prime_factorial_valuation_succ

Exact expanded PA statement

forall p n sn e f g. sn = S n -> ((~(p = 1) /\ forall frm_prime_left_bfv_prime frm_prime_right_bfv_prime. p = frm_prime_left_bfv_prime * frm_prime_right_bfv_prime -> frm_prime_left_bfv_prime = 1 \/ frm_prime_right_bfv_prime = 1)) -> (exists bfv_factorial_bfv_successor_predecessor. ((exists ff_b_bfv_successor_predecessor_factorial ff_c_bfv_successor_predecessor_factorial. ((forall ff_i_bfv_successor_predecessor_factorial_range. (exists ff_lt_bfv_successor_predecessor_factorial_range_bound. ff_lt_bfv_successor_predecessor_factorial_range_bound + S ff_i_bfv_successor_predecessor_factorial_range = n) -> (((exists ff_h_bfv_successor_predecessor_factorial_range_decoded. ff_h_bfv_successor_predecessor_factorial_range_decoded + S (1 + ff_i_bfv_successor_predecessor_factorial_range) = S ((S (ff_i_bfv_successor_predecessor_factorial_range)) * ff_c_bfv_successor_predecessor_factorial)) /\ exists ff_q_bfv_successor_predecessor_factorial_range_decoded. ff_b_bfv_successor_predecessor_factorial = ff_q_bfv_successor_predecessor_factorial_range_decoded * S ((S (ff_i_bfv_successor_predecessor_factorial_range)) * ff_c_bfv_successor_predecessor_factorial) + (1 + ff_i_bfv_successor_predecessor_factorial_range)))) /\ (exists ff_u_bfv_successor_predecessor_factorial_product ff_v_bfv_successor_predecessor_factorial_product. ((((exists ff_h_bfv_successor_predecessor_factorial_product_start. ff_h_bfv_successor_predecessor_factorial_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_predecessor_factorial_product)) /\ exists ff_q_bfv_successor_predecessor_factorial_product_start. ff_u_bfv_successor_predecessor_factorial_product = ff_q_bfv_successor_predecessor_factorial_product_start * S ((S (0)) * ff_v_bfv_successor_predecessor_factorial_product) + (1))) /\ ((((exists ff_h_bfv_successor_predecessor_factorial_product_terminal. ff_h_bfv_successor_predecessor_factorial_product_terminal + S (bfv_factorial_bfv_successor_predecessor) = S ((S (n)) * ff_v_bfv_successor_predecessor_factorial_product)) /\ exists ff_q_bfv_successor_predecessor_factorial_product_terminal. ff_u_bfv_successor_predecessor_factorial_product = ff_q_bfv_successor_predecessor_factorial_product_terminal * S ((S (n)) * ff_v_bfv_successor_predecessor_factorial_product) + (bfv_factorial_bfv_successor_predecessor))) /\ forall ff_i_bfv_successor_predecessor_factorial_product. (exists ff_lt_bfv_successor_predecessor_factorial_product_bound. ff_lt_bfv_successor_predecessor_factorial_product_bound + S ff_i_bfv_successor_predecessor_factorial_product = n) -> exists ff_p_bfv_successor_predecessor_factorial_product ff_r_bfv_successor_predecessor_factorial_product ff_s_bfv_successor_predecessor_factorial_product. ((((exists ff_h_bfv_successor_predecessor_factorial_product_factor. ff_h_bfv_successor_predecessor_factorial_product_factor + S (ff_p_bfv_successor_predecessor_factorial_product) = S ((S (ff_i_bfv_successor_predecessor_factorial_product)) * ff_c_bfv_successor_predecessor_factorial)) /\ exists ff_q_bfv_successor_predecessor_factorial_product_factor. ff_b_bfv_successor_predecessor_factorial = ff_q_bfv_successor_predecessor_factorial_product_factor * S ((S (ff_i_bfv_successor_predecessor_factorial_product)) * ff_c_bfv_successor_predecessor_factorial) + (ff_p_bfv_successor_predecessor_factorial_product))) /\ ((((exists ff_h_bfv_successor_predecessor_factorial_product_partial. ff_h_bfv_successor_predecessor_factorial_product_partial + S (ff_r_bfv_successor_predecessor_factorial_product) = S ((S (ff_i_bfv_successor_predecessor_factorial_product)) * ff_v_bfv_successor_predecessor_factorial_product)) /\ exists ff_q_bfv_successor_predecessor_factorial_product_partial. ff_u_bfv_successor_predecessor_factorial_product = ff_q_bfv_successor_predecessor_factorial_product_partial * S ((S (ff_i_bfv_successor_predecessor_factorial_product)) * ff_v_bfv_successor_predecessor_factorial_product) + (ff_r_bfv_successor_predecessor_factorial_product))) /\ ((((exists ff_h_bfv_successor_predecessor_factorial_product_successor. ff_h_bfv_successor_predecessor_factorial_product_successor + S (ff_s_bfv_successor_predecessor_factorial_product) = S ((S (S ff_i_bfv_successor_predecessor_factorial_product)) * ff_v_bfv_successor_predecessor_factorial_product)) /\ exists ff_q_bfv_successor_predecessor_factorial_product_successor. ff_u_bfv_successor_predecessor_factorial_product = ff_q_bfv_successor_predecessor_factorial_product_successor * S ((S (S ff_i_bfv_successor_predecessor_factorial_product)) * ff_v_bfv_successor_predecessor_factorial_product) + (ff_s_bfv_successor_predecessor_factorial_product))) /\ ff_s_bfv_successor_predecessor_factorial_product = ff_r_bfv_successor_predecessor_factorial_product * ff_p_bfv_successor_predecessor_factorial_product)))))))) /\ (((exists bpv_gap_bfv_successor_predecessor_valuation_exponent_bound. bpv_gap_bfv_successor_predecessor_valuation_exponent_bound + e = bfv_factorial_bfv_successor_predecessor) /\ (exists bpv_result_bfv_successor_predecessor_valuation_selected. ((exists ff_b_bfv_successor_predecessor_valuation_selected_power ff_c_bfv_successor_predecessor_valuation_selected_power. ((forall ff_i_bfv_successor_predecessor_valuation_selected_power_repeat. (exists ff_lt_bfv_successor_predecessor_valuation_selected_power_repeat_bound. ff_lt_bfv_successor_predecessor_valuation_selected_power_repeat_bound + S ff_i_bfv_successor_predecessor_valuation_selected_power_repeat = e) -> (((exists ff_h_bfv_successor_predecessor_valuation_selected_power_repeat_decoded. ff_h_bfv_successor_predecessor_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_successor_predecessor_valuation_selected_power_repeat)) * ff_c_bfv_successor_predecessor_valuation_selected_power)) /\ exists ff_q_bfv_successor_predecessor_valuation_selected_power_repeat_decoded. ff_b_bfv_successor_predecessor_valuation_selected_power = ff_q_bfv_successor_predecessor_valuation_selected_power_repeat_decoded * S ((S (ff_i_bfv_successor_predecessor_valuation_selected_power_repeat)) * ff_c_bfv_successor_predecessor_valuation_selected_power) + (p)))) /\ (exists ff_u_bfv_successor_predecessor_valuation_selected_power_product ff_v_bfv_successor_predecessor_valuation_selected_power_product. ((((exists ff_h_bfv_successor_predecessor_valuation_selected_power_product_start. ff_h_bfv_successor_predecessor_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_selected_power_product_start. ff_u_bfv_successor_predecessor_valuation_selected_power_product = ff_q_bfv_successor_predecessor_valuation_selected_power_product_start * S ((S (0)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bfv_successor_predecessor_valuation_selected_power_product_terminal. ff_h_bfv_successor_predecessor_valuation_selected_power_product_terminal + S (bpv_result_bfv_successor_predecessor_valuation_selected) = S ((S (e)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_selected_power_product_terminal. ff_u_bfv_successor_predecessor_valuation_selected_power_product = ff_q_bfv_successor_predecessor_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product) + (bpv_result_bfv_successor_predecessor_valuation_selected))) /\ forall ff_i_bfv_successor_predecessor_valuation_selected_power_product. (exists ff_lt_bfv_successor_predecessor_valuation_selected_power_product_bound. ff_lt_bfv_successor_predecessor_valuation_selected_power_product_bound + S ff_i_bfv_successor_predecessor_valuation_selected_power_product = e) -> exists ff_p_bfv_successor_predecessor_valuation_selected_power_product ff_r_bfv_successor_predecessor_valuation_selected_power_product ff_s_bfv_successor_predecessor_valuation_selected_power_product. ((((exists ff_h_bfv_successor_predecessor_valuation_selected_power_product_factor. ff_h_bfv_successor_predecessor_valuation_selected_power_product_factor + S (ff_p_bfv_successor_predecessor_valuation_selected_power_product) = S ((S (ff_i_bfv_successor_predecessor_valuation_selected_power_product)) * ff_c_bfv_successor_predecessor_valuation_selected_power)) /\ exists ff_q_bfv_successor_predecessor_valuation_selected_power_product_factor. ff_b_bfv_successor_predecessor_valuation_selected_power = ff_q_bfv_successor_predecessor_valuation_selected_power_product_factor * S ((S (ff_i_bfv_successor_predecessor_valuation_selected_power_product)) * ff_c_bfv_successor_predecessor_valuation_selected_power) + (ff_p_bfv_successor_predecessor_valuation_selected_power_product))) /\ ((((exists ff_h_bfv_successor_predecessor_valuation_selected_power_product_partial. ff_h_bfv_successor_predecessor_valuation_selected_power_product_partial + S (ff_r_bfv_successor_predecessor_valuation_selected_power_product) = S ((S (ff_i_bfv_successor_predecessor_valuation_selected_power_product)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_selected_power_product_partial. ff_u_bfv_successor_predecessor_valuation_selected_power_product = ff_q_bfv_successor_predecessor_valuation_selected_power_product_partial * S ((S (ff_i_bfv_successor_predecessor_valuation_selected_power_product)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product) + (ff_r_bfv_successor_predecessor_valuation_selected_power_product))) /\ ((((exists ff_h_bfv_successor_predecessor_valuation_selected_power_product_successor. ff_h_bfv_successor_predecessor_valuation_selected_power_product_successor + S (ff_s_bfv_successor_predecessor_valuation_selected_power_product) = S ((S (S ff_i_bfv_successor_predecessor_valuation_selected_power_product)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_selected_power_product_successor. ff_u_bfv_successor_predecessor_valuation_selected_power_product = ff_q_bfv_successor_predecessor_valuation_selected_power_product_successor * S ((S (S ff_i_bfv_successor_predecessor_valuation_selected_power_product)) * ff_v_bfv_successor_predecessor_valuation_selected_power_product) + (ff_s_bfv_successor_predecessor_valuation_selected_power_product))) /\ ff_s_bfv_successor_predecessor_valuation_selected_power_product = ff_r_bfv_successor_predecessor_valuation_selected_power_product * ff_p_bfv_successor_predecessor_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bfv_successor_predecessor_valuation_selected_divides. bfv_factorial_bfv_successor_predecessor = bpv_result_bfv_successor_predecessor_valuation_selected * bpv_factor_bfv_successor_predecessor_valuation_selected_divides)))) /\ forall bpv_candidate_bfv_successor_predecessor_valuation. (exists bpv_gap_bfv_successor_predecessor_valuation_candidate_bound. bpv_gap_bfv_successor_predecessor_valuation_candidate_bound + bpv_candidate_bfv_successor_predecessor_valuation = bfv_factorial_bfv_successor_predecessor) -> (exists bpv_result_bfv_successor_predecessor_valuation_candidate. ((exists ff_b_bfv_successor_predecessor_valuation_candidate_power ff_c_bfv_successor_predecessor_valuation_candidate_power. ((forall ff_i_bfv_successor_predecessor_valuation_candidate_power_repeat. (exists ff_lt_bfv_successor_predecessor_valuation_candidate_power_repeat_bound. ff_lt_bfv_successor_predecessor_valuation_candidate_power_repeat_bound + S ff_i_bfv_successor_predecessor_valuation_candidate_power_repeat = bpv_candidate_bfv_successor_predecessor_valuation) -> (((exists ff_h_bfv_successor_predecessor_valuation_candidate_power_repeat_decoded. ff_h_bfv_successor_predecessor_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_successor_predecessor_valuation_candidate_power_repeat)) * ff_c_bfv_successor_predecessor_valuation_candidate_power)) /\ exists ff_q_bfv_successor_predecessor_valuation_candidate_power_repeat_decoded. ff_b_bfv_successor_predecessor_valuation_candidate_power = ff_q_bfv_successor_predecessor_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bfv_successor_predecessor_valuation_candidate_power_repeat)) * ff_c_bfv_successor_predecessor_valuation_candidate_power) + (p)))) /\ (exists ff_u_bfv_successor_predecessor_valuation_candidate_power_product ff_v_bfv_successor_predecessor_valuation_candidate_power_product. ((((exists ff_h_bfv_successor_predecessor_valuation_candidate_power_product_start. ff_h_bfv_successor_predecessor_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_candidate_power_product_start. ff_u_bfv_successor_predecessor_valuation_candidate_power_product = ff_q_bfv_successor_predecessor_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bfv_successor_predecessor_valuation_candidate_power_product_terminal. ff_h_bfv_successor_predecessor_valuation_candidate_power_product_terminal + S (bpv_result_bfv_successor_predecessor_valuation_candidate) = S ((S (bpv_candidate_bfv_successor_predecessor_valuation)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_candidate_power_product_terminal. ff_u_bfv_successor_predecessor_valuation_candidate_power_product = ff_q_bfv_successor_predecessor_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_successor_predecessor_valuation)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product) + (bpv_result_bfv_successor_predecessor_valuation_candidate))) /\ forall ff_i_bfv_successor_predecessor_valuation_candidate_power_product. (exists ff_lt_bfv_successor_predecessor_valuation_candidate_power_product_bound. ff_lt_bfv_successor_predecessor_valuation_candidate_power_product_bound + S ff_i_bfv_successor_predecessor_valuation_candidate_power_product = bpv_candidate_bfv_successor_predecessor_valuation) -> exists ff_p_bfv_successor_predecessor_valuation_candidate_power_product ff_r_bfv_successor_predecessor_valuation_candidate_power_product ff_s_bfv_successor_predecessor_valuation_candidate_power_product. ((((exists ff_h_bfv_successor_predecessor_valuation_candidate_power_product_factor. ff_h_bfv_successor_predecessor_valuation_candidate_power_product_factor + S (ff_p_bfv_successor_predecessor_valuation_candidate_power_product) = S ((S (ff_i_bfv_successor_predecessor_valuation_candidate_power_product)) * ff_c_bfv_successor_predecessor_valuation_candidate_power)) /\ exists ff_q_bfv_successor_predecessor_valuation_candidate_power_product_factor. ff_b_bfv_successor_predecessor_valuation_candidate_power = ff_q_bfv_successor_predecessor_valuation_candidate_power_product_factor * S ((S (ff_i_bfv_successor_predecessor_valuation_candidate_power_product)) * ff_c_bfv_successor_predecessor_valuation_candidate_power) + (ff_p_bfv_successor_predecessor_valuation_candidate_power_product))) /\ ((((exists ff_h_bfv_successor_predecessor_valuation_candidate_power_product_partial. ff_h_bfv_successor_predecessor_valuation_candidate_power_product_partial + S (ff_r_bfv_successor_predecessor_valuation_candidate_power_product) = S ((S (ff_i_bfv_successor_predecessor_valuation_candidate_power_product)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_candidate_power_product_partial. ff_u_bfv_successor_predecessor_valuation_candidate_power_product = ff_q_bfv_successor_predecessor_valuation_candidate_power_product_partial * S ((S (ff_i_bfv_successor_predecessor_valuation_candidate_power_product)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product) + (ff_r_bfv_successor_predecessor_valuation_candidate_power_product))) /\ ((((exists ff_h_bfv_successor_predecessor_valuation_candidate_power_product_successor. ff_h_bfv_successor_predecessor_valuation_candidate_power_product_successor + S (ff_s_bfv_successor_predecessor_valuation_candidate_power_product) = S ((S (S ff_i_bfv_successor_predecessor_valuation_candidate_power_product)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_predecessor_valuation_candidate_power_product_successor. ff_u_bfv_successor_predecessor_valuation_candidate_power_product = ff_q_bfv_successor_predecessor_valuation_candidate_power_product_successor * S ((S (S ff_i_bfv_successor_predecessor_valuation_candidate_power_product)) * ff_v_bfv_successor_predecessor_valuation_candidate_power_product) + (ff_s_bfv_successor_predecessor_valuation_candidate_power_product))) /\ ff_s_bfv_successor_predecessor_valuation_candidate_power_product = ff_r_bfv_successor_predecessor_valuation_candidate_power_product * ff_p_bfv_successor_predecessor_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bfv_successor_predecessor_valuation_candidate_divides. bfv_factorial_bfv_successor_predecessor = bpv_result_bfv_successor_predecessor_valuation_candidate * bpv_factor_bfv_successor_predecessor_valuation_candidate_divides))) -> (exists bpv_gap_bfv_successor_predecessor_valuation_maximal. bpv_gap_bfv_successor_predecessor_valuation_maximal + bpv_candidate_bfv_successor_predecessor_valuation = e)))) -> (((exists bpv_gap_bfv_successor_factor_exponent_bound. bpv_gap_bfv_successor_factor_exponent_bound + f = sn) /\ (exists bpv_result_bfv_successor_factor_selected. ((exists ff_b_bfv_successor_factor_selected_power ff_c_bfv_successor_factor_selected_power. ((forall ff_i_bfv_successor_factor_selected_power_repeat. (exists ff_lt_bfv_successor_factor_selected_power_repeat_bound. ff_lt_bfv_successor_factor_selected_power_repeat_bound + S ff_i_bfv_successor_factor_selected_power_repeat = f) -> (((exists ff_h_bfv_successor_factor_selected_power_repeat_decoded. ff_h_bfv_successor_factor_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_successor_factor_selected_power_repeat)) * ff_c_bfv_successor_factor_selected_power)) /\ exists ff_q_bfv_successor_factor_selected_power_repeat_decoded. ff_b_bfv_successor_factor_selected_power = ff_q_bfv_successor_factor_selected_power_repeat_decoded * S ((S (ff_i_bfv_successor_factor_selected_power_repeat)) * ff_c_bfv_successor_factor_selected_power) + (p)))) /\ (exists ff_u_bfv_successor_factor_selected_power_product ff_v_bfv_successor_factor_selected_power_product. ((((exists ff_h_bfv_successor_factor_selected_power_product_start. ff_h_bfv_successor_factor_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_factor_selected_power_product)) /\ exists ff_q_bfv_successor_factor_selected_power_product_start. ff_u_bfv_successor_factor_selected_power_product = ff_q_bfv_successor_factor_selected_power_product_start * S ((S (0)) * ff_v_bfv_successor_factor_selected_power_product) + (1))) /\ ((((exists ff_h_bfv_successor_factor_selected_power_product_terminal. ff_h_bfv_successor_factor_selected_power_product_terminal + S (bpv_result_bfv_successor_factor_selected) = S ((S (f)) * ff_v_bfv_successor_factor_selected_power_product)) /\ exists ff_q_bfv_successor_factor_selected_power_product_terminal. ff_u_bfv_successor_factor_selected_power_product = ff_q_bfv_successor_factor_selected_power_product_terminal * S ((S (f)) * ff_v_bfv_successor_factor_selected_power_product) + (bpv_result_bfv_successor_factor_selected))) /\ forall ff_i_bfv_successor_factor_selected_power_product. (exists ff_lt_bfv_successor_factor_selected_power_product_bound. ff_lt_bfv_successor_factor_selected_power_product_bound + S ff_i_bfv_successor_factor_selected_power_product = f) -> exists ff_p_bfv_successor_factor_selected_power_product ff_r_bfv_successor_factor_selected_power_product ff_s_bfv_successor_factor_selected_power_product. ((((exists ff_h_bfv_successor_factor_selected_power_product_factor. ff_h_bfv_successor_factor_selected_power_product_factor + S (ff_p_bfv_successor_factor_selected_power_product) = S ((S (ff_i_bfv_successor_factor_selected_power_product)) * ff_c_bfv_successor_factor_selected_power)) /\ exists ff_q_bfv_successor_factor_selected_power_product_factor. ff_b_bfv_successor_factor_selected_power = ff_q_bfv_successor_factor_selected_power_product_factor * S ((S (ff_i_bfv_successor_factor_selected_power_product)) * ff_c_bfv_successor_factor_selected_power) + (ff_p_bfv_successor_factor_selected_power_product))) /\ ((((exists ff_h_bfv_successor_factor_selected_power_product_partial. ff_h_bfv_successor_factor_selected_power_product_partial + S (ff_r_bfv_successor_factor_selected_power_product) = S ((S (ff_i_bfv_successor_factor_selected_power_product)) * ff_v_bfv_successor_factor_selected_power_product)) /\ exists ff_q_bfv_successor_factor_selected_power_product_partial. ff_u_bfv_successor_factor_selected_power_product = ff_q_bfv_successor_factor_selected_power_product_partial * S ((S (ff_i_bfv_successor_factor_selected_power_product)) * ff_v_bfv_successor_factor_selected_power_product) + (ff_r_bfv_successor_factor_selected_power_product))) /\ ((((exists ff_h_bfv_successor_factor_selected_power_product_successor. ff_h_bfv_successor_factor_selected_power_product_successor + S (ff_s_bfv_successor_factor_selected_power_product) = S ((S (S ff_i_bfv_successor_factor_selected_power_product)) * ff_v_bfv_successor_factor_selected_power_product)) /\ exists ff_q_bfv_successor_factor_selected_power_product_successor. ff_u_bfv_successor_factor_selected_power_product = ff_q_bfv_successor_factor_selected_power_product_successor * S ((S (S ff_i_bfv_successor_factor_selected_power_product)) * ff_v_bfv_successor_factor_selected_power_product) + (ff_s_bfv_successor_factor_selected_power_product))) /\ ff_s_bfv_successor_factor_selected_power_product = ff_r_bfv_successor_factor_selected_power_product * ff_p_bfv_successor_factor_selected_power_product)))))))) /\ (exists bpv_factor_bfv_successor_factor_selected_divides. sn = bpv_result_bfv_successor_factor_selected * bpv_factor_bfv_successor_factor_selected_divides)))) /\ forall bpv_candidate_bfv_successor_factor. (exists bpv_gap_bfv_successor_factor_candidate_bound. bpv_gap_bfv_successor_factor_candidate_bound + bpv_candidate_bfv_successor_factor = sn) -> (exists bpv_result_bfv_successor_factor_candidate. ((exists ff_b_bfv_successor_factor_candidate_power ff_c_bfv_successor_factor_candidate_power. ((forall ff_i_bfv_successor_factor_candidate_power_repeat. (exists ff_lt_bfv_successor_factor_candidate_power_repeat_bound. ff_lt_bfv_successor_factor_candidate_power_repeat_bound + S ff_i_bfv_successor_factor_candidate_power_repeat = bpv_candidate_bfv_successor_factor) -> (((exists ff_h_bfv_successor_factor_candidate_power_repeat_decoded. ff_h_bfv_successor_factor_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_successor_factor_candidate_power_repeat)) * ff_c_bfv_successor_factor_candidate_power)) /\ exists ff_q_bfv_successor_factor_candidate_power_repeat_decoded. ff_b_bfv_successor_factor_candidate_power = ff_q_bfv_successor_factor_candidate_power_repeat_decoded * S ((S (ff_i_bfv_successor_factor_candidate_power_repeat)) * ff_c_bfv_successor_factor_candidate_power) + (p)))) /\ (exists ff_u_bfv_successor_factor_candidate_power_product ff_v_bfv_successor_factor_candidate_power_product. ((((exists ff_h_bfv_successor_factor_candidate_power_product_start. ff_h_bfv_successor_factor_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_factor_candidate_power_product)) /\ exists ff_q_bfv_successor_factor_candidate_power_product_start. ff_u_bfv_successor_factor_candidate_power_product = ff_q_bfv_successor_factor_candidate_power_product_start * S ((S (0)) * ff_v_bfv_successor_factor_candidate_power_product) + (1))) /\ ((((exists ff_h_bfv_successor_factor_candidate_power_product_terminal. ff_h_bfv_successor_factor_candidate_power_product_terminal + S (bpv_result_bfv_successor_factor_candidate) = S ((S (bpv_candidate_bfv_successor_factor)) * ff_v_bfv_successor_factor_candidate_power_product)) /\ exists ff_q_bfv_successor_factor_candidate_power_product_terminal. ff_u_bfv_successor_factor_candidate_power_product = ff_q_bfv_successor_factor_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_successor_factor)) * ff_v_bfv_successor_factor_candidate_power_product) + (bpv_result_bfv_successor_factor_candidate))) /\ forall ff_i_bfv_successor_factor_candidate_power_product. (exists ff_lt_bfv_successor_factor_candidate_power_product_bound. ff_lt_bfv_successor_factor_candidate_power_product_bound + S ff_i_bfv_successor_factor_candidate_power_product = bpv_candidate_bfv_successor_factor) -> exists ff_p_bfv_successor_factor_candidate_power_product ff_r_bfv_successor_factor_candidate_power_product ff_s_bfv_successor_factor_candidate_power_product. ((((exists ff_h_bfv_successor_factor_candidate_power_product_factor. ff_h_bfv_successor_factor_candidate_power_product_factor + S (ff_p_bfv_successor_factor_candidate_power_product) = S ((S (ff_i_bfv_successor_factor_candidate_power_product)) * ff_c_bfv_successor_factor_candidate_power)) /\ exists ff_q_bfv_successor_factor_candidate_power_product_factor. ff_b_bfv_successor_factor_candidate_power = ff_q_bfv_successor_factor_candidate_power_product_factor * S ((S (ff_i_bfv_successor_factor_candidate_power_product)) * ff_c_bfv_successor_factor_candidate_power) + (ff_p_bfv_successor_factor_candidate_power_product))) /\ ((((exists ff_h_bfv_successor_factor_candidate_power_product_partial. ff_h_bfv_successor_factor_candidate_power_product_partial + S (ff_r_bfv_successor_factor_candidate_power_product) = S ((S (ff_i_bfv_successor_factor_candidate_power_product)) * ff_v_bfv_successor_factor_candidate_power_product)) /\ exists ff_q_bfv_successor_factor_candidate_power_product_partial. ff_u_bfv_successor_factor_candidate_power_product = ff_q_bfv_successor_factor_candidate_power_product_partial * S ((S (ff_i_bfv_successor_factor_candidate_power_product)) * ff_v_bfv_successor_factor_candidate_power_product) + (ff_r_bfv_successor_factor_candidate_power_product))) /\ ((((exists ff_h_bfv_successor_factor_candidate_power_product_successor. ff_h_bfv_successor_factor_candidate_power_product_successor + S (ff_s_bfv_successor_factor_candidate_power_product) = S ((S (S ff_i_bfv_successor_factor_candidate_power_product)) * ff_v_bfv_successor_factor_candidate_power_product)) /\ exists ff_q_bfv_successor_factor_candidate_power_product_successor. ff_u_bfv_successor_factor_candidate_power_product = ff_q_bfv_successor_factor_candidate_power_product_successor * S ((S (S ff_i_bfv_successor_factor_candidate_power_product)) * ff_v_bfv_successor_factor_candidate_power_product) + (ff_s_bfv_successor_factor_candidate_power_product))) /\ ff_s_bfv_successor_factor_candidate_power_product = ff_r_bfv_successor_factor_candidate_power_product * ff_p_bfv_successor_factor_candidate_power_product)))))))) /\ (exists bpv_factor_bfv_successor_factor_candidate_divides. sn = bpv_result_bfv_successor_factor_candidate * bpv_factor_bfv_successor_factor_candidate_divides))) -> (exists bpv_gap_bfv_successor_factor_maximal. bpv_gap_bfv_successor_factor_maximal + bpv_candidate_bfv_successor_factor = f)) -> (exists bfv_factorial_bfv_successor_value. ((exists ff_b_bfv_successor_value_factorial ff_c_bfv_successor_value_factorial. ((forall ff_i_bfv_successor_value_factorial_range. (exists ff_lt_bfv_successor_value_factorial_range_bound. ff_lt_bfv_successor_value_factorial_range_bound + S ff_i_bfv_successor_value_factorial_range = sn) -> (((exists ff_h_bfv_successor_value_factorial_range_decoded. ff_h_bfv_successor_value_factorial_range_decoded + S (1 + ff_i_bfv_successor_value_factorial_range) = S ((S (ff_i_bfv_successor_value_factorial_range)) * ff_c_bfv_successor_value_factorial)) /\ exists ff_q_bfv_successor_value_factorial_range_decoded. ff_b_bfv_successor_value_factorial = ff_q_bfv_successor_value_factorial_range_decoded * S ((S (ff_i_bfv_successor_value_factorial_range)) * ff_c_bfv_successor_value_factorial) + (1 + ff_i_bfv_successor_value_factorial_range)))) /\ (exists ff_u_bfv_successor_value_factorial_product ff_v_bfv_successor_value_factorial_product. ((((exists ff_h_bfv_successor_value_factorial_product_start. ff_h_bfv_successor_value_factorial_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_value_factorial_product)) /\ exists ff_q_bfv_successor_value_factorial_product_start. ff_u_bfv_successor_value_factorial_product = ff_q_bfv_successor_value_factorial_product_start * S ((S (0)) * ff_v_bfv_successor_value_factorial_product) + (1))) /\ ((((exists ff_h_bfv_successor_value_factorial_product_terminal. ff_h_bfv_successor_value_factorial_product_terminal + S (bfv_factorial_bfv_successor_value) = S ((S (sn)) * ff_v_bfv_successor_value_factorial_product)) /\ exists ff_q_bfv_successor_value_factorial_product_terminal. ff_u_bfv_successor_value_factorial_product = ff_q_bfv_successor_value_factorial_product_terminal * S ((S (sn)) * ff_v_bfv_successor_value_factorial_product) + (bfv_factorial_bfv_successor_value))) /\ forall ff_i_bfv_successor_value_factorial_product. (exists ff_lt_bfv_successor_value_factorial_product_bound. ff_lt_bfv_successor_value_factorial_product_bound + S ff_i_bfv_successor_value_factorial_product = sn) -> exists ff_p_bfv_successor_value_factorial_product ff_r_bfv_successor_value_factorial_product ff_s_bfv_successor_value_factorial_product. ((((exists ff_h_bfv_successor_value_factorial_product_factor. ff_h_bfv_successor_value_factorial_product_factor + S (ff_p_bfv_successor_value_factorial_product) = S ((S (ff_i_bfv_successor_value_factorial_product)) * ff_c_bfv_successor_value_factorial)) /\ exists ff_q_bfv_successor_value_factorial_product_factor. ff_b_bfv_successor_value_factorial = ff_q_bfv_successor_value_factorial_product_factor * S ((S (ff_i_bfv_successor_value_factorial_product)) * ff_c_bfv_successor_value_factorial) + (ff_p_bfv_successor_value_factorial_product))) /\ ((((exists ff_h_bfv_successor_value_factorial_product_partial. ff_h_bfv_successor_value_factorial_product_partial + S (ff_r_bfv_successor_value_factorial_product) = S ((S (ff_i_bfv_successor_value_factorial_product)) * ff_v_bfv_successor_value_factorial_product)) /\ exists ff_q_bfv_successor_value_factorial_product_partial. ff_u_bfv_successor_value_factorial_product = ff_q_bfv_successor_value_factorial_product_partial * S ((S (ff_i_bfv_successor_value_factorial_product)) * ff_v_bfv_successor_value_factorial_product) + (ff_r_bfv_successor_value_factorial_product))) /\ ((((exists ff_h_bfv_successor_value_factorial_product_successor. ff_h_bfv_successor_value_factorial_product_successor + S (ff_s_bfv_successor_value_factorial_product) = S ((S (S ff_i_bfv_successor_value_factorial_product)) * ff_v_bfv_successor_value_factorial_product)) /\ exists ff_q_bfv_successor_value_factorial_product_successor. ff_u_bfv_successor_value_factorial_product = ff_q_bfv_successor_value_factorial_product_successor * S ((S (S ff_i_bfv_successor_value_factorial_product)) * ff_v_bfv_successor_value_factorial_product) + (ff_s_bfv_successor_value_factorial_product))) /\ ff_s_bfv_successor_value_factorial_product = ff_r_bfv_successor_value_factorial_product * ff_p_bfv_successor_value_factorial_product)))))))) /\ (((exists bpv_gap_bfv_successor_value_valuation_exponent_bound. bpv_gap_bfv_successor_value_valuation_exponent_bound + g = bfv_factorial_bfv_successor_value) /\ (exists bpv_result_bfv_successor_value_valuation_selected. ((exists ff_b_bfv_successor_value_valuation_selected_power ff_c_bfv_successor_value_valuation_selected_power. ((forall ff_i_bfv_successor_value_valuation_selected_power_repeat. (exists ff_lt_bfv_successor_value_valuation_selected_power_repeat_bound. ff_lt_bfv_successor_value_valuation_selected_power_repeat_bound + S ff_i_bfv_successor_value_valuation_selected_power_repeat = g) -> (((exists ff_h_bfv_successor_value_valuation_selected_power_repeat_decoded. ff_h_bfv_successor_value_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_successor_value_valuation_selected_power_repeat)) * ff_c_bfv_successor_value_valuation_selected_power)) /\ exists ff_q_bfv_successor_value_valuation_selected_power_repeat_decoded. ff_b_bfv_successor_value_valuation_selected_power = ff_q_bfv_successor_value_valuation_selected_power_repeat_decoded * S ((S (ff_i_bfv_successor_value_valuation_selected_power_repeat)) * ff_c_bfv_successor_value_valuation_selected_power) + (p)))) /\ (exists ff_u_bfv_successor_value_valuation_selected_power_product ff_v_bfv_successor_value_valuation_selected_power_product. ((((exists ff_h_bfv_successor_value_valuation_selected_power_product_start. ff_h_bfv_successor_value_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_value_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_value_valuation_selected_power_product_start. ff_u_bfv_successor_value_valuation_selected_power_product = ff_q_bfv_successor_value_valuation_selected_power_product_start * S ((S (0)) * ff_v_bfv_successor_value_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bfv_successor_value_valuation_selected_power_product_terminal. ff_h_bfv_successor_value_valuation_selected_power_product_terminal + S (bpv_result_bfv_successor_value_valuation_selected) = S ((S (g)) * ff_v_bfv_successor_value_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_value_valuation_selected_power_product_terminal. ff_u_bfv_successor_value_valuation_selected_power_product = ff_q_bfv_successor_value_valuation_selected_power_product_terminal * S ((S (g)) * ff_v_bfv_successor_value_valuation_selected_power_product) + (bpv_result_bfv_successor_value_valuation_selected))) /\ forall ff_i_bfv_successor_value_valuation_selected_power_product. (exists ff_lt_bfv_successor_value_valuation_selected_power_product_bound. ff_lt_bfv_successor_value_valuation_selected_power_product_bound + S ff_i_bfv_successor_value_valuation_selected_power_product = g) -> exists ff_p_bfv_successor_value_valuation_selected_power_product ff_r_bfv_successor_value_valuation_selected_power_product ff_s_bfv_successor_value_valuation_selected_power_product. ((((exists ff_h_bfv_successor_value_valuation_selected_power_product_factor. ff_h_bfv_successor_value_valuation_selected_power_product_factor + S (ff_p_bfv_successor_value_valuation_selected_power_product) = S ((S (ff_i_bfv_successor_value_valuation_selected_power_product)) * ff_c_bfv_successor_value_valuation_selected_power)) /\ exists ff_q_bfv_successor_value_valuation_selected_power_product_factor. ff_b_bfv_successor_value_valuation_selected_power = ff_q_bfv_successor_value_valuation_selected_power_product_factor * S ((S (ff_i_bfv_successor_value_valuation_selected_power_product)) * ff_c_bfv_successor_value_valuation_selected_power) + (ff_p_bfv_successor_value_valuation_selected_power_product))) /\ ((((exists ff_h_bfv_successor_value_valuation_selected_power_product_partial. ff_h_bfv_successor_value_valuation_selected_power_product_partial + S (ff_r_bfv_successor_value_valuation_selected_power_product) = S ((S (ff_i_bfv_successor_value_valuation_selected_power_product)) * ff_v_bfv_successor_value_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_value_valuation_selected_power_product_partial. ff_u_bfv_successor_value_valuation_selected_power_product = ff_q_bfv_successor_value_valuation_selected_power_product_partial * S ((S (ff_i_bfv_successor_value_valuation_selected_power_product)) * ff_v_bfv_successor_value_valuation_selected_power_product) + (ff_r_bfv_successor_value_valuation_selected_power_product))) /\ ((((exists ff_h_bfv_successor_value_valuation_selected_power_product_successor. ff_h_bfv_successor_value_valuation_selected_power_product_successor + S (ff_s_bfv_successor_value_valuation_selected_power_product) = S ((S (S ff_i_bfv_successor_value_valuation_selected_power_product)) * ff_v_bfv_successor_value_valuation_selected_power_product)) /\ exists ff_q_bfv_successor_value_valuation_selected_power_product_successor. ff_u_bfv_successor_value_valuation_selected_power_product = ff_q_bfv_successor_value_valuation_selected_power_product_successor * S ((S (S ff_i_bfv_successor_value_valuation_selected_power_product)) * ff_v_bfv_successor_value_valuation_selected_power_product) + (ff_s_bfv_successor_value_valuation_selected_power_product))) /\ ff_s_bfv_successor_value_valuation_selected_power_product = ff_r_bfv_successor_value_valuation_selected_power_product * ff_p_bfv_successor_value_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bfv_successor_value_valuation_selected_divides. bfv_factorial_bfv_successor_value = bpv_result_bfv_successor_value_valuation_selected * bpv_factor_bfv_successor_value_valuation_selected_divides)))) /\ forall bpv_candidate_bfv_successor_value_valuation. (exists bpv_gap_bfv_successor_value_valuation_candidate_bound. bpv_gap_bfv_successor_value_valuation_candidate_bound + bpv_candidate_bfv_successor_value_valuation = bfv_factorial_bfv_successor_value) -> (exists bpv_result_bfv_successor_value_valuation_candidate. ((exists ff_b_bfv_successor_value_valuation_candidate_power ff_c_bfv_successor_value_valuation_candidate_power. ((forall ff_i_bfv_successor_value_valuation_candidate_power_repeat. (exists ff_lt_bfv_successor_value_valuation_candidate_power_repeat_bound. ff_lt_bfv_successor_value_valuation_candidate_power_repeat_bound + S ff_i_bfv_successor_value_valuation_candidate_power_repeat = bpv_candidate_bfv_successor_value_valuation) -> (((exists ff_h_bfv_successor_value_valuation_candidate_power_repeat_decoded. ff_h_bfv_successor_value_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_successor_value_valuation_candidate_power_repeat)) * ff_c_bfv_successor_value_valuation_candidate_power)) /\ exists ff_q_bfv_successor_value_valuation_candidate_power_repeat_decoded. ff_b_bfv_successor_value_valuation_candidate_power = ff_q_bfv_successor_value_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bfv_successor_value_valuation_candidate_power_repeat)) * ff_c_bfv_successor_value_valuation_candidate_power) + (p)))) /\ (exists ff_u_bfv_successor_value_valuation_candidate_power_product ff_v_bfv_successor_value_valuation_candidate_power_product. ((((exists ff_h_bfv_successor_value_valuation_candidate_power_product_start. ff_h_bfv_successor_value_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_successor_value_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_value_valuation_candidate_power_product_start. ff_u_bfv_successor_value_valuation_candidate_power_product = ff_q_bfv_successor_value_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bfv_successor_value_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bfv_successor_value_valuation_candidate_power_product_terminal. ff_h_bfv_successor_value_valuation_candidate_power_product_terminal + S (bpv_result_bfv_successor_value_valuation_candidate) = S ((S (bpv_candidate_bfv_successor_value_valuation)) * ff_v_bfv_successor_value_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_value_valuation_candidate_power_product_terminal. ff_u_bfv_successor_value_valuation_candidate_power_product = ff_q_bfv_successor_value_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_successor_value_valuation)) * ff_v_bfv_successor_value_valuation_candidate_power_product) + (bpv_result_bfv_successor_value_valuation_candidate))) /\ forall ff_i_bfv_successor_value_valuation_candidate_power_product. (exists ff_lt_bfv_successor_value_valuation_candidate_power_product_bound. ff_lt_bfv_successor_value_valuation_candidate_power_product_bound + S ff_i_bfv_successor_value_valuation_candidate_power_product = bpv_candidate_bfv_successor_value_valuation) -> exists ff_p_bfv_successor_value_valuation_candidate_power_product ff_r_bfv_successor_value_valuation_candidate_power_product ff_s_bfv_successor_value_valuation_candidate_power_product. ((((exists ff_h_bfv_successor_value_valuation_candidate_power_product_factor. ff_h_bfv_successor_value_valuation_candidate_power_product_factor + S (ff_p_bfv_successor_value_valuation_candidate_power_product) = S ((S (ff_i_bfv_successor_value_valuation_candidate_power_product)) * ff_c_bfv_successor_value_valuation_candidate_power)) /\ exists ff_q_bfv_successor_value_valuation_candidate_power_product_factor. ff_b_bfv_successor_value_valuation_candidate_power = ff_q_bfv_successor_value_valuation_candidate_power_product_factor * S ((S (ff_i_bfv_successor_value_valuation_candidate_power_product)) * ff_c_bfv_successor_value_valuation_candidate_power) + (ff_p_bfv_successor_value_valuation_candidate_power_product))) /\ ((((exists ff_h_bfv_successor_value_valuation_candidate_power_product_partial. ff_h_bfv_successor_value_valuation_candidate_power_product_partial + S (ff_r_bfv_successor_value_valuation_candidate_power_product) = S ((S (ff_i_bfv_successor_value_valuation_candidate_power_product)) * ff_v_bfv_successor_value_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_value_valuation_candidate_power_product_partial. ff_u_bfv_successor_value_valuation_candidate_power_product = ff_q_bfv_successor_value_valuation_candidate_power_product_partial * S ((S (ff_i_bfv_successor_value_valuation_candidate_power_product)) * ff_v_bfv_successor_value_valuation_candidate_power_product) + (ff_r_bfv_successor_value_valuation_candidate_power_product))) /\ ((((exists ff_h_bfv_successor_value_valuation_candidate_power_product_successor. ff_h_bfv_successor_value_valuation_candidate_power_product_successor + S (ff_s_bfv_successor_value_valuation_candidate_power_product) = S ((S (S ff_i_bfv_successor_value_valuation_candidate_power_product)) * ff_v_bfv_successor_value_valuation_candidate_power_product)) /\ exists ff_q_bfv_successor_value_valuation_candidate_power_product_successor. ff_u_bfv_successor_value_valuation_candidate_power_product = ff_q_bfv_successor_value_valuation_candidate_power_product_successor * S ((S (S ff_i_bfv_successor_value_valuation_candidate_power_product)) * ff_v_bfv_successor_value_valuation_candidate_power_product) + (ff_s_bfv_successor_value_valuation_candidate_power_product))) /\ ff_s_bfv_successor_value_valuation_candidate_power_product = ff_r_bfv_successor_value_valuation_candidate_power_product * ff_p_bfv_successor_value_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bfv_successor_value_valuation_candidate_divides. bfv_factorial_bfv_successor_value = bpv_result_bfv_successor_value_valuation_candidate * bpv_factor_bfv_successor_value_valuation_candidate_divides))) -> (exists bpv_gap_bfv_successor_value_valuation_maximal. bpv_gap_bfv_successor_value_valuation_maximal + bpv_candidate_bfv_successor_value_valuation = g)))) -> g = e + f