integer_span_signed_product_equal_coefficients

Exact expanded first-order arithmetic statement

forall ab ac db dc b c w r. exists pb pc. (exists ff_pp_mcp_smatrix_ics_equal_coefficients ff_pps_mcp_smatrix_ics_equal_coefficients ff_nn_mcp_smatrix_ics_equal_coefficients ff_nns_mcp_smatrix_ics_equal_coefficients ff_pn_mcp_smatrix_ics_equal_coefficients ff_pns_mcp_smatrix_ics_equal_coefficients ff_np_mcp_smatrix_ics_equal_coefficients ff_nps_mcp_smatrix_ics_equal_coefficients. ((forall ff_index_mcp_prefix_ics_equal_coefficients_pp. (exists mcp_gap_ics_equal_coefficients_pp_index. mcp_gap_ics_equal_coefficients_pp_index + S (ff_index_mcp_prefix_ics_equal_coefficients_pp) = ((r) * (1))) -> exists ff_row_mcp_prefix_ics_equal_coefficients_pp ff_column_mcp_prefix_ics_equal_coefficients_pp ff_value_mcp_prefix_ics_equal_coefficients_pp. ((ff_index_mcp_prefix_ics_equal_coefficients_pp) = (1) * ff_row_mcp_prefix_ics_equal_coefficients_pp + ff_column_mcp_prefix_ics_equal_coefficients_pp /\ ((exists mcp_gap_ics_equal_coefficients_pp_column. mcp_gap_ics_equal_coefficients_pp_column + S (ff_column_mcp_prefix_ics_equal_coefficients_pp) = (1)) /\ ((exists ff_left_mcp_cell_ics_equal_coefficients_pp_cell ff_left_scale_mcp_cell_ics_equal_coefficients_pp_cell ff_right_mcp_cell_ics_equal_coefficients_pp_cell ff_right_scale_mcp_cell_ics_equal_coefficients_pp_cell. ((forall ff_index_mcp_ics_equal_coefficients_pp_cell_row ff_source_mcp_ics_equal_coefficients_pp_cell_row ff_target_mcp_ics_equal_coefficients_pp_cell_row. (exists mcp_gap_ics_equal_coefficients_pp_cell_row_bound. mcp_gap_ics_equal_coefficients_pp_cell_row_bound + S (ff_index_mcp_ics_equal_coefficients_pp_cell_row) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_pp_cell_row_source. fs_h_mcp_ics_equal_coefficients_pp_cell_row_source + S (ff_source_mcp_ics_equal_coefficients_pp_cell_row) = S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_pp) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_pp_cell_row)) * ac)) /\ exists fs_q_mcp_ics_equal_coefficients_pp_cell_row_source. ab = fs_q_mcp_ics_equal_coefficients_pp_cell_row_source * S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_pp) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_pp_cell_row)) * ac) + (ff_source_mcp_ics_equal_coefficients_pp_cell_row))) -> (((exists fs_h_mcp_ics_equal_coefficients_pp_cell_row_target. fs_h_mcp_ics_equal_coefficients_pp_cell_row_target + S (ff_target_mcp_ics_equal_coefficients_pp_cell_row) = S ((S (ff_index_mcp_ics_equal_coefficients_pp_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pp_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_pp_cell_row_target. ff_left_mcp_cell_ics_equal_coefficients_pp_cell = fs_q_mcp_ics_equal_coefficients_pp_cell_row_target * S ((S (ff_index_mcp_ics_equal_coefficients_pp_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pp_cell) + (ff_target_mcp_ics_equal_coefficients_pp_cell_row))) -> ff_target_mcp_ics_equal_coefficients_pp_cell_row = ff_source_mcp_ics_equal_coefficients_pp_cell_row) /\ ((forall ff_index_mcp_ics_equal_coefficients_pp_cell_column ff_source_mcp_ics_equal_coefficients_pp_cell_column ff_target_mcp_ics_equal_coefficients_pp_cell_column. (exists mcp_gap_ics_equal_coefficients_pp_cell_column_bound. mcp_gap_ics_equal_coefficients_pp_cell_column_bound + S (ff_index_mcp_ics_equal_coefficients_pp_cell_column) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_pp_cell_column_source. fs_h_mcp_ics_equal_coefficients_pp_cell_column_source + S (ff_source_mcp_ics_equal_coefficients_pp_cell_column) = S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_pp) + (1) * ff_index_mcp_ics_equal_coefficients_pp_cell_column)) * c)) /\ exists fs_q_mcp_ics_equal_coefficients_pp_cell_column_source. b = fs_q_mcp_ics_equal_coefficients_pp_cell_column_source * S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_pp) + (1) * ff_index_mcp_ics_equal_coefficients_pp_cell_column)) * c) + (ff_source_mcp_ics_equal_coefficients_pp_cell_column))) -> (((exists fs_h_mcp_ics_equal_coefficients_pp_cell_column_target. fs_h_mcp_ics_equal_coefficients_pp_cell_column_target + S (ff_target_mcp_ics_equal_coefficients_pp_cell_column) = S ((S (ff_index_mcp_ics_equal_coefficients_pp_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pp_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_pp_cell_column_target. ff_right_mcp_cell_ics_equal_coefficients_pp_cell = fs_q_mcp_ics_equal_coefficients_pp_cell_column_target * S ((S (ff_index_mcp_ics_equal_coefficients_pp_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pp_cell) + (ff_target_mcp_ics_equal_coefficients_pp_cell_column))) -> ff_target_mcp_ics_equal_coefficients_pp_cell_column = ff_source_mcp_ics_equal_coefficients_pp_cell_column) /\ (exists ff_code_dot_mcp_ics_equal_coefficients_pp_cell_dot ff_scale_dot_mcp_ics_equal_coefficients_pp_cell_dot. ((forall fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise fpmp_left_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise fpmp_right_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise fpmp_target_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise. (exists fpmp_gap_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise. fpmp_gap_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise + S fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise = w) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_left. ff_h_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_left + S (fpmp_left_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pp_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_left. ff_left_mcp_cell_ics_equal_coefficients_pp_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_left * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pp_cell) + (fpmp_left_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_right. ff_h_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_right + S (fpmp_right_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pp_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_right. ff_right_mcp_cell_ics_equal_coefficients_pp_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_right * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pp_cell) + (fpmp_right_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_target. ff_h_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_target + S (fpmp_target_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_pp_cell_dot)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_target. ff_code_dot_mcp_ics_equal_coefficients_pp_cell_dot = ff_q_fpmp_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise_target * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_pp_cell_dot) + (fpmp_target_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise))) -> fpmp_target_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise = fpmp_left_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise * fpmp_right_dot_mcp_ics_equal_coefficients_pp_cell_dot_pointwise) /\ (exists ff_u_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_start. ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_start + S (0) = S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_start. ff_u_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_start * S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) + (0))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_terminal. ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_terminal + S (ff_value_mcp_prefix_ics_equal_coefficients_pp) = S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_terminal. ff_u_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_terminal * S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) + (ff_value_mcp_prefix_ics_equal_coefficients_pp))) /\ forall ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum. (exists ff_lt_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_bound. ff_lt_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_bound + S ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum = w) -> exists ff_a_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum ff_r_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum ff_s_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_summand. ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_summand + S (ff_a_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_pp_cell_dot)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_summand. ff_code_dot_mcp_ics_equal_coefficients_pp_cell_dot = ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_summand * S ((S (ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_pp_cell_dot) + (ff_a_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_partial. ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_partial + S (ff_r_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_partial. ff_u_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_partial * S ((S (ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) + (ff_r_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_successor. ff_h_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_successor + S (ff_s_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) = S ((S (S ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_successor. ff_u_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum_successor * S ((S (S ff_i_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum) + (ff_s_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum))) /\ ff_s_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum = ff_r_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum + ff_a_dot_mcp_ics_equal_coefficients_pp_cell_dot_sum))))))))))) /\ (((exists fs_h_mcp_ics_equal_coefficients_pp_entry. fs_h_mcp_ics_equal_coefficients_pp_entry + S (ff_value_mcp_prefix_ics_equal_coefficients_pp) = S ((S (ff_index_mcp_prefix_ics_equal_coefficients_pp)) * ff_pps_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_pp_entry. ff_pp_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_pp_entry * S ((S (ff_index_mcp_prefix_ics_equal_coefficients_pp)) * ff_pps_mcp_smatrix_ics_equal_coefficients) + (ff_value_mcp_prefix_ics_equal_coefficients_pp))))))) /\ ((forall ff_index_mcp_prefix_ics_equal_coefficients_nn. (exists mcp_gap_ics_equal_coefficients_nn_index. mcp_gap_ics_equal_coefficients_nn_index + S (ff_index_mcp_prefix_ics_equal_coefficients_nn) = ((r) * (1))) -> exists ff_row_mcp_prefix_ics_equal_coefficients_nn ff_column_mcp_prefix_ics_equal_coefficients_nn ff_value_mcp_prefix_ics_equal_coefficients_nn. ((ff_index_mcp_prefix_ics_equal_coefficients_nn) = (1) * ff_row_mcp_prefix_ics_equal_coefficients_nn + ff_column_mcp_prefix_ics_equal_coefficients_nn /\ ((exists mcp_gap_ics_equal_coefficients_nn_column. mcp_gap_ics_equal_coefficients_nn_column + S (ff_column_mcp_prefix_ics_equal_coefficients_nn) = (1)) /\ ((exists ff_left_mcp_cell_ics_equal_coefficients_nn_cell ff_left_scale_mcp_cell_ics_equal_coefficients_nn_cell ff_right_mcp_cell_ics_equal_coefficients_nn_cell ff_right_scale_mcp_cell_ics_equal_coefficients_nn_cell. ((forall ff_index_mcp_ics_equal_coefficients_nn_cell_row ff_source_mcp_ics_equal_coefficients_nn_cell_row ff_target_mcp_ics_equal_coefficients_nn_cell_row. (exists mcp_gap_ics_equal_coefficients_nn_cell_row_bound. mcp_gap_ics_equal_coefficients_nn_cell_row_bound + S (ff_index_mcp_ics_equal_coefficients_nn_cell_row) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_nn_cell_row_source. fs_h_mcp_ics_equal_coefficients_nn_cell_row_source + S (ff_source_mcp_ics_equal_coefficients_nn_cell_row) = S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_nn) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_nn_cell_row)) * dc)) /\ exists fs_q_mcp_ics_equal_coefficients_nn_cell_row_source. db = fs_q_mcp_ics_equal_coefficients_nn_cell_row_source * S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_nn) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_nn_cell_row)) * dc) + (ff_source_mcp_ics_equal_coefficients_nn_cell_row))) -> (((exists fs_h_mcp_ics_equal_coefficients_nn_cell_row_target. fs_h_mcp_ics_equal_coefficients_nn_cell_row_target + S (ff_target_mcp_ics_equal_coefficients_nn_cell_row) = S ((S (ff_index_mcp_ics_equal_coefficients_nn_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_nn_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_nn_cell_row_target. ff_left_mcp_cell_ics_equal_coefficients_nn_cell = fs_q_mcp_ics_equal_coefficients_nn_cell_row_target * S ((S (ff_index_mcp_ics_equal_coefficients_nn_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_nn_cell) + (ff_target_mcp_ics_equal_coefficients_nn_cell_row))) -> ff_target_mcp_ics_equal_coefficients_nn_cell_row = ff_source_mcp_ics_equal_coefficients_nn_cell_row) /\ ((forall ff_index_mcp_ics_equal_coefficients_nn_cell_column ff_source_mcp_ics_equal_coefficients_nn_cell_column ff_target_mcp_ics_equal_coefficients_nn_cell_column. (exists mcp_gap_ics_equal_coefficients_nn_cell_column_bound. mcp_gap_ics_equal_coefficients_nn_cell_column_bound + S (ff_index_mcp_ics_equal_coefficients_nn_cell_column) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_nn_cell_column_source. fs_h_mcp_ics_equal_coefficients_nn_cell_column_source + S (ff_source_mcp_ics_equal_coefficients_nn_cell_column) = S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_nn) + (1) * ff_index_mcp_ics_equal_coefficients_nn_cell_column)) * c)) /\ exists fs_q_mcp_ics_equal_coefficients_nn_cell_column_source. b = fs_q_mcp_ics_equal_coefficients_nn_cell_column_source * S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_nn) + (1) * ff_index_mcp_ics_equal_coefficients_nn_cell_column)) * c) + (ff_source_mcp_ics_equal_coefficients_nn_cell_column))) -> (((exists fs_h_mcp_ics_equal_coefficients_nn_cell_column_target. fs_h_mcp_ics_equal_coefficients_nn_cell_column_target + S (ff_target_mcp_ics_equal_coefficients_nn_cell_column) = S ((S (ff_index_mcp_ics_equal_coefficients_nn_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_nn_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_nn_cell_column_target. ff_right_mcp_cell_ics_equal_coefficients_nn_cell = fs_q_mcp_ics_equal_coefficients_nn_cell_column_target * S ((S (ff_index_mcp_ics_equal_coefficients_nn_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_nn_cell) + (ff_target_mcp_ics_equal_coefficients_nn_cell_column))) -> ff_target_mcp_ics_equal_coefficients_nn_cell_column = ff_source_mcp_ics_equal_coefficients_nn_cell_column) /\ (exists ff_code_dot_mcp_ics_equal_coefficients_nn_cell_dot ff_scale_dot_mcp_ics_equal_coefficients_nn_cell_dot. ((forall fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise fpmp_left_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise fpmp_right_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise fpmp_target_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise. (exists fpmp_gap_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise. fpmp_gap_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise + S fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise = w) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_left. ff_h_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_left + S (fpmp_left_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_nn_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_left. ff_left_mcp_cell_ics_equal_coefficients_nn_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_left * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_nn_cell) + (fpmp_left_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_right. ff_h_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_right + S (fpmp_right_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_nn_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_right. ff_right_mcp_cell_ics_equal_coefficients_nn_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_right * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_nn_cell) + (fpmp_right_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_target. ff_h_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_target + S (fpmp_target_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_nn_cell_dot)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_target. ff_code_dot_mcp_ics_equal_coefficients_nn_cell_dot = ff_q_fpmp_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise_target * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_nn_cell_dot) + (fpmp_target_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise))) -> fpmp_target_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise = fpmp_left_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise * fpmp_right_dot_mcp_ics_equal_coefficients_nn_cell_dot_pointwise) /\ (exists ff_u_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_start. ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_start + S (0) = S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_start. ff_u_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_start * S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) + (0))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_terminal. ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_terminal + S (ff_value_mcp_prefix_ics_equal_coefficients_nn) = S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_terminal. ff_u_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_terminal * S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) + (ff_value_mcp_prefix_ics_equal_coefficients_nn))) /\ forall ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum. (exists ff_lt_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_bound. ff_lt_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_bound + S ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum = w) -> exists ff_a_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum ff_r_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum ff_s_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_summand. ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_summand + S (ff_a_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_nn_cell_dot)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_summand. ff_code_dot_mcp_ics_equal_coefficients_nn_cell_dot = ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_summand * S ((S (ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_nn_cell_dot) + (ff_a_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_partial. ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_partial + S (ff_r_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_partial. ff_u_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_partial * S ((S (ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) + (ff_r_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_successor. ff_h_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_successor + S (ff_s_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) = S ((S (S ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_successor. ff_u_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum_successor * S ((S (S ff_i_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum) + (ff_s_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum))) /\ ff_s_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum = ff_r_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum + ff_a_dot_mcp_ics_equal_coefficients_nn_cell_dot_sum))))))))))) /\ (((exists fs_h_mcp_ics_equal_coefficients_nn_entry. fs_h_mcp_ics_equal_coefficients_nn_entry + S (ff_value_mcp_prefix_ics_equal_coefficients_nn) = S ((S (ff_index_mcp_prefix_ics_equal_coefficients_nn)) * ff_nns_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_nn_entry. ff_nn_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_nn_entry * S ((S (ff_index_mcp_prefix_ics_equal_coefficients_nn)) * ff_nns_mcp_smatrix_ics_equal_coefficients) + (ff_value_mcp_prefix_ics_equal_coefficients_nn))))))) /\ ((forall ff_index_mcp_prefix_ics_equal_coefficients_pn. (exists mcp_gap_ics_equal_coefficients_pn_index. mcp_gap_ics_equal_coefficients_pn_index + S (ff_index_mcp_prefix_ics_equal_coefficients_pn) = ((r) * (1))) -> exists ff_row_mcp_prefix_ics_equal_coefficients_pn ff_column_mcp_prefix_ics_equal_coefficients_pn ff_value_mcp_prefix_ics_equal_coefficients_pn. ((ff_index_mcp_prefix_ics_equal_coefficients_pn) = (1) * ff_row_mcp_prefix_ics_equal_coefficients_pn + ff_column_mcp_prefix_ics_equal_coefficients_pn /\ ((exists mcp_gap_ics_equal_coefficients_pn_column. mcp_gap_ics_equal_coefficients_pn_column + S (ff_column_mcp_prefix_ics_equal_coefficients_pn) = (1)) /\ ((exists ff_left_mcp_cell_ics_equal_coefficients_pn_cell ff_left_scale_mcp_cell_ics_equal_coefficients_pn_cell ff_right_mcp_cell_ics_equal_coefficients_pn_cell ff_right_scale_mcp_cell_ics_equal_coefficients_pn_cell. ((forall ff_index_mcp_ics_equal_coefficients_pn_cell_row ff_source_mcp_ics_equal_coefficients_pn_cell_row ff_target_mcp_ics_equal_coefficients_pn_cell_row. (exists mcp_gap_ics_equal_coefficients_pn_cell_row_bound. mcp_gap_ics_equal_coefficients_pn_cell_row_bound + S (ff_index_mcp_ics_equal_coefficients_pn_cell_row) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_pn_cell_row_source. fs_h_mcp_ics_equal_coefficients_pn_cell_row_source + S (ff_source_mcp_ics_equal_coefficients_pn_cell_row) = S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_pn) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_pn_cell_row)) * ac)) /\ exists fs_q_mcp_ics_equal_coefficients_pn_cell_row_source. ab = fs_q_mcp_ics_equal_coefficients_pn_cell_row_source * S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_pn) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_pn_cell_row)) * ac) + (ff_source_mcp_ics_equal_coefficients_pn_cell_row))) -> (((exists fs_h_mcp_ics_equal_coefficients_pn_cell_row_target. fs_h_mcp_ics_equal_coefficients_pn_cell_row_target + S (ff_target_mcp_ics_equal_coefficients_pn_cell_row) = S ((S (ff_index_mcp_ics_equal_coefficients_pn_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pn_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_pn_cell_row_target. ff_left_mcp_cell_ics_equal_coefficients_pn_cell = fs_q_mcp_ics_equal_coefficients_pn_cell_row_target * S ((S (ff_index_mcp_ics_equal_coefficients_pn_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pn_cell) + (ff_target_mcp_ics_equal_coefficients_pn_cell_row))) -> ff_target_mcp_ics_equal_coefficients_pn_cell_row = ff_source_mcp_ics_equal_coefficients_pn_cell_row) /\ ((forall ff_index_mcp_ics_equal_coefficients_pn_cell_column ff_source_mcp_ics_equal_coefficients_pn_cell_column ff_target_mcp_ics_equal_coefficients_pn_cell_column. (exists mcp_gap_ics_equal_coefficients_pn_cell_column_bound. mcp_gap_ics_equal_coefficients_pn_cell_column_bound + S (ff_index_mcp_ics_equal_coefficients_pn_cell_column) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_pn_cell_column_source. fs_h_mcp_ics_equal_coefficients_pn_cell_column_source + S (ff_source_mcp_ics_equal_coefficients_pn_cell_column) = S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_pn) + (1) * ff_index_mcp_ics_equal_coefficients_pn_cell_column)) * c)) /\ exists fs_q_mcp_ics_equal_coefficients_pn_cell_column_source. b = fs_q_mcp_ics_equal_coefficients_pn_cell_column_source * S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_pn) + (1) * ff_index_mcp_ics_equal_coefficients_pn_cell_column)) * c) + (ff_source_mcp_ics_equal_coefficients_pn_cell_column))) -> (((exists fs_h_mcp_ics_equal_coefficients_pn_cell_column_target. fs_h_mcp_ics_equal_coefficients_pn_cell_column_target + S (ff_target_mcp_ics_equal_coefficients_pn_cell_column) = S ((S (ff_index_mcp_ics_equal_coefficients_pn_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pn_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_pn_cell_column_target. ff_right_mcp_cell_ics_equal_coefficients_pn_cell = fs_q_mcp_ics_equal_coefficients_pn_cell_column_target * S ((S (ff_index_mcp_ics_equal_coefficients_pn_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pn_cell) + (ff_target_mcp_ics_equal_coefficients_pn_cell_column))) -> ff_target_mcp_ics_equal_coefficients_pn_cell_column = ff_source_mcp_ics_equal_coefficients_pn_cell_column) /\ (exists ff_code_dot_mcp_ics_equal_coefficients_pn_cell_dot ff_scale_dot_mcp_ics_equal_coefficients_pn_cell_dot. ((forall fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise fpmp_left_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise fpmp_right_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise fpmp_target_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise. (exists fpmp_gap_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise. fpmp_gap_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise + S fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise = w) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_left. ff_h_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_left + S (fpmp_left_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pn_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_left. ff_left_mcp_cell_ics_equal_coefficients_pn_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_left * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_pn_cell) + (fpmp_left_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_right. ff_h_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_right + S (fpmp_right_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pn_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_right. ff_right_mcp_cell_ics_equal_coefficients_pn_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_right * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_pn_cell) + (fpmp_right_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_target. ff_h_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_target + S (fpmp_target_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_pn_cell_dot)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_target. ff_code_dot_mcp_ics_equal_coefficients_pn_cell_dot = ff_q_fpmp_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise_target * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_pn_cell_dot) + (fpmp_target_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise))) -> fpmp_target_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise = fpmp_left_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise * fpmp_right_dot_mcp_ics_equal_coefficients_pn_cell_dot_pointwise) /\ (exists ff_u_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_start. ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_start + S (0) = S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_start. ff_u_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_start * S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) + (0))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_terminal. ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_terminal + S (ff_value_mcp_prefix_ics_equal_coefficients_pn) = S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_terminal. ff_u_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_terminal * S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) + (ff_value_mcp_prefix_ics_equal_coefficients_pn))) /\ forall ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum. (exists ff_lt_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_bound. ff_lt_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_bound + S ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum = w) -> exists ff_a_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum ff_r_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum ff_s_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_summand. ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_summand + S (ff_a_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_pn_cell_dot)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_summand. ff_code_dot_mcp_ics_equal_coefficients_pn_cell_dot = ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_summand * S ((S (ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_pn_cell_dot) + (ff_a_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_partial. ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_partial + S (ff_r_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_partial. ff_u_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_partial * S ((S (ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) + (ff_r_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_successor. ff_h_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_successor + S (ff_s_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) = S ((S (S ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_successor. ff_u_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum_successor * S ((S (S ff_i_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum) + (ff_s_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum))) /\ ff_s_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum = ff_r_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum + ff_a_dot_mcp_ics_equal_coefficients_pn_cell_dot_sum))))))))))) /\ (((exists fs_h_mcp_ics_equal_coefficients_pn_entry. fs_h_mcp_ics_equal_coefficients_pn_entry + S (ff_value_mcp_prefix_ics_equal_coefficients_pn) = S ((S (ff_index_mcp_prefix_ics_equal_coefficients_pn)) * ff_pns_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_pn_entry. ff_pn_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_pn_entry * S ((S (ff_index_mcp_prefix_ics_equal_coefficients_pn)) * ff_pns_mcp_smatrix_ics_equal_coefficients) + (ff_value_mcp_prefix_ics_equal_coefficients_pn))))))) /\ ((forall ff_index_mcp_prefix_ics_equal_coefficients_np. (exists mcp_gap_ics_equal_coefficients_np_index. mcp_gap_ics_equal_coefficients_np_index + S (ff_index_mcp_prefix_ics_equal_coefficients_np) = ((r) * (1))) -> exists ff_row_mcp_prefix_ics_equal_coefficients_np ff_column_mcp_prefix_ics_equal_coefficients_np ff_value_mcp_prefix_ics_equal_coefficients_np. ((ff_index_mcp_prefix_ics_equal_coefficients_np) = (1) * ff_row_mcp_prefix_ics_equal_coefficients_np + ff_column_mcp_prefix_ics_equal_coefficients_np /\ ((exists mcp_gap_ics_equal_coefficients_np_column. mcp_gap_ics_equal_coefficients_np_column + S (ff_column_mcp_prefix_ics_equal_coefficients_np) = (1)) /\ ((exists ff_left_mcp_cell_ics_equal_coefficients_np_cell ff_left_scale_mcp_cell_ics_equal_coefficients_np_cell ff_right_mcp_cell_ics_equal_coefficients_np_cell ff_right_scale_mcp_cell_ics_equal_coefficients_np_cell. ((forall ff_index_mcp_ics_equal_coefficients_np_cell_row ff_source_mcp_ics_equal_coefficients_np_cell_row ff_target_mcp_ics_equal_coefficients_np_cell_row. (exists mcp_gap_ics_equal_coefficients_np_cell_row_bound. mcp_gap_ics_equal_coefficients_np_cell_row_bound + S (ff_index_mcp_ics_equal_coefficients_np_cell_row) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_np_cell_row_source. fs_h_mcp_ics_equal_coefficients_np_cell_row_source + S (ff_source_mcp_ics_equal_coefficients_np_cell_row) = S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_np) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_np_cell_row)) * dc)) /\ exists fs_q_mcp_ics_equal_coefficients_np_cell_row_source. db = fs_q_mcp_ics_equal_coefficients_np_cell_row_source * S ((S (((ff_row_mcp_prefix_ics_equal_coefficients_np) * (w)) + (1) * ff_index_mcp_ics_equal_coefficients_np_cell_row)) * dc) + (ff_source_mcp_ics_equal_coefficients_np_cell_row))) -> (((exists fs_h_mcp_ics_equal_coefficients_np_cell_row_target. fs_h_mcp_ics_equal_coefficients_np_cell_row_target + S (ff_target_mcp_ics_equal_coefficients_np_cell_row) = S ((S (ff_index_mcp_ics_equal_coefficients_np_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_np_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_np_cell_row_target. ff_left_mcp_cell_ics_equal_coefficients_np_cell = fs_q_mcp_ics_equal_coefficients_np_cell_row_target * S ((S (ff_index_mcp_ics_equal_coefficients_np_cell_row)) * ff_left_scale_mcp_cell_ics_equal_coefficients_np_cell) + (ff_target_mcp_ics_equal_coefficients_np_cell_row))) -> ff_target_mcp_ics_equal_coefficients_np_cell_row = ff_source_mcp_ics_equal_coefficients_np_cell_row) /\ ((forall ff_index_mcp_ics_equal_coefficients_np_cell_column ff_source_mcp_ics_equal_coefficients_np_cell_column ff_target_mcp_ics_equal_coefficients_np_cell_column. (exists mcp_gap_ics_equal_coefficients_np_cell_column_bound. mcp_gap_ics_equal_coefficients_np_cell_column_bound + S (ff_index_mcp_ics_equal_coefficients_np_cell_column) = (w)) -> (((exists fs_h_mcp_ics_equal_coefficients_np_cell_column_source. fs_h_mcp_ics_equal_coefficients_np_cell_column_source + S (ff_source_mcp_ics_equal_coefficients_np_cell_column) = S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_np) + (1) * ff_index_mcp_ics_equal_coefficients_np_cell_column)) * c)) /\ exists fs_q_mcp_ics_equal_coefficients_np_cell_column_source. b = fs_q_mcp_ics_equal_coefficients_np_cell_column_source * S ((S ((ff_column_mcp_prefix_ics_equal_coefficients_np) + (1) * ff_index_mcp_ics_equal_coefficients_np_cell_column)) * c) + (ff_source_mcp_ics_equal_coefficients_np_cell_column))) -> (((exists fs_h_mcp_ics_equal_coefficients_np_cell_column_target. fs_h_mcp_ics_equal_coefficients_np_cell_column_target + S (ff_target_mcp_ics_equal_coefficients_np_cell_column) = S ((S (ff_index_mcp_ics_equal_coefficients_np_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_np_cell)) /\ exists fs_q_mcp_ics_equal_coefficients_np_cell_column_target. ff_right_mcp_cell_ics_equal_coefficients_np_cell = fs_q_mcp_ics_equal_coefficients_np_cell_column_target * S ((S (ff_index_mcp_ics_equal_coefficients_np_cell_column)) * ff_right_scale_mcp_cell_ics_equal_coefficients_np_cell) + (ff_target_mcp_ics_equal_coefficients_np_cell_column))) -> ff_target_mcp_ics_equal_coefficients_np_cell_column = ff_source_mcp_ics_equal_coefficients_np_cell_column) /\ (exists ff_code_dot_mcp_ics_equal_coefficients_np_cell_dot ff_scale_dot_mcp_ics_equal_coefficients_np_cell_dot. ((forall fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise fpmp_left_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise fpmp_right_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise fpmp_target_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise. (exists fpmp_gap_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise. fpmp_gap_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise + S fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise = w) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_left. ff_h_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_left + S (fpmp_left_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_np_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_left. ff_left_mcp_cell_ics_equal_coefficients_np_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_left * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise)) * ff_left_scale_mcp_cell_ics_equal_coefficients_np_cell) + (fpmp_left_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_right. ff_h_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_right + S (fpmp_right_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_np_cell)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_right. ff_right_mcp_cell_ics_equal_coefficients_np_cell = ff_q_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_right * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise)) * ff_right_scale_mcp_cell_ics_equal_coefficients_np_cell) + (fpmp_right_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_target. ff_h_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_target + S (fpmp_target_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_np_cell_dot)) /\ exists ff_q_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_target. ff_code_dot_mcp_ics_equal_coefficients_np_cell_dot = ff_q_fpmp_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise_target * S ((S (fpmp_index_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise)) * ff_scale_dot_mcp_ics_equal_coefficients_np_cell_dot) + (fpmp_target_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise))) -> fpmp_target_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise = fpmp_left_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise * fpmp_right_dot_mcp_ics_equal_coefficients_np_cell_dot_pointwise) /\ (exists ff_u_dot_mcp_ics_equal_coefficients_np_cell_dot_sum ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_start. ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_start + S (0) = S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_start. ff_u_dot_mcp_ics_equal_coefficients_np_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_start * S ((S (0)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) + (0))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_terminal. ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_terminal + S (ff_value_mcp_prefix_ics_equal_coefficients_np) = S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_terminal. ff_u_dot_mcp_ics_equal_coefficients_np_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_terminal * S ((S (w)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) + (ff_value_mcp_prefix_ics_equal_coefficients_np))) /\ forall ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum. (exists ff_lt_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_bound. ff_lt_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_bound + S ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum = w) -> exists ff_a_dot_mcp_ics_equal_coefficients_np_cell_dot_sum ff_r_dot_mcp_ics_equal_coefficients_np_cell_dot_sum ff_s_dot_mcp_ics_equal_coefficients_np_cell_dot_sum. ((((exists ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_summand. ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_summand + S (ff_a_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_np_cell_dot)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_summand. ff_code_dot_mcp_ics_equal_coefficients_np_cell_dot = ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_summand * S ((S (ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) * ff_scale_dot_mcp_ics_equal_coefficients_np_cell_dot) + (ff_a_dot_mcp_ics_equal_coefficients_np_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_partial. ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_partial + S (ff_r_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) = S ((S (ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_partial. ff_u_dot_mcp_ics_equal_coefficients_np_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_partial * S ((S (ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) + (ff_r_dot_mcp_ics_equal_coefficients_np_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_successor. ff_h_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_successor + S (ff_s_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) = S ((S (S ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) /\ exists ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_successor. ff_u_dot_mcp_ics_equal_coefficients_np_cell_dot_sum = ff_q_dot_mcp_ics_equal_coefficients_np_cell_dot_sum_successor * S ((S (S ff_i_dot_mcp_ics_equal_coefficients_np_cell_dot_sum)) * ff_v_dot_mcp_ics_equal_coefficients_np_cell_dot_sum) + (ff_s_dot_mcp_ics_equal_coefficients_np_cell_dot_sum))) /\ ff_s_dot_mcp_ics_equal_coefficients_np_cell_dot_sum = ff_r_dot_mcp_ics_equal_coefficients_np_cell_dot_sum + ff_a_dot_mcp_ics_equal_coefficients_np_cell_dot_sum))))))))))) /\ (((exists fs_h_mcp_ics_equal_coefficients_np_entry. fs_h_mcp_ics_equal_coefficients_np_entry + S (ff_value_mcp_prefix_ics_equal_coefficients_np) = S ((S (ff_index_mcp_prefix_ics_equal_coefficients_np)) * ff_nps_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_np_entry. ff_np_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_np_entry * S ((S (ff_index_mcp_prefix_ics_equal_coefficients_np)) * ff_nps_mcp_smatrix_ics_equal_coefficients) + (ff_value_mcp_prefix_ics_equal_coefficients_np))))))) /\ ((forall ff_index_mcp_add_ics_equal_coefficients_positive ff_left_mcp_add_ics_equal_coefficients_positive ff_right_mcp_add_ics_equal_coefficients_positive ff_target_mcp_add_ics_equal_coefficients_positive. (exists mcp_gap_ics_equal_coefficients_positive_bound. mcp_gap_ics_equal_coefficients_positive_bound + S (ff_index_mcp_add_ics_equal_coefficients_positive) = ((r) * (1))) -> (((exists fs_h_mcp_ics_equal_coefficients_positive_left. fs_h_mcp_ics_equal_coefficients_positive_left + S (ff_left_mcp_add_ics_equal_coefficients_positive) = S ((S (ff_index_mcp_add_ics_equal_coefficients_positive)) * ff_pps_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_positive_left. ff_pp_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_positive_left * S ((S (ff_index_mcp_add_ics_equal_coefficients_positive)) * ff_pps_mcp_smatrix_ics_equal_coefficients) + (ff_left_mcp_add_ics_equal_coefficients_positive))) -> (((exists fs_h_mcp_ics_equal_coefficients_positive_right. fs_h_mcp_ics_equal_coefficients_positive_right + S (ff_right_mcp_add_ics_equal_coefficients_positive) = S ((S (ff_index_mcp_add_ics_equal_coefficients_positive)) * ff_nns_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_positive_right. ff_nn_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_positive_right * S ((S (ff_index_mcp_add_ics_equal_coefficients_positive)) * ff_nns_mcp_smatrix_ics_equal_coefficients) + (ff_right_mcp_add_ics_equal_coefficients_positive))) -> (((exists fs_h_mcp_ics_equal_coefficients_positive_target. fs_h_mcp_ics_equal_coefficients_positive_target + S (ff_target_mcp_add_ics_equal_coefficients_positive) = S ((S (ff_index_mcp_add_ics_equal_coefficients_positive)) * pc)) /\ exists fs_q_mcp_ics_equal_coefficients_positive_target. pb = fs_q_mcp_ics_equal_coefficients_positive_target * S ((S (ff_index_mcp_add_ics_equal_coefficients_positive)) * pc) + (ff_target_mcp_add_ics_equal_coefficients_positive))) -> ff_target_mcp_add_ics_equal_coefficients_positive = ff_left_mcp_add_ics_equal_coefficients_positive + ff_right_mcp_add_ics_equal_coefficients_positive) /\ (forall ff_index_mcp_add_ics_equal_coefficients_negative ff_left_mcp_add_ics_equal_coefficients_negative ff_right_mcp_add_ics_equal_coefficients_negative ff_target_mcp_add_ics_equal_coefficients_negative. (exists mcp_gap_ics_equal_coefficients_negative_bound. mcp_gap_ics_equal_coefficients_negative_bound + S (ff_index_mcp_add_ics_equal_coefficients_negative) = ((r) * (1))) -> (((exists fs_h_mcp_ics_equal_coefficients_negative_left. fs_h_mcp_ics_equal_coefficients_negative_left + S (ff_left_mcp_add_ics_equal_coefficients_negative) = S ((S (ff_index_mcp_add_ics_equal_coefficients_negative)) * ff_pns_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_negative_left. ff_pn_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_negative_left * S ((S (ff_index_mcp_add_ics_equal_coefficients_negative)) * ff_pns_mcp_smatrix_ics_equal_coefficients) + (ff_left_mcp_add_ics_equal_coefficients_negative))) -> (((exists fs_h_mcp_ics_equal_coefficients_negative_right. fs_h_mcp_ics_equal_coefficients_negative_right + S (ff_right_mcp_add_ics_equal_coefficients_negative) = S ((S (ff_index_mcp_add_ics_equal_coefficients_negative)) * ff_nps_mcp_smatrix_ics_equal_coefficients)) /\ exists fs_q_mcp_ics_equal_coefficients_negative_right. ff_np_mcp_smatrix_ics_equal_coefficients = fs_q_mcp_ics_equal_coefficients_negative_right * S ((S (ff_index_mcp_add_ics_equal_coefficients_negative)) * ff_nps_mcp_smatrix_ics_equal_coefficients) + (ff_right_mcp_add_ics_equal_coefficients_negative))) -> (((exists fs_h_mcp_ics_equal_coefficients_negative_target. fs_h_mcp_ics_equal_coefficients_negative_target + S (ff_target_mcp_add_ics_equal_coefficients_negative) = S ((S (ff_index_mcp_add_ics_equal_coefficients_negative)) * pc)) /\ exists fs_q_mcp_ics_equal_coefficients_negative_target. pb = fs_q_mcp_ics_equal_coefficients_negative_target * S ((S (ff_index_mcp_add_ics_equal_coefficients_negative)) * pc) + (ff_target_mcp_add_ics_equal_coefficients_negative))) -> ff_target_mcp_add_ics_equal_coefficients_negative = ff_left_mcp_add_ics_equal_coefficients_negative + ff_right_mcp_add_ics_equal_coefficients_negative)))))))