalization_inverse_sourcescale) + (p) * pfa_offset_right_normalization_inverse_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((exists ff_h_pfp_normalization_inverse_entry. ff_h_pfp_normalization_inverse_entry + S (a) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_inverse_entry. ab = ff_q_pfp_normalization_inverse_entry * S ((S (0)) * ac) + (a))) -> (((~((a) = 0)) /\ ((((exists pfa_gap_normalization_inverse_resultmultiplicationleft. pfa_gap_normalization_inverse_resultmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_normalization_inverse_resultmultiplicationright. pfa_gap_normalization_inverse_resultmultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_inverse_resultmultiplicationresultbound. pfa_gap_normalization_inverse_resultmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_resultmultiplicationresultcongruence pfa_offset_right_normalization_inverse_resultmultiplicationresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_normalization_inverse_resultmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_inverse_resultmultiplicationresultcongruence))))))))))))prime_field_polynomial_monic_normalization_scalar_nonzeroExact checked statement
forall p k ab ac bb bc L. (~((p) = 1) /\ forall pfa_factor_left_normalization_scalar_prime pfa_factor_right_normalization_scalar_prime. (p) = pfa_factor_left_normalization_scalar_prime * pfa_factor_right_normalization_scalar_prime -> pfa_factor_left_normalization_scalar_prime = 1 \/ pfa_factor_right_normalization_scalar_prime = 1) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_scalar_source. ((((exists ff_h_pfp_normalization_scalar_sourcesource. ff_h_pfp_normalization_scalar_sourcesource + S (pfm_leading_normalization_scalar_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_scalar_sourcesource. ab = ff_q_pfp_normalization_scalar_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_scalar_source))) /\ ((((~((pfm_leading_normalization_scalar_source) = 0)) /\ ((((exists pfa_gap_normalization_scalar_sourceinversemultiplicationleft. pfa_gap_normalization_scalar_sourceinversemultiplicationleft + S (pfm_leading_normalization_scalar_source) = (p)) /\ (((exists pfa_gap_normalization_scalar_sourceinversemultiplicationright. pfa_gap_normalization_scalar_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_scalar_sourceinversemultiplicationresultbound. pfa_gap_normalization_scalar_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_scalar_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_scalar_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_scalar_source) * (k)) + (p) * pfa_offset_left_normalization_scalar_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_scalar_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_scalar_sourcescalescalar. pfa_gap_normalization_scalar_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_scalar_sourcescale. (exists pfa_gap_normalization_scalar_sourcescaleindex. pfa_gap_normalization_scalar_sourcescaleindex + S (pfp_index_normalization_scalar_sourcescale) = (L)) -> exists pfp_source_normalization_scalar_sourcescale pfp_value_normalization_scalar_sourcescale. ((((exists ff_h_pfp_normalization_scalar_sourcescalesource. ff_h_pfp_normalization_scalar_sourcescalesource + S (pfp_source_normalization_scalar_sourcescale) = S ((S (pfp_index_normalization_scalar_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_scalar_sourcescalesource. ab = ff_q_pfp_normalization_scalar_sourcescalesource * S ((S (pfp_index_normalization_scalar_sourcescale)) * ac) + (pfp_source_normalization_scalar_sourcescale))) /\ (((((exists ff_h_pfp_normalization_scalar_sourcescaletarget. ff_h_pfp_normalization_scalar_sourcescaletarget + S (pfp_value_normalization_scalar_sourcescale) = S ((S (pfp_index_normalization_scalar_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_scalar_sourcescaletarget. bb = ff_q_pfp_normalization_scalar_sourcescaletarget * S ((S (pfp_index_normalization_scalar_sourcescale)) * bc) + (pfp_value_normalization_scalar_sourcescale))) /\ ((((exists pfa_gap_normalization_scalar_sourcescaleoperationleft. pfa_gap_normalization_scalar_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_scalar_sourcescaleoperationright. pfa_gap_normalization_scalar_sourcescaleoperationright + S (pfp_source_normalization_scalar_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_scalar_sourcescaleoperationresultbound. pfa_gap_normalization_scalar_sourcescaleoperationresultbound + S (pfp_value_normalization_scalar_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_scalar_sourcescaleoperationresultcongruence pfa_offset_right_normalization_scalar_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_scalar_sourcescale)) + (p) * pfa_offset_left_normalization_scalar_sourcescaleoperationresultcongruence = (pfp_value_normalization_scalar_sourcescale) + (p) * pfa_offset_right_normalization_scalar_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> ~(k=0)prime_field_polynomial_monic_normalization_entryExact checked statement
forall p k ab ac bb bc L i a r. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_entry_source. ((((exists ff_h_pfp_normalization_entry_sourcesource. ff_h_pfp_normalization_entry_sourcesource + S (pfm_leading_normalization_entry_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_entry_sourcesource. ab = ff_q_pfp_normalization_entry_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_entry_source))) /\ ((((~((pfm_leading_normalization_entry_source) = 0)) /\ ((((exists pfa_gap_normalization_entry_sourceinversemultiplicationleft. pfa_gap_normalization_entry_sourceinversemultiplicationleft + S (pfm_leading_normalization_entry_source) = (p)) /\ (((exists pfa_gap_normalization_entry_sourceinversemultiplicationright. pfa_gap_normalization_entry_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_entry_sourceinversemultiplicationresultbound. pfa_gap_normalization_entry_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_entry_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_entry_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_entry_source) * (k)) + (p) * pfa_offset_left_normalization_entry_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_entry_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_entry_sourcescalescalar. pfa_gap_normalization_entry_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_entry_sourcescale. (exists pfa_gap_normalization_entry_sourcescaleindex. pfa_gap_normalization_entry_sourcescaleindex + S (pfp_index_normalization_entry_sourcescale) = (L)) -> exists pfp_source_normalization_entry_sourcescale pfp_value_normalization_entry_sourcescale. ((((exists ff_h_pfp_normalization_entry_sourcescalesource. ff_h_pfp_normalization_entry_sourcescalesource + S (pfp_source_normalization_entry_sourcescale) = S ((S (pfp_index_normalization_entry_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_entry_sourcescalesource. ab = ff_q_pfp_normalization_entry_sourcescalesource * S ((S (pfp_index_normalization_entry_sourcescale)) * ac) + (pfp_source_normalization_entry_sourcescale))) /\ (((((exists ff_h_pfp_normalization_entry_sourcescaletarget. ff_h_pfp_normalization_entry_sourcescaletarget + S (pfp_value_normalization_entry_sourcescale) = S ((S (pfp_index_normalization_entry_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_entry_sourcescaletarget. bb = ff_q_pfp_normalization_entry_sourcescaletarget * S ((S (pfp_index_normalization_entry_sourcescale)) * bc) + (pfp_value_normalization_entry_sourcescale))) /\ ((((exists pfa_gap_normalization_entry_sourcescaleoperationleft. pfa_gap_normalization_entry_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_entry_sourcescaleoperationright. pfa_gap_normalization_entry_sourcescaleoperationright + S (pfp_source_normalization_entry_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_entry_sourcescaleoperationresultbound. pfa_gap_normalization_entry_sourcescaleoperationresultbound + S (pfp_value_normalization_entry_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_entry_sourcescaleoperationresultcongruence pfa_offset_right_normalization_entry_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_entry_sourcescale)) + (p) * pfa_offset_left_normalization_entry_sourcescaleoperationresultcongruence = (pfp_value_normalization_entry_sourcescale) + (p) * pfa_offset_right_normalization_entry_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_normalization_entry_bound. pfa_gap_normalization_entry_bound + S (i) = (L)) -> (((exists ff_h_pfp_normalization_entry_input. ff_h_pfp_normalization_entry_input + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_normalization_entry_input. ab = ff_q_pfp_normalization_entry_input * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_normalization_entry_output. ff_h_pfp_normalization_entry_output + S (r) = S ((S (i)) * bc)) /\ exists ff_q_pfp_normalization_entry_output. bb = ff_q_pfp_normalization_entry_output * S ((S (i)) * bc) + (r))) -> (((exists pfa_gap_normalization_entry_multiplyleft. pfa_gap_normalization_entry_multiplyleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_entry_multiplyright. pfa_gap_normalization_entry_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_normalization_entry_multiplyresultbound. pfa_gap_normalization_entry_multiplyresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_normalization_entry_multiplyresultcongruence pfa_offset_right_normalization_entry_multiplyresultcongruence. ((k) * (a)) + (p) * pfa_offset_left_normalization_entry_multiplyresultcongruence = (r) + (p) * pfa_offset_right_normalization_entry_multiplyresultcongruence)))))))))prime_field_polynomial_monic_normalization_boundedExact checked statement
forall p k ab ac bb bc L. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_bound_source. ((((exists ff_h_pfp_normalization_bound_sourcesource. ff_h_pfp_normalization_bound_sourcesource + S (pfm_leading_normalization_bound_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_bound_sourcesource. ab = ff_q_pfp_normalization_bound_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_bound_source))) /\ ((((~((pfm_leading_normalization_bound_source) = 0)) /\ ((((exists pfa_gap_normalization_bound_sourceinversemultiplicationleft. pfa_gap_normalization_bound_sourceinversemultiplicationleft + S (pfm_leading_normalization_bound_source) = (p)) /\ (((exists pfa_gap_normalization_bound_sourceinversemultiplicationright. pfa_gap_normalization_bound_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_bound_sourceinversemultiplicationresultbound. pfa_gap_normalization_bound_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_bound_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_bound_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_bound_source) * (k)) + (p) * pfa_offset_left_normalization_bound_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_bound_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_bound_sourcescalescalar. pfa_gap_normalization_bound_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_bound_sourcescale. (exists pfa_gap_normalization_bound_sourcescaleindex. pfa_gap_normalization_bound_sourcescaleindex + S (pfp_index_normalization_bound_sourcescale) = (L)) -> exists pfp_source_normalization_bound_sourcescale pfp_value_normalization_bound_sourcescale. ((((exists ff_h_pfp_normalization_bound_sourcescalesource. ff_h_pfp_normalization_bound_sourcescalesource + S (pfp_source_normalization_bound_sourcescale) = S ((S (pfp_index_normalization_bound_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_bound_sourcescalesource. ab = ff_q_pfp_normalization_bound_sourcescalesource * S ((S (pfp_index_normalization_bound_sourcescale)) * ac) + (pfp_source_normalization_bound_sourcescale))) /\ (((((exists ff_h_pfp_normalization_bound_sourcescaletarget. ff_h_pfp_normalization_bound_sourcescaletarget + S (pfp_value_normalization_bound_sourcescale) = S ((S (pfp_index_normalization_bound_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_bound_sourcescaletarget. bb = ff_q_pfp_normalization_bound_sourcescaletarget * S ((S (pfp_index_normalization_bound_sourcescale)) * bc) + (pfp_value_normalization_bound_sourcescale))) /\ ((((exists pfa_gap_normalization_bound_sourcescaleoperationleft. pfa_gap_normalization_bound_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_bound_sourcescaleoperationright. pfa_gap_normalization_bound_sourcescaleoperationright + S (pfp_source_normalization_bound_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_bound_sourcescaleoperationresultbound. pfa_gap_normalization_bound_sourcescaleoperationresultbound + S (pfp_value_normalization_bound_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_bound_sourcescaleoperationresultcongruence pfa_offset_right_normalization_bound_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_bound_sourcescale)) + (p) * pfa_offset_left_normalization_bound_sourcescaleoperationresultcongruence = (pfp_value_normalization_bound_sourcescale) + (p) * pfa_offset_right_normalization_bound_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((exists pfa_gap_normalization_bound_scalar. pfa_gap_normalization_bound_scalar + S (k) = (p)) /\ (((forall fom_index_pfp_normalization_bound_input. (exists fom_gap_pfp_normalization_bound_input_index_bound. fom_gap_pfp_normalization_bound_input_index_bound + S (fom_index_pfp_normalization_bound_input) = L) -> exists fom_value_pfp_normalization_bound_input. ((((exists fom_beta_height_pfp_normalization_bound_input_entry. fom_beta_height_pfp_normalization_bound_input_entry + S (fom_value_pfp_normalization_bound_input) = S ((S (fom_index_pfp_normalization_bound_input)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_bound_input_entry. ab = fom_beta_quotient_pfp_normalization_bound_input_entry * S ((S (fom_index_pfp_normalization_bound_input)) * ac) + (fom_value_pfp_normalization_bound_input))) /\ (exists fom_gap_pfp_normalization_bound_input_value_bound. fom_gap_pfp_normalization_bound_input_value_bound + S (fom_value_pfp_normalization_bound_input) = p))) /\ ((forall fom_index_pfp_normalization_bound_output. (exists fom_gap_pfp_normalization_bound_output_index_bound. fom_gap_pfp_normalization_bound_output_index_bound + S (fom_index_pfp_normalization_bound_output) = L) -> exists fom_value_pfp_normalization_bound_output. ((((exists fom_beta_height_pfp_normalization_bound_output_entry. fom_beta_height_pfp_normalization_bound_output_entry + S (fom_value_pfp_normalization_bound_output) = S ((S (fom_index_pfp_normalization_bound_output)) * bc)) /\ exists fom_beta_quotient_pfp_normalization_bound_output_entry. bb = fom_beta_quotient_pfp_normalization_bound_output_entry * S ((S (fom_index_pfp_normalization_bound_output)) * bc) + (fom_value_pfp_normalization_bound_output))) /\ (exists fom_gap_pfp_normalization_bound_output_value_bound. fom_gap_pfp_normalization_bound_output_value_bound + S (fom_value_pfp_normalization_bound_output) = p))))))))prime_field_polynomial_monic_normalization_leadingExact checked statement
forall p k ab ac bb bc L. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_leading_source. ((((exists ff_h_pfp_normalization_leading_sourcesource. ff_h_pfp_normalization_leading_sourcesource + S (pfm_leading_normalization_leading_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_leading_sourcesource. ab = ff_q_pfp_normalization_leading_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_leading_source))) /\ ((((~((pfm_leading_normalization_leading_source) = 0)) /\ ((((exists pfa_gap_normalization_leading_sourceinversemultiplicationleft. pfa_gap_normalization_leading_sourceinversemultiplicationleft + S (pfm_leading_normalization_leading_source) = (p)) /\ (((exists pfa_gap_normalization_leading_sourceinversemultiplicationright. pfa_gap_normalization_leading_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_leading_sourceinversemultiplicationresultbound. pfa_gap_normalization_leading_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_leading_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_leading_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_leading_source) * (k)) + (p) * pfa_offset_left_normalization_leading_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_leading_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_leading_sourcescalescalar. pfa_gap_normalization_leading_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_leading_sourcescale. (exists pfa_gap_normalization_leading_sourcescaleindex. pfa_gap_normalization_leading_sourcescaleindex + S (pfp_index_normalization_leading_sourcescale) = (L)) -> exists pfp_source_normalization_leading_sourcescale pfp_value_normalization_leading_sourcescale. ((((exists ff_h_pfp_normalization_leading_sourcescalesource. ff_h_pfp_normalization_leading_sourcescalesource + S (pfp_source_normalization_leading_sourcescale) = S ((S (pfp_index_normalization_leading_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_leading_sourcescalesource. ab = ff_q_pfp_normalization_leading_sourcescalesource * S ((S (pfp_index_normalization_leading_sourcescale)) * ac) + (pfp_source_normalization_leading_sourcescale))) /\ (((((exists ff_h_pfp_normalization_leading_sourcescaletarget. ff_h_pfp_normalization_leading_sourcescaletarget + S (pfp_value_normalization_leading_sourcescale) = S ((S (pfp_index_normalization_leading_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_leading_sourcescaletarget. bb = ff_q_pfp_normalization_leading_sourcescaletarget * S ((S (pfp_index_normalization_leading_sourcescale)) * bc) + (pfp_value_normalization_leading_sourcescale))) /\ ((((exists pfa_gap_normalization_leading_sourcescaleoperationleft. pfa_gap_normalization_leading_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_leading_sourcescaleoperationright. pfa_gap_normalization_leading_sourcescaleoperationright + S (pfp_source_normalization_leading_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_leading_sourcescaleoperationresultbound. pfa_gap_normalization_leading_sourcescaleoperationresultbound + S (pfp_value_normalization_leading_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_leading_sourcescaleoperationresultcongruence pfa_offset_right_normalization_leading_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_leading_sourcescale)) + (p) * pfa_offset_left_normalization_leading_sourcescaleoperationresultcongruence = (pfp_value_normalization_leading_sourcescale) + (p) * pfa_offset_right_normalization_leading_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((exists ff_h_pfp_normalization_leading_result. ff_h_pfp_normalization_leading_result + S (1) = S ((S (0)) * bc)) /\ exists ff_q_pfp_normalization_leading_result. bb = ff_q_pfp_normalization_leading_result * S ((S (0)) * bc) + (1)))prime_field_polynomial_monic_normalization_monicExact checked statement
forall p k ab ac bb bc L. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_monic_source. ((((exists ff_h_pfp_normalization_monic_sourcesource. ff_h_pfp_normalization_monic_sourcesource + S (pfm_leading_normalization_monic_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_monic_sourcesource. ab = ff_q_pfp_normalization_monic_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_monic_source))) /\ ((((~((pfm_leading_normalization_monic_source) = 0)) /\ ((((exists pfa_gap_normalization_monic_sourceinversemultiplicationleft. pfa_gap_normalization_monic_sourceinversemultiplicationleft + S (pfm_leading_normalization_monic_source) = (p)) /\ (((exists pfa_gap_normalization_monic_sourceinversemultiplicationright. pfa_gap_normalization_monic_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_monic_sourceinversemultiplicationresultbound. pfa_gap_normalization_monic_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_monic_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_monic_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_monic_source) * (k)) + (p) * pfa_offset_left_normalization_monic_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_monic_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_monic_sourcescalescalar. pfa_gap_normalization_monic_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_monic_sourcescale. (exists pfa_gap_normalization_monic_sourcescaleindex. pfa_gap_normalization_monic_sourcescaleindex + S (pfp_index_normalization_monic_sourcescale) = (L)) -> exists pfp_source_normalization_monic_sourcescale pfp_value_normalization_monic_sourcescale. ((((exists ff_h_pfp_normalization_monic_sourcescalesource. ff_h_pfp_normalization_monic_sourcescalesource + S (pfp_source_normalization_monic_sourcescale) = S ((S (pfp_index_normalization_monic_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_monic_sourcescalesource. ab = ff_q_pfp_normalization_monic_sourcescalesource * S ((S (pfp_index_normalization_monic_sourcescale)) * ac) + (pfp_source_normalization_monic_sourcescale))) /\ (((((exists ff_h_pfp_normalization_monic_sourcescaletarget. ff_h_pfp_normalization_monic_sourcescaletarget + S (pfp_value_normalization_monic_sourcescale) = S ((S (pfp_index_normalization_monic_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_monic_sourcescaletarget. bb = ff_q_pfp_normalization_monic_sourcescaletarget * S ((S (pfp_index_normalization_monic_sourcescale)) * bc) + (pfp_value_normalization_monic_sourcescale))) /\ ((((exists pfa_gap_normalization_monic_sourcescaleoperationleft. pfa_gap_normalization_monic_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_monic_sourcescaleoperationright. pfa_gap_normalization_monic_sourcescaleoperationright + S (pfp_source_normalization_monic_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_monic_sourcescaleoperationresultbound. pfa_gap_normalization_monic_sourcescaleoperationresultbound + S (pfp_value_normalization_monic_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_monic_sourcescaleoperationresultcongruence pfa_offset_right_normalization_monic_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_monic_sourcescale)) + (p) * pfa_offset_left_normalization_monic_sourcescaleoperationresultcongruence = (pfp_value_normalization_monic_sourcescale) + (p) * pfa_offset_right_normalization_monic_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_normalization_monic_resultcoefficients. (exists fom_gap_pfp_normalization_monic_resultcoefficients_index_bound. fom_gap_pfp_normalization_monic_resultcoefficients_index_bound + S (fom_index_pfp_normalization_monic_resultcoefficients) = L) -> exists fom_value_pfp_normalization_monic_resultcoefficients. ((((exists fom_beta_height_pfp_normalization_monic_resultcoefficients_entry. fom_beta_height_pfp_normalization_monic_resultcoefficients_entry + S (fom_value_pfp_normalization_monic_resultcoefficients) = S ((S (fom_index_pfp_normalization_monic_resultcoefficients)) * bc)) /\ exists fom_beta_quotient_pfp_normalization_monic_resultcoefficients_entry. bb = fom_beta_quotient_pfp_normalization_monic_resultcoefficients_entry * S ((S (fom_index_pfp_normalization_monic_resultcoefficients)) * bc) + (fom_value_pfp_normalization_monic_resultcoefficients))) /\ (exists fom_gap_pfp_normalization_monic_resultcoefficients_value_bound. fom_gap_pfp_normalization_monic_resultcoefficients_value_bound + S (fom_value_pfp_normalization_monic_resultcoefficients) = p))) /\ ((((exists ff_h_pfp_normalization_monic_resultleading. ff_h_pfp_normalization_monic_resultleading + S (1) = S ((S (0)) * bc)) /\ exists ff_q_pfp_normalization_monic_resultleading. bb = ff_q_pfp_normalization_monic_resultleading * S ((S (0)) * bc) + (1))))))))prime_field_polynomial_monic_normalization_represented_degreeExact checked statement
forall p k ab ac bb bc L d. ((((L)=S (d)) /\ (((forall fom_index_pfp_normalization_degree_inputcoefficients. (exists fom_gap_pfp_normalization_degree_inputcoefficients_index_bound. fom_gap_pfp_normalization_degree_inputcoefficients_index_bound + S (fom_index_pfp_normalization_degree_inputcoefficients) = L) -> exists fom_value_pfp_normalization_degree_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_degree_inputcoefficients_entry. fom_beta_height_pfp_normalization_degree_inputcoefficients_entry + S (fom_value_pfp_normalization_degree_inputcoefficients) = S ((S (fom_index_pfp_normalization_degree_inputcoefficients)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_degree_inputcoefficients_entry. ab = fom_beta_quotient_pfp_normalization_degree_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_degree_inputcoefficients)) * ac) + (fom_value_pfp_normalization_degree_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_degree_inputcoefficients_value_bound. fom_gap_pfp_normalization_degree_inputcoefficients_value_bound + S (fom_value_pfp_normalization_degree_inputcoefficients) = p))) /\ ((exists pfd_leading_normalization_degree_input. ((((exists ff_h_pfp_normalization_degree_inputentry. ff_h_pfp_normalization_degree_inputentry + S (pfd_leading_normalization_degree_input) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_degree_inputentry. ab = ff_q_pfp_normalization_degree_inputentry * S ((S (0)) * ac) + (pfd_leading_normalization_degree_input))) /\ ((~(pfd_leading_normalization_degree_input=0)))))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_degree_source. ((((exists ff_h_pfp_normalization_degree_sourcesource. ff_h_pfp_normalization_degree_sourcesource + S (pfm_leading_normalization_degree_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_degree_sourcesource. ab = ff_q_pfp_normalization_degree_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_degree_source))) /\ ((((~((pfm_leading_normalization_degree_source) = 0)) /\ ((((exists pfa_gap_normalization_degree_sourceinversemultiplicationleft. pfa_gap_normalization_degree_sourceinversemultiplicationleft + S (pfm_leading_normalization_degree_source) = (p)) /\ (((exists pfa_gap_normalization_degree_sourceinversemultiplicationright. pfa_gap_normalization_degree_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_degree_sourceinversemultiplicationresultbound. pfa_gap_normalization_degree_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_degree_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_degree_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_degree_source) * (k)) + (p) * pfa_offset_left_normalization_degree_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_degree_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_degree_sourcescalescalar. pfa_gap_normalization_degree_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_degree_sourcescale. (exists pfa_gap_normalization_degree_sourcescaleindex. pfa_gap_normalization_degree_sourcescaleindex + S (pfp_index_normalization_degree_sourcescale) = (L)) -> exists pfp_source_normalization_degree_sourcescale pfp_value_normalization_degree_sourcescale. ((((exists ff_h_pfp_normalization_degree_sourcescalesource. ff_h_pfp_normalization_degree_sourcescalesource + S (pfp_source_normalization_degree_sourcescale) = S ((S (pfp_index_normalization_degree_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_degree_sourcescalesource. ab = ff_q_pfp_normalization_degree_sourcescalesource * S ((S (pfp_index_normalization_degree_sourcescale)) * ac) + (pfp_source_normalization_degree_sourcescale))) /\ (((((exists ff_h_pfp_normalization_degree_sourcescaletarget. ff_h_pfp_normalization_degree_sourcescaletarget + S (pfp_value_normalization_degree_sourcescale) = S ((S (pfp_index_normalization_degree_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_degree_sourcescaletarget. bb = ff_q_pfp_normalization_degree_sourcescaletarget * S ((S (pfp_index_normalization_degree_sourcescale)) * bc) + (pfp_value_normalization_degree_sourcescale))) /\ ((((exists pfa_gap_normalization_degree_sourcescaleoperationleft. pfa_gap_normalization_degree_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_degree_sourcescaleoperationright. pfa_gap_normalization_degree_sourcescaleoperationright + S (pfp_source_normalization_degree_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_degree_sourcescaleoperationresultbound. pfa_gap_normalization_degree_sourcescaleoperationresultbound + S (pfp_value_normalization_degree_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_degree_sourcescaleoperationresultcongruence pfa_offset_right_normalization_degree_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_degree_sourcescale)) + (p) * pfa_offset_left_normalization_degree_sourcescaleoperationresultcongruence = (pfp_value_normalization_degree_sourcescale) + (p) * pfa_offset_right_normalization_degree_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> ((((L)=S (d)) /\ (((forall fom_index_pfp_normalization_degree_resultcoefficients. (exists fom_gap_pfp_normalization_degree_resultcoefficients_index_bound. fom_gap_pfp_normalization_degree_resultcoefficients_index_bound + S (fom_index_pfp_normalization_degree_resultcoefficients) = L) -> exists fom_value_pfp_normalization_degree_resultcoefficients. ((((exists fom_beta_height_pfp_normalization_degree_resultcoefficients_entry. fom_beta_height_pfp_normalization_degree_resultcoefficients_entry + S (fom_value_pfp_normalization_degree_resultcoefficients) = S ((S (fom_index_pfp_normalization_degree_resultcoefficients)) * bc)) /\ exists fom_beta_quotient_pfp_normalization_degree_resultcoefficients_entry. bb = fom_beta_quotient_pfp_normalization_degree_resultcoefficients_entry * S ((S (fom_index_pfp_normalization_degree_resultcoefficients)) * bc) + (fom_value_pfp_normalization_degree_resultcoefficients))) /\ (exists fom_gap_pfp_normalization_degree_resultcoefficients_value_bound. fom_gap_pfp_normalization_degree_resultcoefficients_value_bound + S (fom_value_pfp_normalization_degree_resultcoefficients) = p))) /\ ((exists pfd_leading_normalization_degree_result. ((((exists ff_h_pfp_normalization_degree_resultentry. ff_h_pfp_normalization_degree_resultentry + S (pfd_leading_normalization_degree_result) = S ((S (0)) * bc)) /\ exists ff_q_pfp_normalization_degree_resultentry. bb = ff_q_pfp_normalization_degree_resultentry * S ((S (0)) * bc) + (pfd_leading_normalization_degree_result))) /\ ((~(pfd_leading_normalization_degree_result=0))))))))))prime_field_polynomial_monic_normalization_existsExact checked statement
forall p ab ac L d. (~((p) = 1) /\ forall pfa_factor_left_normalization_exists_prime pfa_factor_right_normalization_exists_prime. (p) = pfa_factor_left_normalization_exists_prime * pfa_factor_right_normalization_exists_prime -> pfa_factor_left_normalization_exists_prime = 1 \/ pfa_factor_right_normalization_exists_prime = 1) -> ((((L)=S (d)) /\ (((forall fom_index_pfp_normalization_exists_inputcoefficients. (exists fom_gap_pfp_normalization_exists_inputcoefficients_index_bound. fom_gap_pfp_normalization_exists_inputcoefficients_index_bound + S (fom_index_pfp_normalization_exists_inputcoefficients) = L) -> exists fom_value_pfp_normalization_exists_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_exists_inputcoefficients_entry. fom_beta_height_pfp_normalization_exists_inputcoefficients_entry + S (fom_value_pfp_normalization_exists_inputcoefficients) = S ((S (fom_index_pfp_normalization_exists_inputcoefficients)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_exists_inputcoefficients_entry. ab = fom_beta_quotient_pfp_normalization_exists_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_exists_inputcoefficients)) * ac) + (fom_value_pfp_normalization_exists_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_exists_inputcoefficients_value_bound. fom_gap_pfp_normalization_exists_inputcoefficients_value_bound + S (fom_value_pfp_normalization_exists_inputcoefficients) = p))) /\ ((exists pfd_leading_normalization_exists_input. ((((exists ff_h_pfp_normalization_exists_inputentry. ff_h_pfp_normalization_exists_inputentry + S (pfd_leading_normalization_exists_input) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_exists_inputentry. ab = ff_q_pfp_normalization_exists_inputentry * S ((S (0)) * ac) + (pfd_leading_normalization_exists_input))) /\ ((~(pfd_leading_normalization_exists_input=0)))))))))) -> exists k bb bc. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_exists_result. ((((exists ff_h_pfp_normalization_exists_resultsource. ff_h_pfp_normalization_exists_resultsource + S (pfm_leading_normalization_exists_result) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_exists_resultsource. ab = ff_q_pfp_normalization_exists_resultsource * S ((S (0)) * ac) + (pfm_leading_normalization_exists_result))) /\ ((((~((pfm_leading_normalization_exists_result) = 0)) /\ ((((exists pfa_gap_normalization_exists_resultinversemultiplicationleft. pfa_gap_normalization_exists_resultinversemultiplicationleft + S (pfm_leading_normalization_exists_result) = (p)) /\ (((exists pfa_gap_normalization_exists_resultinversemultiplicationright. pfa_gap_normalization_exists_resultinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_exists_resultinversemultiplicationresultbound. pfa_gap_normalization_exists_resultinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_exists_resultinversemultiplicationresultcongruence pfa_offset_right_normalization_exists_resultinversemultiplicationresultcongruence. ((pfm_leading_normalization_exists_result) * (k)) + (p) * pfa_offset_left_normalization_exists_resultinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_exists_resultinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_exists_resultscalescalar. pfa_gap_normalization_exists_resultscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_exists_resultscale. (exists pfa_gap_normalization_exists_resultscaleindex. pfa_gap_normalization_exists_resultscaleindex + S (pfp_index_normalization_exists_resultscale) = (L)) -> exists pfp_source_normalization_exists_resultscale pfp_value_normalization_exists_resultscale. ((((exists ff_h_pfp_normalization_exists_resultscalesource. ff_h_pfp_normalization_exists_resultscalesource + S (pfp_source_normalization_exists_resultscale) = S ((S (pfp_index_normalization_exists_resultscale)) * ac)) /\ exists ff_q_pfp_normalization_exists_resultscalesource. ab = ff_q_pfp_normalization_exists_resultscalesource * S ((S (pfp_index_normalization_exists_resultscale)) * ac) + (pfp_source_normalization_exists_resultscale))) /\ (((((exists ff_h_pfp_normalization_exists_resultscaletarget. ff_h_pfp_normalization_exists_resultscaletarget + S (pfp_value_normalization_exists_resultscale) = S ((S (pfp_index_normalization_exists_resultscale)) * bc)) /\ exists ff_q_pfp_normalization_exists_resultscaletarget. bb = ff_q_pfp_normalization_exists_resultscaletarget * S ((S (pfp_index_normalization_exists_resultscale)) * bc) + (pfp_value_normalization_exists_resultscale))) /\ ((((exists pfa_gap_normalization_exists_resultscaleoperationleft. pfa_gap_normalization_exists_resultscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_exists_resultscaleoperationright. pfa_gap_normalization_exists_resultscaleoperationright + S (pfp_source_normalization_exists_resultscale) = (p)) /\ ((((exists pfa_gap_normalization_exists_resultscaleoperationresultbound. pfa_gap_normalization_exists_resultscaleoperationresultbound + S (pfp_value_normalization_exists_resultscale) = (p)) /\ ((exists pfa_offset_left_normalization_exists_resultscaleoperationresultcongruence pfa_offset_right_normalization_exists_resultscaleoperationresultcongruence. ((k) * (pfp_source_normalization_exists_resultscale)) + (p) * pfa_offset_left_normalization_exists_resultscaleoperationresultcongruence = (pfp_value_normalization_exists_resultscale) + (p) * pfa_offset_right_normalization_exists_resultscaleoperationresultcongruence))))))))))))))))))))))prime_field_polynomial_monic_normalization_scalar_functionalExact checked statement
forall p k j ab ac bb bc cb cc L. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_scalar_first. ((((exists ff_h_pfp_normalization_scalar_firstsource. ff_h_pfp_normalization_scalar_firstsource + S (pfm_leading_normalization_scalar_first) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_scalar_firstsource. ab = ff_q_pfp_normalization_scalar_firstsource * S ((S (0)) * ac) + (pfm_leading_normalization_scalar_first))) /\ ((((~((pfm_leading_normalization_scalar_first) = 0)) /\ ((((exists pfa_gap_normalization_scalar_firstinversemultiplicationleft. pfa_gap_normalization_scalar_firstinversemultiplicationleft + S (pfm_leading_normalization_scalar_first) = (p)) /\ (((exists pfa_gap_normalization_scalar_firstinversemultiplicationright. pfa_gap_normalization_scalar_firstinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_scalar_firstinversemultiplicationresultbound. pfa_gap_normalization_scalar_firstinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_scalar_firstinversemultiplicationresultcongruence pfa_offset_right_normalization_scalar_firstinversemultiplicationresultcongruence. ((pfm_leading_normalization_scalar_first) * (k)) + (p) * pfa_offset_left_normalization_scalar_firstinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_scalar_firstinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_scalar_firstscalescalar. pfa_gap_normalization_scalar_firstscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_scalar_firstscale. (exists pfa_gap_normalization_scalar_firstscaleindex. pfa_gap_normalization_scalar_firstscaleindex + S (pfp_index_normalization_scalar_firstscale) = (L)) -> exists pfp_source_normalization_scalar_firstscale pfp_value_normalization_scalar_firstscale. ((((exists ff_h_pfp_normalization_scalar_firstscalesource. ff_h_pfp_normalization_scalar_firstscalesource + S (pfp_source_normalization_scalar_firstscale) = S ((S (pfp_index_normalization_scalar_firstscale)) * ac)) /\ exists ff_q_pfp_normalization_scalar_firstscalesource. ab = ff_q_pfp_normalization_scalar_firstscalesource * S ((S (pfp_index_normalization_scalar_firstscale)) * ac) + (pfp_source_normalization_scalar_firstscale))) /\ (((((exists ff_h_pfp_normalization_scalar_firstscaletarget. ff_h_pfp_normalization_scalar_firstscaletarget + S (pfp_value_normalization_scalar_firstscale) = S ((S (pfp_index_normalization_scalar_firstscale)) * bc)) /\ exists ff_q_pfp_normalization_scalar_firstscaletarget. bb = ff_q_pfp_normalization_scalar_firstscaletarget * S ((S (pfp_index_normalization_scalar_firstscale)) * bc) + (pfp_value_normalization_scalar_firstscale))) /\ ((((exists pfa_gap_normalization_scalar_firstscaleoperationleft. pfa_gap_normalization_scalar_firstscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_scalar_firstscaleoperationright. pfa_gap_normalization_scalar_firstscaleoperationright + S (pfp_source_normalization_scalar_firstscale) = (p)) /\ ((((exists pfa_gap_normalization_scalar_firstscaleoperationresultbound. pfa_gap_normalization_scalar_firstscaleoperationresultbound + S (pfp_value_normalization_scalar_firstscale) = (p)) /\ ((exists pfa_offset_left_normalization_scalar_firstscaleoperationresultcongruence pfa_offset_right_normalization_scalar_firstscaleoperationresultcongruence. ((k) * (pfp_source_normalization_scalar_firstscale)) + (p) * pfa_offset_left_normalization_scalar_firstscaleoperationresultcongruence = (pfp_value_normalization_scalar_firstscale) + (p) * pfa_offset_right_normalization_scalar_firstscaleoperationresultcongruence)))))))))))))))))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_scalar_second. ((((exists ff_h_pfp_normalization_scalar_secondsource. ff_h_pfp_normalization_scalar_secondsource + S (pfm_leading_normalization_scalar_second) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_scalar_secondsource. ab = ff_q_pfp_normalization_scalar_secondsource * S ((S (0)) * ac) + (pfm_leading_normalization_scalar_second))) /\ ((((~((pfm_leading_normalization_scalar_second) = 0)) /\ ((((exists pfa_gap_normalization_scalar_secondinversemultiplicationleft. pfa_gap_normalization_scalar_secondinversemultiplicationleft + S (pfm_leading_normalization_scalar_second) = (p)) /\ (((exists pfa_gap_normalization_scalar_secondinversemultiplicationright. pfa_gap_normalization_scalar_secondinversemultiplicationright + S (j) = (p)) /\ ((((exists pfa_gap_normalization_scalar_secondinversemultiplicationresultbound. pfa_gap_normalization_scalar_secondinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_scalar_secondinversemultiplicationresultcongruence pfa_offset_right_normalization_scalar_secondinversemultiplicationresultcongruence. ((pfm_leading_normalization_scalar_second) * (j)) + (p) * pfa_offset_left_normalization_scalar_secondinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_scalar_secondinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_scalar_secondscalescalar. pfa_gap_normalization_scalar_secondscalescalar + S (j) = (p)) /\ ((forall pfp_index_normalization_scalar_secondscale. (exists pfa_gap_normalization_scalar_secondscaleindex. pfa_gap_normalization_scalar_secondscaleindex + S (pfp_index_normalization_scalar_secondscale) = (L)) -> exists pfp_source_normalization_scalar_secondscale pfp_value_normalization_scalar_secondscale. ((((exists ff_h_pfp_normalization_scalar_secondscalesource. ff_h_pfp_normalization_scalar_secondscalesource + S (pfp_source_normalization_scalar_secondscale) = S ((S (pfp_index_normalization_scalar_secondscale)) * ac)) /\ exists ff_q_pfp_normalization_scalar_secondscalesource. ab = ff_q_pfp_normalization_scalar_secondscalesource * S ((S (pfp_index_normalization_scalar_secondscale)) * ac) + (pfp_source_normalization_scalar_secondscale))) /\ (((((exists ff_h_pfp_normalization_scalar_secondscaletarget. ff_h_pfp_normalization_scalar_secondscaletarget + S (pfp_value_normalization_scalar_secondscale) = S ((S (pfp_index_normalization_scalar_secondscale)) * cc)) /\ exists ff_q_pfp_normalization_scalar_secondscaletarget. cb = ff_q_pfp_normalization_scalar_secondscaletarget * S ((S (pfp_index_normalization_scalar_secondscale)) * cc) + (pfp_value_normalization_scalar_secondscale))) /\ ((((exists pfa_gap_normalization_scalar_secondscaleoperationleft. pfa_gap_normalization_scalar_secondscaleoperationleft + S (j) = (p)) /\ (((exists pfa_gap_normalization_scalar_secondscaleoperationright. pfa_gap_normalization_scalar_secondscaleoperationright + S (pfp_source_normalization_scalar_secondscale) = (p)) /\ ((((exists pfa_gap_normalization_scalar_secondscaleoperationresultbound. pfa_gap_normalization_scalar_secondscaleoperationresultbound + S (pfp_value_normalization_scalar_secondscale) = (p)) /\ ((exists pfa_offset_left_normalization_scalar_secondscaleoperationresultcongruence pfa_offset_right_normalization_scalar_secondscaleoperationresultcongruence. ((j) * (pfp_source_normalization_scalar_secondscale)) + (p) * pfa_offset_left_normalization_scalar_secondscaleoperationresultcongruence = (pfp_value_normalization_scalar_secondscale) + (p) * pfa_offset_right_normalization_scalar_secondscaleoperationresultcongruence)))))))))))))))))))))) -> k=jprime_field_polynomial_monic_normalization_functionalExact checked statement
forall p k j ab ac bb bc cb cc L. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_function_first. ((((exists ff_h_pfp_normalization_function_firstsource. ff_h_pfp_normalization_function_firstsource + S (pfm_leading_normalization_function_first) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_function_firstsource. ab = ff_q_pfp_normalization_function_firstsource * S ((S (0)) * ac) + (pfm_leading_normalization_function_first))) /\ ((((~((pfm_leading_normalization_function_first) = 0)) /\ ((((exists pfa_gap_normalization_function_firstinversemultiplicationleft. pfa_gap_normalization_function_firstinversemultiplicationleft + S (pfm_leading_normalization_function_first) = (p)) /\ (((exists pfa_gap_normalization_function_firstinversemultiplicationright. pfa_gap_normalization_function_firstinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_function_firstinversemultiplicationresultbound. pfa_gap_normalization_function_firstinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_function_firstinversemultiplicationresultcongruence pfa_offset_right_normalization_function_firstinversemultiplicationresultcongruence. ((pfm_leading_normalization_function_first) * (k)) + (p) * pfa_offset_left_normalization_function_firstinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_function_firstinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_function_firstscalescalar. pfa_gap_normalization_function_firstscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_function_firstscale. (exists pfa_gap_normalization_function_firstscaleindex. pfa_gap_normalization_function_firstscaleindex + S (pfp_index_normalization_function_firstscale) = (L)) -> exists pfp_source_normalization_function_firstscale pfp_value_normalization_function_firstscale. ((((exists ff_h_pfp_normalization_function_firstscalesource. ff_h_pfp_normalization_function_firstscalesource + S (pfp_source_normalization_function_firstscale) = S ((S (pfp_index_normalization_function_firstscale)) * ac)) /\ exists ff_q_pfp_normalization_function_firstscalesource. ab = ff_q_pfp_normalization_function_firstscalesource * S ((S (pfp_index_normalization_function_firstscale)) * ac) + (pfp_source_normalization_function_firstscale))) /\ (((((exists ff_h_pfp_normalization_function_firstscaletarget. ff_h_pfp_normalization_function_firstscaletarget + S (pfp_value_normalization_function_firstscale) = S ((S (pfp_index_normalization_function_firstscale)) * bc)) /\ exists ff_q_pfp_normalization_function_firstscaletarget. bb = ff_q_pfp_normalization_function_firstscaletarget * S ((S (pfp_index_normalization_function_firstscale)) * bc) + (pfp_value_normalization_function_firstscale))) /\ ((((exists pfa_gap_normalization_function_firstscaleoperationleft. pfa_gap_normalization_function_firstscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_function_firstscaleoperationright. pfa_gap_normalization_function_firstscaleoperationright + S (pfp_source_normalization_function_firstscale) = (p)) /\ ((((exists pfa_gap_normalization_function_firstscaleoperationresultbound. pfa_gap_normalization_function_firstscaleoperationresultbound + S (pfp_value_normalization_function_firstscale) = (p)) /\ ((exists pfa_offset_left_normalization_function_firstscaleoperationresultcongruence pfa_offset_right_normalization_function_firstscaleoperationresultcongruence. ((k) * (pfp_source_normalization_function_firstscale)) + (p) * pfa_offset_left_normalization_function_firstscaleoperationresultcongruence = (pfp_value_normalization_function_firstscale) + (p) * pfa_offset_right_normalization_function_firstscaleoperationresultcongruence)))))))))))))))))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_function_second. ((((exists ff_h_pfp_normalization_function_secondsource. ff_h_pfp_normalization_function_secondsource + S (pfm_leading_normalization_function_second) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_function_secondsource. ab = ff_q_pfp_normalization_function_secondsource * S ((S (0)) * ac) + (pfm_leading_normalization_function_second))) /\ ((((~((pfm_leading_normalization_function_second) = 0)) /\ ((((exists pfa_gap_normalization_function_secondinversemultiplicationleft. pfa_gap_normalization_function_secondinversemultiplicationleft + S (pfm_leading_normalization_function_second) = (p)) /\ (((exists pfa_gap_normalization_function_secondinversemultiplicationright. pfa_gap_normalization_function_secondinversemultiplicationright + S (j) = (p)) /\ ((((exists pfa_gap_normalization_function_secondinversemultiplicationresultbound. pfa_gap_normalization_function_secondinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_function_secondinversemultiplicationresultcongruence pfa_offset_right_normalization_function_secondinversemultiplicationresultcongruence. ((pfm_leading_normalization_function_second) * (j)) + (p) * pfa_offset_left_normalization_function_secondinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_function_secondinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_function_secondscalescalar. pfa_gap_normalization_function_secondscalescalar + S (j) = (p)) /\ ((forall pfp_index_normalization_function_secondscale. (exists pfa_gap_normalization_function_secondscaleindex. pfa_gap_normalization_function_secondscaleindex + S (pfp_index_normalization_function_secondscale) = (L)) -> exists pfp_source_normalization_function_secondscale pfp_value_normalization_function_secondscale. ((((exists ff_h_pfp_normalization_function_secondscalesource. ff_h_pfp_normalization_function_secondscalesource + S (pfp_source_normalization_function_secondscale) = S ((S (pfp_index_normalization_function_secondscale)) * ac)) /\ exists ff_q_pfp_normalization_function_secondscalesource. ab = ff_q_pfp_normalization_function_secondscalesource * S ((S (pfp_index_normalization_function_secondscale)) * ac) + (pfp_source_normalization_function_secondscale))) /\ (((((exists ff_h_pfp_normalization_function_secondscaletarget. ff_h_pfp_normalization_function_secondscaletarget + S (pfp_value_normalization_function_secondscale) = S ((S (pfp_index_normalization_function_secondscale)) * cc)) /\ exists ff_q_pfp_normalization_function_secondscaletarget. cb = ff_q_pfp_normalization_function_secondscaletarget * S ((S (pfp_index_normalization_function_secondscale)) * cc) + (pfp_value_normalization_function_secondscale))) /\ ((((exists pfa_gap_normalization_function_secondscaleoperationleft. pfa_gap_normalization_function_secondscaleoperationleft + S (j) = (p)) /\ (((exists pfa_gap_normalization_function_secondscaleoperationright. pfa_gap_normalization_function_secondscaleoperationright + S (pfp_source_normalization_function_secondscale) = (p)) /\ ((((exists pfa_gap_normalization_function_secondscaleoperationresultbound. pfa_gap_normalization_function_secondscaleoperationresultbound + S (pfp_value_normalization_function_secondscale) = (p)) /\ ((exists pfa_offset_left_normalization_function_secondscaleoperationresultcongruence pfa_offset_right_normalization_function_secondscaleoperationresultcongruence. ((j) * (pfp_source_normalization_function_secondscale)) + (p) * pfa_offset_left_normalization_function_secondscaleoperationresultcongruence = (pfp_value_normalization_function_secondscale) + (p) * pfa_offset_right_normalization_function_secondscaleoperationresultcongruence)))))))))))))))))))))) -> (forall mdr_i_pfp_normalization_function_result mdr_a_pfp_normalization_function_result. (exists mdr_gap_pfp_normalization_function_resultb. mdr_gap_pfp_normalization_function_resultb + S (mdr_i_pfp_normalization_function_result) = (L)) -> (((exists ff_h_mdr_pfp_normalization_function_resulto. ff_h_mdr_pfp_normalization_function_resulto + S (mdr_a_pfp_normalization_function_result) = S ((S (mdr_i_pfp_normalization_function_result)) * bc)) /\ exists ff_q_mdr_pfp_normalization_function_resulto. bb = ff_q_mdr_pfp_normalization_function_resulto * S ((S (mdr_i_pfp_normalization_function_result)) * bc) + (mdr_a_pfp_normalization_function_result))) -> (((exists ff_h_mdr_pfp_normalization_function_resultn. ff_h_mdr_pfp_normalization_function_resultn + S (mdr_a_pfp_normalization_function_result) = S ((S (mdr_i_pfp_normalization_function_result)) * cc)) /\ exists ff_q_mdr_pfp_normalization_function_resultn. cb = ff_q_mdr_pfp_normalization_function_resultn * S ((S (mdr_i_pfp_normalization_function_result)) * cc) + (mdr_a_pfp_normalization_function_result))))prime_field_polynomial_monic_normalization_value_functionalExact checked statement
forall p k j ab ac bb bc cb cc L i a b. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_value_first. ((((exists ff_h_pfp_normalization_value_firstsource. ff_h_pfp_normalization_value_firstsource + S (pfm_leading_normalization_value_first) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_value_firstsource. ab = ff_q_pfp_normalization_value_firstsource * S ((S (0)) * ac) + (pfm_leading_normalization_value_first))) /\ ((((~((pfm_leading_normalization_value_first) = 0)) /\ ((((exists pfa_gap_normalization_value_firstinversemultiplicationleft. pfa_gap_normalization_value_firstinversemultiplicationleft + S (pfm_leading_normalization_value_first) = (p)) /\ (((exists pfa_gap_normalization_value_firstinversemultiplicationright. pfa_gap_normalization_value_firstinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_value_firstinversemultiplicationresultbound. pfa_gap_normalization_value_firstinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_value_firstinversemultiplicationresultcongruence pfa_offset_right_normalization_value_firstinversemultiplicationresultcongruence. ((pfm_leading_normalization_value_first) * (k)) + (p) * pfa_offset_left_normalization_value_firstinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_value_firstinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_value_firstscalescalar. pfa_gap_normalization_value_firstscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_value_firstscale. (exists pfa_gap_normalization_value_firstscaleindex. pfa_gap_normalization_value_firstscaleindex + S (pfp_index_normalization_value_firstscale) = (L)) -> exists pfp_source_normalization_value_firstscale pfp_value_normalization_value_firstscale. ((((exists ff_h_pfp_normalization_value_firstscalesource. ff_h_pfp_normalization_value_firstscalesource + S (pfp_source_normalization_value_firstscale) = S ((S (pfp_index_normalization_value_firstscale)) * ac)) /\ exists ff_q_pfp_normalization_value_firstscalesource. ab = ff_q_pfp_normalization_value_firstscalesource * S ((S (pfp_index_normalization_value_firstscale)) * ac) + (pfp_source_normalization_value_firstscale))) /\ (((((exists ff_h_pfp_normalization_value_firstscaletarget. ff_h_pfp_normalization_value_firstscaletarget + S (pfp_value_normalization_value_firstscale) = S ((S (pfp_index_normalization_value_firstscale)) * bc)) /\ exists ff_q_pfp_normalization_value_firstscaletarget. bb = ff_q_pfp_normalization_value_firstscaletarget * S ((S (pfp_index_normalization_value_firstscale)) * bc) + (pfp_value_normalization_value_firstscale))) /\ ((((exists pfa_gap_normalization_value_firstscaleoperationleft. pfa_gap_normalization_value_firstscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_value_firstscaleoperationright. pfa_gap_normalization_value_firstscaleoperationright + S (pfp_source_normalization_value_firstscale) = (p)) /\ ((((exists pfa_gap_normalization_value_firstscaleoperationresultbound. pfa_gap_normalization_value_firstscaleoperationresultbound + S (pfp_value_normalization_value_firstscale) = (p)) /\ ((exists pfa_offset_left_normalization_value_firstscaleoperationresultcongruence pfa_offset_right_normalization_value_firstscaleoperationresultcongruence. ((k) * (pfp_source_normalization_value_firstscale)) + (p) * pfa_offset_left_normalization_value_firstscaleoperationresultcongruence = (pfp_value_normalization_value_firstscale) + (p) * pfa_offset_right_normalization_value_firstscaleoperationresultcongruence)))))))))))))))))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_value_second. ((((exists ff_h_pfp_normalization_value_secondsource. ff_h_pfp_normalization_value_secondsource + S (pfm_leading_normalization_value_second) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_value_secondsource. ab = ff_q_pfp_normalization_value_secondsource * S ((S (0)) * ac) + (pfm_leading_normalization_value_second))) /\ ((((~((pfm_leading_normalization_value_second) = 0)) /\ ((((exists pfa_gap_normalization_value_secondinversemultiplicationleft. pfa_gap_normalization_value_secondinversemultiplicationleft + S (pfm_leading_normalization_value_second) = (p)) /\ (((exists pfa_gap_normalization_value_secondinversemultiplicationright. pfa_gap_normalization_value_secondinversemultiplicationright + S (j) = (p)) /\ ((((exists pfa_gap_normalization_value_secondinversemultiplicationresultbound. pfa_gap_normalization_value_secondinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_value_secondinversemultiplicationresultcongruence pfa_offset_right_normalization_value_secondinversemultiplicationresultcongruence. ((pfm_leading_normalization_value_second) * (j)) + (p) * pfa_offset_left_normalization_value_secondinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_value_secondinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_value_secondscalescalar. pfa_gap_normalization_value_secondscalescalar + S (j) = (p)) /\ ((forall pfp_index_normalization_value_secondscale. (exists pfa_gap_normalization_value_secondscaleindex. pfa_gap_normalization_value_secondscaleindex + S (pfp_index_normalization_value_secondscale) = (L)) -> exists pfp_source_normalization_value_secondscale pfp_value_normalization_value_secondscale. ((((exists ff_h_pfp_normalization_value_secondscalesource. ff_h_pfp_normalization_value_secondscalesource + S (pfp_source_normalization_value_secondscale) = S ((S (pfp_index_normalization_value_secondscale)) * ac)) /\ exists ff_q_pfp_normalization_value_secondscalesource. ab = ff_q_pfp_normalization_value_secondscalesource * S ((S (pfp_index_normalization_value_secondscale)) * ac) + (pfp_source_normalization_value_secondscale))) /\ (((((exists ff_h_pfp_normalization_value_secondscaletarget. ff_h_pfp_normalization_value_secondscaletarget + S (pfp_value_normalization_value_secondscale) = S ((S (pfp_index_normalization_value_secondscale)) * cc)) /\ exists ff_q_pfp_normalization_value_secondscaletarget. cb = ff_q_pfp_normalization_value_secondscaletarget * S ((S (pfp_index_normalization_value_secondscale)) * cc) + (pfp_value_normalization_value_secondscale))) /\ ((((exists pfa_gap_normalization_value_secondscaleoperationleft. pfa_gap_normalization_value_secondscaleoperationleft + S (j) = (p)) /\ (((exists pfa_gap_normalization_value_secondscaleoperationright. pfa_gap_normalization_value_secondscaleoperationright + S (pfp_source_normalization_value_secondscale) = (p)) /\ ((((exists pfa_gap_normalization_value_secondscaleoperationresultbound. pfa_gap_normalization_value_secondscaleoperationresultbound + S (pfp_value_normalization_value_secondscale) = (p)) /\ ((exists pfa_offset_left_normalization_value_secondscaleoperationresultcongruence pfa_offset_right_normalization_value_secondscaleoperationresultcongruence. ((j) * (pfp_source_normalization_value_secondscale)) + (p) * pfa_offset_left_normalization_value_secondscaleoperationresultcongruence = (pfp_value_normalization_value_secondscale) + (p) * pfa_offset_right_normalization_value_secondscaleoperationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_normalization_value_bound. pfa_gap_normalization_value_bound + S (i) = (L)) -> (((exists ff_h_pfp_normalization_value_left. ff_h_pfp_normalization_value_left + S (a) = S ((S (i)) * bc)) /\ exists ff_q_pfp_normalization_value_left. bb = ff_q_pfp_normalization_value_left * S ((S (i)) * bc) + (a))) -> (((exists ff_h_pfp_normalization_value_right. ff_h_pfp_normalization_value_right + S (b) = S ((S (i)) * cc)) /\ exists ff_q_pfp_normalization_value_right. cb = ff_q_pfp_normalization_value_right * S ((S (i)) * cc) + (b))) -> a=bprime_field_polynomial_monic_normalization_transportExact checked statement
forall p k ab ac bb bc AB AC BB BC L. (forall mdr_i_pfp_normalization_recode_input mdr_a_pfp_normalization_recode_input. (exists mdr_gap_pfp_normalization_recode_inputb. mdr_gap_pfp_normalization_recode_inputb + S (mdr_i_pfp_normalization_recode_input) = (L)) -> (((exists ff_h_mdr_pfp_normalization_recode_inputo. ff_h_mdr_pfp_normalization_recode_inputo + S (mdr_a_pfp_normalization_recode_input) = S ((S (mdr_i_pfp_normalization_recode_input)) * ac)) /\ exists ff_q_mdr_pfp_normalization_recode_inputo. ab = ff_q_mdr_pfp_normalization_recode_inputo * S ((S (mdr_i_pfp_normalization_recode_input)) * ac) + (mdr_a_pfp_normalization_recode_input))) -> (((exists ff_h_mdr_pfp_normalization_recode_inputn. ff_h_mdr_pfp_normalization_recode_inputn + S (mdr_a_pfp_normalization_recode_input) = S ((S (mdr_i_pfp_normalization_recode_input)) * AC)) /\ exists ff_q_mdr_pfp_normalization_recode_inputn. AB = ff_q_mdr_pfp_normalization_recode_inputn * S ((S (mdr_i_pfp_normalization_recode_input)) * AC) + (mdr_a_pfp_normalization_recode_input)))) -> (forall mdr_i_pfp_normalization_recode_output mdr_a_pfp_normalization_recode_output. (exists mdr_gap_pfp_normalization_recode_outputb. mdr_gap_pfp_normalization_recode_outputb + S (mdr_i_pfp_normalization_recode_output) = (L)) -> (((exists ff_h_mdr_pfp_normalization_recode_outputo. ff_h_mdr_pfp_normalization_recode_outputo + S (mdr_a_pfp_normalization_recode_output) = S ((S (mdr_i_pfp_normalization_recode_output)) * bc)) /\ exists ff_q_mdr_pfp_normalization_recode_outputo. bb = ff_q_mdr_pfp_normalization_recode_outputo * S ((S (mdr_i_pfp_normalization_recode_output)) * bc) + (mdr_a_pfp_normalization_recode_output))) -> (((exists ff_h_mdr_pfp_normalization_recode_outputn. ff_h_mdr_pfp_normalization_recode_outputn + S (mdr_a_pfp_normalization_recode_output) = S ((S (mdr_i_pfp_normalization_recode_output)) * BC)) /\ exists ff_q_mdr_pfp_normalization_recode_outputn. BB = ff_q_mdr_pfp_normalization_recode_outputn * S ((S (mdr_i_pfp_normalization_recode_output)) * BC) + (mdr_a_pfp_normalization_recode_output)))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_recode_old. ((((exists ff_h_pfp_normalization_recode_oldsource. ff_h_pfp_normalization_recode_oldsource + S (pfm_leading_normalization_recode_old) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_recode_oldsource. ab = ff_q_pfp_normalization_recode_oldsource * S ((S (0)) * ac) + (pfm_leading_normalization_recode_old))) /\ ((((~((pfm_leading_normalization_recode_old) = 0)) /\ ((((exists pfa_gap_normalization_recode_oldinversemultiplicationleft. pfa_gap_normalization_recode_oldinversemultiplicationleft + S (pfm_leading_normalization_recode_old) = (p)) /\ (((exists pfa_gap_normalization_recode_oldinversemultiplicationright. pfa_gap_normalization_recode_oldinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_recode_oldinversemultiplicationresultbound. pfa_gap_normalization_recode_oldinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_recode_oldinversemultiplicationresultcongruence pfa_offset_right_normalization_recode_oldinversemultiplicationresultcongruence. ((pfm_leading_normalization_recode_old) * (k)) + (p) * pfa_offset_left_normalization_recode_oldinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_recode_oldinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_recode_oldscalescalar. pfa_gap_normalization_recode_oldscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_recode_oldscale. (exists pfa_gap_normalization_recode_oldscaleindex. pfa_gap_normalization_recode_oldscaleindex + S (pfp_index_normalization_recode_oldscale) = (L)) -> exists pfp_source_normalization_recode_oldscale pfp_value_normalization_recode_oldscale. ((((exists ff_h_pfp_normalization_recode_oldscalesource. ff_h_pfp_normalization_recode_oldscalesource + S (pfp_source_normalization_recode_oldscale) = S ((S (pfp_index_normalization_recode_oldscale)) * ac)) /\ exists ff_q_pfp_normalization_recode_oldscalesource. ab = ff_q_pfp_normalization_recode_oldscalesource * S ((S (pfp_index_normalization_recode_oldscale)) * ac) + (pfp_source_normalization_recode_oldscale))) /\ (((((exists ff_h_pfp_normalization_recode_oldscaletarget. ff_h_pfp_normalization_recode_oldscaletarget + S (pfp_value_normalization_recode_oldscale) = S ((S (pfp_index_normalization_recode_oldscale)) * bc)) /\ exists ff_q_pfp_normalization_recode_oldscaletarget. bb = ff_q_pfp_normalization_recode_oldscaletarget * S ((S (pfp_index_normalization_recode_oldscale)) * bc) + (pfp_value_normalization_recode_oldscale))) /\ ((((exists pfa_gap_normalization_recode_oldscaleoperationleft. pfa_gap_normalization_recode_oldscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_recode_oldscaleoperationright. pfa_gap_normalization_recode_oldscaleoperationright + S (pfp_source_normalization_recode_oldscale) = (p)) /\ ((((exists pfa_gap_normalization_recode_oldscaleoperationresultbound. pfa_gap_normalization_recode_oldscaleoperationresultbound + S (pfp_value_normalization_recode_oldscale) = (p)) /\ ((exists pfa_offset_left_normalization_recode_oldscaleoperationresultcongruence pfa_offset_right_normalization_recode_oldscaleoperationresultcongruence. ((k) * (pfp_source_normalization_recode_oldscale)) + (p) * pfa_offset_left_normalization_recode_oldscaleoperationresultcongruence = (pfp_value_normalization_recode_oldscale) + (p) * pfa_offset_right_normalization_recode_oldscaleoperationresultcongruence)))))))))))))))))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_recode_new. ((((exists ff_h_pfp_normalization_recode_newsource. ff_h_pfp_normalization_recode_newsource + S (pfm_leading_normalization_recode_new) = S ((S (0)) * AC)) /\ exists ff_q_pfp_normalization_recode_newsource. AB = ff_q_pfp_normalization_recode_newsource * S ((S (0)) * AC) + (pfm_leading_normalization_recode_new))) /\ ((((~((pfm_leading_normalization_recode_new) = 0)) /\ ((((exists pfa_gap_normalization_recode_newinversemultiplicationleft. pfa_gap_normalization_recode_newinversemultiplicationleft + S (pfm_leading_normalization_recode_new) = (p)) /\ (((exists pfa_gap_normalization_recode_newinversemultiplicationright. pfa_gap_normalization_recode_newinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_recode_newinversemultiplicationresultbound. pfa_gap_normalization_recode_newinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_recode_newinversemultiplicationresultcongruence pfa_offset_right_normalization_recode_newinversemultiplicationresultcongruence. ((pfm_leading_normalization_recode_new) * (k)) + (p) * pfa_offset_left_normalization_recode_newinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_recode_newinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_recode_newscalescalar. pfa_gap_normalization_recode_newscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_recode_newscale. (exists pfa_gap_normalization_recode_newscaleindex. pfa_gap_normalization_recode_newscaleindex + S (pfp_index_normalization_recode_newscale) = (L)) -> exists pfp_source_normalization_recode_newscale pfp_value_normalization_recode_newscale. ((((exists ff_h_pfp_normalization_recode_newscalesource. ff_h_pfp_normalization_recode_newscalesource + S (pfp_source_normalization_recode_newscale) = S ((S (pfp_index_normalization_recode_newscale)) * AC)) /\ exists ff_q_pfp_normalization_recode_newscalesource. AB = ff_q_pfp_normalization_recode_newscalesource * S ((S (pfp_index_normalization_recode_newscale)) * AC) + (pfp_source_normalization_recode_newscale))) /\ (((((exists ff_h_pfp_normalization_recode_newscaletarget. ff_h_pfp_normalization_recode_newscaletarget + S (pfp_value_normalization_recode_newscale) = S ((S (pfp_index_normalization_recode_newscale)) * BC)) /\ exists ff_q_pfp_normalization_recode_newscaletarget. BB = ff_q_pfp_normalization_recode_newscaletarget * S ((S (pfp_index_normalization_recode_newscale)) * BC) + (pfp_value_normalization_recode_newscale))) /\ ((((exists pfa_gap_normalization_recode_newscaleoperationleft. pfa_gap_normalization_recode_newscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_recode_newscaleoperationright. pfa_gap_normalization_recode_newscaleoperationright + S (pfp_source_normalization_recode_newscale) = (p)) /\ ((((exists pfa_gap_normalization_recode_newscaleoperationresultbound. pfa_gap_normalization_recode_newscaleoperationresultbound + S (pfp_value_normalization_recode_newscale) = (p)) /\ ((exists pfa_offset_left_normalization_recode_newscaleoperationresultcongruence pfa_offset_right_normalization_recode_newscaleoperationresultcongruence. ((k) * (pfp_source_normalization_recode_newscale)) + (p) * pfa_offset_left_normalization_recode_newscaleoperationresultcongruence = (pfp_value_normalization_recode_newscale) + (p) * pfa_offset_right_normalization_recode_newscaleoperationresultcongruence))))))))))))))))))))))prime_field_polynomial_monic_normalization_fixedExact checked statement
forall p b c L. (~((p) = 1) /\ forall pfa_factor_left_normalization_fixed_prime pfa_factor_right_normalization_fixed_prime. (p) = pfa_factor_left_normalization_fixed_prime * pfa_factor_right_normalization_fixed_prime -> pfa_factor_left_normalization_fixed_prime = 1 \/ pfa_factor_right_normalization_fixed_prime = 1) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_normalization_fixed_inputcoefficients. (exists fom_gap_pfp_normalization_fixed_inputcoefficients_index_bound. fom_gap_pfp_normalization_fixed_inputcoefficients_index_bound + S (fom_index_pfp_normalization_fixed_inputcoefficients) = L) -> exists fom_value_pfp_normalization_fixed_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_fixed_inputcoefficients_entry. fom_beta_height_pfp_normalization_fixed_inputcoefficients_entry + S (fom_value_pfp_normalization_fixed_inputcoefficients) = S ((S (fom_index_pfp_normalization_fixed_inputcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_normalization_fixed_inputcoefficients_entry. b = fom_beta_quotient_pfp_normalization_fixed_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_fixed_inputcoefficients)) * c) + (fom_value_pfp_normalization_fixed_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_fixed_inputcoefficients_value_bound. fom_gap_pfp_normalization_fixed_inputcoefficients_value_bound + S (fom_value_pfp_normalization_fixed_inputcoefficients) = p))) /\ ((((exists ff_h_pfp_normalization_fixed_inputleading. ff_h_pfp_normalization_fixed_inputleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_normalization_fixed_inputleading. b = ff_q_pfp_normalization_fixed_inputleading * S ((S (0)) * c) + (1)))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_fixed_result. ((((exists ff_h_pfp_normalization_fixed_resultsource. ff_h_pfp_normalization_fixed_resultsource + S (pfm_leading_normalization_fixed_result) = S ((S (0)) * c)) /\ exists ff_q_pfp_normalization_fixed_resultsource. b = ff_q_pfp_normalization_fixed_resultsource * S ((S (0)) * c) + (pfm_leading_normalization_fixed_result))) /\ ((((~((pfm_leading_normalization_fixed_result) = 0)) /\ ((((exists pfa_gap_normalization_fixed_resultinversemultiplicationleft. pfa_gap_normalization_fixed_resultinversemultiplicationleft + S (pfm_leading_normalization_fixed_result) = (p)) /\ (((exists pfa_gap_normalization_fixed_resultinversemultiplicationright. pfa_gap_normalization_fixed_resultinversemultiplicationright + S (1) = (p)) /\ ((((exists pfa_gap_normalization_fixed_resultinversemultiplicationresultbound. pfa_gap_normalization_fixed_resultinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_fixed_resultinversemultiplicationresultcongruence pfa_offset_right_normalization_fixed_resultinversemultiplicationresultcongruence. ((pfm_leading_normalization_fixed_result) * (1)) + (p) * pfa_offset_left_normalization_fixed_resultinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_fixed_resultinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_fixed_resultscalescalar. pfa_gap_normalization_fixed_resultscalescalar + S (1) = (p)) /\ ((forall pfp_index_normalization_fixed_resultscale. (exists pfa_gap_normalization_fixed_resultscaleindex. pfa_gap_normalization_fixed_resultscaleindex + S (pfp_index_normalization_fixed_resultscale) = (L)) -> exists pfp_source_normalization_fixed_resultscale pfp_value_normalization_fixed_resultscale. ((((exists ff_h_pfp_normalization_fixed_resultscalesource. ff_h_pfp_normalization_fixed_resultscalesource + S (pfp_source_normalization_fixed_resultscale) = S ((S (pfp_index_normalization_fixed_resultscale)) * c)) /\ exists ff_q_pfp_normalization_fixed_resultscalesource. b = ff_q_pfp_normalization_fixed_resultscalesource * S ((S (pfp_index_normalization_fixed_resultscale)) * c) + (pfp_source_normalization_fixed_resultscale))) /\ (((((exists ff_h_pfp_normalization_fixed_resultscaletarget. ff_h_pfp_normalization_fixed_resultscaletarget + S (pfp_value_normalization_fixed_resultscale) = S ((S (pfp_index_normalization_fixed_resultscale)) * c)) /\ exists ff_q_pfp_normalization_fixed_resultscaletarget. b = ff_q_pfp_normalization_fixed_resultscaletarget * S ((S (pfp_index_normalization_fixed_resultscale)) * c) + (pfp_value_normalization_fixed_resultscale))) /\ ((((exists pfa_gap_normalization_fixed_resultscaleoperationleft. pfa_gap_normalization_fixed_resultscaleoperationleft + S (1) = (p)) /\ (((exists pfa_gap_normalization_fixed_resultscaleoperationright. pfa_gap_normalization_fixed_resultscaleoperationright + S (pfp_source_normalization_fixed_resultscale) = (p)) /\ ((((exists pfa_gap_normalization_fixed_resultscaleoperationresultbound. pfa_gap_normalization_fixed_resultscaleoperationresultbound + S (pfp_value_normalization_fixed_resultscale) = (p)) /\ ((exists pfa_offset_left_normalization_fixed_resultscaleoperationresultcongruence pfa_offset_right_normalization_fixed_resultscaleoperationresultcongruence. ((1) * (pfp_source_normalization_fixed_resultscale)) + (p) * pfa_offset_left_normalization_fixed_resultscaleoperationresultcongruence = (pfp_value_normalization_fixed_resultscale) + (p) * pfa_offset_right_normalization_fixed_resultscaleoperationresultcongruence))))))))))))))))))))))prime_field_polynomial_monic_normalization_constantExact checked statement
forall p k ab ac bb bc. (((~((1) = 0)) /\ (((exists pfm_leading_normalization_constant_source. ((((exists ff_h_pfp_normalization_constant_sourcesource. ff_h_pfp_normalization_constant_sourcesource + S (pfm_leading_normalization_constant_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_constant_sourcesource. ab = ff_q_pfp_normalization_constant_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_constant_source))) /\ ((((~((pfm_leading_normalization_constant_source) = 0)) /\ ((((exists pfa_gap_normalization_constant_sourceinversemultiplicationleft. pfa_gap_normalization_constant_sourceinversemultiplicationleft + S (pfm_leading_normalization_constant_source) = (p)) /\ (((exists pfa_gap_normalization_constant_sourceinversemultiplicationright. pfa_gap_normalization_constant_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_constant_sourceinversemultiplicationresultbound. pfa_gap_normalization_constant_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_constant_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_constant_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_constant_source) * (k)) + (p) * pfa_offset_left_normalization_constant_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_constant_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_constant_sourcescalescalar. pfa_gap_normalization_constant_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_constant_sourcescale. (exists pfa_gap_normalization_constant_sourcescaleindex. pfa_gap_normalization_constant_sourcescaleindex + S (pfp_index_normalization_constant_sourcescale) = (1)) -> exists pfp_source_normalization_constant_sourcescale pfp_value_normalization_constant_sourcescale. ((((exists ff_h_pfp_normalization_constant_sourcescalesource. ff_h_pfp_normalization_constant_sourcescalesource + S (pfp_source_normalization_constant_sourcescale) = S ((S (pfp_index_normalization_constant_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_constant_sourcescalesource. ab = ff_q_pfp_normalization_constant_sourcescalesource * S ((S (pfp_index_normalization_constant_sourcescale)) * ac) + (pfp_source_normalization_constant_sourcescale))) /\ (((((exists ff_h_pfp_normalization_constant_sourcescaletarget. ff_h_pfp_normalization_constant_sourcescaletarget + S (pfp_value_normalization_constant_sourcescale) = S ((S (pfp_index_normalization_constant_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_constant_sourcescaletarget. bb = ff_q_pfp_normalization_constant_sourcescaletarget * S ((S (pfp_index_normalization_constant_sourcescale)) * bc) + (pfp_value_normalization_constant_sourcescale))) /\ ((((exists pfa_gap_normalization_constant_sourcescaleoperationleft. pfa_gap_normalization_constant_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_constant_sourcescaleoperationright. pfa_gap_normalization_constant_sourcescaleoperationright + S (pfp_source_normalization_constant_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_constant_sourcescaleoperationresultbound. pfa_gap_normalization_constant_sourcescaleoperationresultbound + S (pfp_value_normalization_constant_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_constant_sourcescaleoperationresultcongruence pfa_offset_right_normalization_constant_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_constant_sourcescale)) + (p) * pfa_offset_left_normalization_constant_sourcescaleoperationresultcongruence = (pfp_value_normalization_constant_sourcescale) + (p) * pfa_offset_right_normalization_constant_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (forall pfp_repeat_index_normalization_constant_result. (exists pfa_gap_normalization_constant_resultindex. pfa_gap_normalization_constant_resultindex + S (pfp_repeat_index_normalization_constant_result) = (1)) -> (((exists ff_h_pfp_normalization_constant_resultentry. ff_h_pfp_normalization_constant_resultentry + S (1) = S ((S (pfp_repeat_index_normalization_constant_result)) * bc)) /\ exists ff_q_pfp_normalization_constant_resultentry. bb = ff_q_pfp_normalization_constant_resultentry * S ((S (pfp_repeat_index_normalization_constant_result)) * bc) + (1))))prime_field_polynomial_monic_normalization_exists_uniqueExact checked statement
forall p ab ac L d. (~((p) = 1) /\ forall pfa_factor_left_normalization_unique_prime pfa_factor_right_normalization_unique_prime. (p) = pfa_factor_left_normalization_unique_prime * pfa_factor_right_normalization_unique_prime -> pfa_factor_left_normalization_unique_prime = 1 \/ pfa_factor_right_normalization_unique_prime = 1) -> ((((L)=S (d)) /\ (((forall fom_index_pfp_normalization_unique_inputcoefficients. (exists fom_gap_pfp_normalization_unique_inputcoefficients_index_bound. fom_gap_pfp_normalization_unique_inputcoefficients_index_bound + S (fom_index_pfp_normalization_unique_inputcoefficients) = L) -> exists fom_value_pfp_normalization_unique_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_unique_inputcoefficients_entry. fom_beta_height_pfp_normalization_unique_inputcoefficients_entry + S (fom_value_pfp_normalization_unique_inputcoefficients) = S ((S (fom_index_pfp_normalization_unique_inputcoefficients)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_unique_inputcoefficients_entry. ab = fom_beta_quotient_pfp_normalization_unique_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_unique_inputcoefficients)) * ac) + (fom_value_pfp_normalization_unique_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_unique_inputcoefficients_value_bound. fom_gap_pfp_normalization_unique_inputcoefficients_value_bound + S (fom_value_pfp_normalization_unique_inputcoefficients) = p))) /\ ((exists pfd_leading_normalization_unique_input. ((((exists ff_h_pfp_normalization_unique_inputentry. ff_h_pfp_normalization_unique_inputentry + S (pfd_leading_normalization_unique_input) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_unique_inputentry. ab = ff_q_pfp_normalization_unique_inputentry * S ((S (0)) * ac) + (pfd_leading_normalization_unique_input))) /\ ((~(pfd_leading_normalization_unique_input=0)))))))))) -> exists k bb bc. (((((~((L) = 0)) /\ (((exists pfm_leading_normalization_unique_graph. ((((exists ff_h_pfp_normalization_unique_graphsource. ff_h_pfp_normalization_unique_graphsource + S (pfm_leading_normalization_unique_graph) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_unique_graphsource. ab = ff_q_pfp_normalization_unique_graphsource * S ((S (0)) * ac) + (pfm_leading_normalization_unique_graph))) /\ ((((~((pfm_leading_normalization_unique_graph) = 0)) /\ ((((exists pfa_gap_normalization_unique_graphinversemultiplicationleft. pfa_gap_normalization_unique_graphinversemultiplicationleft + S (pfm_leading_normalization_unique_graph) = (p)) /\ (((exists pfa_gap_normalization_unique_graphinversemultiplicationright. pfa_gap_normalization_unique_graphinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_unique_graphinversemultiplicationresultbound. pfa_gap_normalization_unique_graphinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_unique_graphinversemultiplicationresultcongruence pfa_offset_right_normalization_unique_graphinversemultiplicationresultcongruence. ((pfm_leading_normalization_unique_graph) * (k)) + (p) * pfa_offset_left_normalization_unique_graphinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_unique_graphinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_unique_graphscalescalar. pfa_gap_normalization_unique_graphscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_unique_graphscale. (exists pfa_gap_normalization_unique_graphscaleindex. pfa_gap_normalization_unique_graphscaleindex + S (pfp_index_normalization_unique_graphscale) = (L)) -> exists pfp_source_normalization_unique_graphscale pfp_value_normalization_unique_graphscale. ((((exists ff_h_pfp_normalization_unique_graphscalesource. ff_h_pfp_normalization_unique_graphscalesource + S (pfp_source_normalization_unique_graphscale) = S ((S (pfp_index_normalization_unique_graphscale)) * ac)) /\ exists ff_q_pfp_normalization_unique_graphscalesource. ab = ff_q_pfp_normalization_unique_graphscalesource * S ((S (pfp_index_normalization_unique_graphscale)) * ac) + (pfp_source_normalization_unique_graphscale))) /\ (((((exists ff_h_pfp_normalization_unique_graphscaletarget. ff_h_pfp_normalization_unique_graphscaletarget + S (pfp_value_normalization_unique_graphscale) = S ((S (pfp_index_normalization_unique_graphscale)) * bc)) /\ exists ff_q_pfp_normalization_unique_graphscaletarget. bb = ff_q_pfp_normalization_unique_graphscaletarget * S ((S (pfp_index_normalization_unique_graphscale)) * bc) + (pfp_value_normalization_unique_graphscale))) /\ ((((exists pfa_gap_normalization_unique_graphscaleoperationleft. pfa_gap_normalization_unique_graphscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_unique_graphscaleoperationright. pfa_gap_normalization_unique_graphscaleoperationright + S (pfp_source_normalization_unique_graphscale) = (p)) /\ ((((exists pfa_gap_normalization_unique_graphscaleoperationresultbound. pfa_gap_normalization_unique_graphscaleoperationresultbound + S (pfp_value_normalization_unique_graphscale) = (p)) /\ ((exists pfa_offset_left_normalization_unique_graphscaleoperationresultcongruence pfa_offset_right_normalization_unique_graphscaleoperationresultcongruence. ((k) * (pfp_source_normalization_unique_graphscale)) + (p) * pfa_offset_left_normalization_unique_graphscaleoperationresultcongruence = (pfp_value_normalization_unique_graphscale) + (p) * pfa_offset_right_normalization_unique_graphscaleoperationresultcongruence)))))))))))))))))))))) /\ (((((~((L) = 0)) /\ (((forall fom_index_pfp_normalization_unique_moniccoefficients. (exists fom_gap_pfp_normalization_unique_moniccoefficients_index_bound. fom_gap_pfp_normalization_unique_moniccoefficients_index_bound + S (fom_index_pfp_normalization_unique_moniccoefficients) = L) -> exists fom_value_pfp_normalization_unique_moniccoefficients. ((((exists fom_beta_height_pfp_normalization_unique_moniccoefficients_entry. fom_beta_height_pfp_normalization_unique_moniccoefficients_entry + S (fom_value_pfp_normalization_unique_moniccoefficients) = S ((S (fom_index_pfp_normalization_unique_moniccoefficients)) * bc)) /\ exists fom_beta_quotient_pfp_normalization_unique_moniccoefficients_entry. bb = fom_beta_quotient_pfp_normalization_unique_moniccoefficients_entry * S ((S (fom_index_pfp_normalization_unique_moniccoefficients)) * bc) + (fom_value_pfp_normalization_unique_moniccoefficients))) /\ (exists fom_gap_pfp_normalization_unique_moniccoefficients_value_bound. fom_gap_pfp_normalization_unique_moniccoefficients_value_bound + S (fom_value_pfp_normalization_unique_moniccoefficients) = p))) /\ ((((exists ff_h_pfp_normalization_unique_monicleading. ff_h_pfp_normalization_unique_monicleading + S (1) = S ((S (0)) * bc)) /\ exists ff_q_pfp_normalization_unique_monicleading. bb = ff_q_pfp_normalization_unique_monicleading * S ((S (0)) * bc) + (1)))))))) /\ ((((((L)=S (d)) /\ (((forall fom_index_pfp_normalization_unique_degreecoefficients. (exists fom_gap_pfp_normalization_unique_degreecoefficients_index_bound. fom_gap_pfp_normalization_unique_degreecoefficients_index_bound + S (fom_index_pfp_normalization_unique_degreecoefficients) = L) -> exists fom_value_pfp_normalization_unique_degreecoefficients. ((((exists fom_beta_height_pfp_normalization_unique_degreecoefficients_entry. fom_beta_height_pfp_normalization_unique_degreecoefficients_entry + S (fom_value_pfp_normalization_unique_degreecoefficients) = S ((S (fom_index_pfp_normalization_unique_degreecoefficients)) * bc)) /\ exists fom_beta_quotient_pfp_normalization_unique_degreecoefficients_entry. bb = fom_beta_quotient_pfp_normalization_unique_degreecoefficients_entry * S ((S (fom_index_pfp_normalization_unique_degreecoefficients)) * bc) + (fom_value_pfp_normalization_unique_degreecoefficients))) /\ (exists fom_gap_pfp_normalization_unique_degreecoefficients_value_bound. fom_gap_pfp_normalization_unique_degreecoefficients_value_bound + S (fom_value_pfp_normalization_unique_degreecoefficients) = p))) /\ ((exists pfd_leading_normalization_unique_degree. ((((exists ff_h_pfp_normalization_unique_degreeentry. ff_h_pfp_normalization_unique_degreeentry + S (pfd_leading_normalization_unique_degree) = S ((S (0)) * bc)) /\ exists ff_q_pfp_normalization_unique_degreeentry. bb = ff_q_pfp_normalization_unique_degreeentry * S ((S (0)) * bc) + (pfd_leading_normalization_unique_degree))) /\ ((~(pfd_leading_normalization_unique_degree=0)))))))))) /\ ((forall j cb cc. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_uniquecomparison. ((((exists ff_h_pfp_normalization_uniquecomparisonsource. ff_h_pfp_normalization_uniquecomparisonsource + S (pfm_leading_normalization_uniquecomparison) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_uniquecomparisonsource. ab = ff_q_pfp_normalization_uniquecomparisonsource * S ((S (0)) * ac) + (pfm_leading_normalization_uniquecomparison))) /\ ((((~((pfm_leading_normalization_uniquecomparison) = 0)) /\ ((((exists pfa_gap_normalization_uniquecomparisoninversemultiplicationleft. pfa_gap_normalization_uniquecomparisoninversemultiplicationleft + S (pfm_leading_normalization_uniquecomparison) = (p)) /\ (((exists pfa_gap_normalization_uniquecomparisoninversemultiplicationright. pfa_gap_normalization_uniquecomparisoninversemultiplicationright + S (j) = (p)) /\ ((((exists pfa_gap_normalization_uniquecomparisoninversemultiplicationresultbound. pfa_gap_normalization_uniquecomparisoninversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_uniquecomparisoninversemultiplicationresultcongruence pfa_offset_right_normalization_uniquecomparisoninversemultiplicationresultcongruence. ((pfm_leading_normalization_uniquecomparison) * (j)) + (p) * pfa_offset_left_normalization_uniquecomparisoninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_uniquecomparisoninversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_uniquecomparisonscalescalar. pfa_gap_normalization_uniquecomparisonscalescalar + S (j) = (p)) /\ ((forall pfp_index_normalization_uniquecomparisonscale. (exists pfa_gap_normalization_uniquecomparisonscaleindex. pfa_gap_normalization_uniquecomparisonscaleindex + S (pfp_index_normalization_uniquecomparisonscale) = (L)) -> exists pfp_source_normalization_uniquecomparisonscale pfp_value_normalization_uniquecomparisonscale. ((((exists ff_h_pfp_normalization_uniquecomparisonscalesource. ff_h_pfp_normalization_uniquecomparisonscalesource + S (pfp_source_normalization_uniquecomparisonscale) = S ((S (pfp_index_normalization_uniquecomparisonscale)) * ac)) /\ exists ff_q_pfp_normalization_uniquecomparisonscalesource. ab = ff_q_pfp_normalization_uniquecomparisonscalesource * S ((S (pfp_index_normalization_uniquecomparisonscale)) * ac) + (pfp_source_normalization_uniquecomparisonscale))) /\ (((((exists ff_h_pfp_normalization_uniquecomparisonscaletarget. ff_h_pfp_normalization_uniquecomparisonscaletarget + S (pfp_value_normalization_uniquecomparisonscale) = S ((S (pfp_index_normalization_uniquecomparisonscale)) * cc)) /\ exists ff_q_pfp_normalization_uniquecomparisonscaletarget. cb = ff_q_pfp_normalization_uniquecomparisonscaletarget * S ((S (pfp_index_normalization_uniquecomparisonscale)) * cc) + (pfp_value_normalization_uniquecomparisonscale))) /\ ((((exists pfa_gap_normalization_uniquecomparisonscaleoperationleft. pfa_gap_normalization_uniquecomparisonscaleoperationleft + S (j) = (p)) /\ (((exists pfa_gap_normalization_uniquecomparisonscaleoperationright. pfa_gap_normalization_uniquecomparisonscaleoperationright + S (pfp_source_normalization_uniquecomparisonscale) = (p)) /\ ((((exists pfa_gap_normalization_uniquecomparisonscaleoperationresultbound. pfa_gap_normalization_uniquecomparisonscaleoperationresultbound + S (pfp_value_normalization_uniquecomparisonscale) = (p)) /\ ((exists pfa_offset_left_normalization_uniquecomparisonscaleoperationresultcongruence pfa_offset_right_normalization_uniquecomparisonscaleoperationresultcongruence. ((j) * (pfp_source_normalization_uniquecomparisonscale)) + (p) * pfa_offset_left_normalization_uniquecomparisonscaleoperationresultcongruence = (pfp_value_normalization_uniquecomparisonscale) + (p) * pfa_offset_right_normalization_uniquecomparisonscaleoperationresultcongruence)))))))))))))))))))))) -> ((j=(k)) /\ ((forall mdr_i_pfp_normalization_uniqueequal mdr_a_pfp_normalization_uniqueequal. (exists mdr_gap_pfp_normalization_uniqueequalb. mdr_gap_pfp_normalization_uniqueequalb + S (mdr_i_pfp_normalization_uniqueequal) = (L)) -> (((exists ff_h_mdr_pfp_normalization_uniqueequalo. ff_h_mdr_pfp_normalization_uniqueequalo + S (mdr_a_pfp_normalization_uniqueequal) = S ((S (mdr_i_pfp_normalization_uniqueequal)) * cc)) /\ exists ff_q_mdr_pfp_normalization_uniqueequalo. cb = ff_q_mdr_pfp_normalization_uniqueequalo * S ((S (mdr_i_pfp_normalization_uniqueequal)) * cc) + (mdr_a_pfp_normalization_uniqueequal))) -> (((exists ff_h_mdr_pfp_normalization_uniqueequaln. ff_h_mdr_pfp_normalization_uniqueequaln + S (mdr_a_pfp_normalization_uniqueequal) = S ((S (mdr_i_pfp_normalization_uniqueequal)) * bc)) /\ exists ff_q_mdr_pfp_normalization_uniqueequaln. bb = ff_q_mdr_pfp_normalization_uniqueequaln * S ((S (mdr_i_pfp_normalization_uniqueequal)) * bc) + (mdr_a_pfp_normalization_uniqueequal))))))))))))))prime_field_polynomial_monic_normalization_degree_zero_existsExact checked statement
forall p ab ac. (~((p) = 1) /\ forall pfa_factor_left_normalization_zero_prime pfa_factor_right_normalization_zero_prime. (p) = pfa_factor_left_normalization_zero_prime * pfa_factor_right_normalization_zero_prime -> pfa_factor_left_normalization_zero_prime = 1 \/ pfa_factor_right_normalization_zero_prime = 1) -> ((((1)=S (0)) /\ (((forall fom_index_pfp_normalization_zero_inputcoefficients. (exists fom_gap_pfp_normalization_zero_inputcoefficients_index_bound. fom_gap_pfp_normalization_zero_inputcoefficients_index_bound + S (fom_index_pfp_normalization_zero_inputcoefficients) = 1) -> exists fom_value_pfp_normalization_zero_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_zero_inputcoefficients_entry. fom_beta_height_pfp_normalization_zero_inputcoefficients_entry + S (fom_value_pfp_normalization_zero_inputcoefficients) = S ((S (fom_index_pfp_normalization_zero_inputcoefficients)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_zero_inputcoefficients_entry. ab = fom_beta_quotient_pfp_normalization_zero_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_zero_inputcoefficients)) * ac) + (fom_value_pfp_normalization_zero_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_zero_inputcoefficients_value_bound. fom_gap_pfp_normalization_zero_inputcoefficients_value_bound + S (fom_value_pfp_normalization_zero_inputcoefficients) = p))) /\ ((exists pfd_leading_normalization_zero_input. ((((exists ff_h_pfp_normalization_zero_inputentry. ff_h_pfp_normalization_zero_inputentry + S (pfd_leading_normalization_zero_input) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_zero_inputentry. ab = ff_q_pfp_normalization_zero_inputentry * S ((S (0)) * ac) + (pfd_leading_normalization_zero_input))) /\ ((~(pfd_leading_normalization_zero_input=0)))))))))) -> exists k bb bc. ((((~((1) = 0)) /\ (((exists pfm_leading_normalization_zero_graph. ((((exists ff_h_pfp_normalization_zero_graphsource. ff_h_pfp_normalization_zero_graphsource + S (pfm_leading_normalization_zero_graph) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_zero_graphsource. ab = ff_q_pfp_normalization_zero_graphsource * S ((S (0)) * ac) + (pfm_leading_normalization_zero_graph))) /\ ((((~((pfm_leading_normalization_zero_graph) = 0)) /\ ((((exists pfa_gap_normalization_zero_graphinversemultiplicationleft. pfa_gap_normalization_zero_graphinversemultiplicationleft + S (pfm_leading_normalization_zero_graph) = (p)) /\ (((exists pfa_gap_normalization_zero_graphinversemultiplicationright. pfa_gap_normalization_zero_graphinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_zero_graphinversemultiplicationresultbound. pfa_gap_normalization_zero_graphinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_zero_graphinversemultiplicationresultcongruence pfa_offset_right_normalization_zero_graphinversemultiplicationresultcongruence. ((pfm_leading_normalization_zero_graph) * (k)) + (p) * pfa_offset_left_normalization_zero_graphinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_zero_graphinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_zero_graphscalescalar. pfa_gap_normalization_zero_graphscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_zero_graphscale. (exists pfa_gap_normalization_zero_graphscaleindex. pfa_gap_normalization_zero_graphscaleindex + S (pfp_index_normalization_zero_graphscale) = (1)) -> exists pfp_source_normalization_zero_graphscale pfp_value_normalization_zero_graphscale. ((((exists ff_h_pfp_normalization_zero_graphscalesource. ff_h_pfp_normalization_zero_graphscalesource + S (pfp_source_normalization_zero_graphscale) = S ((S (pfp_index_normalization_zero_graphscale)) * ac)) /\ exists ff_q_pfp_normalization_zero_graphscalesource. ab = ff_q_pfp_normalization_zero_graphscalesource * S ((S (pfp_index_normalization_zero_graphscale)) * ac) + (pfp_source_normalization_zero_graphscale))) /\ (((((exists ff_h_pfp_normalization_zero_graphscaletarget. ff_h_pfp_normalization_zero_graphscaletarget + S (pfp_value_normalization_zero_graphscale) = S ((S (pfp_index_normalization_zero_graphscale)) * bc)) /\ exists ff_q_pfp_normalization_zero_graphscaletarget. bb = ff_q_pfp_normalization_zero_graphscaletarget * S ((S (pfp_index_normalization_zero_graphscale)) * bc) + (pfp_value_normalization_zero_graphscale))) /\ ((((exists pfa_gap_normalization_zero_graphscaleoperationleft. pfa_gap_normalization_zero_graphscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_zero_graphscaleoperationright. pfa_gap_normalization_zero_graphscaleoperationright + S (pfp_source_normalization_zero_graphscale) = (p)) /\ ((((exists pfa_gap_normalization_zero_graphscaleoperationresultbound. pfa_gap_normalization_zero_graphscaleoperationresultbound + S (pfp_value_normalization_zero_graphscale) = (p)) /\ ((exists pfa_offset_left_normalization_zero_graphscaleoperationresultcongruence pfa_offset_right_normalization_zero_graphscaleoperationresultcongruence. ((k) * (pfp_source_normalization_zero_graphscale)) + (p) * pfa_offset_left_normalization_zero_graphscaleoperationresultcongruence = (pfp_value_normalization_zero_graphscale) + (p) * pfa_offset_right_normalization_zero_graphscaleoperationresultcongruence)))))))))))))))))))))) /\ ((forall pfp_repeat_index_normalization_zero_output. (exists pfa_gap_normalization_zero_outputindex. pfa_gap_normalization_zero_outputindex + S (pfp_repeat_index_normalization_zero_output) = (1)) -> (((exists ff_h_pfp_normalization_zero_outputentry. ff_h_pfp_normalization_zero_outputentry + S (1) = S ((S (pfp_repeat_index_normalization_zero_output)) * bc)) /\ exists ff_q_pfp_normalization_zero_outputentry. bb = ff_q_pfp_normalization_zero_outputentry * S ((S (pfp_repeat_index_normalization_zero_output)) * bc) + (1))))))prime_field_polynomial_horner_trace_prefixExact checked statement
forall p b c a l r u v n h. (((exists pfa_gap_prefix_tracebase. pfa_gap_prefix_tracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_prefix_traceinitial. ff_h_pfp_prefix_traceinitial + S (0) = S ((S (0)) * v)) /\ exists ff_q_pfp_prefix_traceinitial. u = ff_q_pfp_prefix_traceinitial * S ((S (0)) * v) + (0))) /\ (((((exists ff_h_pfp_prefix_traceterminal. ff_h_pfp_prefix_traceterminal + S (r) = S ((S (l)) * v)) /\ exists ff_q_pfp_prefix_traceterminal. u = ff_q_pfp_prefix_traceterminal * S ((S (l)) * v) + (r))) /\ ((forall pfh_index_prefix_tracesteps. (exists pfa_gap_prefix_tracestepsindex. pfa_gap_prefix_tracestepsindex + S (pfh_index_prefix_tracesteps) = (l)) -> (exists pfh_coefficient_prefix_tracestepsstep pfh_before_prefix_tracestepsstep pfh_after_prefix_tracestepsstep pfh_product_prefix_tracestepsstep. ((((exists ff_h_pfp_prefix_tracestepsstepcoefficient. ff_h_pfp_prefix_tracestepsstepcoefficient + S (pfh_coefficient_prefix_tracestepsstep) = S ((S (pfh_index_prefix_tracesteps)) * c)) /\ exists ff_q_pfp_prefix_tracestepsstepcoefficient. b = ff_q_pfp_prefix_tracestepsstepcoefficient * S ((S (pfh_index_prefix_tracesteps)) * c) + (pfh_coefficient_prefix_tracestepsstep))) /\ (((((exists ff_h_pfp_prefix_tracestepsstepbefore. ff_h_pfp_prefix_tracestepsstepbefore + S (pfh_before_prefix_tracestepsstep) = S ((S (pfh_index_prefix_tracesteps)) * v)) /\ exists ff_q_pfp_prefix_tracestepsstepbefore. u = ff_q_pfp_prefix_tracestepsstepbefore * S ((S (pfh_index_prefix_tracesteps)) * v) + (pfh_before_prefix_tracestepsstep))) /\ (((((exists ff_h_pfp_prefix_tracestepsstepafter. ff_h_pfp_prefix_tracestepsstepafter + S (pfh_after_prefix_tracestepsstep) = S ((S (S (pfh_index_prefix_tracesteps))) * v)) /\ exists ff_q_pfp_prefix_tracestepsstepafter. u = ff_q_pfp_prefix_tracestepsstepafter * S ((S (S (pfh_index_prefix_tracesteps))) * v) + (pfh_after_prefix_tracestepsstep))) /\ (((((exists pfa_gap_prefix_tracestepsstepmultiplyleft. pfa_gap_prefix_tracestepsstepmultiplyleft + S (pfh_before_prefix_tracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_tracestepsstepmultiplyright. pfa_gap_prefix_tracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_prefix_tracestepsstepmultiplyresultbound. pfa_gap_prefix_tracestepsstepmultiplyresultbound + S (pfh_product_prefix_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_tracestepsstepmultiplyresultcongruence pfa_offset_right_prefix_tracestepsstepmultiplyresultcongruence. ((pfh_before_prefix_tracestepsstep) * (a)) + (p) * pfa_offset_left_prefix_tracestepsstepmultiplyresultcongruence = (pfh_product_prefix_tracestepsstep) + (p) * pfa_offset_right_prefix_tracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_prefix_tracestepsstepaddleft. pfa_gap_prefix_tracestepsstepaddleft + S (pfh_product_prefix_tracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_tracestepsstepaddright. pfa_gap_prefix_tracestepsstepaddright + S (pfh_coefficient_prefix_tracestepsstep) = (p)) /\ ((((exists pfa_gap_prefix_tracestepsstepaddresultbound. pfa_gap_prefix_tracestepsstepaddresultbound + S (pfh_after_prefix_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_tracestepsstepaddresultcongruence pfa_offset_right_prefix_tracestepsstepaddresultcongruence. ((pfh_product_prefix_tracestepsstep) + (pfh_coefficient_prefix_tracestepsstep)) + (p) * pfa_offset_left_prefix_tracestepsstepaddresultcongruence = (pfh_after_prefix_tracestepsstep) + (p) * pfa_offset_right_prefix_tracestepsstepaddresultcongruence)))))))))))))))))))))))))) -> (exists pfc_gap_prefix_length. pfc_gap_prefix_length+(n)=(l)) -> (((exists ff_h_pfp_prefix_state. ff_h_pfp_prefix_state + S (h) = S ((S (n)) * v)) /\ exists ff_q_pfp_prefix_state. u = ff_q_pfp_prefix_state * S ((S (n)) * v) + (h))) -> (exists pfh_trace_code_prefix_execution pfh_trace_scale_prefix_execution. (((exists pfa_gap_prefix_executiontracebase. pfa_gap_prefix_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_prefix_executiontraceinitial. ff_h_pfp_prefix_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontraceinitial. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontraceinitial * S ((S (0)) * pfh_trace_scale_prefix_execution) + (0))) /\ (((((exists ff_h_pfp_prefix_executiontraceterminal. ff_h_pfp_prefix_executiontraceterminal + S (h) = S ((S (n)) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontraceterminal. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontraceterminal * S ((S (n)) * pfh_trace_scale_prefix_execution) + (h))) /\ ((forall pfh_index_prefix_executiontracesteps. (exists pfa_gap_prefix_executiontracestepsindex. pfa_gap_prefix_executiontracestepsindex + S (pfh_index_prefix_executiontracesteps) = (n)) -> (exists pfh_coefficient_prefix_executiontracestepsstep pfh_before_prefix_executiontracestepsstep pfh_after_prefix_executiontracestepsstep pfh_product_prefix_executiontracestepsstep. ((((exists ff_h_pfp_prefix_executiontracestepsstepcoefficient. ff_h_pfp_prefix_executiontracestepsstepcoefficient + S (pfh_coefficient_prefix_executiontracestepsstep) = S ((S (pfh_index_prefix_executiontracesteps)) * c)) /\ exists ff_q_pfp_prefix_executiontracestepsstepcoefficient. b = ff_q_pfp_prefix_executiontracestepsstepcoefficient * S ((S (pfh_index_prefix_executiontracesteps)) * c) + (pfh_coefficient_prefix_executiontracestepsstep))) /\ (((((exists ff_h_pfp_prefix_executiontracestepsstepbefore. ff_h_pfp_prefix_executiontracestepsstepbefore + S (pfh_before_prefix_executiontracestepsstep) = S ((S (pfh_index_prefix_executiontracesteps)) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontracestepsstepbefore. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontracestepsstepbefore * S ((S (pfh_index_prefix_executiontracesteps)) * pfh_trace_scale_prefix_execution) + (pfh_before_prefix_executiontracestepsstep))) /\ (((((exists ff_h_pfp_prefix_executiontracestepsstepafter. ff_h_pfp_prefix_executiontracestepsstepafter + S (pfh_after_prefix_executiontracestepsstep) = S ((S (S (pfh_index_prefix_executiontracesteps))) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontracestepsstepafter. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontracestepsstepafter * S ((S (S (pfh_index_prefix_executiontracesteps))) * pfh_trace_scale_prefix_execution) + (pfh_after_prefix_executiontracestepsstep))) /\ (((((exists pfa_gap_prefix_executiontracestepsstepmultiplyleft. pfa_gap_prefix_executiontracestepsstepmultiplyleft + S (pfh_before_prefix_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_executiontracestepsstepmultiplyright. pfa_gap_prefix_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_prefix_executiontracestepsstepmultiplyresultbound. pfa_gap_prefix_executiontracestepsstepmultiplyresultbound + S (pfh_product_prefix_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_prefix_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_prefix_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_prefix_executiontracestepsstepmultiplyresultcongruence = (pfh_product_prefix_executiontracestepsstep) + (p) * pfa_offset_right_prefix_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_prefix_executiontracestepsstepaddleft. pfa_gap_prefix_executiontracestepsstepaddleft + S (pfh_product_prefix_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_executiontracestepsstepaddright. pfa_gap_prefix_executiontracestepsstepaddright + S (pfh_coefficient_prefix_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_prefix_executiontracestepsstepaddresultbound. pfa_gap_prefix_executiontracestepsstepaddresultbound + S (pfh_after_prefix_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_executiontracestepsstepaddresultcongruence pfa_offset_right_prefix_executiontracestepsstepaddresultcongruence. ((pfh_product_prefix_executiontracestepsstep) + (pfh_coefficient_prefix_executiontracestepsstep)) + (p) * pfa_offset_left_prefix_executiontracestepsstepaddresultcongruence = (pfh_after_prefix_executiontracestepsstep) + (p) * pfa_offset_right_prefix_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))prime_field_polynomial_horner_trace_state_boundedExact checked statement
forall p b c a l r u v n h. (~((p) = 1) /\ forall pfa_factor_left_state_prime pfa_factor_right_state_prime. (p) = pfa_factor_left_state_prime * pfa_factor_right_state_prime -> pfa_factor_left_state_prime = 1 \/ pfa_factor_right_state_prime = 1) -> (((exists pfa_gap_state_tracebase. pfa_gap_state_tracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_state_traceinitial. ff_h_pfp_state_traceinitial + S (0) = S ((S (0)) * v)) /\ exists ff_q_pfp_state_traceinitial. u = ff_q_pfp_state_traceinitial * S ((S (0)) * v) + (0))) /\ (((((exists ff_h_pfp_state_traceterminal. ff_h_pfp_state_traceterminal + S (r) = S ((S (l)) * v)) /\ exists ff_q_pfp_state_traceterminal. u = ff_q_pfp_state_traceterminal * S ((S (l)) * v) + (r))) /\ ((forall pfh_index_state_tracesteps. (exists pfa_gap_state_tracestepsindex. pfa_gap_state_tracestepsindex + S (pfh_index_state_tracesteps) = (l)) -> (exists pfh_coefficient_state_tracestepsstep pfh_before_state_tracestepsstep pfh_after_state_tracestepsstep pfh_product_state_tracestepsstep. ((((exists ff_h_pfp_state_tracestepsstepcoefficient. ff_h_pfp_state_tracestepsstepcoefficient + S (pfh_coefficient_state_tracestepsstep) = S ((S (pfh_index_state_tracesteps)) * c)) /\ exists ff_q_pfp_state_tracestepsstepcoefficient. b = ff_q_pfp_state_tracestepsstepcoefficient * S ((S (pfh_index_state_tracesteps)) * c) + (pfh_coefficient_state_tracestepsstep))) /\ (((((exists ff_h_pfp_state_tracestepsstepbefore. ff_h_pfp_state_tracestepsstepbefore + S (pfh_before_state_tracestepsstep) = S ((S (pfh_index_state_tracesteps)) * v)) /\ exists ff_q_pfp_state_tracestepsstepbefore. u = ff_q_pfp_state_tracestepsstepbefore * S ((S (pfh_index_state_tracesteps)) * v) + (pfh_before_state_tracestepsstep))) /\ (((((exists ff_h_pfp_state_tracestepsstepafter. ff_h_pfp_state_tracestepsstepafter + S (pfh_after_state_tracestepsstep) = S ((S (S (pfh_index_state_tracesteps))) * v)) /\ exists ff_q_pfp_state_tracestepsstepafter. u = ff_q_pfp_state_tracestepsstepafter * S ((S (S (pfh_index_state_tracesteps))) * v) + (pfh_after_state_tracestepsstep))) /\ (((((exists pfa_gap_state_tracestepsstepmultiplyleft. pfa_gap_state_tracestepsstepmultiplyleft + S (pfh_before_state_tracestepsstep) = (p)) /\ (((exists pfa_gap_state_tracestepsstepmultiplyright. pfa_gap_state_tracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_state_tracestepsstepmultiplyresultbound. pfa_gap_state_tracestepsstepmultiplyresultbound + S (pfh_product_state_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_state_tracestepsstepmultiplyresultcongruence pfa_offset_right_state_tracestepsstepmultiplyresultcongruence. ((pfh_before_state_tracestepsstep) * (a)) + (p) * pfa_offset_left_state_tracestepsstepmultiplyresultcongruence = (pfh_product_state_tracestepsstep) + (p) * pfa_offset_right_state_tracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_state_tracestepsstepaddleft. pfa_gap_state_tracestepsstepaddleft + S (pfh_product_state_tracestepsstep) = (p)) /\ (((exists pfa_gap_state_tracestepsstepaddright. pfa_gap_state_tracestepsstepaddright + S (pfh_coefficient_state_tracestepsstep) = (p)) /\ ((((exists pfa_gap_state_tracestepsstepaddresultbound. pfa_gap_state_tracestepsstepaddresultbound + S (pfh_after_state_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_state_tracestepsstepaddresultcongruence pfa_offset_right_state_tracestepsstepaddresultcongruence. ((pfh_product_state_tracestepsstep) + (pfh_coefficient_state_tracestepsstep)) + (p) * pfa_offset_left_state_tracestepsstepaddresultcongruence = (pfh_after_state_tracestepsstep) + (p) * pfa_offset_right_state_tracestepsstepaddresultcongruence)))))))))))))))))))))))))) -> (exists pfc_gap_state_length. pfc_gap_state_length+(n)=(l)) -> (((exists ff_h_pfp_state_entry. ff_h_pfp_state_entry + S (h) = S ((S (n)) * v)) /\ exists ff_q_pfp_state_entry. u = ff_q_pfp_state_entry * S ((S (n)) * v) + (h))) -> (exists pfa_gap_state_bound. pfa_gap_state_bound + S (h) = (p))prime_field_polynomial_synthetic_existsExact checked statement
forall p b c a n. (~((p) = 1) /\ forall pfa_factor_left_construct_prime pfa_factor_right_construct_prime. (p) = pfa_factor_left_construct_prime * pfa_factor_right_construct_prime -> pfa_factor_left_construct_prime = 1 \/ pfa_factor_right_construct_prime = 1) -> (forall fom_index_pfp_construct_coefficients. (exists fom_gap_pfp_construct_coefficients_index_bound. fom_gap_pfp_construct_coefficients_index_bound + S (fom_index_pfp_construct_coefficients) = S n) -> exists fom_value_pfp_construct_coefficients. ((((exists fom_beta_height_pfp_construct_coefficients_entry. fom_beta_height_pfp_construct_coefficients_entry + S (fom_value_pfp_construct_coefficients) = S ((S (fom_index_pfp_construct_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_construct_coefficients_entry. b = fom_beta_quotient_pfp_construct_coefficients_entry * S ((S (fom_index_pfp_construct_coefficients)) * c) + (fom_value_pfp_construct_coefficients))) /\ (exists fom_gap_pfp_construct_coefficients_value_bound. fom_gap_pfp_construct_coefficients_value_bound + S (fom_value_pfp_construct_coefficients) = p))) -> (exists pfa_gap_construct_base. pfa_gap_construct_base + S (a) = (p)) -> exists qb qc r. (exists pfs_history_code_construct_result pfs_history_scale_construct_result. ((((exists pfa_gap_construct_resulttracebase. pfa_gap_construct_resulttracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_construct_resulttraceinitial. ff_h_pfp_construct_resulttraceinitial + S (0) = S ((S (0)) * pfs_history_scale_construct_result)) /\ exists ff_q_pfp_construct_resulttraceinitial. pfs_history_code_construct_result = ff_q_pfp_construct_resulttraceinitial * S ((S (0)) * pfs_history_scale_construct_result) + (0))) /\ (((((exists ff_h_pfp_construct_resulttraceterminal. ff_h_pfp_construct_resulttraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_construct_result)) /\ exists ff_q_pfp_construct_resulttraceterminal. pfs_history_code_construct_result = ff_q_pfp_construct_resulttraceterminal * S ((S (S (n))) * pfs_history_scale_construct_result) + (r))) /\ ((forall pfh_index_construct_resulttracesteps. (exists pfa_gap_construct_resulttracestepsindex. pfa_gap_construct_resulttracestepsindex + S (pfh_index_construct_resulttracesteps) = (S (n))) -> (exists pfh_coefficient_construct_resulttracestepsstep pfh_before_construct_resulttracestepsstep pfh_after_construct_resulttracestepsstep pfh_product_construct_resulttracestepsstep. ((((exists ff_h_pfp_construct_resulttracestepsstepcoefficient. ff_h_pfp_construct_resulttracestepsstepcoefficient + S (pfh_coefficient_construct_resulttracestepsstep) = S ((S (pfh_index_construct_resulttracesteps)) * c)) /\ exists ff_q_pfp_construct_resulttracestepsstepcoefficient. b = ff_q_pfp_construct_resulttracestepsstepcoefficient * S ((S (pfh_index_construct_resulttracesteps)) * c) + (pfh_coefficient_construct_resulttracestepsstep))) /\ (((((exists ff_h_pfp_construct_resulttracestepsstepbefore. ff_h_pfp_construct_resulttracestepsstepbefore + S (pfh_before_construct_resulttracestepsstep) = S ((S (pfh_index_construct_resulttracesteps)) * pfs_history_scale_construct_result)) /\ exists ff_q_pfp_construct_resulttracestepsstepbefore. pfs_history_code_construct_result = ff_q_pfp_construct_resulttracestepsstepbefore * S ((S (pfh_index_construct_resulttracesteps)) * pfs_history_scale_construct_result) + (pfh_before_construct_resulttracestepsstep))) /\ (((((exists ff_h_pfp_construct_resulttracestepsstepafter. ff_h_pfp_construct_resulttracestepsstepafter + S (pfh_after_construct_resulttracestepsstep) = S ((S (S (pfh_index_construct_resulttracesteps))) * pfs_history_scale_construct_result)) /\ exists ff_q_pfp_construct_resulttracestepsstepafter. pfs_history_code_construct_result = ff_q_pfp_construct_resulttracestepsstepafter * S ((S (S (pfh_index_construct_resulttracesteps))) * pfs_history_scale_construct_result) + (pfh_after_construct_resulttracestepsstep))) /\ (((((exists pfa_gap_construct_resulttracestepsstepmultiplyleft. pfa_gap_construct_resulttracestepsstepmultiplyleft + S (pfh_before_construct_resulttracestepsstep) = (p)) /\ (((exists pfa_gap_construct_resulttracestepsstepmultiplyright. pfa_gap_construct_resulttracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_construct_resulttracestepsstepmultiplyresultbound. pfa_gap_construct_resulttracestepsstepmultiplyresultbound + S (pfh_product_construct_resulttracestepsstep) = (p)) /\ ((exists pfa_offset_left_construct_resulttracestepsstepmultiplyresultcongruence pfa_offset_right_construct_resulttracestepsstepmultiplyresultcongruence. ((pfh_before_construct_resulttracestepsstep) * (a)) + (p) * pfa_offset_left_construct_resulttracestepsstepmultiplyresultcongruence = (pfh_product_construct_resulttracestepsstep) + (p) * pfa_offset_right_construct_resulttracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_construct_resulttracestepsstepaddleft. pfa_gap_construct_resulttracestepsstepaddleft + S (pfh_product_construct_resulttracestepsstep) = (p)) /\ (((exists pfa_gap_construct_resulttracestepsstepaddright. pfa_gap_construct_resulttracestepsstepaddright + S (pfh_coefficient_construct_resulttracestepsstep) = (p)) /\ ((((exists pfa_gap_construct_resulttracestepsstepaddresultbound. pfa_gap_construct_resulttracestepsstepaddresultbound + S (pfh_after_construct_resulttracestepsstep) = (p)) /\ ((exists pfa_offset_left_construct_resulttracestepsstepaddresultcongruence pfa_offset_right_construct_resulttracestepsstepaddresultcongruence. ((pfh_product_construct_resulttracestepsstep) + (pfh_coefficient_construct_resulttracestepsstep)) + (p) * pfa_offset_left_construct_resulttracestepsstepaddresultcongruence = (pfh_after_construct_resulttracestepsstep) + (p) * pfa_offset_right_construct_resulttracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_construct_resultquotient ff_source_mcp_pfs_construct_resultquotient ff_target_mcp_pfs_construct_resultquotient. (exists mcp_gap_pfs_construct_resultquotient_bound. mcp_gap_pfs_construct_resultquotient_bound + S (ff_index_mcp_pfs_construct_resultquotient) = (n)) -> (((exists fs_h_mcp_pfs_construct_resultquotient_source. fs_h_mcp_pfs_construct_resultquotient_source + S (ff_source_mcp_pfs_construct_resultquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_construct_resultquotient)) * pfs_history_scale_construct_result)) /\ exists fs_q_mcp_pfs_construct_resultquotient_source. pfs_history_code_construct_result = fs_q_mcp_pfs_construct_resultquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_construct_resultquotient)) * pfs_history_scale_construct_result) + (ff_source_mcp_pfs_construct_resultquotient))) -> (((exists fs_h_mcp_pfs_construct_resultquotient_target. fs_h_mcp_pfs_construct_resultquotient_target + S (ff_target_mcp_pfs_construct_resultquotient) = S ((S (ff_index_mcp_pfs_construct_resultquotient)) * qc)) /\ exists fs_q_mcp_pfs_construct_resultquotient_target. qb = fs_q_mcp_pfs_construct_resultquotient_target * S ((S (ff_index_mcp_pfs_construct_resultquotient)) * qc) + (ff_target_mcp_pfs_construct_resultquotient))) -> ff_target_mcp_pfs_construct_resultquotient = ff_source_mcp_pfs_construct_resultquotient))))prime_field_polynomial_synthetic_remainder_executionExact checked statement
forall p b c a n qb qc r. (exists pfs_history_code_remainder_source pfs_history_scale_remainder_source. ((((exists pfa_gap_remainder_sourcetracebase. pfa_gap_remainder_sourcetracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_remainder_sourcetraceinitial. ff_h_pfp_remainder_sourcetraceinitial + S (0) = S ((S (0)) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetraceinitial. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetraceinitial * S ((S (0)) * pfs_history_scale_remainder_source) + (0))) /\ (((((exists ff_h_pfp_remainder_sourcetraceterminal. ff_h_pfp_remainder_sourcetraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetraceterminal. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetraceterminal * S ((S (S (n))) * pfs_history_scale_remainder_source) + (r))) /\ ((forall pfh_index_remainder_sourcetracesteps. (exists pfa_gap_remainder_sourcetracestepsindex. pfa_gap_remainder_sourcetracestepsindex + S (pfh_index_remainder_sourcetracesteps) = (S (n))) -> (exists pfh_coefficient_remainder_sourcetracestepsstep pfh_before_remainder_sourcetracestepsstep pfh_after_remainder_sourcetracestepsstep pfh_product_remainder_sourcetracestepsstep. ((((exists ff_h_pfp_remainder_sourcetracestepsstepcoefficient. ff_h_pfp_remainder_sourcetracestepsstepcoefficient + S (pfh_coefficient_remainder_sourcetracestepsstep) = S ((S (pfh_index_remainder_sourcetracesteps)) * c)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepcoefficient. b = ff_q_pfp_remainder_sourcetracestepsstepcoefficient * S ((S (pfh_index_remainder_sourcetracesteps)) * c) + (pfh_coefficient_remainder_sourcetracestepsstep))) /\ (((((exists ff_h_pfp_remainder_sourcetracestepsstepbefore. ff_h_pfp_remainder_sourcetracestepsstepbefore + S (pfh_before_remainder_sourcetracestepsstep) = S ((S (pfh_index_remainder_sourcetracesteps)) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepbefore. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetracestepsstepbefore * S ((S (pfh_index_remainder_sourcetracesteps)) * pfs_history_scale_remainder_source) + (pfh_before_remainder_sourcetracestepsstep))) /\ (((((exists ff_h_pfp_remainder_sourcetracestepsstepafter. ff_h_pfp_remainder_sourcetracestepsstepafter + S (pfh_after_remainder_sourcetracestepsstep) = S ((S (S (pfh_index_remainder_sourcetracesteps))) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepafter. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetracestepsstepafter * S ((S (S (pfh_index_remainder_sourcetracesteps))) * pfs_history_scale_remainder_source) + (pfh_after_remainder_sourcetracestepsstep))) /\ (((((exists pfa_gap_remainder_sourcetracestepsstepmultiplyleft. pfa_gap_remainder_sourcetracestepsstepmultiplyleft + S (pfh_before_remainder_sourcetracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_sourcetracestepsstepmultiplyright. pfa_gap_remainder_sourcetracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepmultiplyresultbound. pfa_gap_remainder_sourcetracestepsstepmultiplyresultbound + S (pfh_product_remainder_sourcetracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_sourcetracestepsstepmultiplyresultcongruence pfa_offset_right_remainder_sourcetracestepsstepmultiplyresultcongruence. ((pfh_before_remainder_sourcetracestepsstep) * (a)) + (p) * pfa_offset_left_remainder_sourcetracestepsstepmultiplyresultcongruence = (pfh_product_remainder_sourcetracestepsstep) + (p) * pfa_offset_right_remainder_sourcetracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepaddleft. pfa_gap_remainder_sourcetracestepsstepaddleft + S (pfh_product_remainder_sourcetracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_sourcetracestepsstepaddright. pfa_gap_remainder_sourcetracestepsstepaddright + S (pfh_coefficient_remainder_sourcetracestepsstep) = (p)) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepaddresultbound. pfa_gap_remainder_sourcetracestepsstepaddresultbound + S (pfh_after_remainder_sourcetracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_sourcetracestepsstepaddresultcongruence pfa_offset_right_remainder_sourcetracestepsstepaddresultcongruence. ((pfh_product_remainder_sourcetracestepsstep) + (pfh_coefficient_remainder_sourcetracestepsstep)) + (p) * pfa_offset_left_remainder_sourcetracestepsstepaddresultcongruence = (pfh_after_remainder_sourcetracestepsstep) + (p) * pfa_offset_right_remainder_sourcetracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_remainder_sourcequotient ff_source_mcp_pfs_remainder_sourcequotient ff_target_mcp_pfs_remainder_sourcequotient. (exists mcp_gap_pfs_remainder_sourcequotient_bound. mcp_gap_pfs_remainder_sourcequotient_bound + S (ff_index_mcp_pfs_remainder_sourcequotient) = (n)) -> (((exists fs_h_mcp_pfs_remainder_sourcequotient_source. fs_h_mcp_pfs_remainder_sourcequotient_source + S (ff_source_mcp_pfs_remainder_sourcequotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_remainder_sourcequotient)) * pfs_history_scale_remainder_source)) /\ exists fs_q_mcp_pfs_remainder_sourcequotient_source. pfs_history_code_remainder_source = fs_q_mcp_pfs_remainder_sourcequotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_remainder_sourcequotient)) * pfs_history_scale_remainder_source) + (ff_source_mcp_pfs_remainder_sourcequotient))) -> (((exists fs_h_mcp_pfs_remainder_sourcequotient_target. fs_h_mcp_pfs_remainder_sourcequotient_target + S (ff_target_mcp_pfs_remainder_sourcequotient) = S ((S (ff_index_mcp_pfs_remainder_sourcequotient)) * qc)) /\ exists fs_q_mcp_pfs_remainder_sourcequotient_target. qb = fs_q_mcp_pfs_remainder_sourcequotient_target * S ((S (ff_index_mcp_pfs_remainder_sourcequotient)) * qc) + (ff_target_mcp_pfs_remainder_sourcequotient))) -> ff_target_mcp_pfs_remainder_sourcequotient = ff_source_mcp_pfs_remainder_sourcequotient)))) -> (exists pfh_trace_code_remainder_execution pfh_trace_scale_remainder_execution. (((exists pfa_gap_remainder_executiontracebase. pfa_gap_remainder_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_remainder_executiontraceinitial. ff_h_pfp_remainder_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontraceinitial. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontraceinitial * S ((S (0)) * pfh_trace_scale_remainder_execution) + (0))) /\ (((((exists ff_h_pfp_remainder_executiontraceterminal. ff_h_pfp_remainder_executiontraceterminal + S (r) = S ((S (S n)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontraceterminal. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontraceterminal * S ((S (S n)) * pfh_trace_scale_remainder_execution) + (r))) /\ ((forall pfh_index_remainder_executiontracesteps. (exists pfa_gap_remainder_executiontracestepsindex. pfa_gap_remainder_executiontracestepsindex + S (pfh_index_remainder_executiontracesteps) = (S n)) -> (exists pfh_coefficient_remainder_executiontracestepsstep pfh_before_remainder_executiontracestepsstep pfh_after_remainder_executiontracestepsstep pfh_product_remainder_executiontracestepsstep. ((((exists ff_h_pfp_remainder_executiontracestepsstepcoefficient. ff_h_pfp_remainder_executiontracestepsstepcoefficient + S (pfh_coefficient_remainder_executiontracestepsstep) = S ((S (pfh_index_remainder_executiontracesteps)) * c)) /\ exists ff_q_pfp_remainder_executiontracestepsstepcoefficient. b = ff_q_pfp_remainder_executiontracestepsstepcoefficient * S ((S (pfh_index_remainder_executiontracesteps)) * c) + (pfh_coefficient_remainder_executiontracestepsstep))) /\ (((((exists ff_h_pfp_remainder_executiontracestepsstepbefore. ff_h_pfp_remainder_executiontracestepsstepbefore + S (pfh_before_remainder_executiontracestepsstep) = S ((S (pfh_index_remainder_executiontracesteps)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontracestepsstepbefore. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontracestepsstepbefore * S ((S (pfh_index_remainder_executiontracesteps)) * pfh_trace_scale_remainder_execution) + (pfh_before_remainder_executiontracestepsstep))) /\ (((((exists ff_h_pfp_remainder_executiontracestepsstepafter. ff_h_pfp_remainder_executiontracestepsstepafter + S (pfh_after_remainder_executiontracestepsstep) = S ((S (S (pfh_index_remainder_executiontracesteps))) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontracestepsstepafter. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontracestepsstepafter * S ((S (S (pfh_index_remainder_executiontracesteps))) * pfh_trace_scale_remainder_execution) + (pfh_after_remainder_executiontracestepsstep))) /\ (((((exists pfa_gap_remainder_executiontracestepsstepmultiplyleft. pfa_gap_remainder_executiontracestepsstepmultiplyleft + S (pfh_before_remainder_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_executiontracestepsstepmultiplyright. pfa_gap_remainder_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_remainder_executiontracestepsstepmultiplyresultbound. pfa_gap_remainder_executiontracestepsstepmultiplyresultbound + S (pfh_product_remainder_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_remainder_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_remainder_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_remainder_executiontracestepsstepmultiplyresultcongruence = (pfh_product_remainder_executiontracestepsstep) + (p) * pfa_offset_right_remainder_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_remainder_executiontracestepsstepaddleft. pfa_gap_remainder_executiontracestepsstepaddleft + S (pfh_product_remainder_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_executiontracestepsstepaddright. pfa_gap_remainder_executiontracestepsstepaddright + S (pfh_coefficient_remainder_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_remainder_executiontracestepsstepaddresultbound. pfa_gap_remainder_executiontracestepsstepaddresultbound + S (pfh_after_remainder_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_executiontracestepsstepaddresultcongruence pfa_offset_right_remainder_executiontracestepsstepaddresultcongruence. ((pfh_product_remainder_executiontracestepsstep) + (pfh_coefficient_remainder_executiontracestepsstep)) + (p) * pfa_offset_left_remainder_executiontracestepsstepaddresultcongruence = (pfh_after_remainder_executiontracestepsstep) + (p) * pfa_offset_right_remainder_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))prime_field_polynomial_synthetic_quotient_entryExact checked statement
forall p b c a n qb qc r i h. (exists pfs_history_code_entry_division pfs_history_scale_entry_division. ((((exists pfa_gap_entry_divisiontracebase. pfa_gap_entry_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_entry_divisiontraceinitial. ff_h_pfp_entry_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_entry_division)) /\ exists ff_q_pfp_entry_divisiontraceinitial. pfs_history_code_entry_division = ff_q_pfp_entry_divisiontraceinitial * S ((S (0)) * pfs_history_scale_entry_division) + (0))) /\ (((((exists ff_h_pfp_entry_divisiontraceterminal. ff_h_pfp_entry_divisiontraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_entry_division)) /\ exists ff_q_pfp_entry_divisiontraceterminal. pfs_history_code_entry_division = ff_q_pfp_entry_divisiontraceterminal * S ((S (S (n))) * pfs_history_scale_entry_division) + (r))) /\ ((forall pfh_index_entry_divisiontracesteps. (exists pfa_gap_entry_divisiontracestepsindex. pfa_gap_entry_divisiontracestepsindex + S (pfh_index_entry_divisiontracesteps) = (S (n))) -> (exists pfh_coefficient_entry_divisiontracestepsstep pfh_before_entry_divisiontracestepsstep pfh_after_entry_divisiontracestepsstep pfh_product_entry_divisiontracestepsstep. ((((exists ff_h_pfp_entry_divisiontracestepsstepcoefficient. ff_h_pfp_entry_divisiontracestepsstepcoefficient + S (pfh_coefficient_entry_divisiontracestepsstep) = S ((S (pfh_index_entry_divisiontracesteps)) * c)) /\ exists ff_q_pfp_entry_divisiontracestepsstepcoefficient. b = ff_q_pfp_entry_divisiontracestepsstepcoefficient * S ((S (pfh_index_entry_divisiontracesteps)) * c) + (pfh_coefficient_entry_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_entry_divisiontracestepsstepbefore. ff_h_pfp_entry_divisiontracestepsstepbefore + S (pfh_before_entry_divisiontracestepsstep) = S ((S (pfh_index_entry_divisiontracesteps)) * pfs_history_scale_entry_division)) /\ exists ff_q_pfp_entry_divisiontracestepsstepbefore. pfs_history_code_entry_division = ff_q_pfp_entry_divisiontracestepsstepbefore * S ((S (pfh_index_entry_divisiontracesteps)) * pfs_history_scale_entry_division) + (pfh_before_entry_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_entry_divisiontracestepsstepafter. ff_h_pfp_entry_divisiontracestepsstepafter + S (pfh_after_entry_divisiontracestepsstep) = S ((S (S (pfh_index_entry_divisiontracesteps))) * pfs_history_scale_entry_division)) /\ exists ff_q_pfp_entry_divisiontracestepsstepafter. pfs_history_code_entry_division = ff_q_pfp_entry_divisiontracestepsstepafter * S ((S (S (pfh_index_entry_divisiontracesteps))) * pfs_history_scale_entry_division) + (pfh_after_entry_divisiontracestepsstep))) /\ (((((exists pfa_gap_entry_divisiontracestepsstepmultiplyleft. pfa_gap_entry_divisiontracestepsstepmultiplyleft + S (pfh_before_entry_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_entry_divisiontracestepsstepmultiplyright. pfa_gap_entry_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_entry_divisiontracestepsstepmultiplyresultbound. pfa_gap_entry_divisiontracestepsstepmultiplyresultbound + S (pfh_product_entry_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_entry_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_entry_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_entry_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_entry_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_entry_divisiontracestepsstep) + (p) * pfa_offset_right_entry_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_entry_divisiontracestepsstepaddleft. pfa_gap_entry_divisiontracestepsstepaddleft + S (pfh_product_entry_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_entry_divisiontracestepsstepaddright. pfa_gap_entry_divisiontracestepsstepaddright + S (pfh_coefficient_entry_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_entry_divisiontracestepsstepaddresultbound. pfa_gap_entry_divisiontracestepsstepaddresultbound + S (pfh_after_entry_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_entry_divisiontracestepsstepaddresultcongruence pfa_offset_right_entry_divisiontracestepsstepaddresultcongruence. ((pfh_product_entry_divisiontracestepsstep) + (pfh_coefficient_entry_divisiontracestepsstep)) + (p) * pfa_offset_left_entry_divisiontracestepsstepaddresultcongruence = (pfh_after_entry_divisiontracestepsstep) + (p) * pfa_offset_right_entry_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_entry_divisionquotient ff_source_mcp_pfs_entry_divisionquotient ff_target_mcp_pfs_entry_divisionquotient. (exists mcp_gap_pfs_entry_divisionquotient_bound. mcp_gap_pfs_entry_divisionquotient_bound + S (ff_index_mcp_pfs_entry_divisionquotient) = (n)) -> (((exists fs_h_mcp_pfs_entry_divisionquotient_source. fs_h_mcp_pfs_entry_divisionquotient_source + S (ff_source_mcp_pfs_entry_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_entry_divisionquotient)) * pfs_history_scale_entry_division)) /\ exists fs_q_mcp_pfs_entry_divisionquotient_source. pfs_history_code_entry_division = fs_q_mcp_pfs_entry_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_entry_divisionquotient)) * pfs_history_scale_entry_division) + (ff_source_mcp_pfs_entry_divisionquotient))) -> (((exists fs_h_mcp_pfs_entry_divisionquotient_target. fs_h_mcp_pfs_entry_divisionquotient_target + S (ff_target_mcp_pfs_entry_divisionquotient) = S ((S (ff_index_mcp_pfs_entry_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_entry_divisionquotient_target. qb = fs_q_mcp_pfs_entry_divisionquotient_target * S ((S (ff_index_mcp_pfs_entry_divisionquotient)) * qc) + (ff_target_mcp_pfs_entry_divisionquotient))) -> ff_target_mcp_pfs_entry_divisionquotient = ff_source_mcp_pfs_entry_divisionquotient)))) -> (exists pfa_gap_entry_index. pfa_gap_entry_index + S (i) = (n)) -> (((exists ff_h_pfp_entry_quotient. ff_h_pfp_entry_quotient + S (h) = S ((S (i)) * qc)) /\ exists ff_q_pfp_entry_quotient. qb = ff_q_pfp_entry_quotient * S ((S (i)) * qc) + (h))) -> (exists pfh_trace_code_entry_execution pfh_trace_scale_entry_execution. (((exists pfa_gap_entry_executiontracebase. pfa_gap_entry_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_entry_executiontraceinitial. ff_h_pfp_entry_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_entry_execution)) /\ exists ff_q_pfp_entry_executiontraceinitial. pfh_trace_code_entry_execution = ff_q_pfp_entry_executiontraceinitial * S ((S (0)) * pfh_trace_scale_entry_execution) + (0))) /\ (((((exists ff_h_pfp_entry_executiontraceterminal. ff_h_pfp_entry_executiontraceterminal + S (h) = S ((S (S i)) * pfh_trace_scale_entry_execution)) /\ exists ff_q_pfp_entry_executiontraceterminal. pfh_trace_code_entry_execution = ff_q_pfp_entry_executiontraceterminal * S ((S (S i)) * pfh_trace_scale_entry_execution) + (h))) /\ ((forall pfh_index_entry_executiontracesteps. (exists pfa_gap_entry_executiontracestepsindex. pfa_gap_entry_executiontracestepsindex + S (pfh_index_entry_executiontracesteps) = (S i)) -> (exists pfh_coefficient_entry_executiontracestepsstep pfh_before_entry_executiontracestepsstep pfh_after_entry_executiontracestepsstep pfh_product_entry_executiontracestepsstep. ((((exists ff_h_pfp_entry_executiontracestepsstepcoefficient. ff_h_pfp_entry_executiontracestepsstepcoefficient + S (pfh_coefficient_entry_executiontracestepsstep) = S ((S (pfh_index_entry_executiontracesteps)) * c)) /\ exists ff_q_pfp_entry_executiontracestepsstepcoefficient. b = ff_q_pfp_entry_executiontracestepsstepcoefficient * S ((S (pfh_index_entry_executiontracesteps)) * c) + (pfh_coefficient_entry_executiontracestepsstep))) /\ (((((exists ff_h_pfp_entry_executiontracestepsstepbefore. ff_h_pfp_entry_executiontracestepsstepbefore + S (pfh_before_entry_executiontracestepsstep) = S ((S (pfh_index_entry_executiontracesteps)) * pfh_trace_scale_entry_execution)) /\ exists ff_q_pfp_entry_executiontracestepsstepbefore. pfh_trace_code_entry_execution = ff_q_pfp_entry_executiontracestepsstepbefore * S ((S (pfh_index_entry_executiontracesteps)) * pfh_trace_scale_entry_execution) + (pfh_before_entry_executiontracestepsstep))) /\ (((((exists ff_h_pfp_entry_executiontracestepsstepafter. ff_h_pfp_entry_executiontracestepsstepafter + S (pfh_after_entry_executiontracestepsstep) = S ((S (S (pfh_index_entry_executiontracesteps))) * pfh_trace_scale_entry_execution)) /\ exists ff_q_pfp_entry_executiontracestepsstepafter. pfh_trace_code_entry_execution = ff_q_pfp_entry_executiontracestepsstepafter * S ((S (S (pfh_index_entry_executiontracesteps))) * pfh_trace_scale_entry_execution) + (pfh_after_entry_executiontracestepsstep))) /\ (((((exists pfa_gap_entry_executiontracestepsstepmultiplyleft. pfa_gap_entry_executiontracestepsstepmultiplyleft + S (pfh_before_entry_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_entry_executiontracestepsstepmultiplyright. pfa_gap_entry_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_entry_executiontracestepsstepmultiplyresultbound. pfa_gap_entry_executiontracestepsstepmultiplyresultbound + S (pfh_product_entry_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_entry_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_entry_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_entry_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_entry_executiontracestepsstepmultiplyresultcongruence = (pfh_product_entry_executiontracestepsstep) + (p) * pfa_offset_right_entry_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_entry_executiontracestepsstepaddleft. pfa_gap_entry_executiontracestepsstepaddleft + S (pfh_product_entry_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_entry_executiontracestepsstepaddright. pfa_gap_entry_executiontracestepsstepaddright + S (pfh_coefficient_entry_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_entry_executiontracestepsstepaddresultbound. pfa_gap_entry_executiontracestepsstepaddresultbound + S (pfh_after_entry_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_entry_executiontracestepsstepaddresultcongruence pfa_offset_right_entry_executiontracestepsstepaddresultcongruence. ((pfh_product_entry_executiontracestepsstep) + (pfh_coefficient_entry_executiontracestepsstep)) + (p) * pfa_offset_left_entry_executiontracestepsstepaddresultcongruence = (pfh_after_entry_executiontracestepsstep) + (p) * pfa_offset_right_entry_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))prime_field_polynomial_synthetic_quotient_boundedExact checked statement
forall p b c a n qb qc r. (~((p) = 1) /\ forall pfa_factor_left_bounded_prime pfa_factor_right_bounded_prime. (p) = pfa_factor_left_bounded_prime * pfa_factor_right_bounded_prime -> pfa_factor_left_bounded_prime = 1 \/ pfa_factor_right_bounded_prime = 1) -> (exists pfs_history_code_bounded_division pfs_history_scale_bounded_division. ((((exists pfa_gap_bounded_divisiontracebase. pfa_gap_bounded_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_bounded_divisiontraceinitial. ff_h_pfp_bounded_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_bounded_division)) /\ exists ff_q_pfp_bounded_divisiontraceinitial. pfs_history_code_bounded_division = ff_q_pfp_bounded_divisiontraceinitial * S ((S (0)) * pfs_history_scale_bounded_division) + (0))) /\ (((((exists ff_h_pfp_bounded_divisiontraceterminal. ff_h_pfp_bounded_divisiontraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_bounded_division)) /\ exists ff_q_pfp_bounded_divisiontraceterminal. pfs_history_code_bounded_division = ff_q_pfp_bounded_divisiontraceterminal * S ((S (S (n))) * pfs_history_scale_bounded_division) + (r))) /\ ((forall pfh_index_bounded_divisiontracesteps. (exists pfa_gap_bounded_divisiontracestepsindex. pfa_gap_bounded_divisiontracestepsindex + S (pfh_index_bounded_divisiontracesteps) = (S (n))) -> (exists pfh_coefficient_bounded_divisiontracestepsstep pfh_before_bounded_divisiontracestepsstep pfh_after_bounded_divisiontracestepsstep pfh_product_bounded_divisiontracestepsstep. ((((exists ff_h_pfp_bounded_divisiontracestepsstepcoefficient. ff_h_pfp_bounded_divisiontracestepsstepcoefficient + S (pfh_coefficient_bounded_divisiontracestepsstep) = S ((S (pfh_index_bounded_divisiontracesteps)) * c)) /\ exists ff_q_pfp_bounded_divisiontracestepsstepcoefficient. b = ff_q_pfp_bounded_divisiontracestepsstepcoefficient * S ((S (pfh_index_bounded_divisiontracesteps)) * c) + (pfh_coefficient_bounded_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_bounded_divisiontracestepsstepbefore. ff_h_pfp_bounded_divisiontracestepsstepbefore + S (pfh_before_bounded_divisiontracestepsstep) = S ((S (pfh_index_bounded_divisiontracesteps)) * pfs_history_scale_bounded_division)) /\ exists ff_q_pfp_bounded_divisiontracestepsstepbefore. pfs_history_code_bounded_division = ff_q_pfp_bounded_divisiontracestepsstepbefore * S ((S (pfh_index_bounded_divisiontracesteps)) * pfs_history_scale_bounded_division) + (pfh_before_bounded_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_bounded_divisiontracestepsstepafter. ff_h_pfp_bounded_divisiontracestepsstepafter + S (pfh_after_bounded_divisiontracestepsstep) = S ((S (S (pfh_index_bounded_divisiontracesteps))) * pfs_history_scale_bounded_division)) /\ exists ff_q_pfp_bounded_divisiontracestepsstepafter. pfs_history_code_bounded_division = ff_q_pfp_bounded_divisiontracestepsstepafter * S ((S (S (pfh_index_bounded_divisiontracesteps))) * pfs_history_scale_bounded_division) + (pfh_after_bounded_divisiontracestepsstep))) /\ (((((exists pfa_gap_bounded_divisiontracestepsstepmultiplyleft. pfa_gap_bounded_divisiontracestepsstepmultiplyleft + S (pfh_before_bounded_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_bounded_divisiontracestepsstepmultiplyright. pfa_gap_bounded_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_bounded_divisiontracestepsstepmultiplyresultbound. pfa_gap_bounded_divisiontracestepsstepmultiplyresultbound + S (pfh_product_bounded_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_bounded_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_bounded_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_bounded_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_bounded_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_bounded_divisiontracestepsstep) + (p) * pfa_offset_right_bounded_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_bounded_divisiontracestepsstepaddleft. pfa_gap_bounded_divisiontracestepsstepaddleft + S (pfh_product_bounded_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_bounded_divisiontracestepsstepaddright. pfa_gap_bounded_divisiontracestepsstepaddright + S (pfh_coefficient_bounded_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_bounded_divisiontracestepsstepaddresultbound. pfa_gap_bounded_divisiontracestepsstepaddresultbound + S (pfh_after_bounded_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_bounded_divisiontracestepsstepaddresultcongruence pfa_offset_right_bounded_divisiontracestepsstepaddresultcongruence. ((pfh_product_bounded_divisiontracestepsstep) + (pfh_coefficient_bounded_divisiontracestepsstep)) + (p) * pfa_offset_left_bounded_divisiontracestepsstepaddresultcongruence = (pfh_after_bounded_divisiontracestepsstep) + (p) * pfa_offset_right_bounded_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_bounded_divisionquotient ff_source_mcp_pfs_bounded_divisionquotient ff_target_mcp_pfs_bounded_divisionquotient. (exists mcp_gap_pfs_bounded_divisionquotient_bound. mcp_gap_pfs_bounded_divisionquotient_bound + S (ff_index_mcp_pfs_bounded_divisionquotient) = (n)) -> (((exists fs_h_mcp_pfs_bounded_divisionquotient_source. fs_h_mcp_pfs_bounded_divisionquotient_source + S (ff_source_mcp_pfs_bounded_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_bounded_divisionquotient)) * pfs_history_scale_bounded_division)) /\ exists fs_q_mcp_pfs_bounded_divisionquotient_source. pfs_history_code_bounded_division = fs_q_mcp_pfs_bounded_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_bounded_divisionquotient)) * pfs_history_scale_bounded_division) + (ff_source_mcp_pfs_bounded_divisionquotient))) -> (((exists fs_h_mcp_pfs_bounded_divisionquotient_target. fs_h_mcp_pfs_bounded_divisionquotient_target + S (ff_target_mcp_pfs_bounded_divisionquotient) = S ((S (ff_index_mcp_pfs_bounded_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_bounded_divisionquotient_target. qb = fs_q_mcp_pfs_bounded_divisionquotient_target * S ((S (ff_index_mcp_pfs_bounded_divisionquotient)) * qc) + (ff_target_mcp_pfs_bounded_divisionquotient))) -> ff_target_mcp_pfs_bounded_divisionquotient = ff_source_mcp_pfs_bounded_divisionquotient)))) -> (forall fom_index_pfp_bounded_quotient. (exists fom_gap_pfp_bounded_quotient_index_bound. fom_gap_pfp_bounded_quotient_index_bound + S (fom_index_pfp_bounded_quotient) = n) -> exists fom_value_pfp_bounded_quotient. ((((exists fom_beta_height_pfp_bounded_quotient_entry. fom_beta_height_pfp_bounded_quotient_entry + S (fom_value_pfp_bounded_quotient) = S ((S (fom_index_pfp_bounded_quotient)) * qc)) /\ exists fom_beta_quotient_pfp_bounded_quotient_entry. qb = fom_beta_quotient_pfp_bounded_quotient_entry * S ((S (fom_index_pfp_bounded_quotient)) * qc) + (fom_value_pfp_bounded_quotient))) /\ (exists fom_gap_pfp_bounded_quotient_value_bound. fom_gap_pfp_bounded_quotient_value_bound + S (fom_value_pfp_bounded_quotient) = p)))prime_field_polynomial_synthetic_remainder_boundedExact checked statement
forall p b c a n qb qc r. (~((p) = 1) /\ forall pfa_factor_left_rbound_prime pfa_factor_right_rbound_prime. (p) = pfa_factor_left_rbound_prime * pfa_factor_right_rbound_prime -> pfa_factor_left_rbound_prime = 1 \/ pfa_factor_right_rbound_prime = 1) -> (exists pfs_history_code_rbound_division pfs_history_scale_rbound_division. ((((exists pfa_gap_rbound_divisiontracebase. pfa_gap_rbound_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_rbound_divisiontraceinitial. ff_h_pfp_rbound_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_rbound_division)) /\ exists ff_q_pfp_rbound_divisiontraceinitial. pfs_history_code_rbound_division = ff_q_pfp_rbound_divisiontraceinitial * S ((S (0)) * pfs_history_scale_rbound_division) + (0))) /\ (((((exists ff_h_pfp_rbound_divisiontraceterminal. ff_h_pfp_rbound_divisiontraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_rbound_division)) /\ exists ff_q_pfp_rbound_divisiontraceterminal. pfs_history_code_rbound_division = ff_q_pfp_rbound_divisiontraceterminal * S ((S (S (n))) * pfs_history_scale_rbound_division) + (r))) /\ ((forall pfh_index_rbound_divisiontracesteps. (exists pfa_gap_rbound_divisiontracestepsindex. pfa_gap_rbound_divisiontracestepsindex + S (pfh_index_rbound_divisiontracesteps) = (S (n))) -> (exists pfh_coefficient_rbound_divisiontracestepsstep pfh_before_rbound_divisiontracestepsstep pfh_after_rbound_divisiontracestepsstep pfh_product_rbound_divisiontracestepsstep. ((((exists ff_h_pfp_rbound_divisiontracestepsstepcoefficient. ff_h_pfp_rbound_divisiontracestepsstepcoefficient + S (pfh_coefficient_rbound_divisiontracestepsstep) = S ((S (pfh_index_rbound_divisiontracesteps)) * c)) /\ exists ff_q_pfp_rbound_divisiontracestepsstepcoefficient. b = ff_q_pfp_rbound_divisiontracestepsstepcoefficient * S ((S (pfh_index_rbound_divisiontracesteps)) * c) + (pfh_coefficient_rbound_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_rbound_divisiontracestepsstepbefore. ff_h_pfp_rbound_divisiontracestepsstepbefore + S (pfh_before_rbound_divisiontracestepsstep) = S ((S (pfh_index_rbound_divisiontracesteps)) * pfs_history_scale_rbound_division)) /\ exists ff_q_pfp_rbound_divisiontracestepsstepbefore. pfs_history_code_rbound_division = ff_q_pfp_rbound_divisiontracestepsstepbefore * S ((S (pfh_index_rbound_divisiontracesteps)) * pfs_history_scale_rbound_division) + (pfh_before_rbound_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_rbound_divisiontracestepsstepafter. ff_h_pfp_rbound_divisiontracestepsstepafter + S (pfh_after_rbound_divisiontracestepsstep) = S ((S (S (pfh_index_rbound_divisiontracesteps))) * pfs_history_scale_rbound_division)) /\ exists ff_q_pfp_rbound_divisiontracestepsstepafter. pfs_history_code_rbound_division = ff_q_pfp_rbound_divisiontracestepsstepafter * S ((S (S (pfh_index_rbound_divisiontracesteps))) * pfs_history_scale_rbound_division) + (pfh_after_rbound_divisiontracestepsstep))) /\ (((((exists pfa_gap_rbound_divisiontracestepsstepmultiplyleft. pfa_gap_rbound_divisiontracestepsstepmultiplyleft + S (pfh_before_rbound_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_rbound_divisiontracestepsstepmultiplyright. pfa_gap_rbound_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_rbound_divisiontracestepsstepmultiplyresultbound. pfa_gap_rbound_divisiontracestepsstepmultiplyresultbound + S (pfh_product_rbound_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_rbound_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_rbound_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_rbound_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_rbound_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_rbound_divisiontracestepsstep) + (p) * pfa_offset_right_rbound_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_rbound_divisiontracestepsstepaddleft. pfa_gap_rbound_divisiontracestepsstepaddleft + S (pfh_product_rbound_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_rbound_divisiontracestepsstepaddright. pfa_gap_rbound_divisiontracestepsstepaddright + S (pfh_coefficient_rbound_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_rbound_divisiontracestepsstepaddresultbound. pfa_gap_rbound_divisiontracestepsstepaddresultbound + S (pfh_after_rbound_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_rbound_divisiontracestepsstepaddresultcongruence pfa_offset_right_rbound_divisiontracestepsstepaddresultcongruence. ((pfh_product_rbound_divisiontracestepsstep) + (pfh_coefficient_rbound_divisiontracestepsstep)) + (p) * pfa_offset_left_rbound_divisiontracestepsstepaddresultcongruence = (pfh_after_rbound_divisiontracestepsstep) + (p) * pfa_offset_right_rbound_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_rbound_divisionquotient ff_source_mcp_pfs_rbound_divisionquotient ff_target_mcp_pfs_rbound_divisionquotient. (exists mcp_gap_pfs_rbound_divisionquotient_bound. mcp_gap_pfs_rbound_divisionquotient_bound + S (ff_index_mcp_pfs_rbound_divisionquotient) = (n)) -> (((exists fs_h_mcp_pfs_rbound_divisionquotient_source. fs_h_mcp_pfs_rbound_divisionquotient_source + S (ff_source_mcp_pfs_rbound_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_rbound_divisionquotient)) * pfs_history_scale_rbound_division)) /\ exists fs_q_mcp_pfs_rbound_divisionquotient_source. pfs_history_code_rbound_division = fs_q_mcp_pfs_rbound_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_rbound_divisionquotient)) * pfs_history_scale_rbound_division) + (ff_source_mcp_pfs_rbound_divisionquotient))) -> (((exists fs_h_mcp_pfs_rbound_divisionquotient_target. fs_h_mcp_pfs_rbound_divisionquotient_target + S (ff_target_mcp_pfs_rbound_divisionquotient) = S ((S (ff_index_mcp_pfs_rbound_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_rbound_divisionquotient_target. qb = fs_q_mcp_pfs_rbound_divisionquotient_target * S ((S (ff_index_mcp_pfs_rbound_divisionquotient)) * qc) + (ff_target_mcp_pfs_rbound_divisionquotient))) -> ff_target_mcp_pfs_rbound_divisionquotient = ff_source_mcp_pfs_rbound_divisionquotient)))) -> (exists pfa_gap_rbound_result. pfa_gap_rbound_result + S (r) = (p))prime_field_polynomial_synthetic_functionalExact checked statement
forall p b c a n qb qc r Qb Qc 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 pfs_history_code_functional_first pfs_history_scale_functional_first. ((((exists pfa_gap_functional_firsttracebase. pfa_gap_functional_firsttracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_functional_firsttraceinitial. ff_h_pfp_functional_firsttraceinitial + S (0) = S ((S (0)) * pfs_history_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttraceinitial. pfs_history_code_functional_first = ff_q_pfp_functional_firsttraceinitial * S ((S (0)) * pfs_history_scale_functional_first) + (0))) /\ (((((exists ff_h_pfp_functional_firsttraceterminal. ff_h_pfp_functional_firsttraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttraceterminal. pfs_history_code_functional_first = ff_q_pfp_functional_firsttraceterminal * S ((S (S (n))) * pfs_history_scale_functional_first) + (r))) /\ ((forall pfh_index_functional_firsttracesteps. (exists pfa_gap_functional_firsttracestepsindex. pfa_gap_functional_firsttracestepsindex + S (pfh_index_functional_firsttracesteps) = (S (n))) -> (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)) * pfs_history_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttracestepsstepbefore. pfs_history_code_functional_first = ff_q_pfp_functional_firsttracestepsstepbefore * S ((S (pfh_index_functional_firsttracesteps)) * pfs_history_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))) * pfs_history_scale_functional_first)) /\ exists ff_q_pfp_functional_firsttracestepsstepafter. pfs_history_code_functional_first = ff_q_pfp_functional_firsttracestepsstepafter * S ((S (S (pfh_index_functional_firsttracesteps))) * pfs_history_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 (a) = (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) * (a)) + (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)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_functional_firstquotient ff_source_mcp_pfs_functional_firstquotient ff_target_mcp_pfs_functional_firstquotient. (exists mcp_gap_pfs_functional_firstquotient_bound. mcp_gap_pfs_functional_firstquotient_bound + S (ff_index_mcp_pfs_functional_firstquotient) = (n)) -> (((exists fs_h_mcp_pfs_functional_firstquotient_source. fs_h_mcp_pfs_functional_firstquotient_source + S (ff_source_mcp_pfs_functional_firstquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_functional_firstquotient)) * pfs_history_scale_functional_first)) /\ exists fs_q_mcp_pfs_functional_firstquotient_source. pfs_history_code_functional_first = fs_q_mcp_pfs_functional_firstquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_functional_firstquotient)) * pfs_history_scale_functional_first) + (ff_source_mcp_pfs_functional_firstquotient))) -> (((exists fs_h_mcp_pfs_functional_firstquotient_target. fs_h_mcp_pfs_functional_firstquotient_target + S (ff_target_mcp_pfs_functional_firstquotient) = S ((S (ff_index_mcp_pfs_functional_firstquotient)) * qc)) /\ exists fs_q_mcp_pfs_functional_firstquotient_target. qb = fs_q_mcp_pfs_functional_firstquotient_target * S ((S (ff_index_mcp_pfs_functional_firstquotient)) * qc) + (ff_target_mcp_pfs_functional_firstquotient))) -> ff_target_mcp_pfs_functional_firstquotient = ff_source_mcp_pfs_functional_firstquotient)))) -> (exists pfs_history_code_functional_second pfs_history_scale_functional_second. ((((exists pfa_gap_functional_secondtracebase. pfa_gap_functional_secondtracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_functional_secondtraceinitial. ff_h_pfp_functional_secondtraceinitial + S (0) = S ((S (0)) * pfs_history_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtraceinitial. pfs_history_code_functional_second = ff_q_pfp_functional_secondtraceinitial * S ((S (0)) * pfs_history_scale_functional_second) + (0))) /\ (((((exists ff_h_pfp_functional_secondtraceterminal. ff_h_pfp_functional_secondtraceterminal + S (s) = S ((S (S (n))) * pfs_history_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtraceterminal. pfs_history_code_functional_second = ff_q_pfp_functional_secondtraceterminal * S ((S (S (n))) * pfs_history_scale_functional_second) + (s))) /\ ((forall pfh_index_functional_secondtracesteps. (exists pfa_gap_functional_secondtracestepsindex. pfa_gap_functional_secondtracestepsindex + S (pfh_index_functional_secondtracesteps) = (S (n))) -> (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)) * pfs_history_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtracestepsstepbefore. pfs_history_code_functional_second = ff_q_pfp_functional_secondtracestepsstepbefore * S ((S (pfh_index_functional_secondtracesteps)) * pfs_history_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))) * pfs_history_scale_functional_second)) /\ exists ff_q_pfp_functional_secondtracestepsstepafter. pfs_history_code_functional_second = ff_q_pfp_functional_secondtracestepsstepafter * S ((S (S (pfh_index_functional_secondtracesteps))) * pfs_history_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 (a) = (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) * (a)) + (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)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_functional_secondquotient ff_source_mcp_pfs_functional_secondquotient ff_target_mcp_pfs_functional_secondquotient. (exists mcp_gap_pfs_functional_secondquotient_bound. mcp_gap_pfs_functional_secondquotient_bound + S (ff_index_mcp_pfs_functional_secondquotient) = (n)) -> (((exists fs_h_mcp_pfs_functional_secondquotient_source. fs_h_mcp_pfs_functional_secondquotient_source + S (ff_source_mcp_pfs_functional_secondquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_functional_secondquotient)) * pfs_history_scale_functional_second)) /\ exists fs_q_mcp_pfs_functional_secondquotient_source. pfs_history_code_functional_second = fs_q_mcp_pfs_functional_secondquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_functional_secondquotient)) * pfs_history_scale_functional_second) + (ff_source_mcp_pfs_functional_secondquotient))) -> (((exists fs_h_mcp_pfs_functional_secondquotient_target. fs_h_mcp_pfs_functional_secondquotient_target + S (ff_target_mcp_pfs_functional_secondquotient) = S ((S (ff_index_mcp_pfs_functional_secondquotient)) * Qc)) /\ exists fs_q_mcp_pfs_functional_secondquotient_target. Qb = fs_q_mcp_pfs_functional_secondquotient_target * S ((S (ff_index_mcp_pfs_functional_secondquotient)) * Qc) + (ff_target_mcp_pfs_functional_secondquotient))) -> ff_target_mcp_pfs_functional_secondquotient = ff_source_mcp_pfs_functional_secondquotient)))) -> ((r=s) /\ ((forall mdr_i_pfp_functional_values mdr_a_pfp_functional_values. (exists mdr_gap_pfp_functional_valuesb. mdr_gap_pfp_functional_valuesb + S (mdr_i_pfp_functional_values) = (n)) -> (((exists ff_h_mdr_pfp_functional_valueso. ff_h_mdr_pfp_functional_valueso + S (mdr_a_pfp_functional_values) = S ((S (mdr_i_pfp_functional_values)) * qc)) /\ exists ff_q_mdr_pfp_functional_valueso. qb = ff_q_mdr_pfp_functional_valueso * S ((S (mdr_i_pfp_functional_values)) * qc) + (mdr_a_pfp_functional_values))) -> (((exists ff_h_mdr_pfp_functional_valuesn. ff_h_mdr_pfp_functional_valuesn + S (mdr_a_pfp_functional_values) = S ((S (mdr_i_pfp_functional_values)) * Qc)) /\ exists ff_q_mdr_pfp_functional_valuesn. Qb = ff_q_mdr_pfp_functional_valuesn * S ((S (mdr_i_pfp_functional_values)) * Qc) + (mdr_a_pfp_functional_values))))))prime_field_polynomial_horner_constant_valueExact checked statement
forall p b c a r v. (~((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 pfh_trace_code_constant_execution pfh_trace_scale_constant_execution. (((exists pfa_gap_constant_executiontracebase. pfa_gap_constant_executiontracebase + S (a) = (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 (r) = 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) + (r))) /\ ((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 (a) = (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) * (a)) + (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))))))))))))))))))))))))))) -> (((exists ff_h_pfp_constant_actual_coefficient. ff_h_pfp_constant_actual_coefficient + S (v) = S ((S (0)) * c)) /\ exists ff_q_pfp_constant_actual_coefficient. b = ff_q_pfp_constant_actual_coefficient * S ((S (0)) * c) + (v))) -> r=vprime_field_polynomial_horner_transition_valuesExact checked statement
forall p b c a i h v r. (~((p) = 1) /\ forall pfa_factor_left_transition_prime pfa_factor_right_transition_prime. (p) = pfa_factor_left_transition_prime * pfa_factor_right_transition_prime -> pfa_factor_left_transition_prime = 1 \/ pfa_factor_right_transition_prime = 1) -> (exists pfh_trace_code_transition_before pfh_trace_scale_transition_before. (((exists pfa_gap_transition_beforetracebase. pfa_gap_transition_beforetracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_transition_beforetraceinitial. ff_h_pfp_transition_beforetraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_transition_before)) /\ exists ff_q_pfp_transition_beforetraceinitial. pfh_trace_code_transition_before = ff_q_pfp_transition_beforetraceinitial * S ((S (0)) * pfh_trace_scale_transition_before) + (0))) /\ (((((exists ff_h_pfp_transition_beforetraceterminal. ff_h_pfp_transition_beforetraceterminal + S (h) = S ((S (i)) * pfh_trace_scale_transition_before)) /\ exists ff_q_pfp_transition_beforetraceterminal. pfh_trace_code_transition_before = ff_q_pfp_transition_beforetraceterminal * S ((S (i)) * pfh_trace_scale_transition_before) + (h))) /\ ((forall pfh_index_transition_beforetracesteps. (exists pfa_gap_transition_beforetracestepsindex. pfa_gap_transition_beforetracestepsindex + S (pfh_index_transition_beforetracesteps) = (i)) -> (exists pfh_coefficient_transition_beforetracestepsstep pfh_before_transition_beforetracestepsstep pfh_after_transition_beforetracestepsstep pfh_product_transition_beforetracestepsstep. ((((exists ff_h_pfp_transition_beforetracestepsstepcoefficient. ff_h_pfp_transition_beforetracestepsstepcoefficient + S (pfh_coefficient_transition_beforetracestepsstep) = S ((S (pfh_index_transition_beforetracesteps)) * c)) /\ exists ff_q_pfp_transition_beforetracestepsstepcoefficient. b = ff_q_pfp_transition_beforetracestepsstepcoefficient * S ((S (pfh_index_transition_beforetracesteps)) * c) + (pfh_coefficient_transition_beforetracestepsstep))) /\ (((((exists ff_h_pfp_transition_beforetracestepsstepbefore. ff_h_pfp_transition_beforetracestepsstepbefore + S (pfh_before_transition_beforetracestepsstep) = S ((S (pfh_index_transition_beforetracesteps)) * pfh_trace_scale_transition_before)) /\ exists ff_q_pfp_transition_beforetracestepsstepbefore. pfh_trace_code_transition_before = ff_q_pfp_transition_beforetracestepsstepbefore * S ((S (pfh_index_transition_beforetracesteps)) * pfh_trace_scale_transition_before) + (pfh_before_transition_beforetracestepsstep))) /\ (((((exists ff_h_pfp_transition_beforetracestepsstepafter. ff_h_pfp_transition_beforetracestepsstepafter + S (pfh_after_transition_beforetracestepsstep) = S ((S (S (pfh_index_transition_beforetracesteps))) * pfh_trace_scale_transition_before)) /\ exists ff_q_pfp_transition_beforetracestepsstepafter. pfh_trace_code_transition_before = ff_q_pfp_transition_beforetracestepsstepafter * S ((S (S (pfh_index_transition_beforetracesteps))) * pfh_trace_scale_transition_before) + (pfh_after_transition_beforetracestepsstep))) /\ (((((exists pfa_gap_transition_beforetracestepsstepmultiplyleft. pfa_gap_transition_beforetracestepsstepmultiplyleft + S (pfh_before_transition_beforetracestepsstep) = (p)) /\ (((exists pfa_gap_transition_beforetracestepsstepmultiplyright. pfa_gap_transition_beforetracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_transition_beforetracestepsstepmultiplyresultbound. pfa_gap_transition_beforetracestepsstepmultiplyresultbound + S (pfh_product_transition_beforetracestepsstep) = (p)) /\ ((exists pfa_offset_left_transition_beforetracestepsstepmultiplyresultcongruence pfa_offset_right_transition_beforetracestepsstepmultiplyresultcongruence. ((pfh_before_transition_beforetracestepsstep) * (a)) + (p) * pfa_offset_left_transition_beforetracestepsstepmultiplyresultcongruence = (pfh_product_transition_beforetracestepsstep) + (p) * pfa_offset_right_transition_beforetracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_transition_beforetracestepsstepaddleft. pfa_gap_transition_beforetracestepsstepaddleft + S (pfh_product_transition_beforetracestepsstep) = (p)) /\ (((exists pfa_gap_transition_beforetracestepsstepaddright. pfa_gap_transition_beforetracestepsstepaddright + S (pfh_coefficient_transition_beforetracestepsstep) = (p)) /\ ((((exists pfa_gap_transition_beforetracestepsstepaddresultbound. pfa_gap_transition_beforetracestepsstepaddresultbound + S (pfh_after_transition_beforetracestepsstep) = (p)) /\ ((exists pfa_offset_left_transition_beforetracestepsstepaddresultcongruence pfa_offset_right_transition_beforetracestepsstepaddresultcongruence. ((pfh_product_transition_beforetracestepsstep) + (pfh_coefficient_transition_beforetracestepsstep)) + (p) * pfa_offset_left_transition_beforetracestepsstepaddresultcongruence = (pfh_after_transition_beforetracestepsstep) + (p) * pfa_offset_right_transition_beforetracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (exists pfh_trace_code_transition_after pfh_trace_scale_transition_after. (((exists pfa_gap_transition_aftertracebase. pfa_gap_transition_aftertracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_transition_aftertraceinitial. ff_h_pfp_transition_aftertraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_transition_after)) /\ exists ff_q_pfp_transition_aftertraceinitial. pfh_trace_code_transition_after = ff_q_pfp_transition_aftertraceinitial * S ((S (0)) * pfh_trace_scale_transition_after) + (0))) /\ (((((exists ff_h_pfp_transition_aftertraceterminal. ff_h_pfp_transition_aftertraceterminal + S (r) = S ((S (S i)) * pfh_trace_scale_transition_after)) /\ exists ff_q_pfp_transition_aftertraceterminal. pfh_trace_code_transition_after = ff_q_pfp_transition_aftertraceterminal * S ((S (S i)) * pfh_trace_scale_transition_after) + (r))) /\ ((forall pfh_index_transition_aftertracesteps. (exists pfa_gap_transition_aftertracestepsindex. pfa_gap_transition_aftertracestepsindex + S (pfh_index_transition_aftertracesteps) = (S i)) -> (exists pfh_coefficient_transition_aftertracestepsstep pfh_before_transition_aftertracestepsstep pfh_after_transition_aftertracestepsstep pfh_product_transition_aftertracestepsstep. ((((exists ff_h_pfp_transition_aftertracestepsstepcoefficient. ff_h_pfp_transition_aftertracestepsstepcoefficient + S (pfh_coefficient_transition_aftertracestepsstep) = S ((S (pfh_index_transition_aftertracesteps)) * c)) /\ exists ff_q_pfp_transition_aftertracestepsstepcoefficient. b = ff_q_pfp_transition_aftertracestepsstepcoefficient * S ((S (pfh_index_transition_aftertracesteps)) * c) + (pfh_coefficient_transition_aftertracestepsstep))) /\ (((((exists ff_h_pfp_transition_aftertracestepsstepbefore. ff_h_pfp_transition_aftertracestepsstepbefore + S (pfh_before_transition_aftertracestepsstep) = S ((S (pfh_index_transition_aftertracesteps)) * pfh_trace_scale_transition_after)) /\ exists ff_q_pfp_transition_aftertracestepsstepbefore. pfh_trace_code_transition_after = ff_q_pfp_transition_aftertracestepsstepbefore * S ((S (pfh_index_transition_aftertracesteps)) * pfh_trace_scale_transition_after) + (pfh_before_transition_aftertracestepsstep))) /\ (((((exists ff_h_pfp_transition_aftertracestepsstepafter. ff_h_pfp_transition_aftertracestepsstepafter + S (pfh_after_transition_aftertracestepsstep) = S ((S (S (pfh_index_transition_aftertracesteps))) * pfh_trace_scale_transition_after)) /\ exists ff_q_pfp_transition_aftertracestepsstepafter. pfh_trace_code_transition_after = ff_q_pfp_transition_aftertracestepsstepafter * S ((S (S (pfh_index_transition_aftertracesteps))) * pfh_trace_scale_transition_after) + (pfh_after_transition_aftertracestepsstep))) /\ (((((exists pfa_gap_transition_aftertracestepsstepmultiplyleft. pfa_gap_transition_aftertracestepsstepmultiplyleft + S (pfh_before_transition_aftertracestepsstep) = (p)) /\ (((exists pfa_gap_transition_aftertracestepsstepmultiplyright. pfa_gap_transition_aftertracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_transition_aftertracestepsstepmultiplyresultbound. pfa_gap_transition_aftertracestepsstepmultiplyresultbound + S (pfh_product_transition_aftertracestepsstep) = (p)) /\ ((exists pfa_offset_left_transition_aftertracestepsstepmultiplyresultcongruence pfa_offset_right_transition_aftertracestepsstepmultiplyresultcongruence. ((pfh_before_transition_aftertracestepsstep) * (a)) + (p) * pfa_offset_left_transition_aftertracestepsstepmultiplyresultcongruence = (pfh_product_transition_aftertracestepsstep) + (p) * pfa_offset_right_transition_aftertracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_transition_aftertracestepsstepaddleft. pfa_gap_transition_aftertracestepsstepaddleft + S (pfh_product_transition_aftertracestepsstep) = (p)) /\ (((exists pfa_gap_transition_aftertracestepsstepaddright. pfa_gap_transition_aftertracestepsstepaddright + S (pfh_coefficient_transition_aftertracestepsstep) = (p)) /\ ((((exists pfa_gap_transition_aftertracestepsstepaddresultbound. pfa_gap_transition_aftertracestepsstepaddresultbound + S (pfh_after_transition_aftertracestepsstep) = (p)) /\ ((exists pfa_offset_left_transition_aftertracestepsstepaddresultcongruence pfa_offset_right_transition_aftertracestepsstepaddresultcongruence. ((pfh_product_transition_aftertracestepsstep) + (pfh_coefficient_transition_aftertracestepsstep)) + (p) * pfa_offset_left_transition_aftertracestepsstepaddresultcongruence = (pfh_after_transition_aftertracestepsstep) + (p) * pfa_offset_right_transition_aftertracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> (((exists ff_h_pfp_transition_coefficient. ff_h_pfp_transition_coefficient + S (v) = S ((S (i)) * c)) /\ exists ff_q_pfp_transition_coefficient. b = ff_q_pfp_transition_coefficient * S ((S (i)) * c) + (v))) -> exists k. ((((exists pfa_gap_transition_productleft. pfa_gap_transition_productleft + S (h) = (p)) /\ (((exists pfa_gap_transition_productright. pfa_gap_transition_productright + S (a) = (p)) /\ ((((exists pfa_gap_transition_productresultbound. pfa_gap_transition_productresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_transition_productresultcongruence pfa_offset_right_transition_productresultcongruence. ((h) * (a)) + (p) * pfa_offset_left_transition_productresultcongruence = (k) + (p) * pfa_offset_right_transition_productresultcongruence))))))))) /\ ((((exists pfa_gap_transition_sumleft. pfa_gap_transition_sumleft + S (k) = (p)) /\ (((exists pfa_gap_transition_sumright. pfa_gap_transition_sumright + S (v) = (p)) /\ ((((exists pfa_gap_transition_sumresultbound. pfa_gap_transition_sumresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_transition_sumresultcongruence pfa_offset_right_transition_sumresultcongruence. ((k) + (v)) + (p) * pfa_offset_left_transition_sumresultcongruence = (r) + (p) * pfa_offset_right_transition_sumresultcongruence)))))))))))prime_field_polynomial_synthetic_leading_coefficientExact checked statement
forall p b c a n qb qc r v. (~((p) = 1) /\ forall pfa_factor_left_leading_prime pfa_factor_right_leading_prime. (p) = pfa_factor_left_leading_prime * pfa_factor_right_leading_prime -> pfa_factor_left_leading_prime = 1 \/ pfa_factor_right_leading_prime = 1) -> (exists pfs_history_code_leading_division pfs_history_scale_leading_division. ((((exists pfa_gap_leading_divisiontracebase. pfa_gap_leading_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_leading_divisiontraceinitial. ff_h_pfp_leading_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_leading_division)) /\ exists ff_q_pfp_leading_divisiontraceinitial. pfs_history_code_leading_division = ff_q_pfp_leading_divisiontraceinitial * S ((S (0)) * pfs_history_scale_leading_division) + (0))) /\ (((((exists ff_h_pfp_leading_divisiontraceterminal. ff_h_pfp_leading_divisiontraceterminal + S (r) = S ((S (S (S n))) * pfs_history_scale_leading_division)) /\ exists ff_q_pfp_leading_divisiontraceterminal. pfs_history_code_leading_division = ff_q_pfp_leading_divisiontraceterminal * S ((S (S (S n))) * pfs_history_scale_leading_division) + (r))) /\ ((forall pfh_index_leading_divisiontracesteps. (exists pfa_gap_leading_divisiontracestepsindex. pfa_gap_leading_divisiontracestepsindex + S (pfh_index_leading_divisiontracesteps) = (S (S n))) -> (exists pfh_coefficient_leading_divisiontracestepsstep pfh_before_leading_divisiontracestepsstep pfh_after_leading_divisiontracestepsstep pfh_product_leading_divisiontracestepsstep. ((((exists ff_h_pfp_leading_divisiontracestepsstepcoefficient. ff_h_pfp_leading_divisiontracestepsstepcoefficient + S (pfh_coefficient_leading_divisiontracestepsstep) = S ((S (pfh_index_leading_divisiontracesteps)) * c)) /\ exists ff_q_pfp_leading_divisiontracestepsstepcoefficient. b = ff_q_pfp_leading_divisiontracestepsstepcoefficient * S ((S (pfh_index_leading_divisiontracesteps)) * c) + (pfh_coefficient_leading_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_leading_divisiontracestepsstepbefore. ff_h_pfp_leading_divisiontracestepsstepbefore + S (pfh_before_leading_divisiontracestepsstep) = S ((S (pfh_index_leading_divisiontracesteps)) * pfs_history_scale_leading_division)) /\ exists ff_q_pfp_leading_divisiontracestepsstepbefore. pfs_history_code_leading_division = ff_q_pfp_leading_divisiontracestepsstepbefore * S ((S (pfh_index_leading_divisiontracesteps)) * pfs_history_scale_leading_division) + (pfh_before_leading_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_leading_divisiontracestepsstepafter. ff_h_pfp_leading_divisiontracestepsstepafter + S (pfh_after_leading_divisiontracestepsstep) = S ((S (S (pfh_index_leading_divisiontracesteps))) * pfs_history_scale_leading_division)) /\ exists ff_q_pfp_leading_divisiontracestepsstepafter. pfs_history_code_leading_division = ff_q_pfp_leading_divisiontracestepsstepafter * S ((S (S (pfh_index_leading_divisiontracesteps))) * pfs_history_scale_leading_division) + (pfh_after_leading_divisiontracestepsstep))) /\ (((((exists pfa_gap_leading_divisiontracestepsstepmultiplyleft. pfa_gap_leading_divisiontracestepsstepmultiplyleft + S (pfh_before_leading_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_leading_divisiontracestepsstepmultiplyright. pfa_gap_leading_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_leading_divisiontracestepsstepmultiplyresultbound. pfa_gap_leading_divisiontracestepsstepmultiplyresultbound + S (pfh_product_leading_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_leading_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_leading_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_leading_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_leading_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_leading_divisiontracestepsstep) + (p) * pfa_offset_right_leading_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_leading_divisiontracestepsstepaddleft. pfa_gap_leading_divisiontracestepsstepaddleft + S (pfh_product_leading_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_leading_divisiontracestepsstepaddright. pfa_gap_leading_divisiontracestepsstepaddright + S (pfh_coefficient_leading_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_leading_divisiontracestepsstepaddresultbound. pfa_gap_leading_divisiontracestepsstepaddresultbound + S (pfh_after_leading_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_leading_divisiontracestepsstepaddresultcongruence pfa_offset_right_leading_divisiontracestepsstepaddresultcongruence. ((pfh_product_leading_divisiontracestepsstep) + (pfh_coefficient_leading_divisiontracestepsstep)) + (p) * pfa_offset_left_leading_divisiontracestepsstepaddresultcongruence = (pfh_after_leading_divisiontracestepsstep) + (p) * pfa_offset_right_leading_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_leading_divisionquotient ff_source_mcp_pfs_leading_divisionquotient ff_target_mcp_pfs_leading_divisionquotient. (exists mcp_gap_pfs_leading_divisionquotient_bound. mcp_gap_pfs_leading_divisionquotient_bound + S (ff_index_mcp_pfs_leading_divisionquotient) = (S n)) -> (((exists fs_h_mcp_pfs_leading_divisionquotient_source. fs_h_mcp_pfs_leading_divisionquotient_source + S (ff_source_mcp_pfs_leading_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_leading_divisionquotient)) * pfs_history_scale_leading_division)) /\ exists fs_q_mcp_pfs_leading_divisionquotient_source. pfs_history_code_leading_division = fs_q_mcp_pfs_leading_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_leading_divisionquotient)) * pfs_history_scale_leading_division) + (ff_source_mcp_pfs_leading_divisionquotient))) -> (((exists fs_h_mcp_pfs_leading_divisionquotient_target. fs_h_mcp_pfs_leading_divisionquotient_target + S (ff_target_mcp_pfs_leading_divisionquotient) = S ((S (ff_index_mcp_pfs_leading_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_leading_divisionquotient_target. qb = fs_q_mcp_pfs_leading_divisionquotient_target * S ((S (ff_index_mcp_pfs_leading_divisionquotient)) * qc) + (ff_target_mcp_pfs_leading_divisionquotient))) -> ff_target_mcp_pfs_leading_divisionquotient = ff_source_mcp_pfs_leading_divisionquotient)))) -> (((exists ff_h_pfp_leading_input. ff_h_pfp_leading_input + S (v) = S ((S (0)) * c)) /\ exists ff_q_pfp_leading_input. b = ff_q_pfp_leading_input * S ((S (0)) * c) + (v))) -> (((exists ff_h_pfp_leading_output. ff_h_pfp_leading_output + S (v) = S ((S (0)) * qc)) /\ exists ff_q_pfp_leading_output. qb = ff_q_pfp_leading_output * S ((S (0)) * qc) + (v)))prime_field_polynomial_synthetic_middle_coefficientsExact checked statement
forall p b c a n qb qc r i h j v. (~((p) = 1) /\ forall pfa_factor_left_middle_prime pfa_factor_right_middle_prime. (p) = pfa_factor_left_middle_prime * pfa_factor_right_middle_prime -> pfa_factor_left_middle_prime = 1 \/ pfa_factor_right_middle_prime = 1) -> (exists pfs_history_code_middle_division pfs_history_scale_middle_division. ((((exists pfa_gap_middle_divisiontracebase. pfa_gap_middle_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_middle_divisiontraceinitial. ff_h_pfp_middle_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_middle_division)) /\ exists ff_q_pfp_middle_divisiontraceinitial. pfs_history_code_middle_division = ff_q_pfp_middle_divisiontraceinitial * S ((S (0)) * pfs_history_scale_middle_division) + (0))) /\ (((((exists ff_h_pfp_middle_divisiontraceterminal. ff_h_pfp_middle_divisiontraceterminal + S (r) = S ((S (S (S n))) * pfs_history_scale_middle_division)) /\ exists ff_q_pfp_middle_divisiontraceterminal. pfs_history_code_middle_division = ff_q_pfp_middle_divisiontraceterminal * S ((S (S (S n))) * pfs_history_scale_middle_division) + (r))) /\ ((forall pfh_index_middle_divisiontracesteps. (exists pfa_gap_middle_divisiontracestepsindex. pfa_gap_middle_divisiontracestepsindex + S (pfh_index_middle_divisiontracesteps) = (S (S n))) -> (exists pfh_coefficient_middle_divisiontracestepsstep pfh_before_middle_divisiontracestepsstep pfh_after_middle_divisiontracestepsstep pfh_product_middle_divisiontracestepsstep. ((((exists ff_h_pfp_middle_divisiontracestepsstepcoefficient. ff_h_pfp_middle_divisiontracestepsstepcoefficient + S (pfh_coefficient_middle_divisiontracestepsstep) = S ((S (pfh_index_middle_divisiontracesteps)) * c)) /\ exists ff_q_pfp_middle_divisiontracestepsstepcoefficient. b = ff_q_pfp_middle_divisiontracestepsstepcoefficient * S ((S (pfh_index_middle_divisiontracesteps)) * c) + (pfh_coefficient_middle_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_middle_divisiontracestepsstepbefore. ff_h_pfp_middle_divisiontracestepsstepbefore + S (pfh_before_middle_divisiontracestepsstep) = S ((S (pfh_index_middle_divisiontracesteps)) * pfs_history_scale_middle_division)) /\ exists ff_q_pfp_middle_divisiontracestepsstepbefore. pfs_history_code_middle_division = ff_q_pfp_middle_divisiontracestepsstepbefore * S ((S (pfh_index_middle_divisiontracesteps)) * pfs_history_scale_middle_division) + (pfh_before_middle_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_middle_divisiontracestepsstepafter. ff_h_pfp_middle_divisiontracestepsstepafter + S (pfh_after_middle_divisiontracestepsstep) = S ((S (S (pfh_index_middle_divisiontracesteps))) * pfs_history_scale_middle_division)) /\ exists ff_q_pfp_middle_divisiontracestepsstepafter. pfs_history_code_middle_division = ff_q_pfp_middle_divisiontracestepsstepafter * S ((S (S (pfh_index_middle_divisiontracesteps))) * pfs_history_scale_middle_division) + (pfh_after_middle_divisiontracestepsstep))) /\ (((((exists pfa_gap_middle_divisiontracestepsstepmultiplyleft. pfa_gap_middle_divisiontracestepsstepmultiplyleft + S (pfh_before_middle_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_middle_divisiontracestepsstepmultiplyright. pfa_gap_middle_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_middle_divisiontracestepsstepmultiplyresultbound. pfa_gap_middle_divisiontracestepsstepmultiplyresultbound + S (pfh_product_middle_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_middle_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_middle_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_middle_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_middle_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_middle_divisiontracestepsstep) + (p) * pfa_offset_right_middle_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_middle_divisiontracestepsstepaddleft. pfa_gap_middle_divisiontracestepsstepaddleft + S (pfh_product_middle_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_middle_divisiontracestepsstepaddright. pfa_gap_middle_divisiontracestepsstepaddright + S (pfh_coefficient_middle_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_middle_divisiontracestepsstepaddresultbound. pfa_gap_middle_divisiontracestepsstepaddresultbound + S (pfh_after_middle_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_middle_divisiontracestepsstepaddresultcongruence pfa_offset_right_middle_divisiontracestepsstepaddresultcongruence. ((pfh_product_middle_divisiontracestepsstep) + (pfh_coefficient_middle_divisiontracestepsstep)) + (p) * pfa_offset_left_middle_divisiontracestepsstepaddresultcongruence = (pfh_after_middle_divisiontracestepsstep) + (p) * pfa_offset_right_middle_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_middle_divisionquotient ff_source_mcp_pfs_middle_divisionquotient ff_target_mcp_pfs_middle_divisionquotient. (exists mcp_gap_pfs_middle_divisionquotient_bound. mcp_gap_pfs_middle_divisionquotient_bound + S (ff_index_mcp_pfs_middle_divisionquotient) = (S n)) -> (((exists fs_h_mcp_pfs_middle_divisionquotient_source. fs_h_mcp_pfs_middle_divisionquotient_source + S (ff_source_mcp_pfs_middle_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_middle_divisionquotient)) * pfs_history_scale_middle_division)) /\ exists fs_q_mcp_pfs_middle_divisionquotient_source. pfs_history_code_middle_division = fs_q_mcp_pfs_middle_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_middle_divisionquotient)) * pfs_history_scale_middle_division) + (ff_source_mcp_pfs_middle_divisionquotient))) -> (((exists fs_h_mcp_pfs_middle_divisionquotient_target. fs_h_mcp_pfs_middle_divisionquotient_target + S (ff_target_mcp_pfs_middle_divisionquotient) = S ((S (ff_index_mcp_pfs_middle_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_middle_divisionquotient_target. qb = fs_q_mcp_pfs_middle_divisionquotient_target * S ((S (ff_index_mcp_pfs_middle_divisionquotient)) * qc) + (ff_target_mcp_pfs_middle_divisionquotient))) -> ff_target_mcp_pfs_middle_divisionquotient = ff_source_mcp_pfs_middle_divisionquotient)))) -> (exists pfa_gap_middle_index. pfa_gap_middle_index + S (i) = (n)) -> (((exists ff_h_pfp_middle_previous. ff_h_pfp_middle_previous + S (h) = S ((S (i)) * qc)) /\ exists ff_q_pfp_middle_previous. qb = ff_q_pfp_middle_previous * S ((S (i)) * qc) + (h))) -> (((exists ff_h_pfp_middle_next. ff_h_pfp_middle_next + S (j) = S ((S (S i)) * qc)) /\ exists ff_q_pfp_middle_next. qb = ff_q_pfp_middle_next * S ((S (S i)) * qc) + (j))) -> (((exists ff_h_pfp_middle_input. ff_h_pfp_middle_input + S (v) = S ((S (S i)) * c)) /\ exists ff_q_pfp_middle_input. b = ff_q_pfp_middle_input * S ((S (S i)) * c) + (v))) -> exists k. ((((exists pfa_gap_middle_productleft. pfa_gap_middle_productleft + S (h) = (p)) /\ (((exists pfa_gap_middle_productright. pfa_gap_middle_productright + S (a) = (p)) /\ ((((exists pfa_gap_middle_productresultbound. pfa_gap_middle_productresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_middle_productresultcongruence pfa_offset_right_middle_productresultcongruence. ((h) * (a)) + (p) * pfa_offset_left_middle_productresultcongruence = (k) + (p) * pfa_offset_right_middle_productresultcongruence))))))))) /\ ((((exists pfa_gap_middle_sumleft. pfa_gap_middle_sumleft + S (k) = (p)) /\ (((exists pfa_gap_middle_sumright. pfa_gap_middle_sumright + S (v) = (p)) /\ ((((exists pfa_gap_middle_sumresultbound. pfa_gap_middle_sumresultbound + S (j) = (p)) /\ ((exists pfa_offset_left_middle_sumresultcongruence pfa_offset_right_middle_sumresultcongruence. ((k) + (v)) + (p) * pfa_offset_left_middle_sumresultcongruence = (j) + (p) * pfa_offset_right_middle_sumresultcongruence)))))))))))prime_field_polynomial_synthetic_final_coefficientExact checked statement
forall p b c a n qb qc r h v. (~((p) = 1) /\ forall pfa_factor_left_final_prime pfa_factor_right_final_prime. (p) = pfa_factor_left_final_prime * pfa_factor_right_final_prime -> pfa_factor_left_final_prime = 1 \/ pfa_factor_right_final_prime = 1) -> (exists pfs_history_code_final_division pfs_history_scale_final_division. ((((exists pfa_gap_final_divisiontracebase. pfa_gap_final_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_final_divisiontraceinitial. ff_h_pfp_final_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_final_division)) /\ exists ff_q_pfp_final_divisiontraceinitial. pfs_history_code_final_division = ff_q_pfp_final_divisiontraceinitial * S ((S (0)) * pfs_history_scale_final_division) + (0))) /\ (((((exists ff_h_pfp_final_divisiontraceterminal. ff_h_pfp_final_divisiontraceterminal + S (r) = S ((S (S (S n))) * pfs_history_scale_final_division)) /\ exists ff_q_pfp_final_divisiontraceterminal. pfs_history_code_final_division = ff_q_pfp_final_divisiontraceterminal * S ((S (S (S n))) * pfs_history_scale_final_division) + (r))) /\ ((forall pfh_index_final_divisiontracesteps. (exists pfa_gap_final_divisiontracestepsindex. pfa_gap_final_divisiontracestepsindex + S (pfh_index_final_divisiontracesteps) = (S (S n))) -> (exists pfh_coefficient_final_divisiontracestepsstep pfh_before_final_divisiontracestepsstep pfh_after_final_divisiontracestepsstep pfh_product_final_divisiontracestepsstep. ((((exists ff_h_pfp_final_divisiontracestepsstepcoefficient. ff_h_pfp_final_divisiontracestepsstepcoefficient + S (pfh_coefficient_final_divisiontracestepsstep) = S ((S (pfh_index_final_divisiontracesteps)) * c)) /\ exists ff_q_pfp_final_divisiontracestepsstepcoefficient. b = ff_q_pfp_final_divisiontracestepsstepcoefficient * S ((S (pfh_index_final_divisiontracesteps)) * c) + (pfh_coefficient_final_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_final_divisiontracestepsstepbefore. ff_h_pfp_final_divisiontracestepsstepbefore + S (pfh_before_final_divisiontracestepsstep) = S ((S (pfh_index_final_divisiontracesteps)) * pfs_history_scale_final_division)) /\ exists ff_q_pfp_final_divisiontracestepsstepbefore. pfs_history_code_final_division = ff_q_pfp_final_divisiontracestepsstepbefore * S ((S (pfh_index_final_divisiontracesteps)) * pfs_history_scale_final_division) + (pfh_before_final_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_final_divisiontracestepsstepafter. ff_h_pfp_final_divisiontracestepsstepafter + S (pfh_after_final_divisiontracestepsstep) = S ((S (S (pfh_index_final_divisiontracesteps))) * pfs_history_scale_final_division)) /\ exists ff_q_pfp_final_divisiontracestepsstepafter. pfs_history_code_final_division = ff_q_pfp_final_divisiontracestepsstepafter * S ((S (S (pfh_index_final_divisiontracesteps))) * pfs_history_scale_final_division) + (pfh_after_final_divisiontracestepsstep))) /\ (((((exists pfa_gap_final_divisiontracestepsstepmultiplyleft. pfa_gap_final_divisiontracestepsstepmultiplyleft + S (pfh_before_final_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_final_divisiontracestepsstepmultiplyright. pfa_gap_final_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_final_divisiontracestepsstepmultiplyresultbound. pfa_gap_final_divisiontracestepsstepmultiplyresultbound + S (pfh_product_final_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_final_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_final_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_final_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_final_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_final_divisiontracestepsstep) + (p) * pfa_offset_right_final_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_final_divisiontracestepsstepaddleft. pfa_gap_final_divisiontracestepsstepaddleft + S (pfh_product_final_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_final_divisiontracestepsstepaddright. pfa_gap_final_divisiontracestepsstepaddright + S (pfh_coefficient_final_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_final_divisiontracestepsstepaddresultbound. pfa_gap_final_divisiontracestepsstepaddresultbound + S (pfh_after_final_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_final_divisiontracestepsstepaddresultcongruence pfa_offset_right_final_divisiontracestepsstepaddresultcongruence. ((pfh_product_final_divisiontracestepsstep) + (pfh_coefficient_final_divisiontracestepsstep)) + (p) * pfa_offset_left_final_divisiontracestepsstepaddresultcongruence = (pfh_after_final_divisiontracestepsstep) + (p) * pfa_offset_right_final_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_final_divisionquotient ff_source_mcp_pfs_final_divisionquotient ff_target_mcp_pfs_final_divisionquotient. (exists mcp_gap_pfs_final_divisionquotient_bound. mcp_gap_pfs_final_divisionquotient_bound + S (ff_index_mcp_pfs_final_divisionquotient) = (S n)) -> (((exists fs_h_mcp_pfs_final_divisionquotient_source. fs_h_mcp_pfs_final_divisionquotient_source + S (ff_source_mcp_pfs_final_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_final_divisionquotient)) * pfs_history_scale_final_division)) /\ exists fs_q_mcp_pfs_final_divisionquotient_source. pfs_history_code_final_division = fs_q_mcp_pfs_final_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_final_divisionquotient)) * pfs_history_scale_final_division) + (ff_source_mcp_pfs_final_divisionquotient))) -> (((exists fs_h_mcp_pfs_final_divisionquotient_target. fs_h_mcp_pfs_final_divisionquotient_target + S (ff_target_mcp_pfs_final_divisionquotient) = S ((S (ff_index_mcp_pfs_final_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_final_divisionquotient_target. qb = fs_q_mcp_pfs_final_divisionquotient_target * S ((S (ff_index_mcp_pfs_final_divisionquotient)) * qc) + (ff_target_mcp_pfs_final_divisionquotient))) -> ff_target_mcp_pfs_final_divisionquotient = ff_source_mcp_pfs_final_divisionquotient)))) -> (((exists ff_h_pfp_final_quotient. ff_h_pfp_final_quotient + S (h) = S ((S (n)) * qc)) /\ exists ff_q_pfp_final_quotient. qb = ff_q_pfp_final_quotient * S ((S (n)) * qc) + (h))) -> (((exists ff_h_pfp_final_input. ff_h_pfp_final_input + S (v) = S ((S (S n)) * c)) /\ exists ff_q_pfp_final_input. b = ff_q_pfp_final_input * S ((S (S n)) * c) + (v))) -> exists k. ((((exists pfa_gap_final_productleft. pfa_gap_final_productleft + S (h) = (p)) /\ (((exists pfa_gap_final_productright. pfa_gap_final_productright + S (a) = (p)) /\ ((((exists pfa_gap_final_productresultbound. pfa_gap_final_productresultbound + S (k) = (p)) /\ ((exists pfa_offset_left_final_productresultcongruence pfa_offset_right_final_productresultcongruence. ((h) * (a)) + (p) * pfa_offset_left_final_productresultcongruence = (k) + (p) * pfa_offset_right_final_productresultcongruence))))))))) /\ ((((exists pfa_gap_final_sumleft. pfa_gap_final_sumleft + S (k) = (p)) /\ (((exists pfa_gap_final_sumright. pfa_gap_final_sumright + S (v) = (p)) /\ ((((exists pfa_gap_final_sumresultbound. pfa_gap_final_sumresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_final_sumresultcongruence pfa_offset_right_final_sumresultcongruence. ((k) + (v)) + (p) * pfa_offset_left_final_sumresultcongruence = (r) + (p) * pfa_offset_right_final_sumresultcongruence)))))))))))prime_field_polynomial_synthetic_represented_degreeExact checked statement
forall p b c a n qb qc r. (~((p) = 1) /\ forall pfa_factor_left_degree_prime pfa_factor_right_degree_prime. (p) = pfa_factor_left_degree_prime * pfa_factor_right_degree_prime -> pfa_factor_left_degree_prime = 1 \/ pfa_factor_right_degree_prime = 1) -> ((((S (S n))=S (S n)) /\ (((forall fom_index_pfp_degree_inputcoefficients. (exists fom_gap_pfp_degree_inputcoefficients_index_bound. fom_gap_pfp_degree_inputcoefficients_index_bound + S (fom_index_pfp_degree_inputcoefficients) = S (S n)) -> exists fom_value_pfp_degree_inputcoefficients. ((((exists fom_beta_height_pfp_degree_inputcoefficients_entry. fom_beta_height_pfp_degree_inputcoefficients_entry + S (fom_value_pfp_degree_inputcoefficients) = S ((S (fom_index_pfp_degree_inputcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_degree_inputcoefficients_entry. b = fom_beta_quotient_pfp_degree_inputcoefficients_entry * S ((S (fom_index_pfp_degree_inputcoefficients)) * c) + (fom_value_pfp_degree_inputcoefficients))) /\ (exists fom_gap_pfp_degree_inputcoefficients_value_bound. fom_gap_pfp_degree_inputcoefficients_value_bound + S (fom_value_pfp_degree_inputcoefficients) = p))) /\ ((exists pfd_leading_degree_input. ((((exists ff_h_pfp_degree_inputentry. ff_h_pfp_degree_inputentry + S (pfd_leading_degree_input) = S ((S (0)) * c)) /\ exists ff_q_pfp_degree_inputentry. b = ff_q_pfp_degree_inputentry * S ((S (0)) * c) + (pfd_leading_degree_input))) /\ ((~(pfd_leading_degree_input=0)))))))))) -> (exists pfs_history_code_degree_division pfs_history_scale_degree_division. ((((exists pfa_gap_degree_divisiontracebase. pfa_gap_degree_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_degree_divisiontraceinitial. ff_h_pfp_degree_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontraceinitial. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontraceinitial * S ((S (0)) * pfs_history_scale_degree_division) + (0))) /\ (((((exists ff_h_pfp_degree_divisiontraceterminal. ff_h_pfp_degree_divisiontraceterminal + S (r) = S ((S (S (S n))) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontraceterminal. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontraceterminal * S ((S (S (S n))) * pfs_history_scale_degree_division) + (r))) /\ ((forall pfh_index_degree_divisiontracesteps. (exists pfa_gap_degree_divisiontracestepsindex. pfa_gap_degree_divisiontracestepsindex + S (pfh_index_degree_divisiontracesteps) = (S (S n))) -> (exists pfh_coefficient_degree_divisiontracestepsstep pfh_before_degree_divisiontracestepsstep pfh_after_degree_divisiontracestepsstep pfh_product_degree_divisiontracestepsstep. ((((exists ff_h_pfp_degree_divisiontracestepsstepcoefficient. ff_h_pfp_degree_divisiontracestepsstepcoefficient + S (pfh_coefficient_degree_divisiontracestepsstep) = S ((S (pfh_index_degree_divisiontracesteps)) * c)) /\ exists ff_q_pfp_degree_divisiontracestepsstepcoefficient. b = ff_q_pfp_degree_divisiontracestepsstepcoefficient * S ((S (pfh_index_degree_divisiontracesteps)) * c) + (pfh_coefficient_degree_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_degree_divisiontracestepsstepbefore. ff_h_pfp_degree_divisiontracestepsstepbefore + S (pfh_before_degree_divisiontracestepsstep) = S ((S (pfh_index_degree_divisiontracesteps)) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontracestepsstepbefore. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontracestepsstepbefore * S ((S (pfh_index_degree_divisiontracesteps)) * pfs_history_scale_degree_division) + (pfh_before_degree_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_degree_divisiontracestepsstepafter. ff_h_pfp_degree_divisiontracestepsstepafter + S (pfh_after_degree_divisiontracestepsstep) = S ((S (S (pfh_index_degree_divisiontracesteps))) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontracestepsstepafter. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontracestepsstepafter * S ((S (S (pfh_index_degree_divisiontracesteps))) * pfs_history_scale_degree_division) + (pfh_after_degree_divisiontracestepsstep))) /\ (((((exists pfa_gap_degree_divisiontracestepsstepmultiplyleft. pfa_gap_degree_divisiontracestepsstepmultiplyleft + S (pfh_before_degree_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_degree_divisiontracestepsstepmultiplyright. pfa_gap_degree_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_degree_divisiontracestepsstepmultiplyresultbound. pfa_gap_degree_divisiontracestepsstepmultiplyresultbound + S (pfh_product_degree_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_degree_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_degree_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_degree_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_degree_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_degree_divisiontracestepsstep) + (p) * pfa_offset_right_degree_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_degree_divisiontracestepsstepaddleft. pfa_gap_degree_divisiontracestepsstepaddleft + S (pfh_product_degree_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_degree_divisiontracestepsstepaddright. pfa_gap_degree_divisiontracestepsstepaddright + S (pfh_coefficient_degree_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_degree_divisiontracestepsstepaddresultbound. pfa_gap_degree_divisiontracestepsstepaddresultbound + S (pfh_after_degree_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_degree_divisiontracestepsstepaddresultcongruence pfa_offset_right_degree_divisiontracestepsstepaddresultcongruence. ((pfh_product_degree_divisiontracestepsstep) + (pfh_coefficient_degree_divisiontracestepsstep)) + (p) * pfa_offset_left_degree_divisiontracestepsstepaddresultcongruence = (pfh_after_degree_divisiontracestepsstep) + (p) * pfa_offset_right_degree_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_degree_divisionquotient ff_source_mcp_pfs_degree_divisionquotient ff_target_mcp_pfs_degree_divisionquotient. (exists mcp_gap_pfs_degree_divisionquotient_bound. mcp_gap_pfs_degree_divisionquotient_bound + S (ff_index_mcp_pfs_degree_divisionquotient) = (S n)) -> (((exists fs_h_mcp_pfs_degree_divisionquotient_source. fs_h_mcp_pfs_degree_divisionquotient_source + S (ff_source_mcp_pfs_degree_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_degree_divisionquotient)) * pfs_history_scale_degree_division)) /\ exists fs_q_mcp_pfs_degree_divisionquotient_source. pfs_history_code_degree_division = fs_q_mcp_pfs_degree_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_degree_divisionquotient)) * pfs_history_scale_degree_division) + (ff_source_mcp_pfs_degree_divisionquotient))) -> (((exists fs_h_mcp_pfs_degree_divisionquotient_target. fs_h_mcp_pfs_degree_divisionquotient_target + S (ff_target_mcp_pfs_degree_divisionquotient) = S ((S (ff_index_mcp_pfs_degree_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_degree_divisionquotient_target. qb = fs_q_mcp_pfs_degree_divisionquotient_target * S ((S (ff_index_mcp_pfs_degree_divisionquotient)) * qc) + (ff_target_mcp_pfs_degree_divisionquotient))) -> ff_target_mcp_pfs_degree_divisionquotient = ff_source_mcp_pfs_degree_divisionquotient)))) -> ((((S n)=S (n)) /\ (((forall fom_index_pfp_degree_quotientcoefficients. (exists fom_gap_pfp_degree_quotientcoefficients_index_bound. fom_gap_pfp_degree_quotientcoefficients_index_bound + S (fom_index_pfp_degree_quotientcoefficients) = S n) -> exists fom_value_pfp_degree_quotientcoefficients. ((((exists fom_beta_height_pfp_degree_quotientcoefficients_entry. fom_beta_height_pfp_degree_quotientcoefficients_entry + S (fom_value_pfp_degree_quotientcoefficients) = S ((S (fom_index_pfp_degree_quotientcoefficients)) * qc)) /\ exists fom_beta_quotient_pfp_degree_quotientcoefficients_entry. qb = fom_beta_quotient_pfp_degree_quotientcoefficients_entry * S ((S (fom_index_pfp_degree_quotientcoefficients)) * qc) + (fom_value_pfp_degree_quotientcoefficients))) /\ (exists fom_gap_pfp_degree_quotientcoefficients_value_bound. fom_gap_pfp_degree_quotientcoefficients_value_bound + S (fom_value_pfp_degree_quotientcoefficients) = p))) /\ ((exists pfd_leading_degree_quotient. ((((exists ff_h_pfp_degree_quotiententry. ff_h_pfp_degree_quotiententry + S (pfd_leading_degree_quotient) = S ((S (0)) * qc)) /\ exists ff_q_pfp_degree_quotiententry. qb = ff_q_pfp_degree_quotiententry * S ((S (0)) * qc) + (pfd_leading_degree_quotient))) /\ ((~(pfd_leading_degree_quotient=0))))))))))prime_field_polynomial_synthetic_constantExact checked statement
forall p b c a qb qc r v. (~((p) = 1) /\ forall pfa_factor_left_constant_division_prime pfa_factor_right_constant_division_prime. (p) = pfa_factor_left_constant_division_prime * pfa_factor_right_constant_division_prime -> pfa_factor_left_constant_division_prime = 1 \/ pfa_factor_right_constant_division_prime = 1) -> (exists pfs_history_code_constant_division pfs_history_scale_constant_division. ((((exists pfa_gap_constant_divisiontracebase. pfa_gap_constant_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_constant_divisiontraceinitial. ff_h_pfp_constant_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_constant_division)) /\ exists ff_q_pfp_constant_divisiontraceinitial. pfs_history_code_constant_division = ff_q_pfp_constant_divisiontraceinitial * S ((S (0)) * pfs_history_scale_constant_division) + (0))) /\ (((((exists ff_h_pfp_constant_divisiontraceterminal. ff_h_pfp_constant_divisiontraceterminal + S (r) = S ((S (S (0))) * pfs_history_scale_constant_division)) /\ exists ff_q_pfp_constant_divisiontraceterminal. pfs_history_code_constant_division = ff_q_pfp_constant_divisiontraceterminal * S ((S (S (0))) * pfs_history_scale_constant_division) + (r))) /\ ((forall pfh_index_constant_divisiontracesteps. (exists pfa_gap_constant_divisiontracestepsindex. pfa_gap_constant_divisiontracestepsindex + S (pfh_index_constant_divisiontracesteps) = (S (0))) -> (exists pfh_coefficient_constant_divisiontracestepsstep pfh_before_constant_divisiontracestepsstep pfh_after_constant_divisiontracestepsstep pfh_product_constant_divisiontracestepsstep. ((((exists ff_h_pfp_constant_divisiontracestepsstepcoefficient. ff_h_pfp_constant_divisiontracestepsstepcoefficient + S (pfh_coefficient_constant_divisiontracestepsstep) = S ((S (pfh_index_constant_divisiontracesteps)) * c)) /\ exists ff_q_pfp_constant_divisiontracestepsstepcoefficient. b = ff_q_pfp_constant_divisiontracestepsstepcoefficient * S ((S (pfh_index_constant_divisiontracesteps)) * c) + (pfh_coefficient_constant_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_constant_divisiontracestepsstepbefore. ff_h_pfp_constant_divisiontracestepsstepbefore + S (pfh_before_constant_divisiontracestepsstep) = S ((S (pfh_index_constant_divisiontracesteps)) * pfs_history_scale_constant_division)) /\ exists ff_q_pfp_constant_divisiontracestepsstepbefore. pfs_history_code_constant_division = ff_q_pfp_constant_divisiontracestepsstepbefore * S ((S (pfh_index_constant_divisiontracesteps)) * pfs_history_scale_constant_division) + (pfh_before_constant_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_constant_divisiontracestepsstepafter. ff_h_pfp_constant_divisiontracestepsstepafter + S (pfh_after_constant_divisiontracestepsstep) = S ((S (S (pfh_index_constant_divisiontracesteps))) * pfs_history_scale_constant_division)) /\ exists ff_q_pfp_constant_divisiontracestepsstepafter. pfs_history_code_constant_division = ff_q_pfp_constant_divisiontracestepsstepafter * S ((S (S (pfh_index_constant_divisiontracesteps))) * pfs_history_scale_constant_division) + (pfh_after_constant_divisiontracestepsstep))) /\ (((((exists pfa_gap_constant_divisiontracestepsstepmultiplyleft. pfa_gap_constant_divisiontracestepsstepmultiplyleft + S (pfh_before_constant_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_constant_divisiontracestepsstepmultiplyright. pfa_gap_constant_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_constant_divisiontracestepsstepmultiplyresultbound. pfa_gap_constant_divisiontracestepsstepmultiplyresultbound + S (pfh_product_constant_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_constant_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_constant_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_constant_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_constant_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_constant_divisiontracestepsstep) + (p) * pfa_offset_right_constant_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_constant_divisiontracestepsstepaddleft. pfa_gap_constant_divisiontracestepsstepaddleft + S (pfh_product_constant_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_constant_divisiontracestepsstepaddright. pfa_gap_constant_divisiontracestepsstepaddright + S (pfh_coefficient_constant_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_constant_divisiontracestepsstepaddresultbound. pfa_gap_constant_divisiontracestepsstepaddresultbound + S (pfh_after_constant_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_constant_divisiontracestepsstepaddresultcongruence pfa_offset_right_constant_divisiontracestepsstepaddresultcongruence. ((pfh_product_constant_divisiontracestepsstep) + (pfh_coefficient_constant_divisiontracestepsstep)) + (p) * pfa_offset_left_constant_divisiontracestepsstepaddresultcongruence = (pfh_after_constant_divisiontracestepsstep) + (p) * pfa_offset_right_constant_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_constant_divisionquotient ff_source_mcp_pfs_constant_divisionquotient ff_target_mcp_pfs_constant_divisionquotient. (exists mcp_gap_pfs_constant_divisionquotient_bound. mcp_gap_pfs_constant_divisionquotient_bound + S (ff_index_mcp_pfs_constant_divisionquotient) = (0)) -> (((exists fs_h_mcp_pfs_constant_divisionquotient_source. fs_h_mcp_pfs_constant_divisionquotient_source + S (ff_source_mcp_pfs_constant_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_constant_divisionquotient)) * pfs_history_scale_constant_division)) /\ exists fs_q_mcp_pfs_constant_divisionquotient_source. pfs_history_code_constant_division = fs_q_mcp_pfs_constant_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_constant_divisionquotient)) * pfs_history_scale_constant_division) + (ff_source_mcp_pfs_constant_divisionquotient))) -> (((exists fs_h_mcp_pfs_constant_divisionquotient_target. fs_h_mcp_pfs_constant_divisionquotient_target + S (ff_target_mcp_pfs_constant_divisionquotient) = S ((S (ff_index_mcp_pfs_constant_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_constant_divisionquotient_target. qb = fs_q_mcp_pfs_constant_divisionquotient_target * S ((S (ff_index_mcp_pfs_constant_divisionquotient)) * qc) + (ff_target_mcp_pfs_constant_divisionquotient))) -> ff_target_mcp_pfs_constant_divisionquotient = ff_source_mcp_pfs_constant_divisionquotient)))) -> (((exists ff_h_pfp_constant_division_coefficient. ff_h_pfp_constant_division_coefficient + S (v) = S ((S (0)) * c)) /\ exists ff_q_pfp_constant_division_coefficient. b = ff_q_pfp_constant_division_coefficient * S ((S (0)) * c) + (v))) -> r=vprime_field_polynomial_synthetic_exists_uniqueExact checked statement
forall p b c a n. (~((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) = S n) -> 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 (a) = (p)) -> exists qb qc r. ((exists pfs_history_code_unique_execution pfs_history_scale_unique_execution. ((((exists pfa_gap_unique_executiontracebase. pfa_gap_unique_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_unique_executiontraceinitial. ff_h_pfp_unique_executiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_unique_execution)) /\ exists ff_q_pfp_unique_executiontraceinitial. pfs_history_code_unique_execution = ff_q_pfp_unique_executiontraceinitial * S ((S (0)) * pfs_history_scale_unique_execution) + (0))) /\ (((((exists ff_h_pfp_unique_executiontraceterminal. ff_h_pfp_unique_executiontraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_unique_execution)) /\ exists ff_q_pfp_unique_executiontraceterminal. pfs_history_code_unique_execution = ff_q_pfp_unique_executiontraceterminal * S ((S (S (n))) * pfs_history_scale_unique_execution) + (r))) /\ ((forall pfh_index_unique_executiontracesteps. (exists pfa_gap_unique_executiontracestepsindex. pfa_gap_unique_executiontracestepsindex + S (pfh_index_unique_executiontracesteps) = (S (n))) -> (exists pfh_coefficient_unique_executiontracestepsstep pfh_before_unique_executiontracestepsstep pfh_after_unique_executiontracestepsstep pfh_product_unique_executiontracestepsstep. ((((exists ff_h_pfp_unique_executiontracestepsstepcoefficient. ff_h_pfp_unique_executiontracestepsstepcoefficient + S (pfh_coefficient_unique_executiontracestepsstep) = S ((S (pfh_index_unique_executiontracesteps)) * c)) /\ exists ff_q_pfp_unique_executiontracestepsstepcoefficient. b = ff_q_pfp_unique_executiontracestepsstepcoefficient * S ((S (pfh_index_unique_executiontracesteps)) * c) + (pfh_coefficient_unique_executiontracestepsstep))) /\ (((((exists ff_h_pfp_unique_executiontracestepsstepbefore. ff_h_pfp_unique_executiontracestepsstepbefore + S (pfh_before_unique_executiontracestepsstep) = S ((S (pfh_index_unique_executiontracesteps)) * pfs_history_scale_unique_execution)) /\ exists ff_q_pfp_unique_executiontracestepsstepbefore. pfs_history_code_unique_execution = ff_q_pfp_unique_executiontracestepsstepbefore * S ((S (pfh_index_unique_executiontracesteps)) * pfs_history_scale_unique_execution) + (pfh_before_unique_executiontracestepsstep))) /\ (((((exists ff_h_pfp_unique_executiontracestepsstepafter. ff_h_pfp_unique_executiontracestepsstepafter + S (pfh_after_unique_executiontracestepsstep) = S ((S (S (pfh_index_unique_executiontracesteps))) * pfs_history_scale_unique_execution)) /\ exists ff_q_pfp_unique_executiontracestepsstepafter. pfs_history_code_unique_execution = ff_q_pfp_unique_executiontracestepsstepafter * S ((S (S (pfh_index_unique_executiontracesteps))) * pfs_history_scale_unique_execution) + (pfh_after_unique_executiontracestepsstep))) /\ (((((exists pfa_gap_unique_executiontracestepsstepmultiplyleft. pfa_gap_unique_executiontracestepsstepmultiplyleft + S (pfh_before_unique_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_unique_executiontracestepsstepmultiplyright. pfa_gap_unique_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_unique_executiontracestepsstepmultiplyresultbound. pfa_gap_unique_executiontracestepsstepmultiplyresultbound + S (pfh_product_unique_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_unique_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_unique_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_unique_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_unique_executiontracestepsstepmultiplyresultcongruence = (pfh_product_unique_executiontracestepsstep) + (p) * pfa_offset_right_unique_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_unique_executiontracestepsstepaddleft. pfa_gap_unique_executiontracestepsstepaddleft + S (pfh_product_unique_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_unique_executiontracestepsstepaddright. pfa_gap_unique_executiontracestepsstepaddright + S (pfh_coefficient_unique_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_unique_executiontracestepsstepaddresultbound. pfa_gap_unique_executiontracestepsstepaddresultbound + S (pfh_after_unique_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_unique_executiontracestepsstepaddresultcongruence pfa_offset_right_unique_executiontracestepsstepaddresultcongruence. ((pfh_product_unique_executiontracestepsstep) + (pfh_coefficient_unique_executiontracestepsstep)) + (p) * pfa_offset_left_unique_executiontracestepsstepaddresultcongruence = (pfh_after_unique_executiontracestepsstep) + (p) * pfa_offset_right_unique_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_unique_executionquotient ff_source_mcp_pfs_unique_executionquotient ff_target_mcp_pfs_unique_executionquotient. (exists mcp_gap_pfs_unique_executionquotient_bound. mcp_gap_pfs_unique_executionquotient_bound + S (ff_index_mcp_pfs_unique_executionquotient) = (n)) -> (((exists fs_h_mcp_pfs_unique_executionquotient_source. fs_h_mcp_pfs_unique_executionquotient_source + S (ff_source_mcp_pfs_unique_executionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_unique_executionquotient)) * pfs_history_scale_unique_execution)) /\ exists fs_q_mcp_pfs_unique_executionquotient_source. pfs_history_code_unique_execution = fs_q_mcp_pfs_unique_executionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_unique_executionquotient)) * pfs_history_scale_unique_execution) + (ff_source_mcp_pfs_unique_executionquotient))) -> (((exists fs_h_mcp_pfs_unique_executionquotient_target. fs_h_mcp_pfs_unique_executionquotient_target + S (ff_target_mcp_pfs_unique_executionquotient) = S ((S (ff_index_mcp_pfs_unique_executionquotient)) * qc)) /\ exists fs_q_mcp_pfs_unique_executionquotient_target. qb = fs_q_mcp_pfs_unique_executionquotient_target * S ((S (ff_index_mcp_pfs_unique_executionquotient)) * qc) + (ff_target_mcp_pfs_unique_executionquotient))) -> ff_target_mcp_pfs_unique_executionquotient = ff_source_mcp_pfs_unique_executionquotient)))) /\ ((forall Qb Qc s. (exists pfs_history_code_unique_other pfs_history_scale_unique_other. ((((exists pfa_gap_unique_othertracebase. pfa_gap_unique_othertracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_unique_othertraceinitial. ff_h_pfp_unique_othertraceinitial + S (0) = S ((S (0)) * pfs_history_scale_unique_other)) /\ exists ff_q_pfp_unique_othertraceinitial. pfs_history_code_unique_other = ff_q_pfp_unique_othertraceinitial * S ((S (0)) * pfs_history_scale_unique_other) + (0))) /\ (((((exists ff_h_pfp_unique_othertraceterminal. ff_h_pfp_unique_othertraceterminal + S (s) = S ((S (S (n))) * pfs_history_scale_unique_other)) /\ exists ff_q_pfp_unique_othertraceterminal. pfs_history_code_unique_other = ff_q_pfp_unique_othertraceterminal * S ((S (S (n))) * pfs_history_scale_unique_other) + (s))) /\ ((forall pfh_index_unique_othertracesteps. (exists pfa_gap_unique_othertracestepsindex. pfa_gap_unique_othertracestepsindex + S (pfh_index_unique_othertracesteps) = (S (n))) -> (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)) * pfs_history_scale_unique_other)) /\ exists ff_q_pfp_unique_othertracestepsstepbefore. pfs_history_code_unique_other = ff_q_pfp_unique_othertracestepsstepbefore * S ((S (pfh_index_unique_othertracesteps)) * pfs_history_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))) * pfs_history_scale_unique_other)) /\ exists ff_q_pfp_unique_othertracestepsstepafter. pfs_history_code_unique_other = ff_q_pfp_unique_othertracestepsstepafter * S ((S (S (pfh_index_unique_othertracesteps))) * pfs_history_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 (a) = (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) * (a)) + (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)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_unique_otherquotient ff_source_mcp_pfs_unique_otherquotient ff_target_mcp_pfs_unique_otherquotient. (exists mcp_gap_pfs_unique_otherquotient_bound. mcp_gap_pfs_unique_otherquotient_bound + S (ff_index_mcp_pfs_unique_otherquotient) = (n)) -> (((exists fs_h_mcp_pfs_unique_otherquotient_source. fs_h_mcp_pfs_unique_otherquotient_source + S (ff_source_mcp_pfs_unique_otherquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_unique_otherquotient)) * pfs_history_scale_unique_other)) /\ exists fs_q_mcp_pfs_unique_otherquotient_source. pfs_history_code_unique_other = fs_q_mcp_pfs_unique_otherquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_unique_otherquotient)) * pfs_history_scale_unique_other) + (ff_source_mcp_pfs_unique_otherquotient))) -> (((exists fs_h_mcp_pfs_unique_otherquotient_target. fs_h_mcp_pfs_unique_otherquotient_target + S (ff_target_mcp_pfs_unique_otherquotient) = S ((S (ff_index_mcp_pfs_unique_otherquotient)) * Qc)) /\ exists fs_q_mcp_pfs_unique_otherquotient_target. Qb = fs_q_mcp_pfs_unique_otherquotient_target * S ((S (ff_index_mcp_pfs_unique_otherquotient)) * Qc) + (ff_target_mcp_pfs_unique_otherquotient))) -> ff_target_mcp_pfs_unique_otherquotient = ff_source_mcp_pfs_unique_otherquotient)))) -> (((s=r) /\ ((forall mdr_i_pfp_unique_values mdr_a_pfp_unique_values. (exists mdr_gap_pfp_unique_valuesb. mdr_gap_pfp_unique_valuesb + S (mdr_i_pfp_unique_values) = (n)) -> (((exists ff_h_mdr_pfp_unique_valueso. ff_h_mdr_pfp_unique_valueso + S (mdr_a_pfp_unique_values) = S ((S (mdr_i_pfp_unique_values)) * Qc)) /\ exists ff_q_mdr_pfp_unique_valueso. Qb = ff_q_mdr_pfp_unique_valueso * S ((S (mdr_i_pfp_unique_values)) * Qc) + (mdr_a_pfp_unique_values))) -> (((exists ff_h_mdr_pfp_unique_valuesn. ff_h_mdr_pfp_unique_valuesn + S (mdr_a_pfp_unique_values) = S ((S (mdr_i_pfp_unique_values)) * qc)) /\ exists ff_q_mdr_pfp_unique_valuesn. qb = ff_q_mdr_pfp_unique_valuesn * S ((S (mdr_i_pfp_unique_values)) * qc) + (mdr_a_pfp_unique_values))))))))))prime_field_polynomial_synthetic_zero_remainder_iffExact checked statement
forall p b c a n qb qc r. (~((p) = 1) /\ forall pfa_factor_left_root_prime pfa_factor_right_root_prime. (p) = pfa_factor_left_root_prime * pfa_factor_right_root_prime -> pfa_factor_left_root_prime = 1 \/ pfa_factor_right_root_prime = 1) -> (exists pfs_history_code_root_division pfs_history_scale_root_division. ((((exists pfa_gap_root_divisiontracebase. pfa_gap_root_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_root_divisiontraceinitial. ff_h_pfp_root_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_root_division)) /\ exists ff_q_pfp_root_divisiontraceinitial. pfs_history_code_root_division = ff_q_pfp_root_divisiontraceinitial * S ((S (0)) * pfs_history_scale_root_division) + (0))) /\ (((((exists ff_h_pfp_root_divisiontraceterminal. ff_h_pfp_root_divisiontraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_root_division)) /\ exists ff_q_pfp_root_divisiontraceterminal. pfs_history_code_root_division = ff_q_pfp_root_divisiontraceterminal * S ((S (S (n))) * pfs_history_scale_root_division) + (r))) /\ ((forall pfh_index_root_divisiontracesteps. (exists pfa_gap_root_divisiontracestepsindex. pfa_gap_root_divisiontracestepsindex + S (pfh_index_root_divisiontracesteps) = (S (n))) -> (exists pfh_coefficient_root_divisiontracestepsstep pfh_before_root_divisiontracestepsstep pfh_after_root_divisiontracestepsstep pfh_product_root_divisiontracestepsstep. ((((exists ff_h_pfp_root_divisiontracestepsstepcoefficient. ff_h_pfp_root_divisiontracestepsstepcoefficient + S (pfh_coefficient_root_divisiontracestepsstep) = S ((S (pfh_index_root_divisiontracesteps)) * c)) /\ exists ff_q_pfp_root_divisiontracestepsstepcoefficient. b = ff_q_pfp_root_divisiontracestepsstepcoefficient * S ((S (pfh_index_root_divisiontracesteps)) * c) + (pfh_coefficient_root_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_root_divisiontracestepsstepbefore. ff_h_pfp_root_divisiontracestepsstepbefore + S (pfh_before_root_divisiontracestepsstep) = S ((S (pfh_index_root_divisiontracesteps)) * pfs_history_scale_root_division)) /\ exists ff_q_pfp_root_divisiontracestepsstepbefore. pfs_history_code_root_division = ff_q_pfp_root_divisiontracestepsstepbefore * S ((S (pfh_index_root_divisiontracesteps)) * pfs_history_scale_root_division) + (pfh_before_root_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_root_divisiontracestepsstepafter. ff_h_pfp_root_divisiontracestepsstepafter + S (pfh_after_root_divisiontracestepsstep) = S ((S (S (pfh_index_root_divisiontracesteps))) * pfs_history_scale_root_division)) /\ exists ff_q_pfp_root_divisiontracestepsstepafter. pfs_history_code_root_division = ff_q_pfp_root_divisiontracestepsstepafter * S ((S (S (pfh_index_root_divisiontracesteps))) * pfs_history_scale_root_division) + (pfh_after_root_divisiontracestepsstep))) /\ (((((exists pfa_gap_root_divisiontracestepsstepmultiplyleft. pfa_gap_root_divisiontracestepsstepmultiplyleft + S (pfh_before_root_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_root_divisiontracestepsstepmultiplyright. pfa_gap_root_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_root_divisiontracestepsstepmultiplyresultbound. pfa_gap_root_divisiontracestepsstepmultiplyresultbound + S (pfh_product_root_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_root_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_root_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_root_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_root_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_root_divisiontracestepsstep) + (p) * pfa_offset_right_root_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_root_divisiontracestepsstepaddleft. pfa_gap_root_divisiontracestepsstepaddleft + S (pfh_product_root_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_root_divisiontracestepsstepaddright. pfa_gap_root_divisiontracestepsstepaddright + S (pfh_coefficient_root_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_root_divisiontracestepsstepaddresultbound. pfa_gap_root_divisiontracestepsstepaddresultbound + S (pfh_after_root_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_root_divisiontracestepsstepaddresultcongruence pfa_offset_right_root_divisiontracestepsstepaddresultcongruence. ((pfh_product_root_divisiontracestepsstep) + (pfh_coefficient_root_divisiontracestepsstep)) + (p) * pfa_offset_left_root_divisiontracestepsstepaddresultcongruence = (pfh_after_root_divisiontracestepsstep) + (p) * pfa_offset_right_root_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_root_divisionquotient ff_source_mcp_pfs_root_divisionquotient ff_target_mcp_pfs_root_divisionquotient. (exists mcp_gap_pfs_root_divisionquotient_bound. mcp_gap_pfs_root_divisionquotient_bound + S (ff_index_mcp_pfs_root_divisionquotient) = (n)) -> (((exists fs_h_mcp_pfs_root_divisionquotient_source. fs_h_mcp_pfs_root_divisionquotient_source + S (ff_source_mcp_pfs_root_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_root_divisionquotient)) * pfs_history_scale_root_division)) /\ exists fs_q_mcp_pfs_root_divisionquotient_source. pfs_history_code_root_division = fs_q_mcp_pfs_root_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_root_divisionquotient)) * pfs_history_scale_root_division) + (ff_source_mcp_pfs_root_divisionquotient))) -> (((exists fs_h_mcp_pfs_root_divisionquotient_target. fs_h_mcp_pfs_root_divisionquotient_target + S (ff_target_mcp_pfs_root_divisionquotient) = S ((S (ff_index_mcp_pfs_root_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_root_divisionquotient_target. qb = fs_q_mcp_pfs_root_divisionquotient_target * S ((S (ff_index_mcp_pfs_root_divisionquotient)) * qc) + (ff_target_mcp_pfs_root_divisionquotient))) -> ff_target_mcp_pfs_root_divisionquotient = ff_source_mcp_pfs_root_divisionquotient)))) -> ((r=0 -> (exists pfh_trace_code_root_forward pfh_trace_scale_root_forward. (((exists pfa_gap_root_forwardtracebase. pfa_gap_root_forwardtracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_root_forwardtraceinitial. ff_h_pfp_root_forwardtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_root_forward)) /\ exists ff_q_pfp_root_forwardtraceinitial. pfh_trace_code_root_forward = ff_q_pfp_root_forwardtraceinitial * S ((S (0)) * pfh_trace_scale_root_forward) + (0))) /\ (((((exists ff_h_pfp_root_forwardtraceterminal. ff_h_pfp_root_forwardtraceterminal + S (0) = S ((S (S n)) * pfh_trace_scale_root_forward)) /\ exists ff_q_pfp_root_forwardtraceterminal. pfh_trace_code_root_forward = ff_q_pfp_root_forwardtraceterminal * S ((S (S n)) * pfh_trace_scale_root_forward) + (0))) /\ ((forall pfh_index_root_forwardtracesteps. (exists pfa_gap_root_forwardtracestepsindex. pfa_gap_root_forwardtracestepsindex + S (pfh_index_root_forwardtracesteps) = (S n)) -> (exists pfh_coefficient_root_forwardtracestepsstep pfh_before_root_forwardtracestepsstep pfh_after_root_forwardtracestepsstep pfh_product_root_forwardtracestepsstep. ((((exists ff_h_pfp_root_forwardtracestepsstepcoefficient. ff_h_pfp_root_forwardtracestepsstepcoefficient + S (pfh_coefficient_root_forwardtracestepsstep) = S ((S (pfh_index_root_forwardtracesteps)) * c)) /\ exists ff_q_pfp_root_forwardtracestepsstepcoefficient. b = ff_q_pfp_root_forwardtracestepsstepcoefficient * S ((S (pfh_index_root_forwardtracesteps)) * c) + (pfh_coefficient_root_forwardtracestepsstep))) /\ (((((exists ff_h_pfp_root_forwardtracestepsstepbefore. ff_h_pfp_root_forwardtracestepsstepbefore + S (pfh_before_root_forwardtracestepsstep) = S ((S (pfh_index_root_forwardtracesteps)) * pfh_trace_scale_root_forward)) /\ exists ff_q_pfp_root_forwardtracestepsstepbefore. pfh_trace_code_root_forward = ff_q_pfp_root_forwardtracestepsstepbefore * S ((S (pfh_index_root_forwardtracesteps)) * pfh_trace_scale_root_forward) + (pfh_before_root_forwardtracestepsstep))) /\ (((((exists ff_h_pfp_root_forwardtracestepsstepafter. ff_h_pfp_root_forwardtracestepsstepafter + S (pfh_after_root_forwardtracestepsstep) = S ((S (S (pfh_index_root_forwardtracesteps))) * pfh_trace_scale_root_forward)) /\ exists ff_q_pfp_root_forwardtracestepsstepafter. pfh_trace_code_root_forward = ff_q_pfp_root_forwardtracestepsstepafter * S ((S (S (pfh_index_root_forwardtracesteps))) * pfh_trace_scale_root_forward) + (pfh_after_root_forwardtracestepsstep))) /\ (((((exists pfa_gap_root_forwardtracestepsstepmultiplyleft. pfa_gap_root_forwardtracestepsstepmultiplyleft + S (pfh_before_root_forwardtracestepsstep) = (p)) /\ (((exists pfa_gap_root_forwardtracestepsstepmultiplyright. pfa_gap_root_forwardtracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_root_forwardtracestepsstepmultiplyresultbound. pfa_gap_root_forwardtracestepsstepmultiplyresultbound + S (pfh_product_root_forwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_root_forwardtracestepsstepmultiplyresultcongruence pfa_offset_right_root_forwardtracestepsstepmultiplyresultcongruence. ((pfh_before_root_forwardtracestepsstep) * (a)) + (p) * pfa_offset_left_root_forwardtracestepsstepmultiplyresultcongruence = (pfh_product_root_forwardtracestepsstep) + (p) * pfa_offset_right_root_forwardtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_root_forwardtracestepsstepaddleft. pfa_gap_root_forwardtracestepsstepaddleft + S (pfh_product_root_forwardtracestepsstep) = (p)) /\ (((exists pfa_gap_root_forwardtracestepsstepaddright. pfa_gap_root_forwardtracestepsstepaddright + S (pfh_coefficient_root_forwardtracestepsstep) = (p)) /\ ((((exists pfa_gap_root_forwardtracestepsstepaddresultbound. pfa_gap_root_forwardtracestepsstepaddresultbound + S (pfh_after_root_forwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_root_forwardtracestepsstepaddresultcongruence pfa_offset_right_root_forwardtracestepsstepaddresultcongruence. ((pfh_product_root_forwardtracestepsstep) + (pfh_coefficient_root_forwardtracestepsstep)) + (p) * pfa_offset_left_root_forwardtracestepsstepaddresultcongruence = (pfh_after_root_forwardtracestepsstep) + (p) * pfa_offset_right_root_forwardtracestepsstepaddresultcongruence)))))))))))))))))))))))))))) /\ (((exists pfh_trace_code_root_backward pfh_trace_scale_root_backward. (((exists pfa_gap_root_backwardtracebase. pfa_gap_root_backwardtracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_root_backwardtraceinitial. ff_h_pfp_root_backwardtraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_root_backward)) /\ exists ff_q_pfp_root_backwardtraceinitial. pfh_trace_code_root_backward = ff_q_pfp_root_backwardtraceinitial * S ((S (0)) * pfh_trace_scale_root_backward) + (0))) /\ (((((exists ff_h_pfp_root_backwardtraceterminal. ff_h_pfp_root_backwardtraceterminal + S (0) = S ((S (S n)) * pfh_trace_scale_root_backward)) /\ exists ff_q_pfp_root_backwardtraceterminal. pfh_trace_code_root_backward = ff_q_pfp_root_backwardtraceterminal * S ((S (S n)) * pfh_trace_scale_root_backward) + (0))) /\ ((forall pfh_index_root_backwardtracesteps. (exists pfa_gap_root_backwardtracestepsindex. pfa_gap_root_backwardtracestepsindex + S (pfh_index_root_backwardtracesteps) = (S n)) -> (exists pfh_coefficient_root_backwardtracestepsstep pfh_before_root_backwardtracestepsstep pfh_after_root_backwardtracestepsstep pfh_product_root_backwardtracestepsstep. ((((exists ff_h_pfp_root_backwardtracestepsstepcoefficient. ff_h_pfp_root_backwardtracestepsstepcoefficient + S (pfh_coefficient_root_backwardtracestepsstep) = S ((S (pfh_index_root_backwardtracesteps)) * c)) /\ exists ff_q_pfp_root_backwardtracestepsstepcoefficient. b = ff_q_pfp_root_backwardtracestepsstepcoefficient * S ((S (pfh_index_root_backwardtracesteps)) * c) + (pfh_coefficient_root_backwardtracestepsstep))) /\ (((((exists ff_h_pfp_root_backwardtracestepsstepbefore. ff_h_pfp_root_backwardtracestepsstepbefore + S (pfh_before_root_backwardtracestepsstep) = S ((S (pfh_index_root_backwardtracesteps)) * pfh_trace_scale_root_backward)) /\ exists ff_q_pfp_root_backwardtracestepsstepbefore. pfh_trace_code_root_backward = ff_q_pfp_root_backwardtracestepsstepbefore * S ((S (pfh_index_root_backwardtracesteps)) * pfh_trace_scale_root_backward) + (pfh_before_root_backwardtracestepsstep))) /\ (((((exists ff_h_pfp_root_backwardtracestepsstepafter. ff_h_pfp_root_backwardtracestepsstepafter + S (pfh_after_root_backwardtracestepsstep) = S ((S (S (pfh_index_root_backwardtracesteps))) * pfh_trace_scale_root_backward)) /\ exists ff_q_pfp_root_backwardtracestepsstepafter. pfh_trace_code_root_backward = ff_q_pfp_root_backwardtracestepsstepafter * S ((S (S (pfh_index_root_backwardtracesteps))) * pfh_trace_scale_root_backward) + (pfh_after_root_backwardtracestepsstep))) /\ (((((exists pfa_gap_root_backwardtracestepsstepmultiplyleft. pfa_gap_root_backwardtracestepsstepmultiplyleft + S (pfh_before_root_backwardtracestepsstep) = (p)) /\ (((exists pfa_gap_root_backwardtracestepsstepmultiplyright. pfa_gap_root_backwardtracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_root_backwardtracestepsstepmultiplyresultbound. pfa_gap_root_backwardtracestepsstepmultiplyresultbound + S (pfh_product_root_backwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_root_backwardtracestepsstepmultiplyresultcongruence pfa_offset_right_root_backwardtracestepsstepmultiplyresultcongruence. ((pfh_before_root_backwardtracestepsstep) * (a)) + (p) * pfa_offset_left_root_backwardtracestepsstepmultiplyresultcongruence = (pfh_product_root_backwardtracestepsstep) + (p) * pfa_offset_right_root_backwardtracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_root_backwardtracestepsstepaddleft. pfa_gap_root_backwardtracestepsstepaddleft + S (pfh_product_root_backwardtracestepsstep) = (p)) /\ (((exists pfa_gap_root_backwardtracestepsstepaddright. pfa_gap_root_backwardtracestepsstepaddright + S (pfh_coefficient_root_backwardtracestepsstep) = (p)) /\ ((((exists pfa_gap_root_backwardtracestepsstepaddresultbound. pfa_gap_root_backwardtracestepsstepaddresultbound + S (pfh_after_root_backwardtracestepsstep) = (p)) /\ ((exists pfa_offset_left_root_backwardtracestepsstepaddresultcongruence pfa_offset_right_root_backwardtracestepsstepaddresultcongruence. ((pfh_product_root_backwardtracestepsstep) + (pfh_coefficient_root_backwardtracestepsstep)) + (p) * pfa_offset_left_root_backwardtracestepsstepaddresultcongruence = (pfh_after_root_backwardtracestepsstep) + (p) * pfa_offset_right_root_backwardtracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> r=0)))zero_addExact checked statement
forall n. 0 + n = nadd_succ_leftExact checked statement
forall n m. S n + m = S (n + m)add_commExact checked statement
forall n m. n + m = m + nadd_assocExact checked statement
forall n m k. (n + m) + k = n + (m + k)mul_zero_leftExact checked statement
forall n. 0 * n = 0mul_succ_leftExact checked statement
forall n m. S n * m = n * m + mmul_commExact checked statement
forall n m. n * m = m * nmul_addExact checked statement
forall n m k. n * (m + k) = n * m + n * kmul_assocExact checked statement
forall n m k. (n * m) * k = n * (m * k)one_mulExact checked statement
forall n. 1 * n = nmul_oneExact checked statement
forall n. n * 1 = nadd_mulExact checked statement
forall n m k. (n + m) * k = n * k + m * ksucc_ne_zeroExact checked statement
forall n. ~(S n = 0)le_reflExact checked statement
forall n. n <= nle_transExact checked statement
forall n m k. n <= m -> m <= k -> n <= kno_succ_add_fixedExact checked statement
forall p n. S p + n = n -> falsedrop_add_prefix_from_fixedExact checked statement
forall a b n. (b + a) + n = n -> a + n = nantisymm_from_witnessesExact checked statement
forall a b n m. a + n = m -> b + m = n -> n = mle_antisymmExact checked statement
forall n m. n <= m -> m <= n -> n = mle_totalExact checked statement
forall n m. n <= m \/ m <= nadd_eq_zero_rightExact checked statement
forall a b. a + b = 0 -> b = 0mul_eq_zeroExact checked statement
forall n m. n * m = 0 -> n = 0 \/ m = 0zero_or_succExact checked statement
forall n. n = 0 \/ exists k. n = S knonzero_is_succExact checked statement
forall n. ~(n = 0) -> exists k. n = S kadd_right_cancelExact checked statement
forall a b c. a + c = b + c -> a = badd_left_cancelExact checked statement
forall a b c. a + b = a + c -> b = czero_leExact checked statement
forall n. 0 <= nle_succ_selfExact checked statement
forall n. n <= S nle_zeroExact checked statement
forall n. n <= 0 -> n = 0one_le_of_ne_zeroExact checked statement
forall n. ~(n = 0) -> 1 <= nle_add_leftExact checked statement
forall a b. exists k. k + a = b + ale_add_rightExact checked statement
forall a b. exists k. k + a = a + badd_le_add_leftExact checked statement
forall a b c. (exists k. k + a = b) -> exists r. r + (c + a) = c + bsucc_le_succExact checked statement
forall a b. (exists k. k + a = b) -> exists r. r + S a = S ble_of_succ_le_succExact checked statement
forall a b. (exists k. k + S a = S b) -> exists r. r + a = ble_succExact checked statement
forall a b. (exists k. k + a = b) -> exists r. r + a = S blt_to_leExact checked statement
forall a b. (exists k. k + S a = b) -> exists r. r + a = blt_irrefl_expandedExact checked statement
forall n. ~(exists k. k + S n = n)le_eq_or_ltExact checked statement
forall a b. (exists k. k + a = b) -> a = b \/ exists k. k + S a = ble_or_ltExact checked statement
forall a b. (exists k. k + a = b) \/ exists k. k + S b = alt_trichotomyExact checked statement
forall a b. a = b \/ ((exists k. k + S a = b) \/ exists k. k + S b = a)lt_not_leExact checked statement
forall a b. (exists k. k + S a = b) -> ~ (exists k. k + b = a)lt_not_eq_add_middleExact checked statement
forall r m a b. (exists k. k + S r = m) -> ~(r = (a + m) + b)mul_le_mul_rightExact checked statement
forall a b c. (exists k. k + a = b) -> exists r. r + a * c = b * cdivision_remainder_succExact checked statement
forall d n. exists q r. n = S d * q + r /\ S r <= S ddivision_remainder_existsExact checked statement
forall m n. ~(m = 0) -> exists q r. n = m * q + r /\ S r <= mpositive_quotient_gap_impossibleExact checked statement
forall m q q2 r s k. (exists z. z + S r = m) -> S k + q = q2 -> ~(m * q + r = m * q2 + s)division_remainder_uniqueExact checked statement
forall m n q r q2 r2. n = m * q + r -> (exists k. k + S r = m) -> n = m * q2 + r2 -> (exists k. k + S r2 = m) -> q = q2 /\ r = r2add_eq_zero_leftExact checked statement
forall a b. a + b = 0 -> a = 0mul_eq_one_componentsExact checked statement
forall a b. a * b = 1 -> a = 1 /\ b = 1mul_ne_zeroExact checked statement
forall a b. ~(a = 0) -> ~(b = 0) -> ~(a * b = 0)multiple_zeroExact checked statement
forall a. exists q. 0 = a * qmultiple_reflExact checked statement
forall a. exists q. a = a * qmultiple_mul_rightExact checked statement
forall a n m. (exists q. n = a * q) -> exists s. n * m = a * smultiple_transExact checked statement
forall a b n. (exists q. n = a * q) -> (exists r. a = b * r) -> exists s. n = b * sdivisor_le_nonzeroExact checked statement
forall d n. ~(n = 0) -> (exists q. n = d * q) -> exists k. k + d = ndivisor_oneExact checked statement
forall d. (exists y. 1 = d * y) -> d = 1factor_differenceExact checked statement
forall c u v r. c * u = c * v + r -> exists w. r = c * wdivides_remainderExact checked statement
forall c a b q r. (exists u. a = c * u) -> (exists v. b = c * v) -> a = b * q + r -> exists w. r = c * wdivides_linear_stepExact checked statement
forall c b q r. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. b * q + r = c * wis_gcd_zero_rightExact checked statement
forall a. (((exists x. a = a * x) /\ (exists y. 0 = a * y)) /\ forall c. (exists u. a = c * u) -> (exists v. 0 = c * v) -> exists w. a = c * w)is_gcd_euclid_forwardExact checked statement
forall d a b q r. a = b * q + r -> (((exists x. b = d * x) /\ (exists y. r = d * y)) /\ forall c. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. d = c * w) -> (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)gcd_exists_up_toExact checked statement
forall B b. (exists t. t + b = B) -> forall a. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)gcd_exists_relationalExact checked statement
forall a b. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)coprime_one_leftExact checked statement
forall a d. (exists x. 1 = d * x) -> (exists y. a = d * y) -> d = 1is_gcd_one_to_coprimeExact checked statement
forall a b. (((exists x. a = 1 * x) /\ (exists y. b = 1 * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. 1 = d * w) -> forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> c = 1add_permute_outerExact checked statement
forall a b c d. (a + b) + (c + d) = (c + b) + (a + d)balanced_bezout_euclid_stepExact checked statement
forall a b q r d xp yp xn yn. a = b * q + r -> b * xp + r * yp = d + (b * xn + r * yn) -> a * yp + b * (xp + q * yn) = d + (a * yn + b * (xn + q * yp))gcd_balanced_bezout_exists_up_toExact checked statement
forall B b. (exists t. t + b = B) -> forall a. exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))gcd_balanced_bezout_existsExact checked statement
forall a b. exists d. ((((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))balanced_combination_scale_rightExact checked statement
forall a b d xp yp xn yn z. a * xp + b * yp = d + (a * xn + b * yn) -> a * (xp * z) + (b * z) * yp = d * z + (a * (xn * z) + (b * z) * yn)common_divisor_divides_balanced_resultExact checked statement
forall c a b d xp yp xn yn. (exists u. a = c * u) -> (exists v. b = c * v) -> a * xp + b * yp = d + (a * xn + b * yn) -> exists w. d = c * wcoprime_balanced_bezoutExact checked statement
forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> exists xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn)gauss_coprime_cancelExact checked statement
forall a b z. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> (exists q. b * z = a * q) -> exists w. z = a * weq_decidableExact checked statement
forall a b. a = b \/ ~(a = b)prime_nonzeroExact checked statement
forall p. (~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) -> ~(p = 0)prime_divisor_eq_one_or_selfExact checked statement
forall p g. (~(p = 1) /\ forall c d. p = c * d -> c = 1 \/ d = 1) -> (exists x. p = g * x) -> g = 1 \/ p = gmod_eq_reflExact checked statement
forall m a. exists u v. a + m * u = a + m * vmod_eq_symmExact checked statement
forall m a b. (exists u v. a + m * u = b + m * v) -> exists r s. b + m * r = a + m * smod_eq_transExact checked statement
forall m a b c. (exists u v. a + m * u = b + m * v) -> (exists r s. b + m * r = c + m * s) -> exists x y. a + m * x = c + m * ymod_eq_addExact checked statement
forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a + c) + m * x = (b + d) + m * ymod_eq_mul_rightExact checked statement
forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (a * c) + m * r = (b * c) + m * smod_eq_mul_leftExact checked statement
forall m a b c. (exists u v. a + m * u = b + m * v) -> exists r s. (c * a) + m * r = (c * b) + m * smod_eq_mulExact checked statement
forall m a b c d. (exists u v. a + m * u = b + m * v) -> (exists r s. c + m * r = d + m * s) -> exists x y. (a * c) + m * x = (b * d) + m * yremainder_decomposition_to_mod_eqExact checked statement
forall m b q x. b = q * m + x -> exists u v. b + m * u = x + m * vmod_eq_bounded_uniqueExact checked statement
forall m a b. (exists ha. ha + S a = m) -> (exists hb. hb + S b = m) -> (exists u v. a + m * u = b + m * v) -> a = bmod_eq_to_remainder_decompositionExact checked statement
forall m b x. ~(m = 0) -> (exists h. h + S x = m) -> (exists u v. b + m * u = x + m * v) -> exists q. b = q * m + xbeta_modulus_nonzeroExact checked statement
forall c i. ~(S ((S i) * c) = 0)beta_at_self_of_boundExact checked statement
forall c i x. (exists h. h + S x = S ((S i) * c)) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. x = q * S ((S i) * c) + x)beta_at_existsExact checked statement
forall b c i. exists x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)beta_at_uniqueExact checked statement
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 = ybeta_at_of_mod_eq_boundExact checked statement
forall b c i x. (exists h. h + S x = S ((S i) * c)) -> (exists u v. b + S ((S i) * c) * u = x + S ((S i) * c) * v) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)dvd_to_mod_zeroExact checked statement
forall m a. (exists k. a = m * k) -> exists u v. a + m * u = 0 + m * vbezout_mod_leftExact checked statement
forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * vbezout_mod_rightExact checked statement
forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. m * xp + n * u = (1 + m * xn) + n * vmod_eq_predecessor_cancelExact checked statement
forall k a z. exists u v. ((a + z) + k * z) + S k * u = a + S k * vbinary_crtExact checked statement
forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> exists x. (exists u v. x + m * u = a + m * v) /\ (exists r s. x + n * r = b + n * s)beta_modulus_coprime_baseExact checked statement
forall c k d. (exists u. S (k * c) = d * u) -> (exists v. c = d * v) -> d = 1common_divisor_beta_moduli_divides_gap_times_cExact checked statement
forall c i j gap d. j = i + gap -> (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> exists w. gap * c = d * wbeta_moduli_coprime_of_gap_dvdExact checked statement
forall c i j gap. j = i + gap -> (exists k. c = gap * k) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1bounded_common_multiple_stepExact checked statement
forall B c. ~(c = 0) -> (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> exists c2. (~(c2 = 0) /\ forall t. (exists h. S t + S h = S (S B)) -> exists k. c2 = S t * k)bounded_common_multiple_existsExact checked statement
forall B. exists c. (~(c = 0) /\ forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k)beta_moduli_coprime_of_lt_bounded_common_multipleExact checked statement
forall B c i j. (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> (exists g. g + S i = j) -> (exists h. h + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1beta_moduli_pairwise_coprime_boundedExact checked statement
forall B c. (forall t. (exists h. S t + S h = S B) -> exists k. c = S t * k) -> forall i j. ~(i = j) -> (exists hi. hi + i = B) -> (exists hj. hj + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1coprime_mul_leftExact checked statement
forall a b n. (forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1) -> (forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1) -> forall d. (exists x. a * b = d * x) -> (exists y. n = d * y) -> d = 1mod_eq_of_mod_eq_multipleExact checked statement
forall m P x a. (exists k. P = m * k) -> (exists u v. x + P * u = a + P * v) -> exists r s. x + m * r = a + m * sbinary_crt_fold_stepExact checked statement
forall P n x b. ~(P = 0) -> ~(n = 0) -> (forall d. (exists u. P = d * u) -> (exists v. n = d * v) -> d = 1) -> exists z. ((forall m a. (exists k. P = m * k) -> (exists u v. x + m * u = a + m * v) -> exists r s. z + m * r = a + m * s) /\ exists q r. z + n * q = b + n * r)right_factor_divides_productExact checked statement
forall a b. exists k. a * b = b * kbeta_value_le_codeExact checked statement
forall b c i x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> exists h. h + x = bbase_le_beta_modulusExact checked statement
forall c i. exists h. h + c = S ((S i) * c)le_scaled_nonzeroExact checked statement
forall C B. ~(C = 0) -> exists h. h + B = C * Bscaled_bounded_common_multipleExact checked statement
forall N C B. (forall t. (exists h. S t + S h = S N) -> exists q. C = S t * q) -> forall t. (exists h. S t + S h = S N) -> exists q. C * B = S t * qbeta_value_lt_scaled_baseExact checked statement
forall b c i x C s j. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> ~(C = 0) -> exists h. h + S x = S ((S j) * (C * S (b + s)))new_value_lt_scaled_baseExact checked statement
forall b s C j. ~(C = 0) -> exists h. h + S s = S ((S j) * (C * S (b + s)))beta_exclusive_accumulated_product_stepExact checked statement
forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))beta_exclusive_recode_congruence_stepExact checked statement
forall N c b e k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * vbeta_exclusive_recode_invariant_stepExact checked statement
forall N c b e k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S k) * c) = 0) /\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))bounded_beta_exclusive_recode_invariantExact checked statement
forall N c b e. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))beta_prefix_extendExact checked statement
forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\ exists q. z = q * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. z = q * S ((S i) * c) + a))beta_repeat_emptyExact checked statement
forall b c a l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> (((exists ff_h_empty_decoded. ff_h_empty_decoded + S (a) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (a))))beta_repeat_succ_extendExact checked statement
forall b c a l sl. sl = S l -> (forall ff_i_before. (exists ff_lt_before_bound. ff_lt_before_bound + S ff_i_before = l) -> (((exists ff_h_before_decoded. ff_h_before_decoded + S (a) = S ((S (ff_i_before)) * c)) /\ exists ff_q_before_decoded. b = ff_q_before_decoded * S ((S (ff_i_before)) * c) + (a)))) -> exists z d. (forall ff_i_after. (exists ff_lt_after_bound. ff_lt_after_bound + S ff_i_after = sl) -> (((exists ff_h_after_decoded. ff_h_after_decoded + S (a) = S ((S (ff_i_after)) * d)) /\ exists ff_q_after_decoded. z = ff_q_after_decoded * S ((S (ff_i_after)) * d) + (a))))beta_repeat_existsExact checked statement
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))))prime_coprime_or_dividesExact checked statement
forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> (forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1) \/ exists k. a = p * kprime_not_divides_coprimeExact checked statement
forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> forall d. (exists x. p = d * x) -> (exists y. a = d * y) -> d = 1coprime_balanced_mod_inverseExact checked statement
forall a m. (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> exists xp xn u v. a * xp + m * u = (1 + a * xn) + m * vcoprime_mod_inverseExact checked statement
forall a m. ~(m = 0) -> (forall d. (exists x. a = d * x) -> (exists y. m = d * y) -> d = 1) -> exists z u v. a * z + m * u = 1 + m * vprime_mod_inverseExact checked statement
forall p a. (~(p = 1) /\ forall c e. p = c * e -> c = 1 \/ e = 1) -> ~(exists k. a = p * k) -> exists z u v. a * z + p * u = 1 + p * vfinite_lt_succ_eq_or_ltExact checked statement
forall n x. (exists h. h + S x = S n) -> x = n \/ exists h. h + S x = nprime_is_succ_succExact checked statement
forall p. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> exists k. p = S (S k)prime_bounded_nonzero_mod_inverseExact checked statement
forall p a. ((~(p = 1) /\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \/ qrbu_factor_right_prime_p = 1)) -> ~(a = 0) -> (exists qrbu_gap_a_lt_p. qrbu_gap_a_lt_p + S a = p) -> (exists qrbu_inverse_bounded_inverse. (~(qrbu_inverse_bounded_inverse = 0) /\ ((exists qrbu_gap_bounded_inverse_bound. qrbu_gap_bounded_inverse_bound + S qrbu_inverse_bounded_inverse = p) /\ (exists qrbu_mod_left_bounded_inverse_mod qrbu_mod_right_bounded_inverse_mod. a * qrbu_inverse_bounded_inverse + p * qrbu_mod_left_bounded_inverse_mod = 1 + p * qrbu_mod_right_bounded_inverse_mod))))bounded_mod_inverse_uniqueExact checked statement
forall p x y z. (exists wip_strict_gap_unique_y_bound. wip_strict_gap_unique_y_bound + S y = p) -> (exists wip_strict_gap_unique_z_bound. wip_strict_gap_unique_z_bound + S z = p) -> (exists wip_mod_left_unique_xy wip_mod_right_unique_xy. x * y + p * wip_mod_left_unique_xy = 1 + p * wip_mod_right_unique_xy) -> (exists wip_mod_left_unique_xz wip_mod_right_unique_xz. x * z + p * wip_mod_left_unique_xz = 1 + p * wip_mod_right_unique_xz) -> y = zbeta_pointwise_mul_prefix_extendExact checked statement
forall mb mc sb sc tb tc l m s. (forall fpmp_index_recode_before fpmp_left_recode_before fpmp_right_recode_before fpmp_target_recode_before. (exists fpmp_gap_recode_before. fpmp_gap_recode_before + S fpmp_index_recode_before = l) -> (((exists ff_h_fpmp_recode_before_left. ff_h_fpmp_recode_before_left + S (fpmp_left_recode_before) = S ((S (fpmp_index_recode_before)) * mc)) /\ exists ff_q_fpmp_recode_before_left. mb = ff_q_fpmp_recode_before_left * S ((S (fpmp_index_recode_before)) * mc) + (fpmp_left_recode_before))) -> (((exists ff_h_fpmp_recode_before_right. ff_h_fpmp_recode_before_right + S (fpmp_right_recode_before) = S ((S (fpmp_index_recode_before)) * sc)) /\ exists ff_q_fpmp_recode_before_right. sb = ff_q_fpmp_recode_before_right * S ((S (fpmp_index_recode_before)) * sc) + (fpmp_right_recode_before))) -> (((exists ff_h_fpmp_recode_before_target. ff_h_fpmp_recode_before_target + S (fpmp_target_recode_before) = S ((S (fpmp_index_recode_before)) * tc)) /\ exists ff_q_fpmp_recode_before_target. tb = ff_q_fpmp_recode_before_target * S ((S (fpmp_index_recode_before)) * tc) + (fpmp_target_recode_before))) -> fpmp_target_recode_before = fpmp_left_recode_before * fpmp_right_recode_before) -> (((exists ff_h_recode_left_last. ff_h_recode_left_last + S (m) = S ((S (l)) * mc)) /\ exists ff_q_recode_left_last. mb = ff_q_recode_left_last * S ((S (l)) * mc) + (m))) -> (((exists ff_h_recode_right_last. ff_h_recode_right_last + S (s) = S ((S (l)) * sc)) /\ exists ff_q_recode_right_last. sb = ff_q_recode_right_last * S ((S (l)) * sc) + (s))) -> exists z d. (forall fpmp_index_recode_after fpmp_left_recode_after fpmp_right_recode_after fpmp_target_recode_after. (exists fpmp_gap_recode_after. fpmp_gap_recode_after + S fpmp_index_recode_after = S l) -> (((exists ff_h_fpmp_recode_after_left. ff_h_fpmp_recode_after_left + S (fpmp_left_recode_after) = S ((S (fpmp_index_recode_after)) * mc)) /\ exists ff_q_fpmp_recode_after_left. mb = ff_q_fpmp_recode_after_left * S ((S (fpmp_index_recode_after)) * mc) + (fpmp_left_recode_after))) -> (((exists ff_h_fpmp_recode_after_right. ff_h_fpmp_recode_after_right + S (fpmp_right_recode_after) = S ((S (fpmp_index_recode_after)) * sc)) /\ exists ff_q_fpmp_recode_after_right. sb = ff_q_fpmp_recode_after_right * S ((S (fpmp_index_recode_after)) * sc) + (fpmp_right_recode_after))) -> (((exists ff_h_fpmp_recode_after_target. ff_h_fpmp_recode_after_target + S (fpmp_target_recode_after) = S ((S (fpmp_index_recode_after)) * d)) /\ exists ff_q_fpmp_recode_after_target. z = ff_q_fpmp_recode_after_target * S ((S (fpmp_index_recode_after)) * d) + (fpmp_target_recode_after))) -> fpmp_target_recode_after = fpmp_left_recode_after * fpmp_right_recode_after)beta_pointwise_mul_prefix_existsExact checked statement
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)beta_division_prefix_extendExact checked statement
forall p b c qb qc rb rc l. (forall fdp_index_before. (exists gsp_lt_gap_before_index_bound. gsp_lt_gap_before_index_bound + S fdp_index_before = l) -> exists fdp_value_before fdp_quotient_before fdp_remainder_before. (((exists ff_h_fdp_before_source. ff_h_fdp_before_source + S (fdp_value_before) = S ((S (fdp_index_before)) * c)) /\ exists ff_q_fdp_before_source. b = ff_q_fdp_before_source * S ((S (fdp_index_before)) * c) + (fdp_value_before))) /\ ((((exists ff_h_fdp_before_quotient_entry. ff_h_fdp_before_quotient_entry + S (fdp_quotient_before) = S ((S (fdp_index_before)) * qc)) /\ exists ff_q_fdp_before_quotient_entry. qb = ff_q_fdp_before_quotient_entry * S ((S (fdp_index_before)) * qc) + (fdp_quotient_before))) /\ ((((exists ff_h_fdp_before_remainder_entry. ff_h_fdp_before_remainder_entry + S (fdp_remainder_before) = S ((S (fdp_index_before)) * rc)) /\ exists ff_q_fdp_before_remainder_entry. rb = ff_q_fdp_before_remainder_entry * S ((S (fdp_index_before)) * rc) + (fdp_remainder_before))) /\ (fdp_value_before = p * fdp_quotient_before + fdp_remainder_before /\ (exists gsp_lt_gap_before_remainder_bound. gsp_lt_gap_before_remainder_bound + S fdp_remainder_before = p))))) -> (exists x q r. (((exists ff_h_fdp_choice_source. ff_h_fdp_choice_source + S (x) = S ((S (l)) * c)) /\ exists ff_q_fdp_choice_source. b = ff_q_fdp_choice_source * S ((S (l)) * c) + (x))) /\ (x = p * q + r /\ (exists gsp_lt_gap_fdp_choice_remainder_bound. gsp_lt_gap_fdp_choice_remainder_bound + S r = p))) -> exists z d u v. (forall fdp_index_after. (exists gsp_lt_gap_after_index_bound. gsp_lt_gap_after_index_bound + S fdp_index_after = S l) -> exists fdp_value_after fdp_quotient_after fdp_remainder_after. (((exists ff_h_fdp_after_source. ff_h_fdp_after_source + S (fdp_value_after) = S ((S (fdp_index_after)) * c)) /\ exists ff_q_fdp_after_source. b = ff_q_fdp_after_source * S ((S (fdp_index_after)) * c) + (fdp_value_after))) /\ ((((exists ff_h_fdp_after_quotient_entry. ff_h_fdp_after_quotient_entry + S (fdp_quotient_after) = S ((S (fdp_index_after)) * d)) /\ exists ff_q_fdp_after_quotient_entry. z = ff_q_fdp_after_quotient_entry * S ((S (fdp_index_after)) * d) + (fdp_quotient_after))) /\ ((((exists ff_h_fdp_after_remainder_entry. ff_h_fdp_after_remainder_entry + S (fdp_remainder_after) = S ((S (fdp_index_after)) * v)) /\ exists ff_q_fdp_after_remainder_entry. u = ff_q_fdp_after_remainder_entry * S ((S (fdp_index_after)) * v) + (fdp_remainder_after))) /\ (fdp_value_after = p * fdp_quotient_after + fdp_remainder_after /\ (exists gsp_lt_gap_after_remainder_bound. gsp_lt_gap_after_remainder_bound + S fdp_remainder_after = p)))))beta_division_prefix_existsExact checked statement
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)))))mod_eq_add_cancel_leftExact checked statement
forall d a b c. (exists fspm_u_cancel_input fspm_v_cancel_input. (a + b) + d * fspm_u_cancel_input = (a + c) + d * fspm_v_cancel_input) -> (exists fspm_u_cancel_result fspm_v_cancel_result. (b) + d * fspm_u_cancel_result = (c) + d * fspm_v_cancel_result)prime_two_leExact checked statement
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))beta_prefix_horner_trace_existsExact checked statement
forall b c t l. exists ff_u_ph_root_trace ff_v_ph_root_trace. ((((exists fs_h_ph_root_trace_start. fs_h_ph_root_trace_start + S (0) = S ((S (0)) * ff_v_ph_root_trace)) /\ exists fs_q_ph_root_trace_start. ff_u_ph_root_trace = fs_q_ph_root_trace_start * S ((S (0)) * ff_v_ph_root_trace) + (0))) /\ forall ff_i_ph_root_trace_steps. (exists ph_bound_root_trace_steps. ph_bound_root_trace_steps + S ff_i_ph_root_trace_steps = l) -> exists ff_coefficient_ph_root_trace_steps ff_previous_ph_root_trace_steps ff_current_ph_root_trace_steps. ((((exists fs_h_ph_root_trace_steps_coefficient. fs_h_ph_root_trace_steps_coefficient + S (ff_coefficient_ph_root_trace_steps) = S ((S (ff_i_ph_root_trace_steps)) * c)) /\ exists fs_q_ph_root_trace_steps_coefficient. b = fs_q_ph_root_trace_steps_coefficient * S ((S (ff_i_ph_root_trace_steps)) * c) + (ff_coefficient_ph_root_trace_steps))) /\ ((((exists fs_h_ph_root_trace_steps_before. fs_h_ph_root_trace_steps_before + S (ff_previous_ph_root_trace_steps) = S ((S (ff_i_ph_root_trace_steps)) * ff_v_ph_root_trace)) /\ exists fs_q_ph_root_trace_steps_before. ff_u_ph_root_trace = fs_q_ph_root_trace_steps_before * S ((S (ff_i_ph_root_trace_steps)) * ff_v_ph_root_trace) + (ff_previous_ph_root_trace_steps))) /\ ((((exists fs_h_ph_root_trace_steps_after. fs_h_ph_root_trace_steps_after + S (ff_current_ph_root_trace_steps) = S ((S (S ff_i_ph_root_trace_steps)) * ff_v_ph_root_trace)) /\ exists fs_q_ph_root_trace_steps_after. ff_u_ph_root_trace = fs_q_ph_root_trace_steps_after * S ((S (S ff_i_ph_root_trace_steps)) * ff_v_ph_root_trace) + (ff_current_ph_root_trace_steps))) /\ ff_current_ph_root_trace_steps = ff_previous_ph_root_trace_steps * t + ff_coefficient_ph_root_trace_steps))))beta_horner_eval_existsExact checked statement
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))))))beta_horner_eval_emptyExact checked statement
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 = 0beta_horner_eval_successor_decomposeExact checked statement
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))beta_affine_matrix_slice_extendExact checked statement
forall b c s d u v l a. (forall ff_index_mcp_extend_before ff_source_mcp_extend_before ff_target_mcp_extend_before. (exists mcp_gap_extend_before_bound. mcp_gap_extend_before_bound + S (ff_index_mcp_extend_before) = (l)) -> (((exists fs_h_mcp_extend_before_source. fs_h_mcp_extend_before_source + S (ff_source_mcp_extend_before) = S ((S ((s) + (d) * ff_index_mcp_extend_before)) * c)) /\ exists fs_q_mcp_extend_before_source. b = fs_q_mcp_extend_before_source * S ((S ((s) + (d) * ff_index_mcp_extend_before)) * c) + (ff_source_mcp_extend_before))) -> (((exists fs_h_mcp_extend_before_target. fs_h_mcp_extend_before_target + S (ff_target_mcp_extend_before) = S ((S (ff_index_mcp_extend_before)) * v)) /\ exists fs_q_mcp_extend_before_target. u = fs_q_mcp_extend_before_target * S ((S (ff_index_mcp_extend_before)) * v) + (ff_target_mcp_extend_before))) -> ff_target_mcp_extend_before = ff_source_mcp_extend_before) -> (((exists fs_h_mcp_extend_source_last. fs_h_mcp_extend_source_last + S (a) = S ((S (s + d * l)) * c)) /\ exists fs_q_mcp_extend_source_last. b = fs_q_mcp_extend_source_last * S ((S (s + d * l)) * c) + (a))) -> exists z e. (forall ff_index_mcp_extend_after ff_source_mcp_extend_after ff_target_mcp_extend_after. (exists mcp_gap_extend_after_bound. mcp_gap_extend_after_bound + S (ff_index_mcp_extend_after) = (S l)) -> (((exists fs_h_mcp_extend_after_source. fs_h_mcp_extend_after_source + S (ff_source_mcp_extend_after) = S ((S ((s) + (d) * ff_index_mcp_extend_after)) * c)) /\ exists fs_q_mcp_extend_after_source. b = fs_q_mcp_extend_after_source * S ((S ((s) + (d) * ff_index_mcp_extend_after)) * c) + (ff_source_mcp_extend_after))) -> (((exists fs_h_mcp_extend_after_target. fs_h_mcp_extend_after_target + S (ff_target_mcp_extend_after) = S ((S (ff_index_mcp_extend_after)) * e)) /\ exists fs_q_mcp_extend_after_target. z = fs_q_mcp_extend_after_target * S ((S (ff_index_mcp_extend_after)) * e) + (ff_target_mcp_extend_after))) -> ff_target_mcp_extend_after = ff_source_mcp_extend_after)beta_affine_matrix_slice_existsExact checked statement
forall b c s d l. exists u v. (forall ff_index_mcp_exists_result ff_source_mcp_exists_result ff_target_mcp_exists_result. (exists mcp_gap_exists_result_bound. mcp_gap_exists_result_bound + S (ff_index_mcp_exists_result) = (l)) -> (((exists fs_h_mcp_exists_result_source. fs_h_mcp_exists_result_source + S (ff_source_mcp_exists_result) = S ((S ((s) + (d) * ff_index_mcp_exists_result)) * c)) /\ exists fs_q_mcp_exists_result_source. b = fs_q_mcp_exists_result_source * S ((S ((s) + (d) * ff_index_mcp_exists_result)) * c) + (ff_source_mcp_exists_result))) -> (((exists fs_h_mcp_exists_result_target. fs_h_mcp_exists_result_target + S (ff_target_mcp_exists_result) = S ((S (ff_index_mcp_exists_result)) * v)) /\ exists fs_q_mcp_exists_result_target. u = fs_q_mcp_exists_result_target * S ((S (ff_index_mcp_exists_result)) * v) + (ff_target_mcp_exists_result))) -> ff_target_mcp_exists_result = ff_source_mcp_exists_result)binary_canonical_residue_functionalExact checked statement
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 = smatrix_recursive_lt_add_leftExact checked statement
forall a b c. (exists mdr_gap_add_source. mdr_gap_add_source + S (a) = (b)) -> (exists mdr_gap_add_result. mdr_gap_add_result + S (c + a) = (c + b))matrix_rank_bounded_prefix_valueExact checked statement
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))matrix_rank_no_index_below_zeroExact checked statement
forall i. ~(exists mdr_gap_no_index. mdr_gap_no_index + S (i) = (0))matrix_rank_bounded_prefix_transportExact checked statement
forall b c u v l B. (forall mdr_i_bounded_transport mdr_a_bounded_transport. (exists mdr_gap_bounded_transportb. mdr_gap_bounded_transportb + S (mdr_i_bounded_transport) = (l)) -> (((exists ff_h_mdr_bounded_transporto. ff_h_mdr_bounded_transporto + S (mdr_a_bounded_transport) = S ((S (mdr_i_bounded_transport)) * c)) /\ exists ff_q_mdr_bounded_transporto. b = ff_q_mdr_bounded_transporto * S ((S (mdr_i_bounded_transport)) * c) + (mdr_a_bounded_transport))) -> (((exists ff_h_mdr_bounded_transportn. ff_h_mdr_bounded_transportn + S (mdr_a_bounded_transport) = S ((S (mdr_i_bounded_transport)) * v)) /\ exists ff_q_mdr_bounded_transportn. u = ff_q_mdr_bounded_transportn * S ((S (mdr_i_bounded_transport)) * v) + (mdr_a_bounded_transport)))) -> (forall fom_index_mrf_bounded_source. (exists fom_gap_mrf_bounded_source_index_bound. fom_gap_mrf_bounded_source_index_bound + S (fom_index_mrf_bounded_source) = l) -> exists fom_value_mrf_bounded_source. ((((exists fom_beta_height_mrf_bounded_source_entry. fom_beta_height_mrf_bounded_source_entry + S (fom_value_mrf_bounded_source) = S ((S (fom_index_mrf_bounded_source)) * c)) /\ exists fom_beta_quotient_mrf_bounded_source_entry. b = fom_beta_quotient_mrf_bounded_source_entry * S ((S (fom_index_mrf_bounded_source)) * c) + (fom_value_mrf_bounded_source))) /\ (exists fom_gap_mrf_bounded_source_value_bound. fom_gap_mrf_bounded_source_value_bound + S (fom_value_mrf_bounded_source) = B))) -> (forall fom_index_mrf_bounded_target. (exists fom_gap_mrf_bounded_target_index_bound. fom_gap_mrf_bounded_target_index_bound + S (fom_index_mrf_bounded_target) = l) -> exists fom_value_mrf_bounded_target. ((((exists fom_beta_height_mrf_bounded_target_entry. fom_beta_height_mrf_bounded_target_entry + S (fom_value_mrf_bounded_target) = S ((S (fom_index_mrf_bounded_target)) * v)) /\ exists fom_beta_quotient_mrf_bounded_target_entry. u = fom_beta_quotient_mrf_bounded_target_entry * S ((S (fom_index_mrf_bounded_target)) * v) + (fom_value_mrf_bounded_target))) /\ (exists fom_gap_mrf_bounded_target_value_bound. fom_gap_mrf_bounded_target_value_bound + S (fom_value_mrf_bounded_target) = B)))matrix_rank_bounded_prefix_emptyExact checked statement
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)))matrix_rank_bounded_prefix_extendExact checked statement
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)))hensel_canonical_residue_existsExact checked statement
forall m a. ~(m = 0) -> exists r. ((exists hpl_gap_bound. hpl_gap_bound + S (r) = (m)) /\ (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. a + m * hgcrt_mod_left_hpl_mod = r + m * hgcrt_mod_right_hpl_mod))signed_integer_floor_existsExact checked statement
forall xp xn m. ~(m = 0) -> exists qp qn r. (((xp) + (m) * (qn) = ((xn) + (m) * (qp)) + (r) /\ exists sif_gap_pair_total. sif_gap_pair_total + S (r) = (m)))prime_field_mod_of_equalExact checked statement
forall p a b. a = b -> (exists pfa_offset_left_equal pfa_offset_right_equal. (a) + (p) * pfa_offset_left_equal = (b) + (p) * pfa_offset_right_equal)prime_field_zero_below_primeExact checked statement
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))prime_field_residue_reflexiveExact checked statement
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))))prime_field_residue_input_equalExact checked statement
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))))prime_field_residue_bounded_valueExact checked statement
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 = aprime_field_add_existsExact checked statement
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)))))))))prime_field_add_functionalExact checked statement
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 = dprime_field_add_commutativeExact checked statement
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)))))))))prime_field_multiply_existsExact checked statement
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)))))))))prime_field_multiply_functionalExact checked statement
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 = dprime_field_multiply_commutativeExact checked statement
forall p a b c. (((exists pfa_gap_multiplycomm_sourceleft. pfa_gap_multiplycomm_sourceleft + S (a) = (p)) /\ (((exists pfa_gap_multiplycomm_sourceright. pfa_gap_multiplycomm_sourceright + S (b) = (p)) /\ ((((exists pfa_gap_multiplycomm_sourceresultbound. pfa_gap_multiplycomm_sourceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplycomm_sourceresultcongruence pfa_offset_right_multiplycomm_sourceresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplycomm_sourceresultcongruence = (c) + (p) * pfa_offset_right_multiplycomm_sourceresultcongruence))))))))) -> (((exists pfa_gap_multiplycomm_targetleft. pfa_gap_multiplycomm_targetleft + S (b) = (p)) /\ (((exists pfa_gap_multiplycomm_targetright. pfa_gap_multiplycomm_targetright + S (a) = (p)) /\ ((((exists pfa_gap_multiplycomm_targetresultbound. pfa_gap_multiplycomm_targetresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplycomm_targetresultcongruence pfa_offset_right_multiplycomm_targetresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_multiplycomm_targetresultcongruence = (c) + (p) * pfa_offset_right_multiplycomm_targetresultcongruence)))))))))prime_field_add_associativeExact checked statement
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 = vprime_field_add_zero_rightExact checked statement
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)))))))))prime_field_add_zero_leftExact checked statement
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)))))))))prime_field_multiply_one_rightExact checked statement
forall p a. (~((p) = 1) /\ forall pfa_factor_left_multiply_one_domain pfa_factor_right_multiply_one_domain. (p) = pfa_factor_left_multiply_one_domain * pfa_factor_right_multiply_one_domain -> pfa_factor_left_multiply_one_domain = 1 \/ pfa_factor_right_multiply_one_domain = 1) -> (exists pfa_gap_multiply_one_bound. pfa_gap_multiply_one_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_oneleft. pfa_gap_multiply_oneleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_oneright. pfa_gap_multiply_oneright + S (1) = (p)) /\ ((((exists pfa_gap_multiply_oneresultbound. pfa_gap_multiply_oneresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_multiply_oneresultcongruence pfa_offset_right_multiply_oneresultcongruence. ((a) * (1)) + (p) * pfa_offset_left_multiply_oneresultcongruence = (a) + (p) * pfa_offset_right_multiply_oneresultcongruence)))))))))prime_field_multiply_one_leftExact checked statement
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)))))))))prime_field_multiply_zero_rightExact checked statement
forall p a. (~((p) = 1) /\ forall pfa_factor_left_multiply_zero_domain pfa_factor_right_multiply_zero_domain. (p) = pfa_factor_left_multiply_zero_domain * pfa_factor_right_multiply_zero_domain -> pfa_factor_left_multiply_zero_domain = 1 \/ pfa_factor_right_multiply_zero_domain = 1) -> (exists pfa_gap_multiply_zero_bound. pfa_gap_multiply_zero_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_zeroleft. pfa_gap_multiply_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_zeroright. pfa_gap_multiply_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_multiply_zeroresultbound. pfa_gap_multiply_zeroresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_multiply_zeroresultcongruence pfa_offset_right_multiply_zeroresultcongruence. ((a) * (0)) + (p) * pfa_offset_left_multiply_zeroresultcongruence = (0) + (p) * pfa_offset_right_multiply_zeroresultcongruence)))))))))prime_field_multiply_zero_leftExact checked statement
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)))))))))prime_field_add_cancel_leftExact checked statement
forall p a b c z. (((exists pfa_gap_cancel_add_firstleft. pfa_gap_cancel_add_firstleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_add_firstright. pfa_gap_cancel_add_firstright + S (b) = (p)) /\ ((((exists pfa_gap_cancel_add_firstresultbound. pfa_gap_cancel_add_firstresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_add_firstresultcongruence pfa_offset_right_cancel_add_firstresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_cancel_add_firstresultcongruence = (z) + (p) * pfa_offset_right_cancel_add_firstresultcongruence))))))))) -> (((exists pfa_gap_cancel_add_secondleft. pfa_gap_cancel_add_secondleft + S (a) = (p)) /\ (((exists pfa_gap_cancel_add_secondright. pfa_gap_cancel_add_secondright + S (c) = (p)) /\ ((((exists pfa_gap_cancel_add_secondresultbound. pfa_gap_cancel_add_secondresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_cancel_add_secondresultcongruence pfa_offset_right_cancel_add_secondresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_cancel_add_secondresultcongruence = (z) + (p) * pfa_offset_right_cancel_add_secondresultcongruence))))))))) -> b = cprime_field_negate_existsExact checked statement
forall p a. (~((p) = 1) /\ forall pfa_factor_left_negate_domain pfa_factor_right_negate_domain. (p) = pfa_factor_left_negate_domain * pfa_factor_right_negate_domain -> pfa_factor_left_negate_domain = 1 \/ pfa_factor_right_negate_domain = 1) -> (exists pfa_gap_negate_input. pfa_gap_negate_input + S (a) = (p)) -> exists b. (((exists pfa_gap_negate_existsadditionleft. pfa_gap_negate_existsadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_existsadditionright. pfa_gap_negate_existsadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_existsadditionresultbound. pfa_gap_negate_existsadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_existsadditionresultcongruence pfa_offset_right_negate_existsadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_existsadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_existsadditionresultcongruence)))))))))prime_field_negate_functionalExact checked statement
forall p a b c. (((exists pfa_gap_negate_firstadditionleft. pfa_gap_negate_firstadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_firstadditionright. pfa_gap_negate_firstadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_firstadditionresultbound. pfa_gap_negate_firstadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_firstadditionresultcongruence pfa_offset_right_negate_firstadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_firstadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_firstadditionresultcongruence))))))))) -> (((exists pfa_gap_negate_secondadditionleft. pfa_gap_negate_secondadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_secondadditionright. pfa_gap_negate_secondadditionright + S (c) = (p)) /\ ((((exists pfa_gap_negate_secondadditionresultbound. pfa_gap_negate_secondadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_secondadditionresultcongruence pfa_offset_right_negate_secondadditionresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_negate_secondadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_secondadditionresultcongruence))))))))) -> b = cprime_field_inverse_existsExact checked statement
forall p a. (~((p) = 1) /\ forall pfa_factor_left_inverse_domain pfa_factor_right_inverse_domain. (p) = pfa_factor_left_inverse_domain * pfa_factor_right_inverse_domain -> pfa_factor_left_inverse_domain = 1 \/ pfa_factor_right_inverse_domain = 1) -> (exists pfa_gap_inverse_input. pfa_gap_inverse_input + S (a) = (p)) -> ~(a = 0) -> exists b. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_existsmultiplicationleft. pfa_gap_inverse_existsmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_existsmultiplicationright. pfa_gap_inverse_existsmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_existsmultiplicationresultbound. pfa_gap_inverse_existsmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_existsmultiplicationresultcongruence pfa_offset_right_inverse_existsmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_existsmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_existsmultiplicationresultcongruence))))))))))))prime_field_inverse_functionalExact checked statement
forall p a b c. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_firstmultiplicationleft. pfa_gap_inverse_firstmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_firstmultiplicationright. pfa_gap_inverse_firstmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_firstmultiplicationresultbound. pfa_gap_inverse_firstmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_firstmultiplicationresultcongruence pfa_offset_right_inverse_firstmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_firstmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_firstmultiplicationresultcongruence)))))))))))) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_secondmultiplicationleft. pfa_gap_inverse_secondmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_secondmultiplicationright. pfa_gap_inverse_secondmultiplicationright + S (c) = (p)) /\ ((((exists pfa_gap_inverse_secondmultiplicationresultbound. pfa_gap_inverse_secondmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_secondmultiplicationresultcongruence pfa_offset_right_inverse_secondmultiplicationresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_inverse_secondmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_secondmultiplicationresultcongruence)))))))))))) -> b = cprime_field_zero_has_no_multiplicative_inverseExact checked statement
forall p b. (~((p) = 1) /\ forall pfa_factor_left_zero_inverse_domain pfa_factor_right_zero_inverse_domain. (p) = pfa_factor_left_zero_inverse_domain * pfa_factor_right_zero_inverse_domain -> pfa_factor_left_zero_inverse_domain = 1 \/ pfa_factor_right_zero_inverse_domain = 1) -> ~(((exists pfa_gap_zero_inverseleft. pfa_gap_zero_inverseleft + S (0) = (p)) /\ (((exists pfa_gap_zero_inverseright. pfa_gap_zero_inverseright + S (b) = (p)) /\ ((((exists pfa_gap_zero_inverseresultbound. pfa_gap_zero_inverseresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_zero_inverseresultcongruence pfa_offset_right_zero_inverseresultcongruence. ((0) * (b)) + (p) * pfa_offset_left_zero_inverseresultcongruence = (1) + (p) * pfa_offset_right_zero_inverseresultcongruence)))))))))prime_field_inverse_output_nonzeroExact checked statement
forall p a b. (~((p) = 1) /\ forall pfa_factor_left_inverse_output_domain pfa_factor_right_inverse_output_domain. (p) = pfa_factor_left_inverse_output_domain * pfa_factor_right_inverse_output_domain -> pfa_factor_left_inverse_output_domain = 1 \/ pfa_factor_right_inverse_output_domain = 1) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_outputmultiplicationleft. pfa_gap_inverse_outputmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_outputmultiplicationright. pfa_gap_inverse_outputmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_outputmultiplicationresultbound. pfa_gap_inverse_outputmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_outputmultiplicationresultcongruence pfa_offset_right_inverse_outputmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_outputmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_outputmultiplicationresultcongruence)))))))))))) -> ~(b = 0)prime_field_residue_addExact checked statement
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))))prime_field_residue_multiplyExact checked statement
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))))prime_field_polynomial_normalization_from_divisionExact checked statement
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)))))))))prime_field_polynomial_normalization_existsExact checked statement
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)))))))))prime_field_polynomial_normalization_entryExact checked statement
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))))prime_field_polynomial_normalization_reflexiveExact checked statement
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)))))))))prime_field_polynomial_add_entryExact checked statement
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)))))))))prime_field_polynomial_add_functionalExact checked statement
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))))prime_field_polynomial_add_commutativeExact checked statement
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))))))))))))))))prime_field_polynomial_add_zero_rightExact checked statement
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))))))))))))))))prime_field_polynomial_scale_from_normalizationExact checked statement
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)))))))))))))))))prime_field_polynomial_scale_existsExact checked statement
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)))))))))))))))))prime_field_polynomial_scale_entryExact checked statement
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)))))))))prime_field_polynomial_scale_boundedExact checked statement
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)))))prime_field_polynomial_scale_functionalExact checked statement
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))))prime_field_polynomial_scale_transportExact checked statement
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)))))))))))))))))prime_field_polynomial_scale_oneExact checked statement
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)))))))))))))))))prime_field_polynomial_horner_canonical_stepExact checked statement
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)))))))))))prime_field_polynomial_horner_trace_from_normalizationExact checked statement
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))))))prime_field_polynomial_horner_existsExact checked statement
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)))))))))))))))))))))))))))prime_field_polynomial_horner_input_boundsExact checked statement
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)))))prime_field_polynomial_horner_emptyExact checked statement
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=0prime_field_polynomial_horner_successor_decomposeExact checked statement
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)))))))))))))))prime_field_polynomial_horner_normalization_residueExact checked statement
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))))prime_field_polynomial_horner_residueExact checked statement
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))))prime_field_polynomial_horner_functionalExact checked statement
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=sprime_field_polynomial_horner_empty_constructExact checked statement
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)))))))))))))))))))))))))))prime_field_polynomial_horner_successor_constructExact checked statement
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)))))))))))))))))))))))))))prime_field_polynomial_horner_constantExact checked statement
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)))))))))))))))))))))))))))prime_field_polynomial_horner_result_boundedExact checked statement
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))