power_divides_of_exponent_le_valuation

forall p a f k. (((exists bpv_gap_blvb_valuation_exponent_bound. bpv_gap_blvb_valuation_exponent_bound + f = a) /\ (exists bpv_result_blvb_valuation_selected. ((exists ff_b_blvb_valuation_selected_power ff_c_blvb_valuation_selected_power. ((forall ff_i_blvb_valuation_selected_power_repeat. (exists ff_lt_blvb_valuation_selected_power_repeat_bound. ff_lt_blvb_valuation_selected_power_repeat_bound + S ff_i_blvb_valuation_selected_power_repeat = f) -> (((exists ff_h_blvb_valuation_selected_power_repeat_decoded. ff_h_blvb_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_valuation_selected_power_repeat)) * ff_c_blvb_valuation_selected_power)) /\ exists ff_q_blvb_valuation_selected_power_repeat_decoded. ff_b_blvb_valuation_selected_power = ff_q_blvb_valuation_selected_power_repeat_decoded * S ((S (ff_i_blvb_valuation_selected_power_repeat)) * ff_c_blvb_valuation_selected_power) + (p)))) /\ (exists ff_u_blvb_valuation_selected_power_product ff_v_blvb_valuation_selected_power_product. ((((exists ff_h_blvb_valuation_selected_power_product_start. ff_h_blvb_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_valuation_selected_power_product)) /\ exists ff_q_blvb_valuation_selected_power_product_start. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_start * S ((S (0)) * ff_v_blvb_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_blvb_valuation_selected_power_product_terminal. ff_h_blvb_valuation_selected_power_product_terminal + S (bpv_result_blvb_valuation_selected) = S ((S (f)) * ff_v_blvb_valuation_selected_power_product)) /\ exists ff_q_blvb_valuation_selected_power_product_terminal. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_terminal * S ((S (f)) * ff_v_blvb_valuation_selected_power_product) + (bpv_result_blvb_valuation_selected))) /\ forall ff_i_blvb_valuation_selected_power_product. (exists ff_lt_blvb_valuation_selected_power_product_bound. ff_lt_blvb_valuation_selected_power_product_bound + S ff_i_blvb_valuation_selected_power_product = f) -> exists ff_p_blvb_valuation_selected_power_product ff_r_blvb_valuation_selected_power_product ff_s_blvb_valuation_selected_power_product. ((((exists ff_h_blvb_valuation_selected_power_product_factor. ff_h_blvb_valuation_selected_power_product_factor + S (ff_p_blvb_valuation_selected_power_product) = S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_c_blvb_valuation_selected_power)) /\ exists ff_q_blvb_valuation_selected_power_product_factor. ff_b_blvb_valuation_selected_power = ff_q_blvb_valuation_selected_power_product_factor * S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_c_blvb_valuation_selected_power) + (ff_p_blvb_valuation_selected_power_product))) /\ ((((exists ff_h_blvb_valuation_selected_power_product_partial. ff_h_blvb_valuation_selected_power_product_partial + S (ff_r_blvb_valuation_selected_power_product) = S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product)) /\ exists ff_q_blvb_valuation_selected_power_product_partial. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_partial * S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product) + (ff_r_blvb_valuation_selected_power_product))) /\ ((((exists ff_h_blvb_valuation_selected_power_product_successor. ff_h_blvb_valuation_selected_power_product_successor + S (ff_s_blvb_valuation_selected_power_product) = S ((S (S ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product)) /\ exists ff_q_blvb_valuation_selected_power_product_successor. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_successor * S ((S (S ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product) + (ff_s_blvb_valuation_selected_power_product))) /\ ff_s_blvb_valuation_selected_power_product = ff_r_blvb_valuation_selected_power_product * ff_p_blvb_valuation_selected_power_product)))))))) /\ (exists bpv_factor_blvb_valuation_selected_divides. a = bpv_result_blvb_valuation_selected * bpv_factor_blvb_valuation_selected_divides)))) /\ forall bpv_candidate_blvb_valuation. (exists bpv_gap_blvb_valuation_candidate_bound. bpv_gap_blvb_valuation_candidate_bound + bpv_candidate_blvb_valuation = a) -> (exists bpv_result_blvb_valuation_candidate. ((exists ff_b_blvb_valuation_candidate_power ff_c_blvb_valuation_candidate_power. ((forall ff_i_blvb_valuation_candidate_power_repeat. (exists ff_lt_blvb_valuation_candidate_power_repeat_bound. ff_lt_blvb_valuation_candidate_power_repeat_bound + S ff_i_blvb_valuation_candidate_power_repeat = bpv_candidate_blvb_valuation) -> (((exists ff_h_blvb_valuation_candidate_power_repeat_decoded. ff_h_blvb_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_valuation_candidate_power_repeat)) * ff_c_blvb_valuation_candidate_power)) /\ exists ff_q_blvb_valuation_candidate_power_repeat_decoded. ff_b_blvb_valuation_candidate_power = ff_q_blvb_valuation_candidate_power_repeat_decoded * S ((S (ff_i_blvb_valuation_candidate_power_repeat)) * ff_c_blvb_valuation_candidate_power) + (p)))) /\ (exists ff_u_blvb_valuation_candidate_power_product ff_v_blvb_valuation_candidate_power_product. ((((exists ff_h_blvb_valuation_candidate_power_product_start. ff_h_blvb_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_valuation_candidate_power_product)) /\ exists ff_q_blvb_valuation_candidate_power_product_start. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_start * S ((S (0)) * ff_v_blvb_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_blvb_valuation_candidate_power_product_terminal. ff_h_blvb_valuation_candidate_power_product_terminal + S (bpv_result_blvb_valuation_candidate) = S ((S (bpv_candidate_blvb_valuation)) * ff_v_blvb_valuation_candidate_power_product)) /\ exists ff_q_blvb_valuation_candidate_power_product_terminal. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_blvb_valuation)) * ff_v_blvb_valuation_candidate_power_product) + (bpv_result_blvb_valuation_candidate))) /\ forall ff_i_blvb_valuation_candidate_power_product. (exists ff_lt_blvb_valuation_candidate_power_product_bound. ff_lt_blvb_valuation_candidate_power_product_bound + S ff_i_blvb_valuation_candidate_power_product = bpv_candidate_blvb_valuation) -> exists ff_p_blvb_valuation_candidate_power_product ff_r_blvb_valuation_candidate_power_product ff_s_blvb_valuation_candidate_power_product. ((((exists ff_h_blvb_valuation_candidate_power_product_factor. ff_h_blvb_valuation_candidate_power_product_factor + S (ff_p_blvb_valuation_candidate_power_product) = S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_c_blvb_valuation_candidate_power)) /\ exists ff_q_blvb_valuation_candidate_power_product_factor. ff_b_blvb_valuation_candidate_power = ff_q_blvb_valuation_candidate_power_product_factor * S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_c_blvb_valuation_candidate_power) + (ff_p_blvb_valuation_candidate_power_product))) /\ ((((exists ff_h_blvb_valuation_candidate_power_product_partial. ff_h_blvb_valuation_candidate_power_product_partial + S (ff_r_blvb_valuation_candidate_power_product) = S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product)) /\ exists ff_q_blvb_valuation_candidate_power_product_partial. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_partial * S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product) + (ff_r_blvb_valuation_candidate_power_product))) /\ ((((exists ff_h_blvb_valuation_candidate_power_product_successor. ff_h_blvb_valuation_candidate_power_product_successor + S (ff_s_blvb_valuation_candidate_power_product) = S ((S (S ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product)) /\ exists ff_q_blvb_valuation_candidate_power_product_successor. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_successor * S ((S (S ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product) + (ff_s_blvb_valuation_candidate_power_product))) /\ ff_s_blvb_valuation_candidate_power_product = ff_r_blvb_valuation_candidate_power_product * ff_p_blvb_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_blvb_valuation_candidate_divides. a = bpv_result_blvb_valuation_candidate * bpv_factor_blvb_valuation_candidate_divides))) -> (exists bpv_gap_blvb_valuation_maximal. bpv_gap_blvb_valuation_maximal + bpv_candidate_blvb_valuation = f)) -> (exists bpv_gap_blvb_candidate_bound. bpv_gap_blvb_candidate_bound + k = f) -> (exists bpv_result_blvb_candidate_divides. ((exists ff_b_blvb_candidate_divides_power ff_c_blvb_candidate_divides_power. ((forall ff_i_blvb_candidate_divides_power_repeat. (exists ff_lt_blvb_candidate_divides_power_repeat_bound. ff_lt_blvb_candidate_divides_power_repeat_bound + S ff_i_blvb_candidate_divides_power_repeat = k) -> (((exists ff_h_blvb_candidate_divides_power_repeat_decoded. ff_h_blvb_candidate_divides_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_candidate_divides_power_repeat)) * ff_c_blvb_candidate_divides_power)) /\ exists ff_q_blvb_candidate_divides_power_repeat_decoded. ff_b_blvb_candidate_divides_power = ff_q_blvb_candidate_divides_power_repeat_decoded * S ((S (ff_i_blvb_candidate_divides_power_repeat)) * ff_c_blvb_candidate_divides_power) + (p)))) /\ (exists ff_u_blvb_candidate_divides_power_product ff_v_blvb_candidate_divides_power_product. ((((exists ff_h_blvb_candidate_divides_power_product_start. ff_h_blvb_candidate_divides_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_candidate_divides_power_product)) /\ exists ff_q_blvb_candidate_divides_power_product_start. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_start * S ((S (0)) * ff_v_blvb_candidate_divides_power_product) + (1))) /\ ((((exists ff_h_blvb_candidate_divides_power_product_terminal. ff_h_blvb_candidate_divides_power_product_terminal + S (bpv_result_blvb_candidate_divides) = S ((S (k)) * ff_v_blvb_candidate_divides_power_product)) /\ exists ff_q_blvb_candidate_divides_power_product_terminal. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_terminal * S ((S (k)) * ff_v_blvb_candidate_divides_power_product) + (bpv_result_blvb_candidate_divides))) /\ forall ff_i_blvb_candidate_divides_power_product. (exists ff_lt_blvb_candidate_divides_power_product_bound. ff_lt_blvb_candidate_divides_power_product_bound + S ff_i_blvb_candidate_divides_power_product = k) -> exists ff_p_blvb_candidate_divides_power_product ff_r_blvb_candidate_divides_power_product ff_s_blvb_candidate_divides_power_product. ((((exists ff_h_blvb_candidate_divides_power_product_factor. ff_h_blvb_candidate_divides_power_product_factor + S (ff_p_blvb_candidate_divides_power_product) = S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_c_blvb_candidate_divides_power)) /\ exists ff_q_blvb_candidate_divides_power_product_factor. ff_b_blvb_candidate_divides_power = ff_q_blvb_candidate_divides_power_product_factor * S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_c_blvb_candidate_divides_power) + (ff_p_blvb_candidate_divides_power_product))) /\ ((((exists ff_h_blvb_candidate_divides_power_product_partial. ff_h_blvb_candidate_divides_power_product_partial + S (ff_r_blvb_candidate_divides_power_product) = S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product)) /\ exists ff_q_blvb_candidate_divides_power_product_partial. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_partial * S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product) + (ff_r_blvb_candidate_divides_power_product))) /\ ((((exists ff_h_blvb_candidate_divides_power_product_successor. ff_h_blvb_candidate_divides_power_product_successor + S (ff_s_blvb_candidate_divides_power_product) = S ((S (S ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product)) /\ exists ff_q_blvb_candidate_divides_power_product_successor. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_successor * S ((S (S ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product) + (ff_s_blvb_candidate_divides_power_product))) /\ ff_s_blvb_candidate_divides_power_product = ff_r_blvb_candidate_divides_power_product * ff_p_blvb_candidate_divides_power_product)))))))) /\ (exists bpv_factor_blvb_candidate_divides_divides. a = bpv_result_blvb_candidate_divides * bpv_factor_blvb_candidate_divides_divides)))