power_valuation_functional

Exact expanded PA statement

forall p a e f. (((exists bpv_gap_functional_left_exponent_bound. bpv_gap_functional_left_exponent_bound + e = a) /\ (exists bpv_result_functional_left_selected. ((exists ff_b_functional_left_selected_power ff_c_functional_left_selected_power. ((forall ff_i_functional_left_selected_power_repeat. (exists ff_lt_functional_left_selected_power_repeat_bound. ff_lt_functional_left_selected_power_repeat_bound + S ff_i_functional_left_selected_power_repeat = e) -> (((exists ff_h_functional_left_selected_power_repeat_decoded. ff_h_functional_left_selected_power_repeat_decoded + S (p) = S ((S (ff_i_functional_left_selected_power_repeat)) * ff_c_functional_left_selected_power)) /\ exists ff_q_functional_left_selected_power_repeat_decoded. ff_b_functional_left_selected_power = ff_q_functional_left_selected_power_repeat_decoded * S ((S (ff_i_functional_left_selected_power_repeat)) * ff_c_functional_left_selected_power) + (p)))) /\ (exists ff_u_functional_left_selected_power_product ff_v_functional_left_selected_power_product. ((((exists ff_h_functional_left_selected_power_product_start. ff_h_functional_left_selected_power_product_start + S (1) = S ((S (0)) * ff_v_functional_left_selected_power_product)) /\ exists ff_q_functional_left_selected_power_product_start. ff_u_functional_left_selected_power_product = ff_q_functional_left_selected_power_product_start * S ((S (0)) * ff_v_functional_left_selected_power_product) + (1))) /\ ((((exists ff_h_functional_left_selected_power_product_terminal. ff_h_functional_left_selected_power_product_terminal + S (bpv_result_functional_left_selected) = S ((S (e)) * ff_v_functional_left_selected_power_product)) /\ exists ff_q_functional_left_selected_power_product_terminal. ff_u_functional_left_selected_power_product = ff_q_functional_left_selected_power_product_terminal * S ((S (e)) * ff_v_functional_left_selected_power_product) + (bpv_result_functional_left_selected))) /\ forall ff_i_functional_left_selected_power_product. (exists ff_lt_functional_left_selected_power_product_bound. ff_lt_functional_left_selected_power_product_bound + S ff_i_functional_left_selected_power_product = e) -> exists ff_p_functional_left_selected_power_product ff_r_functional_left_selected_power_product ff_s_functional_left_selected_power_product. ((((exists ff_h_functional_left_selected_power_product_factor. ff_h_functional_left_selected_power_product_factor + S (ff_p_functional_left_selected_power_product) = S ((S (ff_i_functional_left_selected_power_product)) * ff_c_functional_left_selected_power)) /\ exists ff_q_functional_left_selected_power_product_factor. ff_b_functional_left_selected_power = ff_q_functional_left_selected_power_product_factor * S ((S (ff_i_functional_left_selected_power_product)) * ff_c_functional_left_selected_power) + (ff_p_functional_left_selected_power_product))) /\ ((((exists ff_h_functional_left_selected_power_product_partial. ff_h_functional_left_selected_power_product_partial + S (ff_r_functional_left_selected_power_product) = S ((S (ff_i_functional_left_selected_power_product)) * ff_v_functional_left_selected_power_product)) /\ exists ff_q_functional_left_selected_power_product_partial. ff_u_functional_left_selected_power_product = ff_q_functional_left_selected_power_product_partial * S ((S (ff_i_functional_left_selected_power_product)) * ff_v_functional_left_selected_power_product) + (ff_r_functional_left_selected_power_product))) /\ ((((exists ff_h_functional_left_selected_power_product_successor. ff_h_functional_left_selected_power_product_successor + S (ff_s_functional_left_selected_power_product) = S ((S (S ff_i_functional_left_selected_power_product)) * ff_v_functional_left_selected_power_product)) /\ exists ff_q_functional_left_selected_power_product_successor. ff_u_functional_left_selected_power_product = ff_q_functional_left_selected_power_product_successor * S ((S (S ff_i_functional_left_selected_power_product)) * ff_v_functional_left_selected_power_product) + (ff_s_functional_left_selected_power_product))) /\ ff_s_functional_left_selected_power_product = ff_r_functional_left_selected_power_product * ff_p_functional_left_selected_power_product)))))))) /\ (exists bpv_factor_functional_left_selected_divides. a = bpv_result_functional_left_selected * bpv_factor_functional_left_selected_divides)))) /\ forall bpv_candidate_functional_left. (exists bpv_gap_functional_left_candidate_bound. bpv_gap_functional_left_candidate_bound + bpv_candidate_functional_left = a) -> (exists bpv_result_functional_left_candidate. ((exists ff_b_functional_left_candidate_power ff_c_functional_left_candidate_power. ((forall ff_i_functional_left_candidate_power_repeat. (exists ff_lt_functional_left_candidate_power_repeat_bound. ff_lt_functional_left_candidate_power_repeat_bound + S ff_i_functional_left_candidate_power_repeat = bpv_candidate_functional_left) -> (((exists ff_h_functional_left_candidate_power_repeat_decoded. ff_h_functional_left_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_functional_left_candidate_power_repeat)) * ff_c_functional_left_candidate_power)) /\ exists ff_q_functional_left_candidate_power_repeat_decoded. ff_b_functional_left_candidate_power = ff_q_functional_left_candidate_power_repeat_decoded * S ((S (ff_i_functional_left_candidate_power_repeat)) * ff_c_functional_left_candidate_power) + (p)))) /\ (exists ff_u_functional_left_candidate_power_product ff_v_functional_left_candidate_power_product. ((((exists ff_h_functional_left_candidate_power_product_start. ff_h_functional_left_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_functional_left_candidate_power_product)) /\ exists ff_q_functional_left_candidate_power_product_start. ff_u_functional_left_candidate_power_product = ff_q_functional_left_candidate_power_product_start * S ((S (0)) * ff_v_functional_left_candidate_power_product) + (1))) /\ ((((exists ff_h_functional_left_candidate_power_product_terminal. ff_h_functional_left_candidate_power_product_terminal + S (bpv_result_functional_left_candidate) = S ((S (bpv_candidate_functional_left)) * ff_v_functional_left_candidate_power_product)) /\ exists ff_q_functional_left_candidate_power_product_terminal. ff_u_functional_left_candidate_power_product = ff_q_functional_left_candidate_power_product_terminal * S ((S (bpv_candidate_functional_left)) * ff_v_functional_left_candidate_power_product) + (bpv_result_functional_left_candidate))) /\ forall ff_i_functional_left_candidate_power_product. (exists ff_lt_functional_left_candidate_power_product_bound. ff_lt_functional_left_candidate_power_product_bound + S ff_i_functional_left_candidate_power_product = bpv_candidate_functional_left) -> exists ff_p_functional_left_candidate_power_product ff_r_functional_left_candidate_power_product ff_s_functional_left_candidate_power_product. ((((exists ff_h_functional_left_candidate_power_product_factor. ff_h_functional_left_candidate_power_product_factor + S (ff_p_functional_left_candidate_power_product) = S ((S (ff_i_functional_left_candidate_power_product)) * ff_c_functional_left_candidate_power)) /\ exists ff_q_functional_left_candidate_power_product_factor. ff_b_functional_left_candidate_power = ff_q_functional_left_candidate_power_product_factor * S ((S (ff_i_functional_left_candidate_power_product)) * ff_c_functional_left_candidate_power) + (ff_p_functional_left_candidate_power_product))) /\ ((((exists ff_h_functional_left_candidate_power_product_partial. ff_h_functional_left_candidate_power_product_partial + S (ff_r_functional_left_candidate_power_product) = S ((S (ff_i_functional_left_candidate_power_product)) * ff_v_functional_left_candidate_power_product)) /\ exists ff_q_functional_left_candidate_power_product_partial. ff_u_functional_left_candidate_power_product = ff_q_functional_left_candidate_power_product_partial * S ((S (ff_i_functional_left_candidate_power_product)) * ff_v_functional_left_candidate_power_product) + (ff_r_functional_left_candidate_power_product))) /\ ((((exists ff_h_functional_left_candidate_power_product_successor. ff_h_functional_left_candidate_power_product_successor + S (ff_s_functional_left_candidate_power_product) = S ((S (S ff_i_functional_left_candidate_power_product)) * ff_v_functional_left_candidate_power_product)) /\ exists ff_q_functional_left_candidate_power_product_successor. ff_u_functional_left_candidate_power_product = ff_q_functional_left_candidate_power_product_successor * S ((S (S ff_i_functional_left_candidate_power_product)) * ff_v_functional_left_candidate_power_product) + (ff_s_functional_left_candidate_power_product))) /\ ff_s_functional_left_candidate_power_product = ff_r_functional_left_candidate_power_product * ff_p_functional_left_candidate_power_product)))))))) /\ (exists bpv_factor_functional_left_candidate_divides. a = bpv_result_functional_left_candidate * bpv_factor_functional_left_candidate_divides))) -> (exists bpv_gap_functional_left_maximal. bpv_gap_functional_left_maximal + bpv_candidate_functional_left = e)) -> (((exists bpv_gap_functional_right_exponent_bound. bpv_gap_functional_right_exponent_bound + f = a) /\ (exists bpv_result_functional_right_selected. ((exists ff_b_functional_right_selected_power ff_c_functional_right_selected_power. ((forall ff_i_functional_right_selected_power_repeat. (exists ff_lt_functional_right_selected_power_repeat_bound. ff_lt_functional_right_selected_power_repeat_bound + S ff_i_functional_right_selected_power_repeat = f) -> (((exists ff_h_functional_right_selected_power_repeat_decoded. ff_h_functional_right_selected_power_repeat_decoded + S (p) = S ((S (ff_i_functional_right_selected_power_repeat)) * ff_c_functional_right_selected_power)) /\ exists ff_q_functional_right_selected_power_repeat_decoded. ff_b_functional_right_selected_power = ff_q_functional_right_selected_power_repeat_decoded * S ((S (ff_i_functional_right_selected_power_repeat)) * ff_c_functional_right_selected_power) + (p)))) /\ (exists ff_u_functional_right_selected_power_product ff_v_functional_right_selected_power_product. ((((exists ff_h_functional_right_selected_power_product_start. ff_h_functional_right_selected_power_product_start + S (1) = S ((S (0)) * ff_v_functional_right_selected_power_product)) /\ exists ff_q_functional_right_selected_power_product_start. ff_u_functional_right_selected_power_product = ff_q_functional_right_selected_power_product_start * S ((S (0)) * ff_v_functional_right_selected_power_product) + (1))) /\ ((((exists ff_h_functional_right_selected_power_product_terminal. ff_h_functional_right_selected_power_product_terminal + S (bpv_result_functional_right_selected) = S ((S (f)) * ff_v_functional_right_selected_power_product)) /\ exists ff_q_functional_right_selected_power_product_terminal. ff_u_functional_right_selected_power_product = ff_q_functional_right_selected_power_product_terminal * S ((S (f)) * ff_v_functional_right_selected_power_product) + (bpv_result_functional_right_selected))) /\ forall ff_i_functional_right_selected_power_product. (exists ff_lt_functional_right_selected_power_product_bound. ff_lt_functional_right_selected_power_product_bound + S ff_i_functional_right_selected_power_product = f) -> exists ff_p_functional_right_selected_power_product ff_r_functional_right_selected_power_product ff_s_functional_right_selected_power_product. ((((exists ff_h_functional_right_selected_power_product_factor. ff_h_functional_right_selected_power_product_factor + S (ff_p_functional_right_selected_power_product) = S ((S (ff_i_functional_right_selected_power_product)) * ff_c_functional_right_selected_power)) /\ exists ff_q_functional_right_selected_power_product_factor. ff_b_functional_right_selected_power = ff_q_functional_right_selected_power_product_factor * S ((S (ff_i_functional_right_selected_power_product)) * ff_c_functional_right_selected_power) + (ff_p_functional_right_selected_power_product))) /\ ((((exists ff_h_functional_right_selected_power_product_partial. ff_h_functional_right_selected_power_product_partial + S (ff_r_functional_right_selected_power_product) = S ((S (ff_i_functional_right_selected_power_product)) * ff_v_functional_right_selected_power_product)) /\ exists ff_q_functional_right_selected_power_product_partial. ff_u_functional_right_selected_power_product = ff_q_functional_right_selected_power_product_partial * S ((S (ff_i_functional_right_selected_power_product)) * ff_v_functional_right_selected_power_product) + (ff_r_functional_right_selected_power_product))) /\ ((((exists ff_h_functional_right_selected_power_product_successor. ff_h_functional_right_selected_power_product_successor + S (ff_s_functional_right_selected_power_product) = S ((S (S ff_i_functional_right_selected_power_product)) * ff_v_functional_right_selected_power_product)) /\ exists ff_q_functional_right_selected_power_product_successor. ff_u_functional_right_selected_power_product = ff_q_functional_right_selected_power_product_successor * S ((S (S ff_i_functional_right_selected_power_product)) * ff_v_functional_right_selected_power_product) + (ff_s_functional_right_selected_power_product))) /\ ff_s_functional_right_selected_power_product = ff_r_functional_right_selected_power_product * ff_p_functional_right_selected_power_product)))))))) /\ (exists bpv_factor_functional_right_selected_divides. a = bpv_result_functional_right_selected * bpv_factor_functional_right_selected_divides)))) /\ forall bpv_candidate_functional_right. (exists bpv_gap_functional_right_candidate_bound. bpv_gap_functional_right_candidate_bound + bpv_candidate_functional_right = a) -> (exists bpv_result_functional_right_candidate. ((exists ff_b_functional_right_candidate_power ff_c_functional_right_candidate_power. ((forall ff_i_functional_right_candidate_power_repeat. (exists ff_lt_functional_right_candidate_power_repeat_bound. ff_lt_functional_right_candidate_power_repeat_bound + S ff_i_functional_right_candidate_power_repeat = bpv_candidate_functional_right) -> (((exists ff_h_functional_right_candidate_power_repeat_decoded. ff_h_functional_right_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_functional_right_candidate_power_repeat)) * ff_c_functional_right_candidate_power)) /\ exists ff_q_functional_right_candidate_power_repeat_decoded. ff_b_functional_right_candidate_power = ff_q_functional_right_candidate_power_repeat_decoded * S ((S (ff_i_functional_right_candidate_power_repeat)) * ff_c_functional_right_candidate_power) + (p)))) /\ (exists ff_u_functional_right_candidate_power_product ff_v_functional_right_candidate_power_product. ((((exists ff_h_functional_right_candidate_power_product_start. ff_h_functional_right_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_functional_right_candidate_power_product)) /\ exists ff_q_functional_right_candidate_power_product_start. ff_u_functional_right_candidate_power_product = ff_q_functional_right_candidate_power_product_start * S ((S (0)) * ff_v_functional_right_candidate_power_product) + (1))) /\ ((((exists ff_h_functional_right_candidate_power_product_terminal. ff_h_functional_right_candidate_power_product_terminal + S (bpv_result_functional_right_candidate) = S ((S (bpv_candidate_functional_right)) * ff_v_functional_right_candidate_power_product)) /\ exists ff_q_functional_right_candidate_power_product_terminal. ff_u_functional_right_candidate_power_product = ff_q_functional_right_candidate_power_product_terminal * S ((S (bpv_candidate_functional_right)) * ff_v_functional_right_candidate_power_product) + (bpv_result_functional_right_candidate))) /\ forall ff_i_functional_right_candidate_power_product. (exists ff_lt_functional_right_candidate_power_product_bound. ff_lt_functional_right_candidate_power_product_bound + S ff_i_functional_right_candidate_power_product = bpv_candidate_functional_right) -> exists ff_p_functional_right_candidate_power_product ff_r_functional_right_candidate_power_product ff_s_functional_right_candidate_power_product. ((((exists ff_h_functional_right_candidate_power_product_factor. ff_h_functional_right_candidate_power_product_factor + S (ff_p_functional_right_candidate_power_product) = S ((S (ff_i_functional_right_candidate_power_product)) * ff_c_functional_right_candidate_power)) /\ exists ff_q_functional_right_candidate_power_product_factor. ff_b_functional_right_candidate_power = ff_q_functional_right_candidate_power_product_factor * S ((S (ff_i_functional_right_candidate_power_product)) * ff_c_functional_right_candidate_power) + (ff_p_functional_right_candidate_power_product))) /\ ((((exists ff_h_functional_right_candidate_power_product_partial. ff_h_functional_right_candidate_power_product_partial + S (ff_r_functional_right_candidate_power_product) = S ((S (ff_i_functional_right_candidate_power_product)) * ff_v_functional_right_candidate_power_product)) /\ exists ff_q_functional_right_candidate_power_product_partial. ff_u_functional_right_candidate_power_product = ff_q_functional_right_candidate_power_product_partial * S ((S (ff_i_functional_right_candidate_power_product)) * ff_v_functional_right_candidate_power_product) + (ff_r_functional_right_candidate_power_product))) /\ ((((exists ff_h_functional_right_candidate_power_product_successor. ff_h_functional_right_candidate_power_product_successor + S (ff_s_functional_right_candidate_power_product) = S ((S (S ff_i_functional_right_candidate_power_product)) * ff_v_functional_right_candidate_power_product)) /\ exists ff_q_functional_right_candidate_power_product_successor. ff_u_functional_right_candidate_power_product = ff_q_functional_right_candidate_power_product_successor * S ((S (S ff_i_functional_right_candidate_power_product)) * ff_v_functional_right_candidate_power_product) + (ff_s_functional_right_candidate_power_product))) /\ ff_s_functional_right_candidate_power_product = ff_r_functional_right_candidate_power_product * ff_p_functional_right_candidate_power_product)))))))) /\ (exists bpv_factor_functional_right_candidate_divides. a = bpv_result_functional_right_candidate * bpv_factor_functional_right_candidate_divides))) -> (exists bpv_gap_functional_right_maximal. bpv_gap_functional_right_maximal + bpv_candidate_functional_right = f)) -> e = f