no_bertrand_high_contribution_interval_eq_one

Exact expanded PA statement

forall n s q r C h w. (forall bpr_prime_candidate_b5nbhcieu_exclusion. ((exists bpr_gap_b5nbhcieu_exclusion_lower. bpr_gap_b5nbhcieu_exclusion_lower + S (n) = bpr_prime_candidate_b5nbhcieu_exclusion) /\ (exists bpr_le_gap_b5nbhcieu_exclusion_upper. bpr_le_gap_b5nbhcieu_exclusion_upper + (bpr_prime_candidate_b5nbhcieu_exclusion) = (n + n))) -> ~((~(bpr_prime_candidate_b5nbhcieu_exclusion = 1) /\ forall bpr_left_b5nbhcieu_exclusion_prime bpr_right_b5nbhcieu_exclusion_prime. bpr_prime_candidate_b5nbhcieu_exclusion = bpr_left_b5nbhcieu_exclusion_prime * bpr_right_b5nbhcieu_exclusion_prime -> bpr_left_b5nbhcieu_exclusion_prime = 1 \/ bpr_right_b5nbhcieu_exclusion_prime = 1))) -> (exists bcf_lt_gap_b5nbhcieu_positive. bcf_lt_gap_b5nbhcieu_positive + S (2) = n) -> (((exists bcs_sqrt_lower_gap_b5nbhcieu_floor. bcs_sqrt_lower_gap_b5nbhcieu_floor + (s) * (s) = (n + n)) /\ exists bcs_sqrt_upper_gap_b5nbhcieu_floor. bcs_sqrt_upper_gap_b5nbhcieu_floor + S (n + n) = S (s) * S (s))) -> (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_b5nbhcieu_division_bound. bcf_lt_gap_b5nbhcieu_division_bound + S (r) = 3))) -> (((exists bcf_lt_gap_b5nbhcieu_central_out_of_range. bcf_lt_gap_b5nbhcieu_central_out_of_range + S (n + n) = n) /\ C = 0) \/ ((exists bcf_le_gap_b5nbhcieu_central_in_range. bcf_le_gap_b5nbhcieu_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_b5nbhcieu_central bcf_row_code_scale_b5nbhcieu_central bcf_row_scale_code_b5nbhcieu_central bcf_row_scale_scale_b5nbhcieu_central bcf_row_code_b5nbhcieu_central bcf_row_scale_b5nbhcieu_central. ((forall bcf_row_index_b5nbhcieu_central_table. (exists bcf_lt_gap_b5nbhcieu_central_table_row_bound. bcf_lt_gap_b5nbhcieu_central_table_row_bound + S (bcf_row_index_b5nbhcieu_central_table) = S (n + n)) -> exists bcf_row_code_b5nbhcieu_central_table bcf_row_scale_b5nbhcieu_central_table. ((((exists bcf_height_b5nbhcieu_central_table_decoded_row_code. bcf_height_b5nbhcieu_central_table_decoded_row_code + S (bcf_row_code_b5nbhcieu_central_table) = S ((S (bcf_row_index_b5nbhcieu_central_table)) * bcf_row_code_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_table_decoded_row_code. bcf_row_code_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_table_decoded_row_code * S ((S (bcf_row_index_b5nbhcieu_central_table)) * bcf_row_code_scale_b5nbhcieu_central) + (bcf_row_code_b5nbhcieu_central_table))) /\ ((((exists bcf_height_b5nbhcieu_central_table_decoded_row_scale. bcf_height_b5nbhcieu_central_table_decoded_row_scale + S (bcf_row_scale_b5nbhcieu_central_table) = S ((S (bcf_row_index_b5nbhcieu_central_table)) * bcf_row_scale_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_table_decoded_row_scale. bcf_row_scale_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_table_decoded_row_scale * S ((S (bcf_row_index_b5nbhcieu_central_table)) * bcf_row_scale_scale_b5nbhcieu_central) + (bcf_row_scale_b5nbhcieu_central_table))) /\ ((bcf_row_index_b5nbhcieu_central_table = 0 /\ (forall bcf_index_b5nbhcieu_central_table_zero_row. (exists bcf_lt_gap_b5nbhcieu_central_table_zero_row_bound. bcf_lt_gap_b5nbhcieu_central_table_zero_row_bound + S (bcf_index_b5nbhcieu_central_table_zero_row) = S (n + n)) -> exists bcf_value_b5nbhcieu_central_table_zero_row. ((((exists bcf_height_b5nbhcieu_central_table_zero_row_entry. bcf_height_b5nbhcieu_central_table_zero_row_entry + S (bcf_value_b5nbhcieu_central_table_zero_row) = S ((S (bcf_index_b5nbhcieu_central_table_zero_row)) * bcf_row_scale_b5nbhcieu_central_table)) /\ exists bcf_quotient_b5nbhcieu_central_table_zero_row_entry. bcf_row_code_b5nbhcieu_central_table = bcf_quotient_b5nbhcieu_central_table_zero_row_entry * S ((S (bcf_index_b5nbhcieu_central_table_zero_row)) * bcf_row_scale_b5nbhcieu_central_table) + (bcf_value_b5nbhcieu_central_table_zero_row))) /\ ((bcf_index_b5nbhcieu_central_table_zero_row = 0 /\ bcf_value_b5nbhcieu_central_table_zero_row = 1) \/ exists bcf_predecessor_b5nbhcieu_central_table_zero_row. bcf_index_b5nbhcieu_central_table_zero_row = S bcf_predecessor_b5nbhcieu_central_table_zero_row /\ bcf_value_b5nbhcieu_central_table_zero_row = 0)))) \/ exists bcf_predecessor_b5nbhcieu_central_table bcf_previous_code_b5nbhcieu_central_table bcf_previous_scale_b5nbhcieu_central_table. bcf_row_index_b5nbhcieu_central_table = S bcf_predecessor_b5nbhcieu_central_table /\ ((((exists bcf_height_b5nbhcieu_central_table_decoded_previous_code. bcf_height_b5nbhcieu_central_table_decoded_previous_code + S (bcf_previous_code_b5nbhcieu_central_table) = S ((S (bcf_predecessor_b5nbhcieu_central_table)) * bcf_row_code_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_table_decoded_previous_code. bcf_row_code_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_table_decoded_previous_code * S ((S (bcf_predecessor_b5nbhcieu_central_table)) * bcf_row_code_scale_b5nbhcieu_central) + (bcf_previous_code_b5nbhcieu_central_table))) /\ ((((exists bcf_height_b5nbhcieu_central_table_decoded_previous_scale. bcf_height_b5nbhcieu_central_table_decoded_previous_scale + S (bcf_previous_scale_b5nbhcieu_central_table) = S ((S (bcf_predecessor_b5nbhcieu_central_table)) * bcf_row_scale_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_table_decoded_previous_scale. bcf_row_scale_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_table_decoded_previous_scale * S ((S (bcf_predecessor_b5nbhcieu_central_table)) * bcf_row_scale_scale_b5nbhcieu_central) + (bcf_previous_scale_b5nbhcieu_central_table))) /\ (forall bcf_index_b5nbhcieu_central_table_row_step. (exists bcf_lt_gap_b5nbhcieu_central_table_row_step_bound. bcf_lt_gap_b5nbhcieu_central_table_row_step_bound + S (bcf_index_b5nbhcieu_central_table_row_step) = S (n + n)) -> exists bcf_value_b5nbhcieu_central_table_row_step. ((((exists bcf_height_b5nbhcieu_central_table_row_step_entry. bcf_height_b5nbhcieu_central_table_row_step_entry + S (bcf_value_b5nbhcieu_central_table_row_step) = S ((S (bcf_index_b5nbhcieu_central_table_row_step)) * bcf_row_scale_b5nbhcieu_central_table)) /\ exists bcf_quotient_b5nbhcieu_central_table_row_step_entry. bcf_row_code_b5nbhcieu_central_table = bcf_quotient_b5nbhcieu_central_table_row_step_entry * S ((S (bcf_index_b5nbhcieu_central_table_row_step)) * bcf_row_scale_b5nbhcieu_central_table) + (bcf_value_b5nbhcieu_central_table_row_step))) /\ ((bcf_index_b5nbhcieu_central_table_row_step = 0 /\ bcf_value_b5nbhcieu_central_table_row_step = 1) \/ exists bcf_predecessor_b5nbhcieu_central_table_row_step bcf_left_b5nbhcieu_central_table_row_step bcf_right_b5nbhcieu_central_table_row_step. bcf_index_b5nbhcieu_central_table_row_step = S bcf_predecessor_b5nbhcieu_central_table_row_step /\ ((((exists bcf_height_b5nbhcieu_central_table_row_step_previous_left. bcf_height_b5nbhcieu_central_table_row_step_previous_left + S (bcf_left_b5nbhcieu_central_table_row_step) = S ((S (bcf_predecessor_b5nbhcieu_central_table_row_step)) * bcf_previous_scale_b5nbhcieu_central_table)) /\ exists bcf_quotient_b5nbhcieu_central_table_row_step_previous_left. bcf_previous_code_b5nbhcieu_central_table = bcf_quotient_b5nbhcieu_central_table_row_step_previous_left * S ((S (bcf_predecessor_b5nbhcieu_central_table_row_step)) * bcf_previous_scale_b5nbhcieu_central_table) + (bcf_left_b5nbhcieu_central_table_row_step))) /\ ((((exists bcf_height_b5nbhcieu_central_table_row_step_previous_right. bcf_height_b5nbhcieu_central_table_row_step_previous_right + S (bcf_right_b5nbhcieu_central_table_row_step) = S ((S (S (bcf_predecessor_b5nbhcieu_central_table_row_step))) * bcf_previous_scale_b5nbhcieu_central_table)) /\ exists bcf_quotient_b5nbhcieu_central_table_row_step_previous_right. bcf_previous_code_b5nbhcieu_central_table = bcf_quotient_b5nbhcieu_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_b5nbhcieu_central_table_row_step))) * bcf_previous_scale_b5nbhcieu_central_table) + (bcf_right_b5nbhcieu_central_table_row_step))) /\ bcf_value_b5nbhcieu_central_table_row_step = bcf_left_b5nbhcieu_central_table_row_step + bcf_right_b5nbhcieu_central_table_row_step))))))))))) /\ ((((exists bcf_height_b5nbhcieu_central_decoded_row_code. bcf_height_b5nbhcieu_central_decoded_row_code + S (bcf_row_code_b5nbhcieu_central) = S ((S (n + n)) * bcf_row_code_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_decoded_row_code. bcf_row_code_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_b5nbhcieu_central) + (bcf_row_code_b5nbhcieu_central))) /\ ((((exists bcf_height_b5nbhcieu_central_decoded_row_scale. bcf_height_b5nbhcieu_central_decoded_row_scale + S (bcf_row_scale_b5nbhcieu_central) = S ((S (n + n)) * bcf_row_scale_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_decoded_row_scale. bcf_row_scale_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_b5nbhcieu_central) + (bcf_row_scale_b5nbhcieu_central))) /\ (((exists bcf_height_b5nbhcieu_central_decoded_value. bcf_height_b5nbhcieu_central_decoded_value + S (C) = S ((S (n)) * bcf_row_scale_b5nbhcieu_central)) /\ exists bcf_quotient_b5nbhcieu_central_decoded_value. bcf_row_code_b5nbhcieu_central = bcf_quotient_b5nbhcieu_central_decoded_value * S ((S (n)) * bcf_row_scale_b5nbhcieu_central) + (C))))))))) -> (exists bpr_code_b5nbhcieu_interval bpr_scale_b5nbhcieu_interval. ((forall bpr_index_b5nbhcieu_interval_prefix. (exists bpr_gap_b5nbhcieu_interval_prefix_bound. bpr_gap_b5nbhcieu_interval_prefix_bound + S (bpr_index_b5nbhcieu_interval_prefix) = h) -> exists bpr_value_b5nbhcieu_interval_prefix. ((((exists bpr_height_b5nbhcieu_interval_prefix_decoded. bpr_height_b5nbhcieu_interval_prefix_decoded + S (bpr_value_b5nbhcieu_interval_prefix) = S ((S (bpr_index_b5nbhcieu_interval_prefix)) * bpr_scale_b5nbhcieu_interval)) /\ exists bpr_quotient_b5nbhcieu_interval_prefix_decoded. bpr_code_b5nbhcieu_interval = bpr_quotient_b5nbhcieu_interval_prefix_decoded * S ((S (bpr_index_b5nbhcieu_interval_prefix)) * bpr_scale_b5nbhcieu_interval) + (bpr_value_b5nbhcieu_interval_prefix))) /\ (((((~(S (q + bpr_index_b5nbhcieu_interval_prefix) = 1) /\ forall bpr_left_b5nbhcieu_interval_prefix_choice_prime bpr_right_b5nbhcieu_interval_prefix_choice_prime. S (q + bpr_index_b5nbhcieu_interval_prefix) = bpr_left_b5nbhcieu_interval_prefix_choice_prime * bpr_right_b5nbhcieu_interval_prefix_choice_prime -> bpr_left_b5nbhcieu_interval_prefix_choice_prime = 1 \/ bpr_right_b5nbhcieu_interval_prefix_choice_prime = 1)) /\ exists bpr_choice_exponent_b5nbhcieu_interval_prefix_choice. ((((exists bpr_le_gap_b5nbhcieu_interval_prefix_choice_valuation_selected_bound. bpr_le_gap_b5nbhcieu_interval_prefix_choice_valuation_selected_bound + (bpr_choice_exponent_b5nbhcieu_interval_prefix_choice) = (C)) /\ (exists bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_selected. ((exists bpr_power_code_b5nbhcieu_interval_prefix_choice_valuation_selected_power bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_selected_power. ((forall bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_selected_power. (exists bpr_gap_b5nbhcieu_interval_prefix_choice_valuation_selected_power_repeat_bound. bpr_gap_b5nbhcieu_interval_prefix_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_selected_power) = bpr_choice_exponent_b5nbhcieu_interval_prefix_choice) -> (((exists bpr_height_b5nbhcieu_interval_prefix_choice_valuation_selected_power_repeat_entry. bpr_height_b5nbhcieu_interval_prefix_choice_valuation_selected_power_repeat_entry + S (S (q + bpr_index_b5nbhcieu_interval_prefix)) = S ((S (bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_selected_power)) /\ exists bpr_quotient_b5nbhcieu_interval_prefix_choice_valuation_selected_power_repeat_entry. bpr_power_code_b5nbhcieu_interval_prefix_choice_valuation_selected_power = bpr_quotient_b5nbhcieu_interval_prefix_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_selected_power)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_selected_power) + (S (q + bpr_index_b5nbhcieu_interval_prefix))))) /\ (exists ff_u_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_start. ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_start. ff_u_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_terminal. ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_terminal + S (bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_selected) = S ((S (bpr_choice_exponent_b5nbhcieu_interval_prefix_choice)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_terminal. ff_u_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_b5nbhcieu_interval_prefix_choice)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) + (bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_selected))) /\ forall ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product. (exists ff_lt_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_bound. ff_lt_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_bound + S ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product = bpr_choice_exponent_b5nbhcieu_interval_prefix_choice) -> exists ff_p_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product ff_r_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product ff_s_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product. ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_factor. ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_factor + S (ff_p_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_selected_power)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_factor. bpr_power_code_b5nbhcieu_interval_prefix_choice_valuation_selected_power = ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_factor * S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_selected_power) + (ff_p_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_partial. ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_partial + S (ff_r_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) = S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_partial. ff_u_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_partial * S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) + (ff_r_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_successor. ff_h_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_successor + S (ff_s_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) = S ((S (S ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_successor. ff_u_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product_successor * S ((S (S ff_i_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product) + (ff_s_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product))) /\ ff_s_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product = ff_r_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product * ff_p_b5nbhcieu_interval_prefix_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_b5nbhcieu_interval_prefix_choice_valuation_selected_divides. C = (bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_selected) * bpr_divides_quotient_b5nbhcieu_interval_prefix_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation. (exists bpr_le_gap_b5nbhcieu_interval_prefix_choice_valuation_candidate_bound. bpr_le_gap_b5nbhcieu_interval_prefix_choice_valuation_candidate_bound + (bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation) = (C)) -> (exists bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_candidate. ((exists bpr_power_code_b5nbhcieu_interval_prefix_choice_valuation_candidate_power bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_candidate_power. ((forall bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_candidate_power. (exists bpr_gap_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_repeat_bound. bpr_gap_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_candidate_power) = bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation) -> (((exists bpr_height_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_repeat_entry. bpr_height_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_repeat_entry + S (S (q + bpr_index_b5nbhcieu_interval_prefix)) = S ((S (bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_candidate_power)) /\ exists bpr_quotient_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_repeat_entry. bpr_power_code_b5nbhcieu_interval_prefix_choice_valuation_candidate_power = bpr_quotient_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_b5nbhcieu_interval_prefix_choice_valuation_candidate_power)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_candidate_power) + (S (q + bpr_index_b5nbhcieu_interval_prefix))))) /\ (exists ff_u_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_start. ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_start. ff_u_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_terminal. ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_terminal. ff_u_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) + (bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_candidate))) /\ forall ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product. (exists ff_lt_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_bound. ff_lt_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_bound + S ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product = bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation) -> exists ff_p_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product ff_r_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product ff_s_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product. ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_factor. ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_factor + S (ff_p_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_candidate_power)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_factor. bpr_power_code_b5nbhcieu_interval_prefix_choice_valuation_candidate_power = ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_factor * S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_valuation_candidate_power) + (ff_p_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_partial. ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_partial + S (ff_r_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) = S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_partial. ff_u_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_partial * S ((S (ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) + (ff_r_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_successor. ff_h_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_successor + S (ff_s_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) = S ((S (S ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_successor. ff_u_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product = ff_q_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product) + (ff_s_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product))) /\ ff_s_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product = ff_r_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product * ff_p_b5nbhcieu_interval_prefix_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_b5nbhcieu_interval_prefix_choice_valuation_candidate_divides. C = (bpr_power_value_b5nbhcieu_interval_prefix_choice_valuation_candidate) * bpr_divides_quotient_b5nbhcieu_interval_prefix_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_b5nbhcieu_interval_prefix_choice_valuation_candidate_below. bpr_le_gap_b5nbhcieu_interval_prefix_choice_valuation_candidate_below + (bpr_valuation_candidate_b5nbhcieu_interval_prefix_choice_valuation) = (bpr_choice_exponent_b5nbhcieu_interval_prefix_choice))) /\ (exists bpr_power_code_b5nbhcieu_interval_prefix_choice_power bpr_power_scale_b5nbhcieu_interval_prefix_choice_power. ((forall bpr_power_index_b5nbhcieu_interval_prefix_choice_power. (exists bpr_gap_b5nbhcieu_interval_prefix_choice_power_repeat_bound. bpr_gap_b5nbhcieu_interval_prefix_choice_power_repeat_bound + S (bpr_power_index_b5nbhcieu_interval_prefix_choice_power) = bpr_choice_exponent_b5nbhcieu_interval_prefix_choice) -> (((exists bpr_height_b5nbhcieu_interval_prefix_choice_power_repeat_entry. bpr_height_b5nbhcieu_interval_prefix_choice_power_repeat_entry + S (S (q + bpr_index_b5nbhcieu_interval_prefix)) = S ((S (bpr_power_index_b5nbhcieu_interval_prefix_choice_power)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_power)) /\ exists bpr_quotient_b5nbhcieu_interval_prefix_choice_power_repeat_entry. bpr_power_code_b5nbhcieu_interval_prefix_choice_power = bpr_quotient_b5nbhcieu_interval_prefix_choice_power_repeat_entry * S ((S (bpr_power_index_b5nbhcieu_interval_prefix_choice_power)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_power) + (S (q + bpr_index_b5nbhcieu_interval_prefix))))) /\ (exists ff_u_b5nbhcieu_interval_prefix_choice_power_product ff_v_b5nbhcieu_interval_prefix_choice_power_product. ((((exists ff_h_b5nbhcieu_interval_prefix_choice_power_product_start. ff_h_b5nbhcieu_interval_prefix_choice_power_product_start + S (1) = S ((S (0)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_power_product_start. ff_u_b5nbhcieu_interval_prefix_choice_power_product = ff_q_b5nbhcieu_interval_prefix_choice_power_product_start * S ((S (0)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product) + (1))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_power_product_terminal. ff_h_b5nbhcieu_interval_prefix_choice_power_product_terminal + S (bpr_value_b5nbhcieu_interval_prefix) = S ((S (bpr_choice_exponent_b5nbhcieu_interval_prefix_choice)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_power_product_terminal. ff_u_b5nbhcieu_interval_prefix_choice_power_product = ff_q_b5nbhcieu_interval_prefix_choice_power_product_terminal * S ((S (bpr_choice_exponent_b5nbhcieu_interval_prefix_choice)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product) + (bpr_value_b5nbhcieu_interval_prefix))) /\ forall ff_i_b5nbhcieu_interval_prefix_choice_power_product. (exists ff_lt_b5nbhcieu_interval_prefix_choice_power_product_bound. ff_lt_b5nbhcieu_interval_prefix_choice_power_product_bound + S ff_i_b5nbhcieu_interval_prefix_choice_power_product = bpr_choice_exponent_b5nbhcieu_interval_prefix_choice) -> exists ff_p_b5nbhcieu_interval_prefix_choice_power_product ff_r_b5nbhcieu_interval_prefix_choice_power_product ff_s_b5nbhcieu_interval_prefix_choice_power_product. ((((exists ff_h_b5nbhcieu_interval_prefix_choice_power_product_factor. ff_h_b5nbhcieu_interval_prefix_choice_power_product_factor + S (ff_p_b5nbhcieu_interval_prefix_choice_power_product) = S ((S (ff_i_b5nbhcieu_interval_prefix_choice_power_product)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_power)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_power_product_factor. bpr_power_code_b5nbhcieu_interval_prefix_choice_power = ff_q_b5nbhcieu_interval_prefix_choice_power_product_factor * S ((S (ff_i_b5nbhcieu_interval_prefix_choice_power_product)) * bpr_power_scale_b5nbhcieu_interval_prefix_choice_power) + (ff_p_b5nbhcieu_interval_prefix_choice_power_product))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_power_product_partial. ff_h_b5nbhcieu_interval_prefix_choice_power_product_partial + S (ff_r_b5nbhcieu_interval_prefix_choice_power_product) = S ((S (ff_i_b5nbhcieu_interval_prefix_choice_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_power_product_partial. ff_u_b5nbhcieu_interval_prefix_choice_power_product = ff_q_b5nbhcieu_interval_prefix_choice_power_product_partial * S ((S (ff_i_b5nbhcieu_interval_prefix_choice_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product) + (ff_r_b5nbhcieu_interval_prefix_choice_power_product))) /\ ((((exists ff_h_b5nbhcieu_interval_prefix_choice_power_product_successor. ff_h_b5nbhcieu_interval_prefix_choice_power_product_successor + S (ff_s_b5nbhcieu_interval_prefix_choice_power_product) = S ((S (S ff_i_b5nbhcieu_interval_prefix_choice_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product)) /\ exists ff_q_b5nbhcieu_interval_prefix_choice_power_product_successor. ff_u_b5nbhcieu_interval_prefix_choice_power_product = ff_q_b5nbhcieu_interval_prefix_choice_power_product_successor * S ((S (S ff_i_b5nbhcieu_interval_prefix_choice_power_product)) * ff_v_b5nbhcieu_interval_prefix_choice_power_product) + (ff_s_b5nbhcieu_interval_prefix_choice_power_product))) /\ ff_s_b5nbhcieu_interval_prefix_choice_power_product = ff_r_b5nbhcieu_interval_prefix_choice_power_product * ff_p_b5nbhcieu_interval_prefix_choice_power_product)))))))))) \/ (~((~(S (q + bpr_index_b5nbhcieu_interval_prefix) = 1) /\ forall bpr_left_b5nbhcieu_interval_prefix_choice_prime bpr_right_b5nbhcieu_interval_prefix_choice_prime. S (q + bpr_index_b5nbhcieu_interval_prefix) = bpr_left_b5nbhcieu_interval_prefix_choice_prime * bpr_right_b5nbhcieu_interval_prefix_choice_prime -> bpr_left_b5nbhcieu_interval_prefix_choice_prime = 1 \/ bpr_right_b5nbhcieu_interval_prefix_choice_prime = 1)) /\ bpr_value_b5nbhcieu_interval_prefix = 1))))) /\ (exists ff_u_b5nbhcieu_interval_product ff_v_b5nbhcieu_interval_product. ((((exists ff_h_b5nbhcieu_interval_product_start. ff_h_b5nbhcieu_interval_product_start + S (1) = S ((S (0)) * ff_v_b5nbhcieu_interval_product)) /\ exists ff_q_b5nbhcieu_interval_product_start. ff_u_b5nbhcieu_interval_product = ff_q_b5nbhcieu_interval_product_start * S ((S (0)) * ff_v_b5nbhcieu_interval_product) + (1))) /\ ((((exists ff_h_b5nbhcieu_interval_product_terminal. ff_h_b5nbhcieu_interval_product_terminal + S (w) = S ((S (h)) * ff_v_b5nbhcieu_interval_product)) /\ exists ff_q_b5nbhcieu_interval_product_terminal. ff_u_b5nbhcieu_interval_product = ff_q_b5nbhcieu_interval_product_terminal * S ((S (h)) * ff_v_b5nbhcieu_interval_product) + (w))) /\ forall ff_i_b5nbhcieu_interval_product. (exists ff_lt_b5nbhcieu_interval_product_bound. ff_lt_b5nbhcieu_interval_product_bound + S ff_i_b5nbhcieu_interval_product = h) -> exists ff_p_b5nbhcieu_interval_product ff_r_b5nbhcieu_interval_product ff_s_b5nbhcieu_interval_product. ((((exists ff_h_b5nbhcieu_interval_product_factor. ff_h_b5nbhcieu_interval_product_factor + S (ff_p_b5nbhcieu_interval_product) = S ((S (ff_i_b5nbhcieu_interval_product)) * bpr_scale_b5nbhcieu_interval)) /\ exists ff_q_b5nbhcieu_interval_product_factor. bpr_code_b5nbhcieu_interval = ff_q_b5nbhcieu_interval_product_factor * S ((S (ff_i_b5nbhcieu_interval_product)) * bpr_scale_b5nbhcieu_interval) + (ff_p_b5nbhcieu_interval_product))) /\ ((((exists ff_h_b5nbhcieu_interval_product_partial. ff_h_b5nbhcieu_interval_product_partial + S (ff_r_b5nbhcieu_interval_product) = S ((S (ff_i_b5nbhcieu_interval_product)) * ff_v_b5nbhcieu_interval_product)) /\ exists ff_q_b5nbhcieu_interval_product_partial. ff_u_b5nbhcieu_interval_product = ff_q_b5nbhcieu_interval_product_partial * S ((S (ff_i_b5nbhcieu_interval_product)) * ff_v_b5nbhcieu_interval_product) + (ff_r_b5nbhcieu_interval_product))) /\ ((((exists ff_h_b5nbhcieu_interval_product_successor. ff_h_b5nbhcieu_interval_product_successor + S (ff_s_b5nbhcieu_interval_product) = S ((S (S ff_i_b5nbhcieu_interval_product)) * ff_v_b5nbhcieu_interval_product)) /\ exists ff_q_b5nbhcieu_interval_product_successor. ff_u_b5nbhcieu_interval_product = ff_q_b5nbhcieu_interval_product_successor * S ((S (S ff_i_b5nbhcieu_interval_product)) * ff_v_b5nbhcieu_interval_product) + (ff_s_b5nbhcieu_interval_product))) /\ ff_s_b5nbhcieu_interval_product = ff_r_b5nbhcieu_interval_product * ff_p_b5nbhcieu_interval_product)))))))) -> w = 1