PV0005

prime_power_valuation_pow

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Construct the actual maximal valuation graph of a witnessed power, not merely an equation between supplied output valuations.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall p a k e z. (~((p) = 1) /\ forall pvs_left_pow_construct_domain pvs_right_pow_construct_domain. (p) = pvs_left_pow_construct_domain * pvs_right_pow_construct_domain -> pvs_left_pow_construct_domain = 1 \/ pvs_right_pow_construct_domain = 1) -> ~(a = 0) -> (((exists bpd_gap_pvs_pow_construct_base_selected_bound. bpd_gap_pvs_pow_construct_base_selected_bound + (e) = (a)) /\ (exists bpvi_result_pvs_pow_construct_base_selected. ((exists bpvi_b_pvs_pow_construct_base_selected_power bpvi_c_pvs_pow_construct_base_selected_power. ((forall bpvi_i_pvs_pow_construct_base_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_base_selected_power. bpvi_repeat_gap_pvs_pow_construct_base_selected_power + S bpvi_i_pvs_pow_construct_base_selected_power = e) -> (((exists bpvi_h_pvs_pow_construct_base_selected_power_repeat. bpvi_h_pvs_pow_construct_base_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power)) /\ exists bpvi_q_pvs_pow_construct_base_selected_power_repeat. bpvi_b_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power) + (p)))) /\ (exists bpvi_u_pvs_pow_construct_base_selected_power bpvi_v_pvs_pow_construct_base_selected_power. ((((exists bpvi_h_pvs_pow_construct_base_selected_power_start. bpvi_h_pvs_pow_construct_base_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\ exists bpvi_q_pvs_pow_construct_base_selected_power_start. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_base_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_terminal. bpvi_h_pvs_pow_construct_base_selected_power_terminal + S (bpvi_result_pvs_pow_construct_base_selected) = S ((S (e)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\ exists bpvi_q_pvs_pow_construct_base_selected_power_terminal. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_result_pvs_pow_construct_base_selected))) /\ forall bpvi_j_pvs_pow_construct_base_selected_power. (exists bpvi_product_gap_pvs_pow_construct_base_selected_power. bpvi_product_gap_pvs_pow_construct_base_selected_power + S bpvi_j_pvs_pow_construct_base_selected_power = e) -> exists bpvi_factor_pvs_pow_construct_base_selected_power bpvi_partial_pvs_pow_construct_base_selected_power bpvi_successor_pvs_pow_construct_base_selected_power. ((((exists bpvi_h_pvs_pow_construct_base_selected_power_factor. bpvi_h_pvs_pow_construct_base_selected_power_factor + S (bpvi_factor_pvs_pow_construct_base_selected_power) = S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power)) /\ exists bpvi_q_pvs_pow_construct_base_selected_power_factor. bpvi_b_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power) + (bpvi_factor_pvs_pow_construct_base_selected_power))) /\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_partial. bpvi_h_pvs_pow_construct_base_selected_power_partial + S (bpvi_partial_pvs_pow_construct_base_selected_power) = S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\ exists bpvi_q_pvs_pow_construct_base_selected_power_partial. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_partial_pvs_pow_construct_base_selected_power))) /\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_successor. bpvi_h_pvs_pow_construct_base_selected_power_successor + S (bpvi_successor_pvs_pow_construct_base_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\ exists bpvi_q_pvs_pow_construct_base_selected_power_successor. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_successor_pvs_pow_construct_base_selected_power))) /\ bpvi_successor_pvs_pow_construct_base_selected_power = bpvi_partial_pvs_pow_construct_base_selected_power * bpvi_factor_pvs_pow_construct_base_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_pow_construct_base_selected. a = bpvi_result_pvs_pow_construct_base_selected * bpvi_divisor_factor_pvs_pow_construct_base_selected))) /\ forall bpd_candidate_pvs_pow_construct_base. (exists bpd_gap_pvs_pow_construct_base_candidate_bound. bpd_gap_pvs_pow_construct_base_candidate_bound + (bpd_candidate_pvs_pow_construct_base) = (a)) -> (exists bpvi_result_pvs_pow_construct_base_candidate. ((exists bpvi_b_pvs_pow_construct_base_candidate_power bpvi_c_pvs_pow_construct_base_candidate_power. ((forall bpvi_i_pvs_pow_construct_base_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_base_candidate_power. bpvi_repeat_gap_pvs_pow_construct_base_candidate_power + S bpvi_i_pvs_pow_construct_base_candidate_power = bpd_candidate_pvs_pow_construct_base) -> (((exists bpvi_h_pvs_pow_construct_base_candidate_power_repeat. bpvi_h_pvs_pow_construct_base_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_base_candidate_power_repeat. bpvi_b_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_pow_construct_base_candidate_power bpvi_v_pvs_pow_construct_base_candidate_power. ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_start. bpvi_h_pvs_pow_construct_base_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_base_candidate_power_start. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_terminal. bpvi_h_pvs_pow_construct_base_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_base_candidate) = S ((S (bpd_candidate_pvs_pow_construct_base)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_base_candidate_power_terminal. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_base)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_result_pvs_pow_construct_base_candidate))) /\ forall bpvi_j_pvs_pow_construct_base_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_base_candidate_power. bpvi_product_gap_pvs_pow_construct_base_candidate_power + S bpvi_j_pvs_pow_construct_base_candidate_power = bpd_candidate_pvs_pow_construct_base) -> exists bpvi_factor_pvs_pow_construct_base_candidate_power bpvi_partial_pvs_pow_construct_base_candidate_power bpvi_successor_pvs_pow_construct_base_candidate_power. ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_factor. bpvi_h_pvs_pow_construct_base_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_base_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_base_candidate_power_factor. bpvi_b_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power) + (bpvi_factor_pvs_pow_construct_base_candidate_power))) /\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_partial. bpvi_h_pvs_pow_construct_base_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_base_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_base_candidate_power_partial. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_partial_pvs_pow_construct_base_candidate_power))) /\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_successor. bpvi_h_pvs_pow_construct_base_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_base_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_base_candidate_power_successor. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_successor_pvs_pow_construct_base_candidate_power))) /\ bpvi_successor_pvs_pow_construct_base_candidate_power = bpvi_partial_pvs_pow_construct_base_candidate_power * bpvi_factor_pvs_pow_construct_base_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_pow_construct_base_candidate. a = bpvi_result_pvs_pow_construct_base_candidate * bpvi_divisor_factor_pvs_pow_construct_base_candidate)) -> (exists bpd_gap_pvs_pow_construct_base_maximal. bpd_gap_pvs_pow_construct_base_maximal + (bpd_candidate_pvs_pow_construct_base) = (e))) -> (exists pa_b_pvs_pow_construct_source pa_c_pvs_pow_construct_source. ((forall pa_i_pvs_pow_construct_source_repeat. (exists pa_lt_pvs_pow_construct_source_repeat_bound. pa_lt_pvs_pow_construct_source_repeat_bound + S pa_i_pvs_pow_construct_source_repeat = k) -> (((exists pa_h_pvs_pow_construct_source_repeat_decoded. pa_h_pvs_pow_construct_source_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_construct_source_repeat)) * pa_c_pvs_pow_construct_source)) /\ exists pa_q_pvs_pow_construct_source_repeat_decoded. pa_b_pvs_pow_construct_source = pa_q_pvs_pow_construct_source_repeat_decoded * S ((S (pa_i_pvs_pow_construct_source_repeat)) * pa_c_pvs_pow_construct_source) + (a)))) /\ (exists pa_u_pvs_pow_construct_source_product pa_v_pvs_pow_construct_source_product. ((((exists pa_h_pvs_pow_construct_source_product_start. pa_h_pvs_pow_construct_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_construct_source_product)) /\ exists pa_q_pvs_pow_construct_source_product_start. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_start * S ((S (0)) * pa_v_pvs_pow_construct_source_product) + (1))) /\ ((((exists pa_h_pvs_pow_construct_source_product_terminal. pa_h_pvs_pow_construct_source_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_pow_construct_source_product)) /\ exists pa_q_pvs_pow_construct_source_product_terminal. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_terminal * S ((S (k)) * pa_v_pvs_pow_construct_source_product) + (z))) /\ forall pa_i_pvs_pow_construct_source_product. (exists pa_lt_pvs_pow_construct_source_product_bound. pa_lt_pvs_pow_construct_source_product_bound + S pa_i_pvs_pow_construct_source_product = k) -> exists pa_p_pvs_pow_construct_source_product pa_r_pvs_pow_construct_source_product pa_s_pvs_pow_construct_source_product. ((((exists pa_h_pvs_pow_construct_source_product_factor. pa_h_pvs_pow_construct_source_product_factor + S (pa_p_pvs_pow_construct_source_product) = S ((S (pa_i_pvs_pow_construct_source_product)) * pa_c_pvs_pow_construct_source)) /\ exists pa_q_pvs_pow_construct_source_product_factor. pa_b_pvs_pow_construct_source = pa_q_pvs_pow_construct_source_product_factor * S ((S (pa_i_pvs_pow_construct_source_product)) * pa_c_pvs_pow_construct_source) + (pa_p_pvs_pow_construct_source_product))) /\ ((((exists pa_h_pvs_pow_construct_source_product_partial. pa_h_pvs_pow_construct_source_product_partial + S (pa_r_pvs_pow_construct_source_product) = S ((S (pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product)) /\ exists pa_q_pvs_pow_construct_source_product_partial. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_partial * S ((S (pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product) + (pa_r_pvs_pow_construct_source_product))) /\ ((((exists pa_h_pvs_pow_construct_source_product_successor. pa_h_pvs_pow_construct_source_product_successor + S (pa_s_pvs_pow_construct_source_product) = S ((S (S pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product)) /\ exists pa_q_pvs_pow_construct_source_product_successor. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_successor * S ((S (S pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product) + (pa_s_pvs_pow_construct_source_product))) /\ pa_s_pvs_pow_construct_source_product = pa_r_pvs_pow_construct_source_product * pa_p_pvs_pow_construct_source_product)))))))) -> (((exists bpd_gap_pvs_pow_construct_output_selected_bound. bpd_gap_pvs_pow_construct_output_selected_bound + (k * e) = (z)) /\ (exists bpvi_result_pvs_pow_construct_output_selected. ((exists bpvi_b_pvs_pow_construct_output_selected_power bpvi_c_pvs_pow_construct_output_selected_power. ((forall bpvi_i_pvs_pow_construct_output_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_output_selected_power. bpvi_repeat_gap_pvs_pow_construct_output_selected_power + S bpvi_i_pvs_pow_construct_output_selected_power = k * e) -> (((exists bpvi_h_pvs_pow_construct_output_selected_power_repeat. bpvi_h_pvs_pow_construct_output_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power)) /\ exists bpvi_q_pvs_pow_construct_output_selected_power_repeat. bpvi_b_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power) + (p)))) /\ (exists bpvi_u_pvs_pow_construct_output_selected_power bpvi_v_pvs_pow_construct_output_selected_power. ((((exists bpvi_h_pvs_pow_construct_output_selected_power_start. bpvi_h_pvs_pow_construct_output_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\ exists bpvi_q_pvs_pow_construct_output_selected_power_start. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_output_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_terminal. bpvi_h_pvs_pow_construct_output_selected_power_terminal + S (bpvi_result_pvs_pow_construct_output_selected) = S ((S (k * e)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\ exists bpvi_q_pvs_pow_construct_output_selected_power_terminal. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_terminal * S ((S (k * e)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_result_pvs_pow_construct_output_selected))) /\ forall bpvi_j_pvs_pow_construct_output_selected_power. (exists bpvi_product_gap_pvs_pow_construct_output_selected_power. bpvi_product_gap_pvs_pow_construct_output_selected_power + S bpvi_j_pvs_pow_construct_output_selected_power = k * e) -> exists bpvi_factor_pvs_pow_construct_output_selected_power bpvi_partial_pvs_pow_construct_output_selected_power bpvi_successor_pvs_pow_construct_output_selected_power. ((((exists bpvi_h_pvs_pow_construct_output_selected_power_factor. bpvi_h_pvs_pow_construct_output_selected_power_factor + S (bpvi_factor_pvs_pow_construct_output_selected_power) = S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power)) /\ exists bpvi_q_pvs_pow_construct_output_selected_power_factor. bpvi_b_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power) + (bpvi_factor_pvs_pow_construct_output_selected_power))) /\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_partial. bpvi_h_pvs_pow_construct_output_selected_power_partial + S (bpvi_partial_pvs_pow_construct_output_selected_power) = S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\ exists bpvi_q_pvs_pow_construct_output_selected_power_partial. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_partial_pvs_pow_construct_output_selected_power))) /\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_successor. bpvi_h_pvs_pow_construct_output_selected_power_successor + S (bpvi_successor_pvs_pow_construct_output_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\ exists bpvi_q_pvs_pow_construct_output_selected_power_successor. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_successor_pvs_pow_construct_output_selected_power))) /\ bpvi_successor_pvs_pow_construct_output_selected_power = bpvi_partial_pvs_pow_construct_output_selected_power * bpvi_factor_pvs_pow_construct_output_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_pow_construct_output_selected. z = bpvi_result_pvs_pow_construct_output_selected * bpvi_divisor_factor_pvs_pow_construct_output_selected))) /\ forall bpd_candidate_pvs_pow_construct_output. (exists bpd_gap_pvs_pow_construct_output_candidate_bound. bpd_gap_pvs_pow_construct_output_candidate_bound + (bpd_candidate_pvs_pow_construct_output) = (z)) -> (exists bpvi_result_pvs_pow_construct_output_candidate. ((exists bpvi_b_pvs_pow_construct_output_candidate_power bpvi_c_pvs_pow_construct_output_candidate_power. ((forall bpvi_i_pvs_pow_construct_output_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_output_candidate_power. bpvi_repeat_gap_pvs_pow_construct_output_candidate_power + S bpvi_i_pvs_pow_construct_output_candidate_power = bpd_candidate_pvs_pow_construct_output) -> (((exists bpvi_h_pvs_pow_construct_output_candidate_power_repeat. bpvi_h_pvs_pow_construct_output_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_output_candidate_power_repeat. bpvi_b_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_pow_construct_output_candidate_power bpvi_v_pvs_pow_construct_output_candidate_power. ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_start. bpvi_h_pvs_pow_construct_output_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_output_candidate_power_start. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_terminal. bpvi_h_pvs_pow_construct_output_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_output_candidate) = S ((S (bpd_candidate_pvs_pow_construct_output)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_output_candidate_power_terminal. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_output)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_result_pvs_pow_construct_output_candidate))) /\ forall bpvi_j_pvs_pow_construct_output_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_output_candidate_power. bpvi_product_gap_pvs_pow_construct_output_candidate_power + S bpvi_j_pvs_pow_construct_output_candidate_power = bpd_candidate_pvs_pow_construct_output) -> exists bpvi_factor_pvs_pow_construct_output_candidate_power bpvi_partial_pvs_pow_construct_output_candidate_power bpvi_successor_pvs_pow_construct_output_candidate_power. ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_factor. bpvi_h_pvs_pow_construct_output_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_output_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_output_candidate_power_factor. bpvi_b_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power) + (bpvi_factor_pvs_pow_construct_output_candidate_power))) /\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_partial. bpvi_h_pvs_pow_construct_output_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_output_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_output_candidate_power_partial. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_partial_pvs_pow_construct_output_candidate_power))) /\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_successor. bpvi_h_pvs_pow_construct_output_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_output_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_output_candidate_power_successor. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_successor_pvs_pow_construct_output_candidate_power))) /\ bpvi_successor_pvs_pow_construct_output_candidate_power = bpvi_partial_pvs_pow_construct_output_candidate_power * bpvi_factor_pvs_pow_construct_output_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_pow_construct_output_candidate. z = bpvi_result_pvs_pow_construct_output_candidate * bpvi_divisor_factor_pvs_pow_construct_output_candidate)) -> (exists bpd_gap_pvs_pow_construct_output_maximal. bpd_gap_pvs_pow_construct_output_maximal + (bpd_candidate_pvs_pow_construct_output) = (k * e)))

Constructive proof overview

Generated structural guide

Construct the actual maximal valuation graph of a witnessed power, not merely an equation between supplied output valuations.

The unchanged tactic script uses 3 declared prerequisites and contains 32 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

power_valuation_exists Alpha theorem; checked-use authorized PV0004 prime_power_valuation_pow_value PV0001 prime_valuation_exponent_eq_transport

Direct dependents

PV0007 prime_valuation_distinct_prime_power_zero

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

32 script commands · 5 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (2)
prime power valuation pow valueprime valuation exponent eq transport

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–9

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro k
  4. L4
    intro e
  5. L5
    intro z
  6. L6
    intro hp
  7. L7
    intro ha
  8. L8
    intro hbase
  9. L9
    intro hpow
02Establish hvL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply power valuation exists.

  1. L10
    have hv : ∃ f. BoundedPowerValuation(p,z,z,f)Definitions: BoundedPowerValuation
  2. L11
    specialize power_valuation_exists (p)
  3. L12
    specialize power_valuation_exists (z)
  4. L13
    apply power_valuation_exists
03Separate the logical casesL14–14

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    cases hv
04Use earlier factsL15–24

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L15
    specialize prime_valuation_exponent_eq_transport (p)
  2. L16
    specialize prime_valuation_exponent_eq_transport (z)
  3. L17
    specialize prime_valuation_exponent_eq_transport (x)
  4. L18
    specialize prime_valuation_exponent_eq_transport (k * e)
  5. L19
    apply prime_valuation_exponent_eq_transport
  6. L20
    specialize prime_power_valuation_pow_value (p)
  7. L21
    specialize prime_power_valuation_pow_value (a)
  8. L22
    specialize prime_power_valuation_pow_value (k)
  9. L23
    specialize prime_power_valuation_pow_value (e)
  10. L24
    specialize prime_power_valuation_pow_value (z)
05Use earlier factsL25–32

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L25
    specialize prime_power_valuation_pow_value (x)
  2. L26
    apply prime_power_valuation_pow_value
  3. L27
    exact hp
  4. L28
    exact ha
  5. L29
    exact hbase
  6. L30
    exact hpow
  7. L31
    exact hv_witness
  8. L32
    exact hv_witness

Library-wide reading audit

Original exact command ledger · 32 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro k
  4. 0004intro e
  5. 0005intro z
  6. 0006intro hp
  7. 0007intro ha
  8. 0008intro hbase
  9. 0009intro hpow
  10. 0010have hv : exists f. (((exists bpd_gap_pvs_pow_construct_exists_selected_bound. bpd_gap_pvs_pow_construct_exists_selected_bound + (f) = (z)) /\ (exists bpvi_result_pvs_pow_construct_exists_selected. ((exists bpvi_b_pvs_pow_construct_exists_selected_power bpvi_c_pvs_pow_construct_exists_selected_power. ((forall bpvi_i_pvs_pow_construct_exists_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_exists_selected_power. bpvi_repeat_gap_pvs_pow_construct_exists_selected_power + S bpvi_i_pvs_pow_construct_exists_selected_power = f) -> (((exists bpvi_h_pvs_pow_construct_exists_selected_power_repeat. bpvi_h_pvs_pow_construct_exists_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power)) /\ exists bpvi_q_pvs_pow_construct_exists_selected_power_repeat. bpvi_b_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power) + (p)))) /\ (exists bpvi_u_pvs_pow_construct_exists_selected_power bpvi_v_pvs_pow_construct_exists_selected_power. ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_start. bpvi_h_pvs_pow_construct_exists_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\ exists bpvi_q_pvs_pow_construct_exists_selected_power_start. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_terminal. bpvi_h_pvs_pow_construct_exists_selected_power_terminal + S (bpvi_result_pvs_pow_construct_exists_selected) = S ((S (f)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\ exists bpvi_q_pvs_pow_construct_exists_selected_power_terminal. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_result_pvs_pow_construct_exists_selected))) /\ forall bpvi_j_pvs_pow_construct_exists_selected_power. (exists bpvi_product_gap_pvs_pow_construct_exists_selected_power. bpvi_product_gap_pvs_pow_construct_exists_selected_power + S bpvi_j_pvs_pow_construct_exists_selected_power = f) -> exists bpvi_factor_pvs_pow_construct_exists_selected_power bpvi_partial_pvs_pow_construct_exists_selected_power bpvi_successor_pvs_pow_construct_exists_selected_power. ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_factor. bpvi_h_pvs_pow_construct_exists_selected_power_factor + S (bpvi_factor_pvs_pow_construct_exists_selected_power) = S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power)) /\ exists bpvi_q_pvs_pow_construct_exists_selected_power_factor. bpvi_b_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power) + (bpvi_factor_pvs_pow_construct_exists_selected_power))) /\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_partial. bpvi_h_pvs_pow_construct_exists_selected_power_partial + S (bpvi_partial_pvs_pow_construct_exists_selected_power) = S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\ exists bpvi_q_pvs_pow_construct_exists_selected_power_partial. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_partial_pvs_pow_construct_exists_selected_power))) /\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_successor. bpvi_h_pvs_pow_construct_exists_selected_power_successor + S (bpvi_successor_pvs_pow_construct_exists_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\ exists bpvi_q_pvs_pow_construct_exists_selected_power_successor. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_successor_pvs_pow_construct_exists_selected_power))) /\ bpvi_successor_pvs_pow_construct_exists_selected_power = bpvi_partial_pvs_pow_construct_exists_selected_power * bpvi_factor_pvs_pow_construct_exists_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_pow_construct_exists_selected. z = bpvi_result_pvs_pow_construct_exists_selected * bpvi_divisor_factor_pvs_pow_construct_exists_selected))) /\ forall bpd_candidate_pvs_pow_construct_exists. (exists bpd_gap_pvs_pow_construct_exists_candidate_bound. bpd_gap_pvs_pow_construct_exists_candidate_bound + (bpd_candidate_pvs_pow_construct_exists) = (z)) -> (exists bpvi_result_pvs_pow_construct_exists_candidate. ((exists bpvi_b_pvs_pow_construct_exists_candidate_power bpvi_c_pvs_pow_construct_exists_candidate_power. ((forall bpvi_i_pvs_pow_construct_exists_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_exists_candidate_power. bpvi_repeat_gap_pvs_pow_construct_exists_candidate_power + S bpvi_i_pvs_pow_construct_exists_candidate_power = bpd_candidate_pvs_pow_construct_exists) -> (((exists bpvi_h_pvs_pow_construct_exists_candidate_power_repeat. bpvi_h_pvs_pow_construct_exists_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_repeat. bpvi_b_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_pow_construct_exists_candidate_power bpvi_v_pvs_pow_construct_exists_candidate_power. ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_start. bpvi_h_pvs_pow_construct_exists_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_start. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_terminal. bpvi_h_pvs_pow_construct_exists_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_exists_candidate) = S ((S (bpd_candidate_pvs_pow_construct_exists)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_terminal. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_exists)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_result_pvs_pow_construct_exists_candidate))) /\ forall bpvi_j_pvs_pow_construct_exists_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_exists_candidate_power. bpvi_product_gap_pvs_pow_construct_exists_candidate_power + S bpvi_j_pvs_pow_construct_exists_candidate_power = bpd_candidate_pvs_pow_construct_exists) -> exists bpvi_factor_pvs_pow_construct_exists_candidate_power bpvi_partial_pvs_pow_construct_exists_candidate_power bpvi_successor_pvs_pow_construct_exists_candidate_power. ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_factor. bpvi_h_pvs_pow_construct_exists_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_exists_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_factor. bpvi_b_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power) + (bpvi_factor_pvs_pow_construct_exists_candidate_power))) /\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_partial. bpvi_h_pvs_pow_construct_exists_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_exists_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_partial. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_partial_pvs_pow_construct_exists_candidate_power))) /\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_successor. bpvi_h_pvs_pow_construct_exists_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_exists_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_successor. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_successor_pvs_pow_construct_exists_candidate_power))) /\ bpvi_successor_pvs_pow_construct_exists_candidate_power = bpvi_partial_pvs_pow_construct_exists_candidate_power * bpvi_factor_pvs_pow_construct_exists_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_pow_construct_exists_candidate. z = bpvi_result_pvs_pow_construct_exists_candidate * bpvi_divisor_factor_pvs_pow_construct_exists_candidate)) -> (exists bpd_gap_pvs_pow_construct_exists_maximal. bpd_gap_pvs_pow_construct_exists_maximal + (bpd_candidate_pvs_pow_construct_exists) = (f)))
  11. 0011specialize power_valuation_exists (p)
  12. 0012specialize power_valuation_exists (z)
  13. 0013apply power_valuation_exists
  14. 0014cases hv
  15. 0015specialize prime_valuation_exponent_eq_transport (p)
  16. 0016specialize prime_valuation_exponent_eq_transport (z)
  17. 0017specialize prime_valuation_exponent_eq_transport (x)
  18. 0018specialize prime_valuation_exponent_eq_transport (k * e)
  19. 0019apply prime_valuation_exponent_eq_transport
  20. 0020specialize prime_power_valuation_pow_value (p)
  21. 0021specialize prime_power_valuation_pow_value (a)
  22. 0022specialize prime_power_valuation_pow_value (k)
  23. 0023specialize prime_power_valuation_pow_value (e)
  24. 0024specialize prime_power_valuation_pow_value (z)
  25. 0025specialize prime_power_valuation_pow_value (x)
  26. 0026apply prime_power_valuation_pow_value
  27. 0027exact hp
  28. 0028exact ha
  29. 0029exact hbase
  30. 0030exact hpow
  31. 0031exact hv_witness
  32. 0032exact hv_witness