prime_field_polynomial_aligned_subtract_from_fixed

Exact expanded first-order arithmetic statement

forall p ab ac bb bc rb rc K. (forall pfs_index_sub_fixed_input. (exists pfa_gap_sub_fixed_inputindex. pfa_gap_sub_fixed_inputindex + S (pfs_index_sub_fixed_input) = (K)) -> exists pfs_left_sub_fixed_input pfs_right_sub_fixed_input pfs_result_sub_fixed_input. ((((exists ff_h_pfp_sub_fixed_inputleft. ff_h_pfp_sub_fixed_inputleft + S (pfs_left_sub_fixed_input) = S ((S (pfs_index_sub_fixed_input)) * ac)) /\ exists ff_q_pfp_sub_fixed_inputleft. ab = ff_q_pfp_sub_fixed_inputleft * S ((S (pfs_index_sub_fixed_input)) * ac) + (pfs_left_sub_fixed_input))) /\ (((((exists ff_h_pfp_sub_fixed_inputright. ff_h_pfp_sub_fixed_inputright + S (pfs_right_sub_fixed_input) = S ((S (pfs_index_sub_fixed_input)) * bc)) /\ exists ff_q_pfp_sub_fixed_inputright. bb = ff_q_pfp_sub_fixed_inputright * S ((S (pfs_index_sub_fixed_input)) * bc) + (pfs_right_sub_fixed_input))) /\ (((((exists ff_h_pfp_sub_fixed_inputresult. ff_h_pfp_sub_fixed_inputresult + S (pfs_result_sub_fixed_input) = S ((S (pfs_index_sub_fixed_input)) * rc)) /\ exists ff_q_pfp_sub_fixed_inputresult. rb = ff_q_pfp_sub_fixed_inputresult * S ((S (pfs_index_sub_fixed_input)) * rc) + (pfs_result_sub_fixed_input))) /\ ((((exists pfa_gap_sub_fixed_inputoperationleft. pfa_gap_sub_fixed_inputoperationleft + S (pfs_right_sub_fixed_input) = (p)) /\ (((exists pfa_gap_sub_fixed_inputoperationright. pfa_gap_sub_fixed_inputoperationright + S (pfs_result_sub_fixed_input) = (p)) /\ ((((exists pfa_gap_sub_fixed_inputoperationresultbound. pfa_gap_sub_fixed_inputoperationresultbound + S (pfs_left_sub_fixed_input) = (p)) /\ ((exists pfa_offset_left_sub_fixed_inputoperationresultcongruence pfa_offset_right_sub_fixed_inputoperationresultcongruence. ((pfs_right_sub_fixed_input) + (pfs_result_sub_fixed_input)) + (p) * pfa_offset_left_sub_fixed_inputoperationresultcongruence = (pfs_left_sub_fixed_input) + (p) * pfa_offset_right_sub_fixed_inputoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_sub_fixed_result_left_bounded. (exists fom_gap_pfp_sub_fixed_result_left_bounded_index_bound. fom_gap_pfp_sub_fixed_result_left_bounded_index_bound + S (fom_index_pfp_sub_fixed_result_left_bounded) = K) -> exists fom_value_pfp_sub_fixed_result_left_bounded. ((((exists fom_beta_height_pfp_sub_fixed_result_left_bounded_entry. fom_beta_height_pfp_sub_fixed_result_left_bounded_entry + S (fom_value_pfp_sub_fixed_result_left_bounded) = S ((S (fom_index_pfp_sub_fixed_result_left_bounded)) * bc)) /\ exists fom_beta_quotient_pfp_sub_fixed_result_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_fixed_result_left_bounded_entry * S ((S (fom_index_pfp_sub_fixed_result_left_bounded)) * bc) + (fom_value_pfp_sub_fixed_result_left_bounded))) /\ (exists fom_gap_pfp_sub_fixed_result_left_bounded_value_bound. fom_gap_pfp_sub_fixed_result_left_bounded_value_bound + S (fom_value_pfp_sub_fixed_result_left_bounded) = p))) /\ (((forall fom_index_pfp_sub_fixed_result_right_bounded. (exists fom_gap_pfp_sub_fixed_result_right_bounded_index_bound. fom_gap_pfp_sub_fixed_result_right_bounded_index_bound + S (fom_index_pfp_sub_fixed_result_right_bounded) = K) -> exists fom_value_pfp_sub_fixed_result_right_bounded. ((((exists fom_beta_height_pfp_sub_fixed_result_right_bounded_entry. fom_beta_height_pfp_sub_fixed_result_right_bounded_entry + S (fom_value_pfp_sub_fixed_result_right_bounded) = S ((S (fom_index_pfp_sub_fixed_result_right_bounded)) * rc)) /\ exists fom_beta_quotient_pfp_sub_fixed_result_right_bounded_entry. rb = fom_beta_quotient_pfp_sub_fixed_result_right_bounded_entry * S ((S (fom_index_pfp_sub_fixed_result_right_bounded)) * rc) + (fom_value_pfp_sub_fixed_result_right_bounded))) /\ (exists fom_gap_pfp_sub_fixed_result_right_bounded_value_bound. fom_gap_pfp_sub_fixed_result_right_bounded_value_bound + S (fom_value_pfp_sub_fixed_result_right_bounded) = p))) /\ (((forall fom_index_pfp_sub_fixed_result_result_bounded. (exists fom_gap_pfp_sub_fixed_result_result_bounded_index_bound. fom_gap_pfp_sub_fixed_result_result_bounded_index_bound + S (fom_index_pfp_sub_fixed_result_result_bounded) = K) -> exists fom_value_pfp_sub_fixed_result_result_bounded. ((((exists fom_beta_height_pfp_sub_fixed_result_result_bounded_entry. fom_beta_height_pfp_sub_fixed_result_result_bounded_entry + S (fom_value_pfp_sub_fixed_result_result_bounded) = S ((S (fom_index_pfp_sub_fixed_result_result_bounded)) * ac)) /\ exists fom_beta_quotient_pfp_sub_fixed_result_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_fixed_result_result_bounded_entry * S ((S (fom_index_pfp_sub_fixed_result_result_bounded)) * ac) + (fom_value_pfp_sub_fixed_result_result_bounded))) /\ (exists fom_gap_pfp_sub_fixed_result_result_bounded_value_bound. fom_gap_pfp_sub_fixed_result_result_bounded_value_bound + S (fom_value_pfp_sub_fixed_result_result_bounded) = p))) /\ ((exists pfaa_left_b_sub_fixed_result pfaa_left_c_sub_fixed_result pfaa_right_b_sub_fixed_result pfaa_right_c_sub_fixed_result pfaa_sum_b_sub_fixed_result pfaa_sum_c_sub_fixed_result pfaa_length_sub_fixed_result. ((((forall pfrep_power_sub_fixed_result_witness_common_left pfrep_left_sub_fixed_result_witness_common_left pfrep_right_sub_fixed_result_witness_common_left. ((exists pfrep_position_sub_fixed_result_witness_common_leftfirst. ((pfrep_position_sub_fixed_result_witness_common_leftfirst+S (pfrep_power_sub_fixed_result_witness_common_left)=(K)) /\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_leftfirstentry. ff_h_pfp_sub_fixed_result_witness_common_leftfirstentry + S (pfrep_left_sub_fixed_result_witness_common_left) = S ((S (pfrep_position_sub_fixed_result_witness_common_leftfirst)) * bc)) /\ exists ff_q_pfp_sub_fixed_result_witness_common_leftfirstentry. bb = ff_q_pfp_sub_fixed_result_witness_common_leftfirstentry * S ((S (pfrep_position_sub_fixed_result_witness_common_leftfirst)) * bc) + (pfrep_left_sub_fixed_result_witness_common_left)))))) \/ (((exists pfrep_gap_sub_fixed_result_witness_common_leftfirstoutside. pfrep_gap_sub_fixed_result_witness_common_leftfirstoutside+(K)=(pfrep_power_sub_fixed_result_witness_common_left)) /\ (((pfrep_left_sub_fixed_result_witness_common_left)=0))))) -> ((exists pfrep_position_sub_fixed_result_witness_common_leftsecond. ((pfrep_position_sub_fixed_result_witness_common_leftsecond+S (pfrep_power_sub_fixed_result_witness_common_left)=(pfaa_length_sub_fixed_result)) /\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_leftsecondentry. ff_h_pfp_sub_fixed_result_witness_common_leftsecondentry + S (pfrep_right_sub_fixed_result_witness_common_left) = S ((S (pfrep_position_sub_fixed_result_witness_common_leftsecond)) * pfaa_left_c_sub_fixed_result)) /\ exists ff_q_pfp_sub_fixed_result_witness_common_leftsecondentry. pfaa_left_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_common_leftsecondentry * S ((S (pfrep_position_sub_fixed_result_witness_common_leftsecond)) * pfaa_left_c_sub_fixed_result) + (pfrep_right_sub_fixed_result_witness_common_left)))))) \/ (((exists pfrep_gap_sub_fixed_result_witness_common_leftsecondoutside. pfrep_gap_sub_fixed_result_witness_common_leftsecondoutside+(pfaa_length_sub_fixed_result)=(pfrep_power_sub_fixed_result_witness_common_left)) /\ (((pfrep_right_sub_fixed_result_witness_common_left)=0))))) -> pfrep_left_sub_fixed_result_witness_common_left=pfrep_right_sub_fixed_result_witness_common_left) /\ ((forall pfrep_power_sub_fixed_result_witness_common_right pfrep_left_sub_fixed_result_witness_common_right pfrep_right_sub_fixed_result_witness_common_right. ((exists pfrep_position_sub_fixed_result_witness_common_rightfirst. ((pfrep_position_sub_fixed_result_witness_common_rightfirst+S (pfrep_power_sub_fixed_result_witness_common_right)=(K)) /\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_rightfirstentry. ff_h_pfp_sub_fixed_result_witness_common_rightfirstentry + S (pfrep_left_sub_fixed_result_witness_common_right) = S ((S (pfrep_position_sub_fixed_result_witness_common_rightfirst)) * rc)) /\ exists ff_q_pfp_sub_fixed_result_witness_common_rightfirstentry. rb = ff_q_pfp_sub_fixed_result_witness_common_rightfirstentry * S ((S (pfrep_position_sub_fixed_result_witness_common_rightfirst)) * rc) + (pfrep_left_sub_fixed_result_witness_common_right)))))) \/ (((exists pfrep_gap_sub_fixed_result_witness_common_rightfirstoutside. pfrep_gap_sub_fixed_result_witness_common_rightfirstoutside+(K)=(pfrep_power_sub_fixed_result_witness_common_right)) /\ (((pfrep_left_sub_fixed_result_witness_common_right)=0))))) -> ((exists pfrep_position_sub_fixed_result_witness_common_rightsecond. ((pfrep_position_sub_fixed_result_witness_common_rightsecond+S (pfrep_power_sub_fixed_result_witness_common_right)=(pfaa_length_sub_fixed_result)) /\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_rightsecondentry. ff_h_pfp_sub_fixed_result_witness_common_rightsecondentry + S (pfrep_right_sub_fixed_result_witness_common_right) = S ((S (pfrep_position_sub_fixed_result_witness_common_rightsecond)) * pfaa_right_c_sub_fixed_result)) /\ exists ff_q_pfp_sub_fixed_result_witness_common_rightsecondentry. pfaa_right_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_common_rightsecondentry * S ((S (pfrep_position_sub_fixed_result_witness_common_rightsecond)) * pfaa_right_c_sub_fixed_result) + (pfrep_right_sub_fixed_result_witness_common_right)))))) \/ (((exists pfrep_gap_sub_fixed_result_witness_common_rightsecondoutside. pfrep_gap_sub_fixed_result_witness_common_rightsecondoutside+(pfaa_length_sub_fixed_result)=(pfrep_power_sub_fixed_result_witness_common_right)) /\ (((pfrep_right_sub_fixed_result_witness_common_right)=0))))) -> pfrep_left_sub_fixed_result_witness_common_right=pfrep_right_sub_fixed_result_witness_common_right)))) /\ (((forall pfp_index_sub_fixed_result_witness_operation. (exists pfa_gap_sub_fixed_result_witness_operationindex. pfa_gap_sub_fixed_result_witness_operationindex + S (pfp_index_sub_fixed_result_witness_operation) = (pfaa_length_sub_fixed_result)) -> exists pfp_left_sub_fixed_result_witness_operation pfp_right_sub_fixed_result_witness_operation pfp_value_sub_fixed_result_witness_operation. ((((exists ff_h_pfp_sub_fixed_result_witness_operationleft. ff_h_pfp_sub_fixed_result_witness_operationleft + S (pfp_left_sub_fixed_result_witness_operation) = S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_left_c_sub_fixed_result)) /\ exists ff_q_pfp_sub_fixed_result_witness_operationleft. pfaa_left_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_operationleft * S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_left_c_sub_fixed_result) + (pfp_left_sub_fixed_result_witness_operation))) /\ (((((exists ff_h_pfp_sub_fixed_result_witness_operationright. ff_h_pfp_sub_fixed_result_witness_operationright + S (pfp_right_sub_fixed_result_witness_operation) = S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_right_c_sub_fixed_result)) /\ exists ff_q_pfp_sub_fixed_result_witness_operationright. pfaa_right_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_operationright * S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_right_c_sub_fixed_result) + (pfp_right_sub_fixed_result_witness_operation))) /\ (((((exists ff_h_pfp_sub_fixed_result_witness_operationtarget. ff_h_pfp_sub_fixed_result_witness_operationtarget + S (pfp_value_sub_fixed_result_witness_operation) = S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_sum_c_sub_fixed_result)) /\ exists ff_q_pfp_sub_fixed_result_witness_operationtarget. pfaa_sum_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_operationtarget * S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_sum_c_sub_fixed_result) + (pfp_value_sub_fixed_result_witness_operation))) /\ ((((exists pfa_gap_sub_fixed_result_witness_operationoperationleft. pfa_gap_sub_fixed_result_witness_operationoperationleft + S (pfp_left_sub_fixed_result_witness_operation) = (p)) /\ (((exists pfa_gap_sub_fixed_result_witness_operationoperationright. pfa_gap_sub_fixed_result_witness_operationoperationright + S (pfp_right_sub_fixed_result_witness_operation) = (p)) /\ ((((exists pfa_gap_sub_fixed_result_witness_operationoperationresultbound. pfa_gap_sub_fixed_result_witness_operationoperationresultbound + S (pfp_value_sub_fixed_result_witness_operation) = (p)) /\ ((exists pfa_offset_left_sub_fixed_result_witness_operationoperationresultcongruence pfa_offset_right_sub_fixed_result_witness_operationoperationresultcongruence. ((pfp_left_sub_fixed_result_witness_operation) + (pfp_right_sub_fixed_result_witness_operation)) + (p) * pfa_offset_left_sub_fixed_result_witness_operationoperationresultcongruence = (pfp_value_sub_fixed_result_witness_operation) + (p) * pfa_offset_right_sub_fixed_result_witness_operationoperationresultcongruence)))))))))))))))) /\ ((forall pfrep_power_sub_fixed_result_witness_output pfrep_left_sub_fixed_result_witness_output pfrep_right_sub_fixed_result_witness_output. ((exists pfrep_position_sub_fixed_result_witness_outputfirst. ((pfrep_position_sub_fixed_result_witness_outputfirst+S (pfrep_power_sub_fixed_result_witness_output)=(pfaa_length_sub_fixed_result)) /\ ((((exists ff_h_pfp_sub_fixed_result_witness_outputfirstentry. ff_h_pfp_sub_fixed_result_witness_outputfirstentry + S (pfrep_left_sub_fixed_result_witness_output) = S ((S (pfrep_position_sub_fixed_result_witness_outputfirst)) * pfaa_sum_c_sub_fixed_result)) /\ exists ff_q_pfp_sub_fixed_result_witness_outputfirstentry. pfaa_sum_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_outputfirstentry * S ((S (pfrep_position_sub_fixed_result_witness_outputfirst)) * pfaa_sum_c_sub_fixed_result) + (pfrep_left_sub_fixed_result_witness_output)))))) \/ (((exists pfrep_gap_sub_fixed_result_witness_outputfirstoutside. pfrep_gap_sub_fixed_result_witness_outputfirstoutside+(pfaa_length_sub_fixed_result)=(pfrep_power_sub_fixed_result_witness_output)) /\ (((pfrep_left_sub_fixed_result_witness_output)=0))))) -> ((exists pfrep_position_sub_fixed_result_witness_outputsecond. ((pfrep_position_sub_fixed_result_witness_outputsecond+S (pfrep_power_sub_fixed_result_witness_output)=(K)) /\ ((((exists ff_h_pfp_sub_fixed_result_witness_outputsecondentry. ff_h_pfp_sub_fixed_result_witness_outputsecondentry + S (pfrep_right_sub_fixed_result_witness_output) = S ((S (pfrep_position_sub_fixed_result_witness_outputsecond)) * ac)) /\ exists ff_q_pfp_sub_fixed_result_witness_outputsecondentry. ab = ff_q_pfp_sub_fixed_result_witness_outputsecondentry * S ((S (pfrep_position_sub_fixed_result_witness_outputsecond)) * ac) + (pfrep_right_sub_fixed_result_witness_output)))))) \/ (((exists pfrep_gap_sub_fixed_result_witness_outputsecondoutside. pfrep_gap_sub_fixed_result_witness_outputsecondoutside+(K)=(pfrep_power_sub_fixed_result_witness_output)) /\ (((pfrep_right_sub_fixed_result_witness_output)=0))))) -> pfrep_left_sub_fixed_result_witness_output=pfrep_right_sub_fixed_result_witness_output)))))))))))))