beta_valuation_prefix_extend

forall p b c vb vc l a e. (forall mkm_index_extend_source. (exists mkm_lt_extend_source_bound. mkm_lt_extend_source_bound + S (mkm_index_extend_source) = (l)) -> (exists mkm_value_extend_source_point mkm_exponent_extend_source_point. (((exists fs_h_mkm_extend_source_point_source. fs_h_mkm_extend_source_point_source + S (mkm_value_extend_source_point) = S ((S (mkm_index_extend_source)) * c)) /\ exists fs_q_mkm_extend_source_point_source. b = fs_q_mkm_extend_source_point_source * S ((S (mkm_index_extend_source)) * c) + (mkm_value_extend_source_point))) /\ ((((exists fs_h_mkm_extend_source_point_decoded. fs_h_mkm_extend_source_point_decoded + S (mkm_exponent_extend_source_point) = S ((S (mkm_index_extend_source)) * vc)) /\ exists fs_q_mkm_extend_source_point_decoded. vb = fs_q_mkm_extend_source_point_decoded * S ((S (mkm_index_extend_source)) * vc) + (mkm_exponent_extend_source_point))) /\ (((exists bpv_gap_mkm_extend_source_point_valuation_exponent_bound. bpv_gap_mkm_extend_source_point_valuation_exponent_bound + mkm_exponent_extend_source_point = (mkm_value_extend_source_point)) /\ (exists bpv_result_mkm_extend_source_point_valuation_selected. ((exists ff_b_mkm_extend_source_point_valuation_selected_power ff_c_mkm_extend_source_point_valuation_selected_power. ((forall ff_i_mkm_extend_source_point_valuation_selected_power_repeat. (exists ff_lt_mkm_extend_source_point_valuation_selected_power_repeat_bound. ff_lt_mkm_extend_source_point_valuation_selected_power_repeat_bound + S ff_i_mkm_extend_source_point_valuation_selected_power_repeat = mkm_exponent_extend_source_point) -> (((exists ff_h_mkm_extend_source_point_valuation_selected_power_repeat_decoded. ff_h_mkm_extend_source_point_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_extend_source_point_valuation_selected_power_repeat)) * ff_c_mkm_extend_source_point_valuation_selected_power)) /\ exists ff_q_mkm_extend_source_point_valuation_selected_power_repeat_decoded. ff_b_mkm_extend_source_point_valuation_selected_power = ff_q_mkm_extend_source_point_valuation_selected_power_repeat_decoded * S ((S (ff_i_mkm_extend_source_point_valuation_selected_power_repeat)) * ff_c_mkm_extend_source_point_valuation_selected_power) + (p)))) /\ (exists ff_u_mkm_extend_source_point_valuation_selected_power_product ff_v_mkm_extend_source_point_valuation_selected_power_product. ((((exists ff_h_mkm_extend_source_point_valuation_selected_power_product_start. ff_h_mkm_extend_source_point_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_extend_source_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_selected_power_product_start. ff_u_mkm_extend_source_point_valuation_selected_power_product = ff_q_mkm_extend_source_point_valuation_selected_power_product_start * S ((S (0)) * ff_v_mkm_extend_source_point_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_mkm_extend_source_point_valuation_selected_power_product_terminal. ff_h_mkm_extend_source_point_valuation_selected_power_product_terminal + S (bpv_result_mkm_extend_source_point_valuation_selected) = S ((S (mkm_exponent_extend_source_point)) * ff_v_mkm_extend_source_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_selected_power_product_terminal. ff_u_mkm_extend_source_point_valuation_selected_power_product = ff_q_mkm_extend_source_point_valuation_selected_power_product_terminal * S ((S (mkm_exponent_extend_source_point)) * ff_v_mkm_extend_source_point_valuation_selected_power_product) + (bpv_result_mkm_extend_source_point_valuation_selected))) /\ forall ff_i_mkm_extend_source_point_valuation_selected_power_product. (exists ff_lt_mkm_extend_source_point_valuation_selected_power_product_bound. ff_lt_mkm_extend_source_point_valuation_selected_power_product_bound + S ff_i_mkm_extend_source_point_valuation_selected_power_product = mkm_exponent_extend_source_point) -> exists ff_p_mkm_extend_source_point_valuation_selected_power_product ff_r_mkm_extend_source_point_valuation_selected_power_product ff_s_mkm_extend_source_point_valuation_selected_power_product. ((((exists ff_h_mkm_extend_source_point_valuation_selected_power_product_factor. ff_h_mkm_extend_source_point_valuation_selected_power_product_factor + S (ff_p_mkm_extend_source_point_valuation_selected_power_product) = S ((S (ff_i_mkm_extend_source_point_valuation_selected_power_product)) * ff_c_mkm_extend_source_point_valuation_selected_power)) /\ exists ff_q_mkm_extend_source_point_valuation_selected_power_product_factor. ff_b_mkm_extend_source_point_valuation_selected_power = ff_q_mkm_extend_source_point_valuation_selected_power_product_factor * S ((S (ff_i_mkm_extend_source_point_valuation_selected_power_product)) * ff_c_mkm_extend_source_point_valuation_selected_power) + (ff_p_mkm_extend_source_point_valuation_selected_power_product))) /\ ((((exists ff_h_mkm_extend_source_point_valuation_selected_power_product_partial. ff_h_mkm_extend_source_point_valuation_selected_power_product_partial + S (ff_r_mkm_extend_source_point_valuation_selected_power_product) = S ((S (ff_i_mkm_extend_source_point_valuation_selected_power_product)) * ff_v_mkm_extend_source_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_selected_power_product_partial. ff_u_mkm_extend_source_point_valuation_selected_power_product = ff_q_mkm_extend_source_point_valuation_selected_power_product_partial * S ((S (ff_i_mkm_extend_source_point_valuation_selected_power_product)) * ff_v_mkm_extend_source_point_valuation_selected_power_product) + (ff_r_mkm_extend_source_point_valuation_selected_power_product))) /\ ((((exists ff_h_mkm_extend_source_point_valuation_selected_power_product_successor. ff_h_mkm_extend_source_point_valuation_selected_power_product_successor + S (ff_s_mkm_extend_source_point_valuation_selected_power_product) = S ((S (S ff_i_mkm_extend_source_point_valuation_selected_power_product)) * ff_v_mkm_extend_source_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_selected_power_product_successor. ff_u_mkm_extend_source_point_valuation_selected_power_product = ff_q_mkm_extend_source_point_valuation_selected_power_product_successor * S ((S (S ff_i_mkm_extend_source_point_valuation_selected_power_product)) * ff_v_mkm_extend_source_point_valuation_selected_power_product) + (ff_s_mkm_extend_source_point_valuation_selected_power_product))) /\ ff_s_mkm_extend_source_point_valuation_selected_power_product = ff_r_mkm_extend_source_point_valuation_selected_power_product * ff_p_mkm_extend_source_point_valuation_selected_power_product)))))))) /\ (exists bpv_factor_mkm_extend_source_point_valuation_selected_divides. (mkm_value_extend_source_point) = bpv_result_mkm_extend_source_point_valuation_selected * bpv_factor_mkm_extend_source_point_valuation_selected_divides)))) /\ forall bpv_candidate_mkm_extend_source_point_valuation. (exists bpv_gap_mkm_extend_source_point_valuation_candidate_bound. bpv_gap_mkm_extend_source_point_valuation_candidate_bound + bpv_candidate_mkm_extend_source_point_valuation = (mkm_value_extend_source_point)) -> (exists bpv_result_mkm_extend_source_point_valuation_candidate. ((exists ff_b_mkm_extend_source_point_valuation_candidate_power ff_c_mkm_extend_source_point_valuation_candidate_power. ((forall ff_i_mkm_extend_source_point_valuation_candidate_power_repeat. (exists ff_lt_mkm_extend_source_point_valuation_candidate_power_repeat_bound. ff_lt_mkm_extend_source_point_valuation_candidate_power_repeat_bound + S ff_i_mkm_extend_source_point_valuation_candidate_power_repeat = bpv_candidate_mkm_extend_source_point_valuation) -> (((exists ff_h_mkm_extend_source_point_valuation_candidate_power_repeat_decoded. ff_h_mkm_extend_source_point_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_extend_source_point_valuation_candidate_power_repeat)) * ff_c_mkm_extend_source_point_valuation_candidate_power)) /\ exists ff_q_mkm_extend_source_point_valuation_candidate_power_repeat_decoded. ff_b_mkm_extend_source_point_valuation_candidate_power = ff_q_mkm_extend_source_point_valuation_candidate_power_repeat_decoded * S ((S (ff_i_mkm_extend_source_point_valuation_candidate_power_repeat)) * ff_c_mkm_extend_source_point_valuation_candidate_power) + (p)))) /\ (exists ff_u_mkm_extend_source_point_valuation_candidate_power_product ff_v_mkm_extend_source_point_valuation_candidate_power_product. ((((exists ff_h_mkm_extend_source_point_valuation_candidate_power_product_start. ff_h_mkm_extend_source_point_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_candidate_power_product_start. ff_u_mkm_extend_source_point_valuation_candidate_power_product = ff_q_mkm_extend_source_point_valuation_candidate_power_product_start * S ((S (0)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_mkm_extend_source_point_valuation_candidate_power_product_terminal. ff_h_mkm_extend_source_point_valuation_candidate_power_product_terminal + S (bpv_result_mkm_extend_source_point_valuation_candidate) = S ((S (bpv_candidate_mkm_extend_source_point_valuation)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_candidate_power_product_terminal. ff_u_mkm_extend_source_point_valuation_candidate_power_product = ff_q_mkm_extend_source_point_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_mkm_extend_source_point_valuation)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product) + (bpv_result_mkm_extend_source_point_valuation_candidate))) /\ forall ff_i_mkm_extend_source_point_valuation_candidate_power_product. (exists ff_lt_mkm_extend_source_point_valuation_candidate_power_product_bound. ff_lt_mkm_extend_source_point_valuation_candidate_power_product_bound + S ff_i_mkm_extend_source_point_valuation_candidate_power_product = bpv_candidate_mkm_extend_source_point_valuation) -> exists ff_p_mkm_extend_source_point_valuation_candidate_power_product ff_r_mkm_extend_source_point_valuation_candidate_power_product ff_s_mkm_extend_source_point_valuation_candidate_power_product. ((((exists ff_h_mkm_extend_source_point_valuation_candidate_power_product_factor. ff_h_mkm_extend_source_point_valuation_candidate_power_product_factor + S (ff_p_mkm_extend_source_point_valuation_candidate_power_product) = S ((S (ff_i_mkm_extend_source_point_valuation_candidate_power_product)) * ff_c_mkm_extend_source_point_valuation_candidate_power)) /\ exists ff_q_mkm_extend_source_point_valuation_candidate_power_product_factor. ff_b_mkm_extend_source_point_valuation_candidate_power = ff_q_mkm_extend_source_point_valuation_candidate_power_product_factor * S ((S (ff_i_mkm_extend_source_point_valuation_candidate_power_product)) * ff_c_mkm_extend_source_point_valuation_candidate_power) + (ff_p_mkm_extend_source_point_valuation_candidate_power_product))) /\ ((((exists ff_h_mkm_extend_source_point_valuation_candidate_power_product_partial. ff_h_mkm_extend_source_point_valuation_candidate_power_product_partial + S (ff_r_mkm_extend_source_point_valuation_candidate_power_product) = S ((S (ff_i_mkm_extend_source_point_valuation_candidate_power_product)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_candidate_power_product_partial. ff_u_mkm_extend_source_point_valuation_candidate_power_product = ff_q_mkm_extend_source_point_valuation_candidate_power_product_partial * S ((S (ff_i_mkm_extend_source_point_valuation_candidate_power_product)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product) + (ff_r_mkm_extend_source_point_valuation_candidate_power_product))) /\ ((((exists ff_h_mkm_extend_source_point_valuation_candidate_power_product_successor. ff_h_mkm_extend_source_point_valuation_candidate_power_product_successor + S (ff_s_mkm_extend_source_point_valuation_candidate_power_product) = S ((S (S ff_i_mkm_extend_source_point_valuation_candidate_power_product)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_source_point_valuation_candidate_power_product_successor. ff_u_mkm_extend_source_point_valuation_candidate_power_product = ff_q_mkm_extend_source_point_valuation_candidate_power_product_successor * S ((S (S ff_i_mkm_extend_source_point_valuation_candidate_power_product)) * ff_v_mkm_extend_source_point_valuation_candidate_power_product) + (ff_s_mkm_extend_source_point_valuation_candidate_power_product))) /\ ff_s_mkm_extend_source_point_valuation_candidate_power_product = ff_r_mkm_extend_source_point_valuation_candidate_power_product * ff_p_mkm_extend_source_point_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_mkm_extend_source_point_valuation_candidate_divides. (mkm_value_extend_source_point) = bpv_result_mkm_extend_source_point_valuation_candidate * bpv_factor_mkm_extend_source_point_valuation_candidate_divides))) -> (exists bpv_gap_mkm_extend_source_point_valuation_maximal. bpv_gap_mkm_extend_source_point_valuation_maximal + bpv_candidate_mkm_extend_source_point_valuation = mkm_exponent_extend_source_point))))) -> (((exists fs_h_mkm_extend_a. fs_h_mkm_extend_a + S (a) = S ((S (l)) * c)) /\ exists fs_q_mkm_extend_a. b = fs_q_mkm_extend_a * S ((S (l)) * c) + (a))) -> (((exists bpv_gap_mkm_extend_val_exponent_bound. bpv_gap_mkm_extend_val_exponent_bound + e = (a)) /\ (exists bpv_result_mkm_extend_val_selected. ((exists ff_b_mkm_extend_val_selected_power ff_c_mkm_extend_val_selected_power. ((forall ff_i_mkm_extend_val_selected_power_repeat. (exists ff_lt_mkm_extend_val_selected_power_repeat_bound. ff_lt_mkm_extend_val_selected_power_repeat_bound + S ff_i_mkm_extend_val_selected_power_repeat = e) -> (((exists ff_h_mkm_extend_val_selected_power_repeat_decoded. ff_h_mkm_extend_val_selected_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_extend_val_selected_power_repeat)) * ff_c_mkm_extend_val_selected_power)) /\ exists ff_q_mkm_extend_val_selected_power_repeat_decoded. ff_b_mkm_extend_val_selected_power = ff_q_mkm_extend_val_selected_power_repeat_decoded * S ((S (ff_i_mkm_extend_val_selected_power_repeat)) * ff_c_mkm_extend_val_selected_power) + (p)))) /\ (exists ff_u_mkm_extend_val_selected_power_product ff_v_mkm_extend_val_selected_power_product. ((((exists ff_h_mkm_extend_val_selected_power_product_start. ff_h_mkm_extend_val_selected_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_extend_val_selected_power_product)) /\ exists ff_q_mkm_extend_val_selected_power_product_start. ff_u_mkm_extend_val_selected_power_product = ff_q_mkm_extend_val_selected_power_product_start * S ((S (0)) * ff_v_mkm_extend_val_selected_power_product) + (1))) /\ ((((exists ff_h_mkm_extend_val_selected_power_product_terminal. ff_h_mkm_extend_val_selected_power_product_terminal + S (bpv_result_mkm_extend_val_selected) = S ((S (e)) * ff_v_mkm_extend_val_selected_power_product)) /\ exists ff_q_mkm_extend_val_selected_power_product_terminal. ff_u_mkm_extend_val_selected_power_product = ff_q_mkm_extend_val_selected_power_product_terminal * S ((S (e)) * ff_v_mkm_extend_val_selected_power_product) + (bpv_result_mkm_extend_val_selected))) /\ forall ff_i_mkm_extend_val_selected_power_product. (exists ff_lt_mkm_extend_val_selected_power_product_bound. ff_lt_mkm_extend_val_selected_power_product_bound + S ff_i_mkm_extend_val_selected_power_product = e) -> exists ff_p_mkm_extend_val_selected_power_product ff_r_mkm_extend_val_selected_power_product ff_s_mkm_extend_val_selected_power_product. ((((exists ff_h_mkm_extend_val_selected_power_product_factor. ff_h_mkm_extend_val_selected_power_product_factor + S (ff_p_mkm_extend_val_selected_power_product) = S ((S (ff_i_mkm_extend_val_selected_power_product)) * ff_c_mkm_extend_val_selected_power)) /\ exists ff_q_mkm_extend_val_selected_power_product_factor. ff_b_mkm_extend_val_selected_power = ff_q_mkm_extend_val_selected_power_product_factor * S ((S (ff_i_mkm_extend_val_selected_power_product)) * ff_c_mkm_extend_val_selected_power) + (ff_p_mkm_extend_val_selected_power_product))) /\ ((((exists ff_h_mkm_extend_val_selected_power_product_partial. ff_h_mkm_extend_val_selected_power_product_partial + S (ff_r_mkm_extend_val_selected_power_product) = S ((S (ff_i_mkm_extend_val_selected_power_product)) * ff_v_mkm_extend_val_selected_power_product)) /\ exists ff_q_mkm_extend_val_selected_power_product_partial. ff_u_mkm_extend_val_selected_power_product = ff_q_mkm_extend_val_selected_power_product_partial * S ((S (ff_i_mkm_extend_val_selected_power_product)) * ff_v_mkm_extend_val_selected_power_product) + (ff_r_mkm_extend_val_selected_power_product))) /\ ((((exists ff_h_mkm_extend_val_selected_power_product_successor. ff_h_mkm_extend_val_selected_power_product_successor + S (ff_s_mkm_extend_val_selected_power_product) = S ((S (S ff_i_mkm_extend_val_selected_power_product)) * ff_v_mkm_extend_val_selected_power_product)) /\ exists ff_q_mkm_extend_val_selected_power_product_successor. ff_u_mkm_extend_val_selected_power_product = ff_q_mkm_extend_val_selected_power_product_successor * S ((S (S ff_i_mkm_extend_val_selected_power_product)) * ff_v_mkm_extend_val_selected_power_product) + (ff_s_mkm_extend_val_selected_power_product))) /\ ff_s_mkm_extend_val_selected_power_product = ff_r_mkm_extend_val_selected_power_product * ff_p_mkm_extend_val_selected_power_product)))))))) /\ (exists bpv_factor_mkm_extend_val_selected_divides. (a) = bpv_result_mkm_extend_val_selected * bpv_factor_mkm_extend_val_selected_divides)))) /\ forall bpv_candidate_mkm_extend_val. (exists bpv_gap_mkm_extend_val_candidate_bound. bpv_gap_mkm_extend_val_candidate_bound + bpv_candidate_mkm_extend_val = (a)) -> (exists bpv_result_mkm_extend_val_candidate. ((exists ff_b_mkm_extend_val_candidate_power ff_c_mkm_extend_val_candidate_power. ((forall ff_i_mkm_extend_val_candidate_power_repeat. (exists ff_lt_mkm_extend_val_candidate_power_repeat_bound. ff_lt_mkm_extend_val_candidate_power_repeat_bound + S ff_i_mkm_extend_val_candidate_power_repeat = bpv_candidate_mkm_extend_val) -> (((exists ff_h_mkm_extend_val_candidate_power_repeat_decoded. ff_h_mkm_extend_val_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_extend_val_candidate_power_repeat)) * ff_c_mkm_extend_val_candidate_power)) /\ exists ff_q_mkm_extend_val_candidate_power_repeat_decoded. ff_b_mkm_extend_val_candidate_power = ff_q_mkm_extend_val_candidate_power_repeat_decoded * S ((S (ff_i_mkm_extend_val_candidate_power_repeat)) * ff_c_mkm_extend_val_candidate_power) + (p)))) /\ (exists ff_u_mkm_extend_val_candidate_power_product ff_v_mkm_extend_val_candidate_power_product. ((((exists ff_h_mkm_extend_val_candidate_power_product_start. ff_h_mkm_extend_val_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_extend_val_candidate_power_product)) /\ exists ff_q_mkm_extend_val_candidate_power_product_start. ff_u_mkm_extend_val_candidate_power_product = ff_q_mkm_extend_val_candidate_power_product_start * S ((S (0)) * ff_v_mkm_extend_val_candidate_power_product) + (1))) /\ ((((exists ff_h_mkm_extend_val_candidate_power_product_terminal. ff_h_mkm_extend_val_candidate_power_product_terminal + S (bpv_result_mkm_extend_val_candidate) = S ((S (bpv_candidate_mkm_extend_val)) * ff_v_mkm_extend_val_candidate_power_product)) /\ exists ff_q_mkm_extend_val_candidate_power_product_terminal. ff_u_mkm_extend_val_candidate_power_product = ff_q_mkm_extend_val_candidate_power_product_terminal * S ((S (bpv_candidate_mkm_extend_val)) * ff_v_mkm_extend_val_candidate_power_product) + (bpv_result_mkm_extend_val_candidate))) /\ forall ff_i_mkm_extend_val_candidate_power_product. (exists ff_lt_mkm_extend_val_candidate_power_product_bound. ff_lt_mkm_extend_val_candidate_power_product_bound + S ff_i_mkm_extend_val_candidate_power_product = bpv_candidate_mkm_extend_val) -> exists ff_p_mkm_extend_val_candidate_power_product ff_r_mkm_extend_val_candidate_power_product ff_s_mkm_extend_val_candidate_power_product. ((((exists ff_h_mkm_extend_val_candidate_power_product_factor. ff_h_mkm_extend_val_candidate_power_product_factor + S (ff_p_mkm_extend_val_candidate_power_product) = S ((S (ff_i_mkm_extend_val_candidate_power_product)) * ff_c_mkm_extend_val_candidate_power)) /\ exists ff_q_mkm_extend_val_candidate_power_product_factor. ff_b_mkm_extend_val_candidate_power = ff_q_mkm_extend_val_candidate_power_product_factor * S ((S (ff_i_mkm_extend_val_candidate_power_product)) * ff_c_mkm_extend_val_candidate_power) + (ff_p_mkm_extend_val_candidate_power_product))) /\ ((((exists ff_h_mkm_extend_val_candidate_power_product_partial. ff_h_mkm_extend_val_candidate_power_product_partial + S (ff_r_mkm_extend_val_candidate_power_product) = S ((S (ff_i_mkm_extend_val_candidate_power_product)) * ff_v_mkm_extend_val_candidate_power_product)) /\ exists ff_q_mkm_extend_val_candidate_power_product_partial. ff_u_mkm_extend_val_candidate_power_product = ff_q_mkm_extend_val_candidate_power_product_partial * S ((S (ff_i_mkm_extend_val_candidate_power_product)) * ff_v_mkm_extend_val_candidate_power_product) + (ff_r_mkm_extend_val_candidate_power_product))) /\ ((((exists ff_h_mkm_extend_val_candidate_power_product_successor. ff_h_mkm_extend_val_candidate_power_product_successor + S (ff_s_mkm_extend_val_candidate_power_product) = S ((S (S ff_i_mkm_extend_val_candidate_power_product)) * ff_v_mkm_extend_val_candidate_power_product)) /\ exists ff_q_mkm_extend_val_candidate_power_product_successor. ff_u_mkm_extend_val_candidate_power_product = ff_q_mkm_extend_val_candidate_power_product_successor * S ((S (S ff_i_mkm_extend_val_candidate_power_product)) * ff_v_mkm_extend_val_candidate_power_product) + (ff_s_mkm_extend_val_candidate_power_product))) /\ ff_s_mkm_extend_val_candidate_power_product = ff_r_mkm_extend_val_candidate_power_product * ff_p_mkm_extend_val_candidate_power_product)))))))) /\ (exists bpv_factor_mkm_extend_val_candidate_divides. (a) = bpv_result_mkm_extend_val_candidate * bpv_factor_mkm_extend_val_candidate_divides))) -> (exists bpv_gap_mkm_extend_val_maximal. bpv_gap_mkm_extend_val_maximal + bpv_candidate_mkm_extend_val = e)) -> exists wb wc. (forall mkm_index_extend_target. (exists mkm_lt_extend_target_bound. mkm_lt_extend_target_bound + S (mkm_index_extend_target) = (S l)) -> (exists mkm_value_extend_target_point mkm_exponent_extend_target_point. (((exists fs_h_mkm_extend_target_point_source. fs_h_mkm_extend_target_point_source + S (mkm_value_extend_target_point) = S ((S (mkm_index_extend_target)) * c)) /\ exists fs_q_mkm_extend_target_point_source. b = fs_q_mkm_extend_target_point_source * S ((S (mkm_index_extend_target)) * c) + (mkm_value_extend_target_point))) /\ ((((exists fs_h_mkm_extend_target_point_decoded. fs_h_mkm_extend_target_point_decoded + S (mkm_exponent_extend_target_point) = S ((S (mkm_index_extend_target)) * wc)) /\ exists fs_q_mkm_extend_target_point_decoded. wb = fs_q_mkm_extend_target_point_decoded * S ((S (mkm_index_extend_target)) * wc) + (mkm_exponent_extend_target_point))) /\ (((exists bpv_gap_mkm_extend_target_point_valuation_exponent_bound. bpv_gap_mkm_extend_target_point_valuation_exponent_bound + mkm_exponent_extend_target_point = (mkm_value_extend_target_point)) /\ (exists bpv_result_mkm_extend_target_point_valuation_selected. ((exists ff_b_mkm_extend_target_point_valuation_selected_power ff_c_mkm_extend_target_point_valuation_selected_power. ((forall ff_i_mkm_extend_target_point_valuation_selected_power_repeat. (exists ff_lt_mkm_extend_target_point_valuation_selected_power_repeat_bound. ff_lt_mkm_extend_target_point_valuation_selected_power_repeat_bound + S ff_i_mkm_extend_target_point_valuation_selected_power_repeat = mkm_exponent_extend_target_point) -> (((exists ff_h_mkm_extend_target_point_valuation_selected_power_repeat_decoded. ff_h_mkm_extend_target_point_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_extend_target_point_valuation_selected_power_repeat)) * ff_c_mkm_extend_target_point_valuation_selected_power)) /\ exists ff_q_mkm_extend_target_point_valuation_selected_power_repeat_decoded. ff_b_mkm_extend_target_point_valuation_selected_power = ff_q_mkm_extend_target_point_valuation_selected_power_repeat_decoded * S ((S (ff_i_mkm_extend_target_point_valuation_selected_power_repeat)) * ff_c_mkm_extend_target_point_valuation_selected_power) + (p)))) /\ (exists ff_u_mkm_extend_target_point_valuation_selected_power_product ff_v_mkm_extend_target_point_valuation_selected_power_product. ((((exists ff_h_mkm_extend_target_point_valuation_selected_power_product_start. ff_h_mkm_extend_target_point_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_extend_target_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_selected_power_product_start. ff_u_mkm_extend_target_point_valuation_selected_power_product = ff_q_mkm_extend_target_point_valuation_selected_power_product_start * S ((S (0)) * ff_v_mkm_extend_target_point_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_mkm_extend_target_point_valuation_selected_power_product_terminal. ff_h_mkm_extend_target_point_valuation_selected_power_product_terminal + S (bpv_result_mkm_extend_target_point_valuation_selected) = S ((S (mkm_exponent_extend_target_point)) * ff_v_mkm_extend_target_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_selected_power_product_terminal. ff_u_mkm_extend_target_point_valuation_selected_power_product = ff_q_mkm_extend_target_point_valuation_selected_power_product_terminal * S ((S (mkm_exponent_extend_target_point)) * ff_v_mkm_extend_target_point_valuation_selected_power_product) + (bpv_result_mkm_extend_target_point_valuation_selected))) /\ forall ff_i_mkm_extend_target_point_valuation_selected_power_product. (exists ff_lt_mkm_extend_target_point_valuation_selected_power_product_bound. ff_lt_mkm_extend_target_point_valuation_selected_power_product_bound + S ff_i_mkm_extend_target_point_valuation_selected_power_product = mkm_exponent_extend_target_point) -> exists ff_p_mkm_extend_target_point_valuation_selected_power_product ff_r_mkm_extend_target_point_valuation_selected_power_product ff_s_mkm_extend_target_point_valuation_selected_power_product. ((((exists ff_h_mkm_extend_target_point_valuation_selected_power_product_factor. ff_h_mkm_extend_target_point_valuation_selected_power_product_factor + S (ff_p_mkm_extend_target_point_valuation_selected_power_product) = S ((S (ff_i_mkm_extend_target_point_valuation_selected_power_product)) * ff_c_mkm_extend_target_point_valuation_selected_power)) /\ exists ff_q_mkm_extend_target_point_valuation_selected_power_product_factor. ff_b_mkm_extend_target_point_valuation_selected_power = ff_q_mkm_extend_target_point_valuation_selected_power_product_factor * S ((S (ff_i_mkm_extend_target_point_valuation_selected_power_product)) * ff_c_mkm_extend_target_point_valuation_selected_power) + (ff_p_mkm_extend_target_point_valuation_selected_power_product))) /\ ((((exists ff_h_mkm_extend_target_point_valuation_selected_power_product_partial. ff_h_mkm_extend_target_point_valuation_selected_power_product_partial + S (ff_r_mkm_extend_target_point_valuation_selected_power_product) = S ((S (ff_i_mkm_extend_target_point_valuation_selected_power_product)) * ff_v_mkm_extend_target_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_selected_power_product_partial. ff_u_mkm_extend_target_point_valuation_selected_power_product = ff_q_mkm_extend_target_point_valuation_selected_power_product_partial * S ((S (ff_i_mkm_extend_target_point_valuation_selected_power_product)) * ff_v_mkm_extend_target_point_valuation_selected_power_product) + (ff_r_mkm_extend_target_point_valuation_selected_power_product))) /\ ((((exists ff_h_mkm_extend_target_point_valuation_selected_power_product_successor. ff_h_mkm_extend_target_point_valuation_selected_power_product_successor + S (ff_s_mkm_extend_target_point_valuation_selected_power_product) = S ((S (S ff_i_mkm_extend_target_point_valuation_selected_power_product)) * ff_v_mkm_extend_target_point_valuation_selected_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_selected_power_product_successor. ff_u_mkm_extend_target_point_valuation_selected_power_product = ff_q_mkm_extend_target_point_valuation_selected_power_product_successor * S ((S (S ff_i_mkm_extend_target_point_valuation_selected_power_product)) * ff_v_mkm_extend_target_point_valuation_selected_power_product) + (ff_s_mkm_extend_target_point_valuation_selected_power_product))) /\ ff_s_mkm_extend_target_point_valuation_selected_power_product = ff_r_mkm_extend_target_point_valuation_selected_power_product * ff_p_mkm_extend_target_point_valuation_selected_power_product)))))))) /\ (exists bpv_factor_mkm_extend_target_point_valuation_selected_divides. (mkm_value_extend_target_point) = bpv_result_mkm_extend_target_point_valuation_selected * bpv_factor_mkm_extend_target_point_valuation_selected_divides)))) /\ forall bpv_candidate_mkm_extend_target_point_valuation. (exists bpv_gap_mkm_extend_target_point_valuation_candidate_bound. bpv_gap_mkm_extend_target_point_valuation_candidate_bound + bpv_candidate_mkm_extend_target_point_valuation = (mkm_value_extend_target_point)) -> (exists bpv_result_mkm_extend_target_point_valuation_candidate. ((exists ff_b_mkm_extend_target_point_valuation_candidate_power ff_c_mkm_extend_target_point_valuation_candidate_power. ((forall ff_i_mkm_extend_target_point_valuation_candidate_power_repeat. (exists ff_lt_mkm_extend_target_point_valuation_candidate_power_repeat_bound. ff_lt_mkm_extend_target_point_valuation_candidate_power_repeat_bound + S ff_i_mkm_extend_target_point_valuation_candidate_power_repeat = bpv_candidate_mkm_extend_target_point_valuation) -> (((exists ff_h_mkm_extend_target_point_valuation_candidate_power_repeat_decoded. ff_h_mkm_extend_target_point_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_mkm_extend_target_point_valuation_candidate_power_repeat)) * ff_c_mkm_extend_target_point_valuation_candidate_power)) /\ exists ff_q_mkm_extend_target_point_valuation_candidate_power_repeat_decoded. ff_b_mkm_extend_target_point_valuation_candidate_power = ff_q_mkm_extend_target_point_valuation_candidate_power_repeat_decoded * S ((S (ff_i_mkm_extend_target_point_valuation_candidate_power_repeat)) * ff_c_mkm_extend_target_point_valuation_candidate_power) + (p)))) /\ (exists ff_u_mkm_extend_target_point_valuation_candidate_power_product ff_v_mkm_extend_target_point_valuation_candidate_power_product. ((((exists ff_h_mkm_extend_target_point_valuation_candidate_power_product_start. ff_h_mkm_extend_target_point_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_candidate_power_product_start. ff_u_mkm_extend_target_point_valuation_candidate_power_product = ff_q_mkm_extend_target_point_valuation_candidate_power_product_start * S ((S (0)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_mkm_extend_target_point_valuation_candidate_power_product_terminal. ff_h_mkm_extend_target_point_valuation_candidate_power_product_terminal + S (bpv_result_mkm_extend_target_point_valuation_candidate) = S ((S (bpv_candidate_mkm_extend_target_point_valuation)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_candidate_power_product_terminal. ff_u_mkm_extend_target_point_valuation_candidate_power_product = ff_q_mkm_extend_target_point_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_mkm_extend_target_point_valuation)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product) + (bpv_result_mkm_extend_target_point_valuation_candidate))) /\ forall ff_i_mkm_extend_target_point_valuation_candidate_power_product. (exists ff_lt_mkm_extend_target_point_valuation_candidate_power_product_bound. ff_lt_mkm_extend_target_point_valuation_candidate_power_product_bound + S ff_i_mkm_extend_target_point_valuation_candidate_power_product = bpv_candidate_mkm_extend_target_point_valuation) -> exists ff_p_mkm_extend_target_point_valuation_candidate_power_product ff_r_mkm_extend_target_point_valuation_candidate_power_product ff_s_mkm_extend_target_point_valuation_candidate_power_product. ((((exists ff_h_mkm_extend_target_point_valuation_candidate_power_product_factor. ff_h_mkm_extend_target_point_valuation_candidate_power_product_factor + S (ff_p_mkm_extend_target_point_valuation_candidate_power_product) = S ((S (ff_i_mkm_extend_target_point_valuation_candidate_power_product)) * ff_c_mkm_extend_target_point_valuation_candidate_power)) /\ exists ff_q_mkm_extend_target_point_valuation_candidate_power_product_factor. ff_b_mkm_extend_target_point_valuation_candidate_power = ff_q_mkm_extend_target_point_valuation_candidate_power_product_factor * S ((S (ff_i_mkm_extend_target_point_valuation_candidate_power_product)) * ff_c_mkm_extend_target_point_valuation_candidate_power) + (ff_p_mkm_extend_target_point_valuation_candidate_power_product))) /\ ((((exists ff_h_mkm_extend_target_point_valuation_candidate_power_product_partial. ff_h_mkm_extend_target_point_valuation_candidate_power_product_partial + S (ff_r_mkm_extend_target_point_valuation_candidate_power_product) = S ((S (ff_i_mkm_extend_target_point_valuation_candidate_power_product)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_candidate_power_product_partial. ff_u_mkm_extend_target_point_valuation_candidate_power_product = ff_q_mkm_extend_target_point_valuation_candidate_power_product_partial * S ((S (ff_i_mkm_extend_target_point_valuation_candidate_power_product)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product) + (ff_r_mkm_extend_target_point_valuation_candidate_power_product))) /\ ((((exists ff_h_mkm_extend_target_point_valuation_candidate_power_product_successor. ff_h_mkm_extend_target_point_valuation_candidate_power_product_successor + S (ff_s_mkm_extend_target_point_valuation_candidate_power_product) = S ((S (S ff_i_mkm_extend_target_point_valuation_candidate_power_product)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product)) /\ exists ff_q_mkm_extend_target_point_valuation_candidate_power_product_successor. ff_u_mkm_extend_target_point_valuation_candidate_power_product = ff_q_mkm_extend_target_point_valuation_candidate_power_product_successor * S ((S (S ff_i_mkm_extend_target_point_valuation_candidate_power_product)) * ff_v_mkm_extend_target_point_valuation_candidate_power_product) + (ff_s_mkm_extend_target_point_valuation_candidate_power_product))) /\ ff_s_mkm_extend_target_point_valuation_candidate_power_product = ff_r_mkm_extend_target_point_valuation_candidate_power_product * ff_p_mkm_extend_target_point_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_mkm_extend_target_point_valuation_candidate_divides. (mkm_value_extend_target_point) = bpv_result_mkm_extend_target_point_valuation_candidate * bpv_factor_mkm_extend_target_point_valuation_candidate_divides))) -> (exists bpv_gap_mkm_extend_target_point_valuation_maximal. bpv_gap_mkm_extend_target_point_valuation_maximal + bpv_candidate_mkm_extend_target_point_valuation = mkm_exponent_extend_target_point)))))