prime_power_valuation_square_factor_preserves_evenness

Exact expanded first-order arithmetic statement

forall p z n e f g k. ((~(p = 1) /\ forall frm_prime_left_ftsv_prime frm_prime_right_ftsv_prime. p = frm_prime_left_ftsv_prime * frm_prime_right_ftsv_prime -> frm_prime_left_ftsv_prime = 1 \/ frm_prime_right_ftsv_prime = 1)) -> ~(z = 0) -> ~(n = 0) -> (((exists bpv_gap_ftsv_shift_factor_exponent_bound. bpv_gap_ftsv_shift_factor_exponent_bound + e = z) /\ (exists bpv_result_ftsv_shift_factor_selected. ((exists ff_b_ftsv_shift_factor_selected_power ff_c_ftsv_shift_factor_selected_power. ((forall ff_i_ftsv_shift_factor_selected_power_repeat. (exists ff_lt_ftsv_shift_factor_selected_power_repeat_bound. ff_lt_ftsv_shift_factor_selected_power_repeat_bound + S ff_i_ftsv_shift_factor_selected_power_repeat = e) -> (((exists ff_h_ftsv_shift_factor_selected_power_repeat_decoded. ff_h_ftsv_shift_factor_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_shift_factor_selected_power_repeat)) * ff_c_ftsv_shift_factor_selected_power)) /\ exists ff_q_ftsv_shift_factor_selected_power_repeat_decoded. ff_b_ftsv_shift_factor_selected_power = ff_q_ftsv_shift_factor_selected_power_repeat_decoded * S ((S (ff_i_ftsv_shift_factor_selected_power_repeat)) * ff_c_ftsv_shift_factor_selected_power) + (p)))) /\ (exists ff_u_ftsv_shift_factor_selected_power_product ff_v_ftsv_shift_factor_selected_power_product. ((((exists ff_h_ftsv_shift_factor_selected_power_product_start. ff_h_ftsv_shift_factor_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_shift_factor_selected_power_product)) /\ exists ff_q_ftsv_shift_factor_selected_power_product_start. ff_u_ftsv_shift_factor_selected_power_product = ff_q_ftsv_shift_factor_selected_power_product_start * S ((S (0)) * ff_v_ftsv_shift_factor_selected_power_product) + (1))) /\ ((((exists ff_h_ftsv_shift_factor_selected_power_product_terminal. ff_h_ftsv_shift_factor_selected_power_product_terminal + S (bpv_result_ftsv_shift_factor_selected) = S ((S (e)) * ff_v_ftsv_shift_factor_selected_power_product)) /\ exists ff_q_ftsv_shift_factor_selected_power_product_terminal. ff_u_ftsv_shift_factor_selected_power_product = ff_q_ftsv_shift_factor_selected_power_product_terminal * S ((S (e)) * ff_v_ftsv_shift_factor_selected_power_product) + (bpv_result_ftsv_shift_factor_selected))) /\ forall ff_i_ftsv_shift_factor_selected_power_product. (exists ff_lt_ftsv_shift_factor_selected_power_product_bound. ff_lt_ftsv_shift_factor_selected_power_product_bound + S ff_i_ftsv_shift_factor_selected_power_product = e) -> exists ff_p_ftsv_shift_factor_selected_power_product ff_r_ftsv_shift_factor_selected_power_product ff_s_ftsv_shift_factor_selected_power_product. ((((exists ff_h_ftsv_shift_factor_selected_power_product_factor. ff_h_ftsv_shift_factor_selected_power_product_factor + S (ff_p_ftsv_shift_factor_selected_power_product) = S ((S (ff_i_ftsv_shift_factor_selected_power_product)) * ff_c_ftsv_shift_factor_selected_power)) /\ exists ff_q_ftsv_shift_factor_selected_power_product_factor. ff_b_ftsv_shift_factor_selected_power = ff_q_ftsv_shift_factor_selected_power_product_factor * S ((S (ff_i_ftsv_shift_factor_selected_power_product)) * ff_c_ftsv_shift_factor_selected_power) + (ff_p_ftsv_shift_factor_selected_power_product))) /\ ((((exists ff_h_ftsv_shift_factor_selected_power_product_partial. ff_h_ftsv_shift_factor_selected_power_product_partial + S (ff_r_ftsv_shift_factor_selected_power_product) = S ((S (ff_i_ftsv_shift_factor_selected_power_product)) * ff_v_ftsv_shift_factor_selected_power_product)) /\ exists ff_q_ftsv_shift_factor_selected_power_product_partial. ff_u_ftsv_shift_factor_selected_power_product = ff_q_ftsv_shift_factor_selected_power_product_partial * S ((S (ff_i_ftsv_shift_factor_selected_power_product)) * ff_v_ftsv_shift_factor_selected_power_product) + (ff_r_ftsv_shift_factor_selected_power_product))) /\ ((((exists ff_h_ftsv_shift_factor_selected_power_product_successor. ff_h_ftsv_shift_factor_selected_power_product_successor + S (ff_s_ftsv_shift_factor_selected_power_product) = S ((S (S ff_i_ftsv_shift_factor_selected_power_product)) * ff_v_ftsv_shift_factor_selected_power_product)) /\ exists ff_q_ftsv_shift_factor_selected_power_product_successor. ff_u_ftsv_shift_factor_selected_power_product = ff_q_ftsv_shift_factor_selected_power_product_successor * S ((S (S ff_i_ftsv_shift_factor_selected_power_product)) * ff_v_ftsv_shift_factor_selected_power_product) + (ff_s_ftsv_shift_factor_selected_power_product))) /\ ff_s_ftsv_shift_factor_selected_power_product = ff_r_ftsv_shift_factor_selected_power_product * ff_p_ftsv_shift_factor_selected_power_product)))))))) /\ (exists bpv_factor_ftsv_shift_factor_selected_divides. z = bpv_result_ftsv_shift_factor_selected * bpv_factor_ftsv_shift_factor_selected_divides)))) /\ forall bpv_candidate_ftsv_shift_factor. (exists bpv_gap_ftsv_shift_factor_candidate_bound. bpv_gap_ftsv_shift_factor_candidate_bound + bpv_candidate_ftsv_shift_factor = z) -> (exists bpv_result_ftsv_shift_factor_candidate. ((exists ff_b_ftsv_shift_factor_candidate_power ff_c_ftsv_shift_factor_candidate_power. ((forall ff_i_ftsv_shift_factor_candidate_power_repeat. (exists ff_lt_ftsv_shift_factor_candidate_power_repeat_bound. ff_lt_ftsv_shift_factor_candidate_power_repeat_bound + S ff_i_ftsv_shift_factor_candidate_power_repeat = bpv_candidate_ftsv_shift_factor) -> (((exists ff_h_ftsv_shift_factor_candidate_power_repeat_decoded. ff_h_ftsv_shift_factor_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_shift_factor_candidate_power_repeat)) * ff_c_ftsv_shift_factor_candidate_power)) /\ exists ff_q_ftsv_shift_factor_candidate_power_repeat_decoded. ff_b_ftsv_shift_factor_candidate_power = ff_q_ftsv_shift_factor_candidate_power_repeat_decoded * S ((S (ff_i_ftsv_shift_factor_candidate_power_repeat)) * ff_c_ftsv_shift_factor_candidate_power) + (p)))) /\ (exists ff_u_ftsv_shift_factor_candidate_power_product ff_v_ftsv_shift_factor_candidate_power_product. ((((exists ff_h_ftsv_shift_factor_candidate_power_product_start. ff_h_ftsv_shift_factor_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_shift_factor_candidate_power_product)) /\ exists ff_q_ftsv_shift_factor_candidate_power_product_start. ff_u_ftsv_shift_factor_candidate_power_product = ff_q_ftsv_shift_factor_candidate_power_product_start * S ((S (0)) * ff_v_ftsv_shift_factor_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsv_shift_factor_candidate_power_product_terminal. ff_h_ftsv_shift_factor_candidate_power_product_terminal + S (bpv_result_ftsv_shift_factor_candidate) = S ((S (bpv_candidate_ftsv_shift_factor)) * ff_v_ftsv_shift_factor_candidate_power_product)) /\ exists ff_q_ftsv_shift_factor_candidate_power_product_terminal. ff_u_ftsv_shift_factor_candidate_power_product = ff_q_ftsv_shift_factor_candidate_power_product_terminal * S ((S (bpv_candidate_ftsv_shift_factor)) * ff_v_ftsv_shift_factor_candidate_power_product) + (bpv_result_ftsv_shift_factor_candidate))) /\ forall ff_i_ftsv_shift_factor_candidate_power_product. (exists ff_lt_ftsv_shift_factor_candidate_power_product_bound. ff_lt_ftsv_shift_factor_candidate_power_product_bound + S ff_i_ftsv_shift_factor_candidate_power_product = bpv_candidate_ftsv_shift_factor) -> exists ff_p_ftsv_shift_factor_candidate_power_product ff_r_ftsv_shift_factor_candidate_power_product ff_s_ftsv_shift_factor_candidate_power_product. ((((exists ff_h_ftsv_shift_factor_candidate_power_product_factor. ff_h_ftsv_shift_factor_candidate_power_product_factor + S (ff_p_ftsv_shift_factor_candidate_power_product) = S ((S (ff_i_ftsv_shift_factor_candidate_power_product)) * ff_c_ftsv_shift_factor_candidate_power)) /\ exists ff_q_ftsv_shift_factor_candidate_power_product_factor. ff_b_ftsv_shift_factor_candidate_power = ff_q_ftsv_shift_factor_candidate_power_product_factor * S ((S (ff_i_ftsv_shift_factor_candidate_power_product)) * ff_c_ftsv_shift_factor_candidate_power) + (ff_p_ftsv_shift_factor_candidate_power_product))) /\ ((((exists ff_h_ftsv_shift_factor_candidate_power_product_partial. ff_h_ftsv_shift_factor_candidate_power_product_partial + S (ff_r_ftsv_shift_factor_candidate_power_product) = S ((S (ff_i_ftsv_shift_factor_candidate_power_product)) * ff_v_ftsv_shift_factor_candidate_power_product)) /\ exists ff_q_ftsv_shift_factor_candidate_power_product_partial. ff_u_ftsv_shift_factor_candidate_power_product = ff_q_ftsv_shift_factor_candidate_power_product_partial * S ((S (ff_i_ftsv_shift_factor_candidate_power_product)) * ff_v_ftsv_shift_factor_candidate_power_product) + (ff_r_ftsv_shift_factor_candidate_power_product))) /\ ((((exists ff_h_ftsv_shift_factor_candidate_power_product_successor. ff_h_ftsv_shift_factor_candidate_power_product_successor + S (ff_s_ftsv_shift_factor_candidate_power_product) = S ((S (S ff_i_ftsv_shift_factor_candidate_power_product)) * ff_v_ftsv_shift_factor_candidate_power_product)) /\ exists ff_q_ftsv_shift_factor_candidate_power_product_successor. ff_u_ftsv_shift_factor_candidate_power_product = ff_q_ftsv_shift_factor_candidate_power_product_successor * S ((S (S ff_i_ftsv_shift_factor_candidate_power_product)) * ff_v_ftsv_shift_factor_candidate_power_product) + (ff_s_ftsv_shift_factor_candidate_power_product))) /\ ff_s_ftsv_shift_factor_candidate_power_product = ff_r_ftsv_shift_factor_candidate_power_product * ff_p_ftsv_shift_factor_candidate_power_product)))))))) /\ (exists bpv_factor_ftsv_shift_factor_candidate_divides. z = bpv_result_ftsv_shift_factor_candidate * bpv_factor_ftsv_shift_factor_candidate_divides))) -> (exists bpv_gap_ftsv_shift_factor_maximal. bpv_gap_ftsv_shift_factor_maximal + bpv_candidate_ftsv_shift_factor = e)) -> (((exists bpv_gap_ftsv_shift_value_exponent_bound. bpv_gap_ftsv_shift_value_exponent_bound + f = n) /\ (exists bpv_result_ftsv_shift_value_selected. ((exists ff_b_ftsv_shift_value_selected_power ff_c_ftsv_shift_value_selected_power. ((forall ff_i_ftsv_shift_value_selected_power_repeat. (exists ff_lt_ftsv_shift_value_selected_power_repeat_bound. ff_lt_ftsv_shift_value_selected_power_repeat_bound + S ff_i_ftsv_shift_value_selected_power_repeat = f) -> (((exists ff_h_ftsv_shift_value_selected_power_repeat_decoded. ff_h_ftsv_shift_value_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_shift_value_selected_power_repeat)) * ff_c_ftsv_shift_value_selected_power)) /\ exists ff_q_ftsv_shift_value_selected_power_repeat_decoded. ff_b_ftsv_shift_value_selected_power = ff_q_ftsv_shift_value_selected_power_repeat_decoded * S ((S (ff_i_ftsv_shift_value_selected_power_repeat)) * ff_c_ftsv_shift_value_selected_power) + (p)))) /\ (exists ff_u_ftsv_shift_value_selected_power_product ff_v_ftsv_shift_value_selected_power_product. ((((exists ff_h_ftsv_shift_value_selected_power_product_start. ff_h_ftsv_shift_value_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_shift_value_selected_power_product)) /\ exists ff_q_ftsv_shift_value_selected_power_product_start. ff_u_ftsv_shift_value_selected_power_product = ff_q_ftsv_shift_value_selected_power_product_start * S ((S (0)) * ff_v_ftsv_shift_value_selected_power_product) + (1))) /\ ((((exists ff_h_ftsv_shift_value_selected_power_product_terminal. ff_h_ftsv_shift_value_selected_power_product_terminal + S (bpv_result_ftsv_shift_value_selected) = S ((S (f)) * ff_v_ftsv_shift_value_selected_power_product)) /\ exists ff_q_ftsv_shift_value_selected_power_product_terminal. ff_u_ftsv_shift_value_selected_power_product = ff_q_ftsv_shift_value_selected_power_product_terminal * S ((S (f)) * ff_v_ftsv_shift_value_selected_power_product) + (bpv_result_ftsv_shift_value_selected))) /\ forall ff_i_ftsv_shift_value_selected_power_product. (exists ff_lt_ftsv_shift_value_selected_power_product_bound. ff_lt_ftsv_shift_value_selected_power_product_bound + S ff_i_ftsv_shift_value_selected_power_product = f) -> exists ff_p_ftsv_shift_value_selected_power_product ff_r_ftsv_shift_value_selected_power_product ff_s_ftsv_shift_value_selected_power_product. ((((exists ff_h_ftsv_shift_value_selected_power_product_factor. ff_h_ftsv_shift_value_selected_power_product_factor + S (ff_p_ftsv_shift_value_selected_power_product) = S ((S (ff_i_ftsv_shift_value_selected_power_product)) * ff_c_ftsv_shift_value_selected_power)) /\ exists ff_q_ftsv_shift_value_selected_power_product_factor. ff_b_ftsv_shift_value_selected_power = ff_q_ftsv_shift_value_selected_power_product_factor * S ((S (ff_i_ftsv_shift_value_selected_power_product)) * ff_c_ftsv_shift_value_selected_power) + (ff_p_ftsv_shift_value_selected_power_product))) /\ ((((exists ff_h_ftsv_shift_value_selected_power_product_partial. ff_h_ftsv_shift_value_selected_power_product_partial + S (ff_r_ftsv_shift_value_selected_power_product) = S ((S (ff_i_ftsv_shift_value_selected_power_product)) * ff_v_ftsv_shift_value_selected_power_product)) /\ exists ff_q_ftsv_shift_value_selected_power_product_partial. ff_u_ftsv_shift_value_selected_power_product = ff_q_ftsv_shift_value_selected_power_product_partial * S ((S (ff_i_ftsv_shift_value_selected_power_product)) * ff_v_ftsv_shift_value_selected_power_product) + (ff_r_ftsv_shift_value_selected_power_product))) /\ ((((exists ff_h_ftsv_shift_value_selected_power_product_successor. ff_h_ftsv_shift_value_selected_power_product_successor + S (ff_s_ftsv_shift_value_selected_power_product) = S ((S (S ff_i_ftsv_shift_value_selected_power_product)) * ff_v_ftsv_shift_value_selected_power_product)) /\ exists ff_q_ftsv_shift_value_selected_power_product_successor. ff_u_ftsv_shift_value_selected_power_product = ff_q_ftsv_shift_value_selected_power_product_successor * S ((S (S ff_i_ftsv_shift_value_selected_power_product)) * ff_v_ftsv_shift_value_selected_power_product) + (ff_s_ftsv_shift_value_selected_power_product))) /\ ff_s_ftsv_shift_value_selected_power_product = ff_r_ftsv_shift_value_selected_power_product * ff_p_ftsv_shift_value_selected_power_product)))))))) /\ (exists bpv_factor_ftsv_shift_value_selected_divides. n = bpv_result_ftsv_shift_value_selected * bpv_factor_ftsv_shift_value_selected_divides)))) /\ forall bpv_candidate_ftsv_shift_value. (exists bpv_gap_ftsv_shift_value_candidate_bound. bpv_gap_ftsv_shift_value_candidate_bound + bpv_candidate_ftsv_shift_value = n) -> (exists bpv_result_ftsv_shift_value_candidate. ((exists ff_b_ftsv_shift_value_candidate_power ff_c_ftsv_shift_value_candidate_power. ((forall ff_i_ftsv_shift_value_candidate_power_repeat. (exists ff_lt_ftsv_shift_value_candidate_power_repeat_bound. ff_lt_ftsv_shift_value_candidate_power_repeat_bound + S ff_i_ftsv_shift_value_candidate_power_repeat = bpv_candidate_ftsv_shift_value) -> (((exists ff_h_ftsv_shift_value_candidate_power_repeat_decoded. ff_h_ftsv_shift_value_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_shift_value_candidate_power_repeat)) * ff_c_ftsv_shift_value_candidate_power)) /\ exists ff_q_ftsv_shift_value_candidate_power_repeat_decoded. ff_b_ftsv_shift_value_candidate_power = ff_q_ftsv_shift_value_candidate_power_repeat_decoded * S ((S (ff_i_ftsv_shift_value_candidate_power_repeat)) * ff_c_ftsv_shift_value_candidate_power) + (p)))) /\ (exists ff_u_ftsv_shift_value_candidate_power_product ff_v_ftsv_shift_value_candidate_power_product. ((((exists ff_h_ftsv_shift_value_candidate_power_product_start. ff_h_ftsv_shift_value_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_shift_value_candidate_power_product)) /\ exists ff_q_ftsv_shift_value_candidate_power_product_start. ff_u_ftsv_shift_value_candidate_power_product = ff_q_ftsv_shift_value_candidate_power_product_start * S ((S (0)) * ff_v_ftsv_shift_value_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsv_shift_value_candidate_power_product_terminal. ff_h_ftsv_shift_value_candidate_power_product_terminal + S (bpv_result_ftsv_shift_value_candidate) = S ((S (bpv_candidate_ftsv_shift_value)) * ff_v_ftsv_shift_value_candidate_power_product)) /\ exists ff_q_ftsv_shift_value_candidate_power_product_terminal. ff_u_ftsv_shift_value_candidate_power_product = ff_q_ftsv_shift_value_candidate_power_product_terminal * S ((S (bpv_candidate_ftsv_shift_value)) * ff_v_ftsv_shift_value_candidate_power_product) + (bpv_result_ftsv_shift_value_candidate))) /\ forall ff_i_ftsv_shift_value_candidate_power_product. (exists ff_lt_ftsv_shift_value_candidate_power_product_bound. ff_lt_ftsv_shift_value_candidate_power_product_bound + S ff_i_ftsv_shift_value_candidate_power_product = bpv_candidate_ftsv_shift_value) -> exists ff_p_ftsv_shift_value_candidate_power_product ff_r_ftsv_shift_value_candidate_power_product ff_s_ftsv_shift_value_candidate_power_product. ((((exists ff_h_ftsv_shift_value_candidate_power_product_factor. ff_h_ftsv_shift_value_candidate_power_product_factor + S (ff_p_ftsv_shift_value_candidate_power_product) = S ((S (ff_i_ftsv_shift_value_candidate_power_product)) * ff_c_ftsv_shift_value_candidate_power)) /\ exists ff_q_ftsv_shift_value_candidate_power_product_factor. ff_b_ftsv_shift_value_candidate_power = ff_q_ftsv_shift_value_candidate_power_product_factor * S ((S (ff_i_ftsv_shift_value_candidate_power_product)) * ff_c_ftsv_shift_value_candidate_power) + (ff_p_ftsv_shift_value_candidate_power_product))) /\ ((((exists ff_h_ftsv_shift_value_candidate_power_product_partial. ff_h_ftsv_shift_value_candidate_power_product_partial + S (ff_r_ftsv_shift_value_candidate_power_product) = S ((S (ff_i_ftsv_shift_value_candidate_power_product)) * ff_v_ftsv_shift_value_candidate_power_product)) /\ exists ff_q_ftsv_shift_value_candidate_power_product_partial. ff_u_ftsv_shift_value_candidate_power_product = ff_q_ftsv_shift_value_candidate_power_product_partial * S ((S (ff_i_ftsv_shift_value_candidate_power_product)) * ff_v_ftsv_shift_value_candidate_power_product) + (ff_r_ftsv_shift_value_candidate_power_product))) /\ ((((exists ff_h_ftsv_shift_value_candidate_power_product_successor. ff_h_ftsv_shift_value_candidate_power_product_successor + S (ff_s_ftsv_shift_value_candidate_power_product) = S ((S (S ff_i_ftsv_shift_value_candidate_power_product)) * ff_v_ftsv_shift_value_candidate_power_product)) /\ exists ff_q_ftsv_shift_value_candidate_power_product_successor. ff_u_ftsv_shift_value_candidate_power_product = ff_q_ftsv_shift_value_candidate_power_product_successor * S ((S (S ff_i_ftsv_shift_value_candidate_power_product)) * ff_v_ftsv_shift_value_candidate_power_product) + (ff_s_ftsv_shift_value_candidate_power_product))) /\ ff_s_ftsv_shift_value_candidate_power_product = ff_r_ftsv_shift_value_candidate_power_product * ff_p_ftsv_shift_value_candidate_power_product)))))))) /\ (exists bpv_factor_ftsv_shift_value_candidate_divides. n = bpv_result_ftsv_shift_value_candidate * bpv_factor_ftsv_shift_value_candidate_divides))) -> (exists bpv_gap_ftsv_shift_value_maximal. bpv_gap_ftsv_shift_value_maximal + bpv_candidate_ftsv_shift_value = f)) -> (((exists bpv_gap_ftsv_shift_product_exponent_bound. bpv_gap_ftsv_shift_product_exponent_bound + g = ((z * z) * n)) /\ (exists bpv_result_ftsv_shift_product_selected. ((exists ff_b_ftsv_shift_product_selected_power ff_c_ftsv_shift_product_selected_power. ((forall ff_i_ftsv_shift_product_selected_power_repeat. (exists ff_lt_ftsv_shift_product_selected_power_repeat_bound. ff_lt_ftsv_shift_product_selected_power_repeat_bound + S ff_i_ftsv_shift_product_selected_power_repeat = g) -> (((exists ff_h_ftsv_shift_product_selected_power_repeat_decoded. ff_h_ftsv_shift_product_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_shift_product_selected_power_repeat)) * ff_c_ftsv_shift_product_selected_power)) /\ exists ff_q_ftsv_shift_product_selected_power_repeat_decoded. ff_b_ftsv_shift_product_selected_power = ff_q_ftsv_shift_product_selected_power_repeat_decoded * S ((S (ff_i_ftsv_shift_product_selected_power_repeat)) * ff_c_ftsv_shift_product_selected_power) + (p)))) /\ (exists ff_u_ftsv_shift_product_selected_power_product ff_v_ftsv_shift_product_selected_power_product. ((((exists ff_h_ftsv_shift_product_selected_power_product_start. ff_h_ftsv_shift_product_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_shift_product_selected_power_product)) /\ exists ff_q_ftsv_shift_product_selected_power_product_start. ff_u_ftsv_shift_product_selected_power_product = ff_q_ftsv_shift_product_selected_power_product_start * S ((S (0)) * ff_v_ftsv_shift_product_selected_power_product) + (1))) /\ ((((exists ff_h_ftsv_shift_product_selected_power_product_terminal. ff_h_ftsv_shift_product_selected_power_product_terminal + S (bpv_result_ftsv_shift_product_selected) = S ((S (g)) * ff_v_ftsv_shift_product_selected_power_product)) /\ exists ff_q_ftsv_shift_product_selected_power_product_terminal. ff_u_ftsv_shift_product_selected_power_product = ff_q_ftsv_shift_product_selected_power_product_terminal * S ((S (g)) * ff_v_ftsv_shift_product_selected_power_product) + (bpv_result_ftsv_shift_product_selected))) /\ forall ff_i_ftsv_shift_product_selected_power_product. (exists ff_lt_ftsv_shift_product_selected_power_product_bound. ff_lt_ftsv_shift_product_selected_power_product_bound + S ff_i_ftsv_shift_product_selected_power_product = g) -> exists ff_p_ftsv_shift_product_selected_power_product ff_r_ftsv_shift_product_selected_power_product ff_s_ftsv_shift_product_selected_power_product. ((((exists ff_h_ftsv_shift_product_selected_power_product_factor. ff_h_ftsv_shift_product_selected_power_product_factor + S (ff_p_ftsv_shift_product_selected_power_product) = S ((S (ff_i_ftsv_shift_product_selected_power_product)) * ff_c_ftsv_shift_product_selected_power)) /\ exists ff_q_ftsv_shift_product_selected_power_product_factor. ff_b_ftsv_shift_product_selected_power = ff_q_ftsv_shift_product_selected_power_product_factor * S ((S (ff_i_ftsv_shift_product_selected_power_product)) * ff_c_ftsv_shift_product_selected_power) + (ff_p_ftsv_shift_product_selected_power_product))) /\ ((((exists ff_h_ftsv_shift_product_selected_power_product_partial. ff_h_ftsv_shift_product_selected_power_product_partial + S (ff_r_ftsv_shift_product_selected_power_product) = S ((S (ff_i_ftsv_shift_product_selected_power_product)) * ff_v_ftsv_shift_product_selected_power_product)) /\ exists ff_q_ftsv_shift_product_selected_power_product_partial. ff_u_ftsv_shift_product_selected_power_product = ff_q_ftsv_shift_product_selected_power_product_partial * S ((S (ff_i_ftsv_shift_product_selected_power_product)) * ff_v_ftsv_shift_product_selected_power_product) + (ff_r_ftsv_shift_product_selected_power_product))) /\ ((((exists ff_h_ftsv_shift_product_selected_power_product_successor. ff_h_ftsv_shift_product_selected_power_product_successor + S (ff_s_ftsv_shift_product_selected_power_product) = S ((S (S ff_i_ftsv_shift_product_selected_power_product)) * ff_v_ftsv_shift_product_selected_power_product)) /\ exists ff_q_ftsv_shift_product_selected_power_product_successor. ff_u_ftsv_shift_product_selected_power_product = ff_q_ftsv_shift_product_selected_power_product_successor * S ((S (S ff_i_ftsv_shift_product_selected_power_product)) * ff_v_ftsv_shift_product_selected_power_product) + (ff_s_ftsv_shift_product_selected_power_product))) /\ ff_s_ftsv_shift_product_selected_power_product = ff_r_ftsv_shift_product_selected_power_product * ff_p_ftsv_shift_product_selected_power_product)))))))) /\ (exists bpv_factor_ftsv_shift_product_selected_divides. ((z * z) * n) = bpv_result_ftsv_shift_product_selected * bpv_factor_ftsv_shift_product_selected_divides)))) /\ forall bpv_candidate_ftsv_shift_product. (exists bpv_gap_ftsv_shift_product_candidate_bound. bpv_gap_ftsv_shift_product_candidate_bound + bpv_candidate_ftsv_shift_product = ((z * z) * n)) -> (exists bpv_result_ftsv_shift_product_candidate. ((exists ff_b_ftsv_shift_product_candidate_power ff_c_ftsv_shift_product_candidate_power. ((forall ff_i_ftsv_shift_product_candidate_power_repeat. (exists ff_lt_ftsv_shift_product_candidate_power_repeat_bound. ff_lt_ftsv_shift_product_candidate_power_repeat_bound + S ff_i_ftsv_shift_product_candidate_power_repeat = bpv_candidate_ftsv_shift_product) -> (((exists ff_h_ftsv_shift_product_candidate_power_repeat_decoded. ff_h_ftsv_shift_product_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsv_shift_product_candidate_power_repeat)) * ff_c_ftsv_shift_product_candidate_power)) /\ exists ff_q_ftsv_shift_product_candidate_power_repeat_decoded. ff_b_ftsv_shift_product_candidate_power = ff_q_ftsv_shift_product_candidate_power_repeat_decoded * S ((S (ff_i_ftsv_shift_product_candidate_power_repeat)) * ff_c_ftsv_shift_product_candidate_power) + (p)))) /\ (exists ff_u_ftsv_shift_product_candidate_power_product ff_v_ftsv_shift_product_candidate_power_product. ((((exists ff_h_ftsv_shift_product_candidate_power_product_start. ff_h_ftsv_shift_product_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsv_shift_product_candidate_power_product)) /\ exists ff_q_ftsv_shift_product_candidate_power_product_start. ff_u_ftsv_shift_product_candidate_power_product = ff_q_ftsv_shift_product_candidate_power_product_start * S ((S (0)) * ff_v_ftsv_shift_product_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsv_shift_product_candidate_power_product_terminal. ff_h_ftsv_shift_product_candidate_power_product_terminal + S (bpv_result_ftsv_shift_product_candidate) = S ((S (bpv_candidate_ftsv_shift_product)) * ff_v_ftsv_shift_product_candidate_power_product)) /\ exists ff_q_ftsv_shift_product_candidate_power_product_terminal. ff_u_ftsv_shift_product_candidate_power_product = ff_q_ftsv_shift_product_candidate_power_product_terminal * S ((S (bpv_candidate_ftsv_shift_product)) * ff_v_ftsv_shift_product_candidate_power_product) + (bpv_result_ftsv_shift_product_candidate))) /\ forall ff_i_ftsv_shift_product_candidate_power_product. (exists ff_lt_ftsv_shift_product_candidate_power_product_bound. ff_lt_ftsv_shift_product_candidate_power_product_bound + S ff_i_ftsv_shift_product_candidate_power_product = bpv_candidate_ftsv_shift_product) -> exists ff_p_ftsv_shift_product_candidate_power_product ff_r_ftsv_shift_product_candidate_power_product ff_s_ftsv_shift_product_candidate_power_product. ((((exists ff_h_ftsv_shift_product_candidate_power_product_factor. ff_h_ftsv_shift_product_candidate_power_product_factor + S (ff_p_ftsv_shift_product_candidate_power_product) = S ((S (ff_i_ftsv_shift_product_candidate_power_product)) * ff_c_ftsv_shift_product_candidate_power)) /\ exists ff_q_ftsv_shift_product_candidate_power_product_factor. ff_b_ftsv_shift_product_candidate_power = ff_q_ftsv_shift_product_candidate_power_product_factor * S ((S (ff_i_ftsv_shift_product_candidate_power_product)) * ff_c_ftsv_shift_product_candidate_power) + (ff_p_ftsv_shift_product_candidate_power_product))) /\ ((((exists ff_h_ftsv_shift_product_candidate_power_product_partial. ff_h_ftsv_shift_product_candidate_power_product_partial + S (ff_r_ftsv_shift_product_candidate_power_product) = S ((S (ff_i_ftsv_shift_product_candidate_power_product)) * ff_v_ftsv_shift_product_candidate_power_product)) /\ exists ff_q_ftsv_shift_product_candidate_power_product_partial. ff_u_ftsv_shift_product_candidate_power_product = ff_q_ftsv_shift_product_candidate_power_product_partial * S ((S (ff_i_ftsv_shift_product_candidate_power_product)) * ff_v_ftsv_shift_product_candidate_power_product) + (ff_r_ftsv_shift_product_candidate_power_product))) /\ ((((exists ff_h_ftsv_shift_product_candidate_power_product_successor. ff_h_ftsv_shift_product_candidate_power_product_successor + S (ff_s_ftsv_shift_product_candidate_power_product) = S ((S (S ff_i_ftsv_shift_product_candidate_power_product)) * ff_v_ftsv_shift_product_candidate_power_product)) /\ exists ff_q_ftsv_shift_product_candidate_power_product_successor. ff_u_ftsv_shift_product_candidate_power_product = ff_q_ftsv_shift_product_candidate_power_product_successor * S ((S (S ff_i_ftsv_shift_product_candidate_power_product)) * ff_v_ftsv_shift_product_candidate_power_product) + (ff_s_ftsv_shift_product_candidate_power_product))) /\ ff_s_ftsv_shift_product_candidate_power_product = ff_r_ftsv_shift_product_candidate_power_product * ff_p_ftsv_shift_product_candidate_power_product)))))))) /\ (exists bpv_factor_ftsv_shift_product_candidate_divides. ((z * z) * n) = bpv_result_ftsv_shift_product_candidate * bpv_factor_ftsv_shift_product_candidate_divides))) -> (exists bpv_gap_ftsv_shift_product_maximal. bpv_gap_ftsv_shift_product_maximal + bpv_candidate_ftsv_shift_product = g)) -> f = k + k -> exists h. g = h + h