prime_contribution_product_exists

Exact expanded PA statement

forall n m. exists z. (exists bpr_product_code_bpc_product_exists bpr_product_scale_bpc_product_exists. ((forall bpr_prefix_index_bpc_product_exists_prefix. (exists bpr_gap_bpc_product_exists_prefix_bound. bpr_gap_bpc_product_exists_prefix_bound + S (bpr_prefix_index_bpc_product_exists_prefix) = m) -> exists bpr_prefix_value_bpc_product_exists_prefix. ((((exists bpr_height_bpc_product_exists_prefix_decoded. bpr_height_bpc_product_exists_prefix_decoded + S (bpr_prefix_value_bpc_product_exists_prefix) = S ((S (bpr_prefix_index_bpc_product_exists_prefix)) * bpr_product_scale_bpc_product_exists)) /\ exists bpr_quotient_bpc_product_exists_prefix_decoded. bpr_product_code_bpc_product_exists = bpr_quotient_bpc_product_exists_prefix_decoded * S ((S (bpr_prefix_index_bpc_product_exists_prefix)) * bpr_product_scale_bpc_product_exists) + (bpr_prefix_value_bpc_product_exists_prefix))) /\ (((((~(S (bpr_prefix_index_bpc_product_exists_prefix) = 1) /\ forall bpr_left_bpc_product_exists_prefix_choice_prime bpr_right_bpc_product_exists_prefix_choice_prime. S (bpr_prefix_index_bpc_product_exists_prefix) = bpr_left_bpc_product_exists_prefix_choice_prime * bpr_right_bpc_product_exists_prefix_choice_prime -> bpr_left_bpc_product_exists_prefix_choice_prime = 1 \/ bpr_right_bpc_product_exists_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_bpc_product_exists_prefix_choice. ((((exists bpr_le_gap_bpc_product_exists_prefix_choice_valuation_selected_bound. bpr_le_gap_bpc_product_exists_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_bpc_product_exists_prefix_choice) = (n)) /\ (exists bpr_power_value_bpc_product_exists_prefix_choice_valuation_selected. ((exists bpr_power_code_bpc_product_exists_prefix_choice_valuation_selected_power bpr_power_scale_bpc_product_exists_prefix_choice_valuation_selected_power. ((forall bpr_power_index_bpc_product_exists_prefix_choice_valuation_selected_power. (exists bpr_gap_bpc_product_exists_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_bpc_product_exists_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpc_product_exists_prefix_choice_valuation_selected_power) = bpr_choice_exponent_bpc_product_exists_prefix_choice) -> (((exists bpr_height_bpc_product_exists_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_bpc_product_exists_prefix_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpc_product_exists_prefix)) = S ((S (bpr_power_index_bpc_product_exists_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_bpc_product_exists_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpc_product_exists_prefix_choice_valuation_selected_power = bpr_quotient_bpc_product_exists_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpc_product_exists_prefix_choice_valuation_selected_power)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_selected_power) + (S (bpr_prefix_index_bpc_product_exists_prefix))))) /\ (exists ff_u_bpc_product_exists_prefix_choice_valuation_selected_power_product ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_start. ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_start. ff_u_bpc_product_exists_prefix_choice_valuation_selected_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_terminal. ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpc_product_exists_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpc_product_exists_prefix_choice)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_terminal. ff_u_bpc_product_exists_prefix_choice_valuation_selected_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpc_product_exists_prefix_choice)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product) + (bpr_power_value_bpc_product_exists_prefix_choice_valuation_selected))) /\ forall ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product. (exists ff_lt_bpc_product_exists_prefix_choice_valuation_selected_power_product_bound. ff_lt_bpc_product_exists_prefix_choice_valuation_selected_power_product_bound + S ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_bpc_product_exists_prefix_choice) -> exists ff_p_bpc_product_exists_prefix_choice_valuation_selected_power_product ff_r_bpc_product_exists_prefix_choice_valuation_selected_power_product ff_s_bpc_product_exists_prefix_choice_valuation_selected_power_product. ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_factor. ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_factor + S (ff_p_bpc_product_exists_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_selected_power)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_bpc_product_exists_prefix_choice_valuation_selected_power = ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_selected_power) + (ff_p_bpc_product_exists_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_partial. ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_partial + S (ff_r_bpc_product_exists_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_partial. ff_u_bpc_product_exists_prefix_choice_valuation_selected_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product) + (ff_r_bpc_product_exists_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_successor. ff_h_bpc_product_exists_prefix_choice_valuation_selected_power_product_successor + S (ff_s_bpc_product_exists_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_successor. ff_u_bpc_product_exists_prefix_choice_valuation_selected_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpc_product_exists_prefix_choice_valuation_selected_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_selected_power_product) + (ff_s_bpc_product_exists_prefix_choice_valuation_selected_power_product))) /\ ff_s_bpc_product_exists_prefix_choice_valuation_selected_power_product = ff_r_bpc_product_exists_prefix_choice_valuation_selected_power_product * ff_p_bpc_product_exists_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpc_product_exists_prefix_choice_valuation_selected_divides. n = (bpr_power_value_bpc_product_exists_prefix_choice_valuation_selected) * bpr_divides_quotient_bpc_product_exists_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation. (exists bpr_le_gap_bpc_product_exists_prefix_choice_valuation_candidate_bound. bpr_le_gap_bpc_product_exists_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation) = (n)) -> (exists bpr_power_value_bpc_product_exists_prefix_choice_valuation_candidate. ((exists bpr_power_code_bpc_product_exists_prefix_choice_valuation_candidate_power bpr_power_scale_bpc_product_exists_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_bpc_product_exists_prefix_choice_valuation_candidate_power. (exists bpr_gap_bpc_product_exists_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpc_product_exists_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpc_product_exists_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation) -> (((exists bpr_height_bpc_product_exists_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_bpc_product_exists_prefix_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpc_product_exists_prefix)) = S ((S (bpr_power_index_bpc_product_exists_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpc_product_exists_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpc_product_exists_prefix_choice_valuation_candidate_power = bpr_quotient_bpc_product_exists_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpc_product_exists_prefix_choice_valuation_candidate_power)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpc_product_exists_prefix))))) /\ (exists ff_u_bpc_product_exists_prefix_choice_valuation_candidate_power_product ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_start. ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_start. ff_u_bpc_product_exists_prefix_choice_valuation_candidate_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_terminal. ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpc_product_exists_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_terminal. ff_u_bpc_product_exists_prefix_choice_valuation_candidate_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_bpc_product_exists_prefix_choice_valuation_candidate))) /\ forall ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product. (exists ff_lt_bpc_product_exists_prefix_choice_valuation_candidate_power_product_bound. ff_lt_bpc_product_exists_prefix_choice_valuation_candidate_power_product_bound + S ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation) -> exists ff_p_bpc_product_exists_prefix_choice_valuation_candidate_power_product ff_r_bpc_product_exists_prefix_choice_valuation_candidate_power_product ff_s_bpc_product_exists_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_factor. ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_bpc_product_exists_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_candidate_power)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_bpc_product_exists_prefix_choice_valuation_candidate_power = ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_bpc_product_exists_prefix_choice_valuation_candidate_power) + (ff_p_bpc_product_exists_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_partial. ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_bpc_product_exists_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_partial. ff_u_bpc_product_exists_prefix_choice_valuation_candidate_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product) + (ff_r_bpc_product_exists_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_successor. ff_h_bpc_product_exists_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_bpc_product_exists_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_successor. ff_u_bpc_product_exists_prefix_choice_valuation_candidate_power_product = ff_q_bpc_product_exists_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpc_product_exists_prefix_choice_valuation_candidate_power_product)) * ff_v_bpc_product_exists_prefix_choice_valuation_candidate_power_product) + (ff_s_bpc_product_exists_prefix_choice_valuation_candidate_power_product))) /\ ff_s_bpc_product_exists_prefix_choice_valuation_candidate_power_product = ff_r_bpc_product_exists_prefix_choice_valuation_candidate_power_product * ff_p_bpc_product_exists_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpc_product_exists_prefix_choice_valuation_candidate_divides. n = (bpr_power_value_bpc_product_exists_prefix_choice_valuation_candidate) * bpr_divides_quotient_bpc_product_exists_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpc_product_exists_prefix_choice_valuation_candidate_below. bpr_le_gap_bpc_product_exists_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_bpc_product_exists_prefix_choice_valuation) = (bpr_choice_exponent_bpc_product_exists_prefix_choice))) /\ (exists bpr_power_code_bpc_product_exists_prefix_choice_power bpr_power_scale_bpc_product_exists_prefix_choice_power. ((forall bpr_power_index_bpc_product_exists_prefix_choice_power. (exists bpr_gap_bpc_product_exists_prefix_choice_power_repeat_bound. bpr_gap_bpc_product_exists_prefix_choice_power_repeat_bound + S (bpr_power_index_bpc_product_exists_prefix_choice_power) = bpr_choice_exponent_bpc_product_exists_prefix_choice) -> (((exists bpr_height_bpc_product_exists_prefix_choice_power_repeat_entry. bpr_height_bpc_product_exists_prefix_choice_power_repeat_entry + S (S (bpr_prefix_index_bpc_product_exists_prefix)) = S ((S (bpr_power_index_bpc_product_exists_prefix_choice_power)) * bpr_power_scale_bpc_product_exists_prefix_choice_power)) /\ exists bpr_quotient_bpc_product_exists_prefix_choice_power_repeat_entry. bpr_power_code_bpc_product_exists_prefix_choice_power = bpr_quotient_bpc_product_exists_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_bpc_product_exists_prefix_choice_power)) * bpr_power_scale_bpc_product_exists_prefix_choice_power) + (S (bpr_prefix_index_bpc_product_exists_prefix))))) /\ (exists ff_u_bpc_product_exists_prefix_choice_power_product ff_v_bpc_product_exists_prefix_choice_power_product. ((((exists ff_h_bpc_product_exists_prefix_choice_power_product_start. ff_h_bpc_product_exists_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpc_product_exists_prefix_choice_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_power_product_start. ff_u_bpc_product_exists_prefix_choice_power_product = ff_q_bpc_product_exists_prefix_choice_power_product_start * S ((S (0)) * ff_v_bpc_product_exists_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_power_product_terminal. ff_h_bpc_product_exists_prefix_choice_power_product_terminal + S (bpr_prefix_value_bpc_product_exists_prefix) = S ((S (bpr_choice_exponent_bpc_product_exists_prefix_choice)) * ff_v_bpc_product_exists_prefix_choice_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_power_product_terminal. ff_u_bpc_product_exists_prefix_choice_power_product = ff_q_bpc_product_exists_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpc_product_exists_prefix_choice)) * ff_v_bpc_product_exists_prefix_choice_power_product) + (bpr_prefix_value_bpc_product_exists_prefix))) /\ forall ff_i_bpc_product_exists_prefix_choice_power_product. (exists ff_lt_bpc_product_exists_prefix_choice_power_product_bound. ff_lt_bpc_product_exists_prefix_choice_power_product_bound + S ff_i_bpc_product_exists_prefix_choice_power_product = bpr_choice_exponent_bpc_product_exists_prefix_choice) -> exists ff_p_bpc_product_exists_prefix_choice_power_product ff_r_bpc_product_exists_prefix_choice_power_product ff_s_bpc_product_exists_prefix_choice_power_product. ((((exists ff_h_bpc_product_exists_prefix_choice_power_product_factor. ff_h_bpc_product_exists_prefix_choice_power_product_factor + S (ff_p_bpc_product_exists_prefix_choice_power_product) = S ((S (ff_i_bpc_product_exists_prefix_choice_power_product)) * bpr_power_scale_bpc_product_exists_prefix_choice_power)) /\ exists ff_q_bpc_product_exists_prefix_choice_power_product_factor. bpr_power_code_bpc_product_exists_prefix_choice_power = ff_q_bpc_product_exists_prefix_choice_power_product_factor * S ((S (ff_i_bpc_product_exists_prefix_choice_power_product)) * bpr_power_scale_bpc_product_exists_prefix_choice_power) + (ff_p_bpc_product_exists_prefix_choice_power_product))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_power_product_partial. ff_h_bpc_product_exists_prefix_choice_power_product_partial + S (ff_r_bpc_product_exists_prefix_choice_power_product) = S ((S (ff_i_bpc_product_exists_prefix_choice_power_product)) * ff_v_bpc_product_exists_prefix_choice_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_power_product_partial. ff_u_bpc_product_exists_prefix_choice_power_product = ff_q_bpc_product_exists_prefix_choice_power_product_partial * S ((S (ff_i_bpc_product_exists_prefix_choice_power_product)) * ff_v_bpc_product_exists_prefix_choice_power_product) + (ff_r_bpc_product_exists_prefix_choice_power_product))) /\ ((((exists ff_h_bpc_product_exists_prefix_choice_power_product_successor. ff_h_bpc_product_exists_prefix_choice_power_product_successor + S (ff_s_bpc_product_exists_prefix_choice_power_product) = S ((S (S ff_i_bpc_product_exists_prefix_choice_power_product)) * ff_v_bpc_product_exists_prefix_choice_power_product)) /\ exists ff_q_bpc_product_exists_prefix_choice_power_product_successor. ff_u_bpc_product_exists_prefix_choice_power_product = ff_q_bpc_product_exists_prefix_choice_power_product_successor * S ((S (S ff_i_bpc_product_exists_prefix_choice_power_product)) * ff_v_bpc_product_exists_prefix_choice_power_product) + (ff_s_bpc_product_exists_prefix_choice_power_product))) /\ ff_s_bpc_product_exists_prefix_choice_power_product = ff_r_bpc_product_exists_prefix_choice_power_product * ff_p_bpc_product_exists_prefix_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpc_product_exists_prefix) = 1) /\ forall bpr_left_bpc_product_exists_prefix_choice_prime bpr_right_bpc_product_exists_prefix_choice_prime. S (bpr_prefix_index_bpc_product_exists_prefix) = bpr_left_bpc_product_exists_prefix_choice_prime * bpr_right_bpc_product_exists_prefix_choice_prime -> bpr_left_bpc_product_exists_prefix_choice_prime = 1 \/ bpr_right_bpc_product_exists_prefix_choice_prime = 1)) /\ bpr_prefix_value_bpc_product_exists_prefix = 1))))) /\ (exists ff_u_bpc_product_exists_product ff_v_bpc_product_exists_product. ((((exists ff_h_bpc_product_exists_product_start. ff_h_bpc_product_exists_product_start + S (1) = S ((S (0)) * ff_v_bpc_product_exists_product)) /\ exists ff_q_bpc_product_exists_product_start. ff_u_bpc_product_exists_product = ff_q_bpc_product_exists_product_start * S ((S (0)) * ff_v_bpc_product_exists_product) + (1))) /\ ((((exists ff_h_bpc_product_exists_product_terminal. ff_h_bpc_product_exists_product_terminal + S (z) = S ((S (m)) * ff_v_bpc_product_exists_product)) /\ exists ff_q_bpc_product_exists_product_terminal. ff_u_bpc_product_exists_product = ff_q_bpc_product_exists_product_terminal * S ((S (m)) * ff_v_bpc_product_exists_product) + (z))) /\ forall ff_i_bpc_product_exists_product. (exists ff_lt_bpc_product_exists_product_bound. ff_lt_bpc_product_exists_product_bound + S ff_i_bpc_product_exists_product = m) -> exists ff_p_bpc_product_exists_product ff_r_bpc_product_exists_product ff_s_bpc_product_exists_product. ((((exists ff_h_bpc_product_exists_product_factor. ff_h_bpc_product_exists_product_factor + S (ff_p_bpc_product_exists_product) = S ((S (ff_i_bpc_product_exists_product)) * bpr_product_scale_bpc_product_exists)) /\ exists ff_q_bpc_product_exists_product_factor. bpr_product_code_bpc_product_exists = ff_q_bpc_product_exists_product_factor * S ((S (ff_i_bpc_product_exists_product)) * bpr_product_scale_bpc_product_exists) + (ff_p_bpc_product_exists_product))) /\ ((((exists ff_h_bpc_product_exists_product_partial. ff_h_bpc_product_exists_product_partial + S (ff_r_bpc_product_exists_product) = S ((S (ff_i_bpc_product_exists_product)) * ff_v_bpc_product_exists_product)) /\ exists ff_q_bpc_product_exists_product_partial. ff_u_bpc_product_exists_product = ff_q_bpc_product_exists_product_partial * S ((S (ff_i_bpc_product_exists_product)) * ff_v_bpc_product_exists_product) + (ff_r_bpc_product_exists_product))) /\ ((((exists ff_h_bpc_product_exists_product_successor. ff_h_bpc_product_exists_product_successor + S (ff_s_bpc_product_exists_product) = S ((S (S ff_i_bpc_product_exists_product)) * ff_v_bpc_product_exists_product)) /\ exists ff_q_bpc_product_exists_product_successor. ff_u_bpc_product_exists_product = ff_q_bpc_product_exists_product_successor * S ((S (S ff_i_bpc_product_exists_product)) * ff_v_bpc_product_exists_product) + (ff_s_bpc_product_exists_product))) /\ ff_s_bpc_product_exists_product = ff_r_bpc_product_exists_product * ff_p_bpc_product_exists_product))))))))