prime_field_polynomial_aligned_subtract_functional

Exact expanded first-order arithmetic statement

forall p ab ac L bb bc M rb rc N sb sc J. (~((p) = 1) /\ forall pfa_factor_left_sub_function_prime pfa_factor_right_sub_function_prime. (p) = pfa_factor_left_sub_function_prime * pfa_factor_right_sub_function_prime -> pfa_factor_left_sub_function_prime = 1 \/ pfa_factor_right_sub_function_prime = 1) -> (((forall fom_index_pfp_sub_function_first_left_bounded. (exists fom_gap_pfp_sub_function_first_left_bounded_index_bound. fom_gap_pfp_sub_function_first_left_bounded_index_bound + S (fom_index_pfp_sub_function_first_left_bounded) = M) -> exists fom_value_pfp_sub_function_first_left_bounded. ((((exists fom_beta_height_pfp_sub_function_first_left_bounded_entry. fom_beta_height_pfp_sub_function_first_left_bounded_entry + S (fom_value_pfp_sub_function_first_left_bounded) = S ((S (fom_index_pfp_sub_function_first_left_bounded)) * bc)) /\ exists fom_beta_quotient_pfp_sub_function_first_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_function_first_left_bounded_entry * S ((S (fom_index_pfp_sub_function_first_left_bounded)) * bc) + (fom_value_pfp_sub_function_first_left_bounded))) /\ (exists fom_gap_pfp_sub_function_first_left_bounded_value_bound. fom_gap_pfp_sub_function_first_left_bounded_value_bound + S (fom_value_pfp_sub_function_first_left_bounded) = p))) /\ (((forall fom_index_pfp_sub_function_first_right_bounded. (exists fom_gap_pfp_sub_function_first_right_bounded_index_bound. fom_gap_pfp_sub_function_first_right_bounded_index_bound + S (fom_index_pfp_sub_function_first_right_bounded) = N) -> exists fom_value_pfp_sub_function_first_right_bounded. ((((exists fom_beta_height_pfp_sub_function_first_right_bounded_entry. fom_beta_height_pfp_sub_function_first_right_bounded_entry + S (fom_value_pfp_sub_function_first_right_bounded) = S ((S (fom_index_pfp_sub_function_first_right_bounded)) * rc)) /\ exists fom_beta_quotient_pfp_sub_function_first_right_bounded_entry. rb = fom_beta_quotient_pfp_sub_function_first_right_bounded_entry * S ((S (fom_index_pfp_sub_function_first_right_bounded)) * rc) + (fom_value_pfp_sub_function_first_right_bounded))) /\ (exists fom_gap_pfp_sub_function_first_right_bounded_value_bound. fom_gap_pfp_sub_function_first_right_bounded_value_bound + S (fom_value_pfp_sub_function_first_right_bounded) = p))) /\ (((forall fom_index_pfp_sub_function_first_result_bounded. (exists fom_gap_pfp_sub_function_first_result_bounded_index_bound. fom_gap_pfp_sub_function_first_result_bounded_index_bound + S (fom_index_pfp_sub_function_first_result_bounded) = L) -> exists fom_value_pfp_sub_function_first_result_bounded. ((((exists fom_beta_height_pfp_sub_function_first_result_bounded_entry. fom_beta_height_pfp_sub_function_first_result_bounded_entry + S (fom_value_pfp_sub_function_first_result_bounded) = S ((S (fom_index_pfp_sub_function_first_result_bounded)) * ac)) /\ exists fom_beta_quotient_pfp_sub_function_first_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_function_first_result_bounded_entry * S ((S (fom_index_pfp_sub_function_first_result_bounded)) * ac) + (fom_value_pfp_sub_function_first_result_bounded))) /\ (exists fom_gap_pfp_sub_function_first_result_bounded_value_bound. fom_gap_pfp_sub_function_first_result_bounded_value_bound + S (fom_value_pfp_sub_function_first_result_bounded) = p))) /\ ((exists pfaa_left_b_sub_function_first pfaa_left_c_sub_function_first pfaa_right_b_sub_function_first pfaa_right_c_sub_function_first pfaa_sum_b_sub_function_first pfaa_sum_c_sub_function_first pfaa_length_sub_function_first. ((((forall pfrep_power_sub_function_first_witness_common_left pfrep_left_sub_function_first_witness_common_left pfrep_right_sub_function_first_witness_common_left. ((exists pfrep_position_sub_function_first_witness_common_leftfirst. ((pfrep_position_sub_function_first_witness_common_leftfirst+S (pfrep_power_sub_function_first_witness_common_left)=(M)) /\ ((((exists ff_h_pfp_sub_function_first_witness_common_leftfirstentry. ff_h_pfp_sub_function_first_witness_common_leftfirstentry + S (pfrep_left_sub_function_first_witness_common_left) = S ((S (pfrep_position_sub_function_first_witness_common_leftfirst)) * bc)) /\ exists ff_q_pfp_sub_function_first_witness_common_leftfirstentry. bb = ff_q_pfp_sub_function_first_witness_common_leftfirstentry * S ((S (pfrep_position_sub_function_first_witness_common_leftfirst)) * bc) + (pfrep_left_sub_function_first_witness_common_left)))))) \/ (((exists pfrep_gap_sub_function_first_witness_common_leftfirstoutside. pfrep_gap_sub_function_first_witness_common_leftfirstoutside+(M)=(pfrep_power_sub_function_first_witness_common_left)) /\ (((pfrep_left_sub_function_first_witness_common_left)=0))))) -> ((exists pfrep_position_sub_function_first_witness_common_leftsecond. ((pfrep_position_sub_function_first_witness_common_leftsecond+S (pfrep_power_sub_function_first_witness_common_left)=(pfaa_length_sub_function_first)) /\ ((((exists ff_h_pfp_sub_function_first_witness_common_leftsecondentry. ff_h_pfp_sub_function_first_witness_common_leftsecondentry + S (pfrep_right_sub_function_first_witness_common_left) = S ((S (pfrep_position_sub_function_first_witness_common_leftsecond)) * pfaa_left_c_sub_function_first)) /\ exists ff_q_pfp_sub_function_first_witness_common_leftsecondentry. pfaa_left_b_sub_function_first = ff_q_pfp_sub_function_first_witness_common_leftsecondentry * S ((S (pfrep_position_sub_function_first_witness_common_leftsecond)) * pfaa_left_c_sub_function_first) + (pfrep_right_sub_function_first_witness_common_left)))))) \/ (((exists pfrep_gap_sub_function_first_witness_common_leftsecondoutside. pfrep_gap_sub_function_first_witness_common_leftsecondoutside+(pfaa_length_sub_function_first)=(pfrep_power_sub_function_first_witness_common_left)) /\ (((pfrep_right_sub_function_first_witness_common_left)=0))))) -> pfrep_left_sub_function_first_witness_common_left=pfrep_right_sub_function_first_witness_common_left) /\ ((forall pfrep_power_sub_function_first_witness_common_right pfrep_left_sub_function_first_witness_common_right pfrep_right_sub_function_first_witness_common_right. ((exists pfrep_position_sub_function_first_witness_common_rightfirst. ((pfrep_position_sub_function_first_witness_common_rightfirst+S (pfrep_power_sub_function_first_witness_common_right)=(N)) /\ ((((exists ff_h_pfp_sub_function_first_witness_common_rightfirstentry. ff_h_pfp_sub_function_first_witness_common_rightfirstentry + S (pfrep_left_sub_function_first_witness_common_right) = S ((S (pfrep_position_sub_function_first_witness_common_rightfirst)) * rc)) /\ exists ff_q_pfp_sub_function_first_witness_common_rightfirstentry. rb = ff_q_pfp_sub_function_first_witness_common_rightfirstentry * S ((S (pfrep_position_sub_function_first_witness_common_rightfirst)) * rc) + (pfrep_left_sub_function_first_witness_common_right)))))) \/ (((exists pfrep_gap_sub_function_first_witness_common_rightfirstoutside. pfrep_gap_sub_function_first_witness_common_rightfirstoutside+(N)=(pfrep_power_sub_function_first_witness_common_right)) /\ (((pfrep_left_sub_function_first_witness_common_right)=0))))) -> ((exists pfrep_position_sub_function_first_witness_common_rightsecond. ((pfrep_position_sub_function_first_witness_common_rightsecond+S (pfrep_power_sub_function_first_witness_common_right)=(pfaa_length_sub_function_first)) /\ ((((exists ff_h_pfp_sub_function_first_witness_common_rightsecondentry. ff_h_pfp_sub_function_first_witness_common_rightsecondentry + S (pfrep_right_sub_function_first_witness_common_right) = S ((S (pfrep_position_sub_function_first_witness_common_rightsecond)) * pfaa_right_c_sub_function_first)) /\ exists ff_q_pfp_sub_function_first_witness_common_rightsecondentry. pfaa_right_b_sub_function_first = ff_q_pfp_sub_function_first_witness_common_rightsecondentry * S ((S (pfrep_position_sub_function_first_witness_common_rightsecond)) * pfaa_right_c_sub_function_first) + (pfrep_right_sub_function_first_witness_common_right)))))) \/ (((exists pfrep_gap_sub_function_first_witness_common_rightsecondoutside. pfrep_gap_sub_function_first_witness_common_rightsecondoutside+(pfaa_length_sub_function_first)=(pfrep_power_sub_function_first_witness_common_right)) /\ (((pfrep_right_sub_function_first_witness_common_right)=0))))) -> pfrep_left_sub_function_first_witness_common_right=pfrep_right_sub_function_first_witness_common_right)))) /\ (((forall pfp_index_sub_function_first_witness_operation. (exists pfa_gap_sub_function_first_witness_operationindex. pfa_gap_sub_function_first_witness_operationindex + S (pfp_index_sub_function_first_witness_operation) = (pfaa_length_sub_function_first)) -> exists pfp_left_sub_function_first_witness_operation pfp_right_sub_function_first_witness_operation pfp_value_sub_function_first_witness_operation. ((((exists ff_h_pfp_sub_function_first_witness_operationleft. ff_h_pfp_sub_function_first_witness_operationleft + S (pfp_left_sub_function_first_witness_operation) = S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_left_c_sub_function_first)) /\ exists ff_q_pfp_sub_function_first_witness_operationleft. pfaa_left_b_sub_function_first = ff_q_pfp_sub_function_first_witness_operationleft * S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_left_c_sub_function_first) + (pfp_left_sub_function_first_witness_operation))) /\ (((((exists ff_h_pfp_sub_function_first_witness_operationright. ff_h_pfp_sub_function_first_witness_operationright + S (pfp_right_sub_function_first_witness_operation) = S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_right_c_sub_function_first)) /\ exists ff_q_pfp_sub_function_first_witness_operationright. pfaa_right_b_sub_function_first = ff_q_pfp_sub_function_first_witness_operationright * S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_right_c_sub_function_first) + (pfp_right_sub_function_first_witness_operation))) /\ (((((exists ff_h_pfp_sub_function_first_witness_operationtarget. ff_h_pfp_sub_function_first_witness_operationtarget + S (pfp_value_sub_function_first_witness_operation) = S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_sum_c_sub_function_first)) /\ exists ff_q_pfp_sub_function_first_witness_operationtarget. pfaa_sum_b_sub_function_first = ff_q_pfp_sub_function_first_witness_operationtarget * S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_sum_c_sub_function_first) + (pfp_value_sub_function_first_witness_operation))) /\ ((((exists pfa_gap_sub_function_first_witness_operationoperationleft. pfa_gap_sub_function_first_witness_operationoperationleft + S (pfp_left_sub_function_first_witness_operation) = (p)) /\ (((exists pfa_gap_sub_function_first_witness_operationoperationright. pfa_gap_sub_function_first_witness_operationoperationright + S (pfp_right_sub_function_first_witness_operation) = (p)) /\ ((((exists pfa_gap_sub_function_first_witness_operationoperationresultbound. pfa_gap_sub_function_first_witness_operationoperationresultbound + S (pfp_value_sub_function_first_witness_operation) = (p)) /\ ((exists pfa_offset_left_sub_function_first_witness_operationoperationresultcongruence pfa_offset_right_sub_function_first_witness_operationoperationresultcongruence. ((pfp_left_sub_function_first_witness_operation) + (pfp_right_sub_function_first_witness_operation)) + (p) * pfa_offset_left_sub_function_first_witness_operationoperationresultcongruence = (pfp_value_sub_function_first_witness_operation) + (p) * pfa_offset_right_sub_function_first_witness_operationoperationresultcongruence)))))))))))))))) /\ ((forall pfrep_power_sub_function_first_witness_output pfrep_left_sub_function_first_witness_output pfrep_right_sub_function_first_witness_output. ((exists pfrep_position_sub_function_first_witness_outputfirst. ((pfrep_position_sub_function_first_witness_outputfirst+S (pfrep_power_sub_function_first_witness_output)=(pfaa_length_sub_function_first)) /\ ((((exists ff_h_pfp_sub_function_first_witness_outputfirstentry. ff_h_pfp_sub_function_first_witness_outputfirstentry + S (pfrep_left_sub_function_first_witness_output) = S ((S (pfrep_position_sub_function_first_witness_outputfirst)) * pfaa_sum_c_sub_function_first)) /\ exists ff_q_pfp_sub_function_first_witness_outputfirstentry. pfaa_sum_b_sub_function_first = ff_q_pfp_sub_function_first_witness_outputfirstentry * S ((S (pfrep_position_sub_function_first_witness_outputfirst)) * pfaa_sum_c_sub_function_first) + (pfrep_left_sub_function_first_witness_output)))))) \/ (((exists pfrep_gap_sub_function_first_witness_outputfirstoutside. pfrep_gap_sub_function_first_witness_outputfirstoutside+(pfaa_length_sub_function_first)=(pfrep_power_sub_function_first_witness_output)) /\ (((pfrep_left_sub_function_first_witness_output)=0))))) -> ((exists pfrep_position_sub_function_first_witness_outputsecond. ((pfrep_position_sub_function_first_witness_outputsecond+S (pfrep_power_sub_function_first_witness_output)=(L)) /\ ((((exists ff_h_pfp_sub_function_first_witness_outputsecondentry. ff_h_pfp_sub_function_first_witness_outputsecondentry + S (pfrep_right_sub_function_first_witness_output) = S ((S (pfrep_position_sub_function_first_witness_outputsecond)) * ac)) /\ exists ff_q_pfp_sub_function_first_witness_outputsecondentry. ab = ff_q_pfp_sub_function_first_witness_outputsecondentry * S ((S (pfrep_position_sub_function_first_witness_outputsecond)) * ac) + (pfrep_right_sub_function_first_witness_output)))))) \/ (((exists pfrep_gap_sub_function_first_witness_outputsecondoutside. pfrep_gap_sub_function_first_witness_outputsecondoutside+(L)=(pfrep_power_sub_function_first_witness_output)) /\ (((pfrep_right_sub_function_first_witness_output)=0))))) -> pfrep_left_sub_function_first_witness_output=pfrep_right_sub_function_first_witness_output))))))))))))) -> (((forall fom_index_pfp_sub_function_second_left_bounded. (exists fom_gap_pfp_sub_function_second_left_bounded_index_bound. fom_gap_pfp_sub_function_second_left_bounded_index_bound + S (fom_index_pfp_sub_function_second_left_bounded) = M) -> exists fom_value_pfp_sub_function_second_left_bounded. ((((exists fom_beta_height_pfp_sub_function_second_left_bounded_entry. fom_beta_height_pfp_sub_function_second_left_bounded_entry + S (fom_value_pfp_sub_function_second_left_bounded) = S ((S (fom_index_pfp_sub_function_second_left_bounded)) * bc)) /\ exists fom_beta_quotient_pfp_sub_function_second_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_function_second_left_bounded_entry * S ((S (fom_index_pfp_sub_function_second_left_bounded)) * bc) + (fom_value_pfp_sub_function_second_left_bounded))) /\ (exists fom_gap_pfp_sub_function_second_left_bounded_value_bound. fom_gap_pfp_sub_function_second_left_bounded_value_bound + S (fom_value_pfp_sub_function_second_left_bounded) = p))) /\ (((forall fom_index_pfp_sub_function_second_right_bounded. (exists fom_gap_pfp_sub_function_second_right_bounded_index_bound. fom_gap_pfp_sub_function_second_right_bounded_index_bound + S (fom_index_pfp_sub_function_second_right_bounded) = J) -> exists fom_value_pfp_sub_function_second_right_bounded. ((((exists fom_beta_height_pfp_sub_function_second_right_bounded_entry. fom_beta_height_pfp_sub_function_second_right_bounded_entry + S (fom_value_pfp_sub_function_second_right_bounded) = S ((S (fom_index_pfp_sub_function_second_right_bounded)) * sc)) /\ exists fom_beta_quotient_pfp_sub_function_second_right_bounded_entry. sb = fom_beta_quotient_pfp_sub_function_second_right_bounded_entry * S ((S (fom_index_pfp_sub_function_second_right_bounded)) * sc) + (fom_value_pfp_sub_function_second_right_bounded))) /\ (exists fom_gap_pfp_sub_function_second_right_bounded_value_bound. fom_gap_pfp_sub_function_second_right_bounded_value_bound + S (fom_value_pfp_sub_function_second_right_bounded) = p))) /\ (((forall fom_index_pfp_sub_function_second_result_bounded. (exists fom_gap_pfp_sub_function_second_result_bounded_index_bound. fom_gap_pfp_sub_function_second_result_bounded_index_bound + S (fom_index_pfp_sub_function_second_result_bounded) = L) -> exists fom_value_pfp_sub_function_second_result_bounded. ((((exists fom_beta_height_pfp_sub_function_second_result_bounded_entry. fom_beta_height_pfp_sub_function_second_result_bounded_entry + S (fom_value_pfp_sub_function_second_result_bounded) = S ((S (fom_index_pfp_sub_function_second_result_bounded)) * ac)) /\ exists fom_beta_quotient_pfp_sub_function_second_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_function_second_result_bounded_entry * S ((S (fom_index_pfp_sub_function_second_result_bounded)) * ac) + (fom_value_pfp_sub_function_second_result_bounded))) /\ (exists fom_gap_pfp_sub_function_second_result_bounded_value_bound. fom_gap_pfp_sub_function_second_result_bounded_value_bound + S (fom_value_pfp_sub_function_second_result_bounded) = p))) /\ ((exists pfaa_left_b_sub_function_second pfaa_left_c_sub_function_second pfaa_right_b_sub_function_second pfaa_right_c_sub_function_second pfaa_sum_b_sub_function_second pfaa_sum_c_sub_function_second pfaa_length_sub_function_second. ((((forall pfrep_power_sub_function_second_witness_common_left pfrep_left_sub_function_second_witness_common_left pfrep_right_sub_function_second_witness_common_left. ((exists pfrep_position_sub_function_second_witness_common_leftfirst. ((pfrep_position_sub_function_second_witness_common_leftfirst+S (pfrep_power_sub_function_second_witness_common_left)=(M)) /\ ((((exists ff_h_pfp_sub_function_second_witness_common_leftfirstentry. ff_h_pfp_sub_function_second_witness_common_leftfirstentry + S (pfrep_left_sub_function_second_witness_common_left) = S ((S (pfrep_position_sub_function_second_witness_common_leftfirst)) * bc)) /\ exists ff_q_pfp_sub_function_second_witness_common_leftfirstentry. bb = ff_q_pfp_sub_function_second_witness_common_leftfirstentry * S ((S (pfrep_position_sub_function_second_witness_common_leftfirst)) * bc) + (pfrep_left_sub_function_second_witness_common_left)))))) \/ (((exists pfrep_gap_sub_function_second_witness_common_leftfirstoutside. pfrep_gap_sub_function_second_witness_common_leftfirstoutside+(M)=(pfrep_power_sub_function_second_witness_common_left)) /\ (((pfrep_left_sub_function_second_witness_common_left)=0))))) -> ((exists pfrep_position_sub_function_second_witness_common_leftsecond. ((pfrep_position_sub_function_second_witness_common_leftsecond+S (pfrep_power_sub_function_second_witness_common_left)=(pfaa_length_sub_function_second)) /\ ((((exists ff_h_pfp_sub_function_second_witness_common_leftsecondentry. ff_h_pfp_sub_function_second_witness_common_leftsecondentry + S (pfrep_right_sub_function_second_witness_common_left) = S ((S (pfrep_position_sub_function_second_witness_common_leftsecond)) * pfaa_left_c_sub_function_second)) /\ exists ff_q_pfp_sub_function_second_witness_common_leftsecondentry. pfaa_left_b_sub_function_second = ff_q_pfp_sub_function_second_witness_common_leftsecondentry * S ((S (pfrep_position_sub_function_second_witness_common_leftsecond)) * pfaa_left_c_sub_function_second) + (pfrep_right_sub_function_second_witness_common_left)))))) \/ (((exists pfrep_gap_sub_function_second_witness_common_leftsecondoutside. pfrep_gap_sub_function_second_witness_common_leftsecondoutside+(pfaa_length_sub_function_second)=(pfrep_power_sub_function_second_witness_common_left)) /\ (((pfrep_right_sub_function_second_witness_common_left)=0))))) -> pfrep_left_sub_function_second_witness_common_left=pfrep_right_sub_function_second_witness_common_left) /\ ((forall pfrep_power_sub_function_second_witness_common_right pfrep_left_sub_function_second_witness_common_right pfrep_right_sub_function_second_witness_common_right. ((exists pfrep_position_sub_function_second_witness_common_rightfirst. ((pfrep_position_sub_function_second_witness_common_rightfirst+S (pfrep_power_sub_function_second_witness_common_right)=(J)) /\ ((((exists ff_h_pfp_sub_function_second_witness_common_rightfirstentry. ff_h_pfp_sub_function_second_witness_common_rightfirstentry + S (pfrep_left_sub_function_second_witness_common_right) = S ((S (pfrep_position_sub_function_second_witness_common_rightfirst)) * sc)) /\ exists ff_q_pfp_sub_function_second_witness_common_rightfirstentry. sb = ff_q_pfp_sub_function_second_witness_common_rightfirstentry * S ((S (pfrep_position_sub_function_second_witness_common_rightfirst)) * sc) + (pfrep_left_sub_function_second_witness_common_right)))))) \/ (((exists pfrep_gap_sub_function_second_witness_common_rightfirstoutside. pfrep_gap_sub_function_second_witness_common_rightfirstoutside+(J)=(pfrep_power_sub_function_second_witness_common_right)) /\ (((pfrep_left_sub_function_second_witness_common_right)=0))))) -> ((exists pfrep_position_sub_function_second_witness_common_rightsecond. ((pfrep_position_sub_function_second_witness_common_rightsecond+S (pfrep_power_sub_function_second_witness_common_right)=(pfaa_length_sub_function_second)) /\ ((((exists ff_h_pfp_sub_function_second_witness_common_rightsecondentry. ff_h_pfp_sub_function_second_witness_common_rightsecondentry + S (pfrep_right_sub_function_second_witness_common_right) = S ((S (pfrep_position_sub_function_second_witness_common_rightsecond)) * pfaa_right_c_sub_function_second)) /\ exists ff_q_pfp_sub_function_second_witness_common_rightsecondentry. pfaa_right_b_sub_function_second = ff_q_pfp_sub_function_second_witness_common_rightsecondentry * S ((S (pfrep_position_sub_function_second_witness_common_rightsecond)) * pfaa_right_c_sub_function_second) + (pfrep_right_sub_function_second_witness_common_right)))))) \/ (((exists pfrep_gap_sub_function_second_witness_common_rightsecondoutside. pfrep_gap_sub_function_second_witness_common_rightsecondoutside+(pfaa_length_sub_function_second)=(pfrep_power_sub_function_second_witness_common_right)) /\ (((pfrep_right_sub_function_second_witness_common_right)=0))))) -> pfrep_left_sub_function_second_witness_common_right=pfrep_right_sub_function_second_witness_common_right)))) /\ (((forall pfp_index_sub_function_second_witness_operation. (exists pfa_gap_sub_function_second_witness_operationindex. pfa_gap_sub_function_second_witness_operationindex + S (pfp_index_sub_function_second_witness_operation) = (pfaa_length_sub_function_second)) -> exists pfp_left_sub_function_second_witness_operation pfp_right_sub_function_second_witness_operation pfp_value_sub_function_second_witness_operation. ((((exists ff_h_pfp_sub_function_second_witness_operationleft. ff_h_pfp_sub_function_second_witness_operationleft + S (pfp_left_sub_function_second_witness_operation) = S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_left_c_sub_function_second)) /\ exists ff_q_pfp_sub_function_second_witness_operationleft. pfaa_left_b_sub_function_second = ff_q_pfp_sub_function_second_witness_operationleft * S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_left_c_sub_function_second) + (pfp_left_sub_function_second_witness_operation))) /\ (((((exists ff_h_pfp_sub_function_second_witness_operationright. ff_h_pfp_sub_function_second_witness_operationright + S (pfp_right_sub_function_second_witness_operation) = S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_right_c_sub_function_second)) /\ exists ff_q_pfp_sub_function_second_witness_operationright. pfaa_right_b_sub_function_second = ff_q_pfp_sub_function_second_witness_operationright * S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_right_c_sub_function_second) + (pfp_right_sub_function_second_witness_operation))) /\ (((((exists ff_h_pfp_sub_function_second_witness_operationtarget. ff_h_pfp_sub_function_second_witness_operationtarget + S (pfp_value_sub_function_second_witness_operation) = S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_sum_c_sub_function_second)) /\ exists ff_q_pfp_sub_function_second_witness_operationtarget. pfaa_sum_b_sub_function_second = ff_q_pfp_sub_function_second_witness_operationtarget * S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_sum_c_sub_function_second) + (pfp_value_sub_function_second_witness_operation))) /\ ((((exists pfa_gap_sub_function_second_witness_operationoperationleft. pfa_gap_sub_function_second_witness_operationoperationleft + S (pfp_left_sub_function_second_witness_operation) = (p)) /\ (((exists pfa_gap_sub_function_second_witness_operationoperationright. pfa_gap_sub_function_second_witness_operationoperationright + S (pfp_right_sub_function_second_witness_operation) = (p)) /\ ((((exists pfa_gap_sub_function_second_witness_operationoperationresultbound. pfa_gap_sub_function_second_witness_operationoperationresultbound + S (pfp_value_sub_function_second_witness_operation) = (p)) /\ ((exists pfa_offset_left_sub_function_second_witness_operationoperationresultcongruence pfa_offset_right_sub_function_second_witness_operationoperationresultcongruence. ((pfp_left_sub_function_second_witness_operation) + (pfp_right_sub_function_second_witness_operation)) + (p) * pfa_offset_left_sub_function_second_witness_operationoperationresultcongruence = (pfp_value_sub_function_second_witness_operation) + (p) * pfa_offset_right_sub_function_second_witness_operationoperationresultcongruence)))))))))))))))) /\ ((forall pfrep_power_sub_function_second_witness_output pfrep_left_sub_function_second_witness_output pfrep_right_sub_function_second_witness_output. ((exists pfrep_position_sub_function_second_witness_outputfirst. ((pfrep_position_sub_function_second_witness_outputfirst+S (pfrep_power_sub_function_second_witness_output)=(pfaa_length_sub_function_second)) /\ ((((exists ff_h_pfp_sub_function_second_witness_outputfirstentry. ff_h_pfp_sub_function_second_witness_outputfirstentry + S (pfrep_left_sub_function_second_witness_output) = S ((S (pfrep_position_sub_function_second_witness_outputfirst)) * pfaa_sum_c_sub_function_second)) /\ exists ff_q_pfp_sub_function_second_witness_outputfirstentry. pfaa_sum_b_sub_function_second = ff_q_pfp_sub_function_second_witness_outputfirstentry * S ((S (pfrep_position_sub_function_second_witness_outputfirst)) * pfaa_sum_c_sub_function_second) + (pfrep_left_sub_function_second_witness_output)))))) \/ (((exists pfrep_gap_sub_function_second_witness_outputfirstoutside. pfrep_gap_sub_function_second_witness_outputfirstoutside+(pfaa_length_sub_function_second)=(pfrep_power_sub_function_second_witness_output)) /\ (((pfrep_left_sub_function_second_witness_output)=0))))) -> ((exists pfrep_position_sub_function_second_witness_outputsecond. ((pfrep_position_sub_function_second_witness_outputsecond+S (pfrep_power_sub_function_second_witness_output)=(L)) /\ ((((exists ff_h_pfp_sub_function_second_witness_outputsecondentry. ff_h_pfp_sub_function_second_witness_outputsecondentry + S (pfrep_right_sub_function_second_witness_output) = S ((S (pfrep_position_sub_function_second_witness_outputsecond)) * ac)) /\ exists ff_q_pfp_sub_function_second_witness_outputsecondentry. ab = ff_q_pfp_sub_function_second_witness_outputsecondentry * S ((S (pfrep_position_sub_function_second_witness_outputsecond)) * ac) + (pfrep_right_sub_function_second_witness_output)))))) \/ (((exists pfrep_gap_sub_function_second_witness_outputsecondoutside. pfrep_gap_sub_function_second_witness_outputsecondoutside+(L)=(pfrep_power_sub_function_second_witness_output)) /\ (((pfrep_right_sub_function_second_witness_output)=0))))) -> pfrep_left_sub_function_second_witness_output=pfrep_right_sub_function_second_witness_output))))))))))))) -> (forall pfrep_power_sub_function_result pfrep_left_sub_function_result pfrep_right_sub_function_result. ((exists pfrep_position_sub_function_resultfirst. ((pfrep_position_sub_function_resultfirst+S (pfrep_power_sub_function_result)=(N)) /\ ((((exists ff_h_pfp_sub_function_resultfirstentry. ff_h_pfp_sub_function_resultfirstentry + S (pfrep_left_sub_function_result) = S ((S (pfrep_position_sub_function_resultfirst)) * rc)) /\ exists ff_q_pfp_sub_function_resultfirstentry. rb = ff_q_pfp_sub_function_resultfirstentry * S ((S (pfrep_position_sub_function_resultfirst)) * rc) + (pfrep_left_sub_function_result)))))) \/ (((exists pfrep_gap_sub_function_resultfirstoutside. pfrep_gap_sub_function_resultfirstoutside+(N)=(pfrep_power_sub_function_result)) /\ (((pfrep_left_sub_function_result)=0))))) -> ((exists pfrep_position_sub_function_resultsecond. ((pfrep_position_sub_function_resultsecond+S (pfrep_power_sub_function_result)=(J)) /\ ((((exists ff_h_pfp_sub_function_resultsecondentry. ff_h_pfp_sub_function_resultsecondentry + S (pfrep_right_sub_function_result) = S ((S (pfrep_position_sub_function_resultsecond)) * sc)) /\ exists ff_q_pfp_sub_function_resultsecondentry. sb = ff_q_pfp_sub_function_resultsecondentry * S ((S (pfrep_position_sub_function_resultsecond)) * sc) + (pfrep_right_sub_function_result)))))) \/ (((exists pfrep_gap_sub_function_resultsecondoutside. pfrep_gap_sub_function_resultsecondoutside+(J)=(pfrep_power_sub_function_result)) /\ (((pfrep_right_sub_function_result)=0))))) -> pfrep_left_sub_function_result=pfrep_right_sub_function_result)