BT00QP — power_valuation_exact

Exact expanded PA statement

forall p a e. ((~(p = 1) /\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (exists bpd_result_valuation_exact bpd_cofactor_valuation_exact. ((exists ff_b_valuation_exact_power ff_c_valuation_exact_power. ((forall ff_i_valuation_exact_power_repeat. (exists ff_lt_valuation_exact_power_repeat_bound. ff_lt_valuation_exact_power_repeat_bound + S ff_i_valuation_exact_power_repeat = e) -> (((exists ff_h_valuation_exact_power_repeat_decoded. ff_h_valuation_exact_power_repeat_decoded + S (p) = S ((S (ff_i_valuation_exact_power_repeat)) * ff_c_valuation_exact_power)) /\ exists ff_q_valuation_exact_power_repeat_decoded. ff_b_valuation_exact_power = ff_q_valuation_exact_power_repeat_decoded * S ((S (ff_i_valuation_exact_power_repeat)) * ff_c_valuation_exact_power) + (p)))) /\ (exists ff_u_valuation_exact_power_product ff_v_valuation_exact_power_product. ((((exists ff_h_valuation_exact_power_product_start. ff_h_valuation_exact_power_product_start + S (1) = S ((S (0)) * ff_v_valuation_exact_power_product)) /\ exists ff_q_valuation_exact_power_product_start. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_start * S ((S (0)) * ff_v_valuation_exact_power_product) + (1))) /\ ((((exists ff_h_valuation_exact_power_product_terminal. ff_h_valuation_exact_power_product_terminal + S (bpd_result_valuation_exact) = S ((S (e)) * ff_v_valuation_exact_power_product)) /\ exists ff_q_valuation_exact_power_product_terminal. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_terminal * S ((S (e)) * ff_v_valuation_exact_power_product) + (bpd_result_valuation_exact))) /\ forall ff_i_valuation_exact_power_product. (exists ff_lt_valuation_exact_power_product_bound. ff_lt_valuation_exact_power_product_bound + S ff_i_valuation_exact_power_product = e) -> exists ff_p_valuation_exact_power_product ff_r_valuation_exact_power_product ff_s_valuation_exact_power_product. ((((exists ff_h_valuation_exact_power_product_factor. ff_h_valuation_exact_power_product_factor + S (ff_p_valuation_exact_power_product) = S ((S (ff_i_valuation_exact_power_product)) * ff_c_valuation_exact_power)) /\ exists ff_q_valuation_exact_power_product_factor. ff_b_valuation_exact_power = ff_q_valuation_exact_power_product_factor * S ((S (ff_i_valuation_exact_power_product)) * ff_c_valuation_exact_power) + (ff_p_valuation_exact_power_product))) /\ ((((exists ff_h_valuation_exact_power_product_partial. ff_h_valuation_exact_power_product_partial + S (ff_r_valuation_exact_power_product) = S ((S (ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product)) /\ exists ff_q_valuation_exact_power_product_partial. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_partial * S ((S (ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product) + (ff_r_valuation_exact_power_product))) /\ ((((exists ff_h_valuation_exact_power_product_successor. ff_h_valuation_exact_power_product_successor + S (ff_s_valuation_exact_power_product) = S ((S (S ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product)) /\ exists ff_q_valuation_exact_power_product_successor. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_successor * S ((S (S ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product) + (ff_s_valuation_exact_power_product))) /\ ff_s_valuation_exact_power_product = ff_r_valuation_exact_power_product * ff_p_valuation_exact_power_product)))))))) /\ ((a = bpd_result_valuation_exact * bpd_cofactor_valuation_exact) /\ ((~(bpd_cofactor_valuation_exact = 0)) /\ (~(exists bpd_factor_valuation_exact_prime. bpd_cofactor_valuation_exact = (p) * bpd_factor_valuation_exact_prime))))))

Structural proof guide

A prime valuation extracts a nonzero cofactor not divisible by its prime.

Direct prerequisites: power_valuation_selected_and_successor_not_divides, power_divides_successor_of_cofactor. The authored body proceeds by case analysis (4), intermediate claims (1), equality transport (1).

Proof neighborhood

Direct dependencies

BT00QI power_valuation_selected_and_successor_not_divides BT00QM power_divides_successor_of_cofactor

Direct dependents

BT00QQ power_valuation_mul_successor_not_divides

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

0001intro p
0002intro a
0003intro e
0004intro hp
0005intro ha
0006intro hvaluation
0007have hcharacterization : (exists bpv_result_bpd_exact_selected. ((exists ff_b_bpd_exact_selected_power ff_c_bpd_exact_selected_power. ((forall ff_i_bpd_exact_selected_power_repeat. (exists ff_lt_bpd_exact_selected_power_repeat_bound. ff_lt_bpd_exact_selected_power_repeat_bound + S ff_i_bpd_exact_selected_power_repeat = e) -> (((exists ff_h_bpd_exact_selected_power_repeat_decoded. ff_h_bpd_exact_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_exact_selected_power_repeat)) * ff_c_bpd_exact_selected_power)) /\ exists ff_q_bpd_exact_selected_power_repeat_decoded. ff_b_bpd_exact_selected_power = ff_q_bpd_exact_selected_power_repeat_decoded * S ((S (ff_i_bpd_exact_selected_power_repeat)) * ff_c_bpd_exact_selected_power) + (p)))) /\ (exists ff_u_bpd_exact_selected_power_product ff_v_bpd_exact_selected_power_product. ((((exists ff_h_bpd_exact_selected_power_product_start. ff_h_bpd_exact_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_exact_selected_power_product)) /\ exists ff_q_bpd_exact_selected_power_product_start. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_start * S ((S (0)) * ff_v_bpd_exact_selected_power_product) + (1))) /\ ((((exists ff_h_bpd_exact_selected_power_product_terminal. ff_h_bpd_exact_selected_power_product_terminal + S (bpv_result_bpd_exact_selected) = S ((S (e)) * ff_v_bpd_exact_selected_power_product)) /\ exists ff_q_bpd_exact_selected_power_product_terminal. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_exact_selected_power_product) + (bpv_result_bpd_exact_selected))) /\ forall ff_i_bpd_exact_selected_power_product. (exists ff_lt_bpd_exact_selected_power_product_bound. ff_lt_bpd_exact_selected_power_product_bound + S ff_i_bpd_exact_selected_power_product = e) -> exists ff_p_bpd_exact_selected_power_product ff_r_bpd_exact_selected_power_product ff_s_bpd_exact_selected_power_product. ((((exists ff_h_bpd_exact_selected_power_product_factor. ff_h_bpd_exact_selected_power_product_factor + S (ff_p_bpd_exact_selected_power_product) = S ((S (ff_i_bpd_exact_selected_power_product)) * ff_c_bpd_exact_selected_power)) /\ exists ff_q_bpd_exact_selected_power_product_factor. ff_b_bpd_exact_selected_power = ff_q_bpd_exact_selected_power_product_factor * S ((S (ff_i_bpd_exact_selected_power_product)) * ff_c_bpd_exact_selected_power) + (ff_p_bpd_exact_selected_power_product))) /\ ((((exists ff_h_bpd_exact_selected_power_product_partial. ff_h_bpd_exact_selected_power_product_partial + S (ff_r_bpd_exact_selected_power_product) = S ((S (ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product)) /\ exists ff_q_bpd_exact_selected_power_product_partial. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_partial * S ((S (ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product) + (ff_r_bpd_exact_selected_power_product))) /\ ((((exists ff_h_bpd_exact_selected_power_product_successor. ff_h_bpd_exact_selected_power_product_successor + S (ff_s_bpd_exact_selected_power_product) = S ((S (S ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product)) /\ exists ff_q_bpd_exact_selected_power_product_successor. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_successor * S ((S (S ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product) + (ff_s_bpd_exact_selected_power_product))) /\ ff_s_bpd_exact_selected_power_product = ff_r_bpd_exact_selected_power_product * ff_p_bpd_exact_selected_power_product)))))))) /\ (exists bpv_factor_bpd_exact_selected_divides. a = bpv_result_bpd_exact_selected * bpv_factor_bpd_exact_selected_divides))) /\ ~(exists bpvi_result_bpd_exact_successor. ((exists bpvi_b_bpd_exact_successor_power bpvi_c_bpd_exact_successor_power. ((forall bpvi_i_bpd_exact_successor_power. (exists bpvi_repeat_gap_bpd_exact_successor_power. bpvi_repeat_gap_bpd_exact_successor_power + S bpvi_i_bpd_exact_successor_power = S e) -> (((exists bpvi_h_bpd_exact_successor_power_repeat. bpvi_h_bpd_exact_successor_power_repeat + S (p) = S ((S (bpvi_i_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power)) /\ exists bpvi_q_bpd_exact_successor_power_repeat. bpvi_b_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_repeat * S ((S (bpvi_i_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power) + (p)))) /\ (exists bpvi_u_bpd_exact_successor_power bpvi_v_bpd_exact_successor_power. ((((exists bpvi_h_bpd_exact_successor_power_start. bpvi_h_bpd_exact_successor_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_exact_successor_power)) /\ exists bpvi_q_bpd_exact_successor_power_start. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_start * S ((S (0)) * bpvi_v_bpd_exact_successor_power) + (1))) /\ ((((exists bpvi_h_bpd_exact_successor_power_terminal. bpvi_h_bpd_exact_successor_power_terminal + S (bpvi_result_bpd_exact_successor) = S ((S (S e)) * bpvi_v_bpd_exact_successor_power)) /\ exists bpvi_q_bpd_exact_successor_power_terminal. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_terminal * S ((S (S e)) * bpvi_v_bpd_exact_successor_power) + (bpvi_result_bpd_exact_successor))) /\ forall bpvi_j_bpd_exact_successor_power. (exists bpvi_product_gap_bpd_exact_successor_power. bpvi_product_gap_bpd_exact_successor_power + S bpvi_j_bpd_exact_successor_power = S e) -> exists bpvi_factor_bpd_exact_successor_power bpvi_partial_bpd_exact_successor_power bpvi_successor_bpd_exact_successor_power. ((((exists bpvi_h_bpd_exact_successor_power_factor. bpvi_h_bpd_exact_successor_power_factor + S (bpvi_factor_bpd_exact_successor_power) = S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power)) /\ exists bpvi_q_bpd_exact_successor_power_factor. bpvi_b_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_factor * S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power) + (bpvi_factor_bpd_exact_successor_power))) /\ ((((exists bpvi_h_bpd_exact_successor_power_partial. bpvi_h_bpd_exact_successor_power_partial + S (bpvi_partial_bpd_exact_successor_power) = S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power)) /\ exists bpvi_q_bpd_exact_successor_power_partial. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_partial * S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power) + (bpvi_partial_bpd_exact_successor_power))) /\ ((((exists bpvi_h_bpd_exact_successor_power_successor. bpvi_h_bpd_exact_successor_power_successor + S (bpvi_successor_bpd_exact_successor_power) = S ((S (S bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power)) /\ exists bpvi_q_bpd_exact_successor_power_successor. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_successor * S ((S (S bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power) + (bpvi_successor_bpd_exact_successor_power))) /\ bpvi_successor_bpd_exact_successor_power = bpvi_partial_bpd_exact_successor_power * bpvi_factor_bpd_exact_successor_power)))))))) /\ exists bpvi_divisor_factor_bpd_exact_successor. a = bpvi_result_bpd_exact_successor * bpvi_divisor_factor_bpd_exact_successor))
0008specialize power_valuation_selected_and_successor_not_divides p
0009specialize power_valuation_selected_and_successor_not_divides a
0010specialize power_valuation_selected_and_successor_not_divides e
0011apply power_valuation_selected_and_successor_not_divides
0012exact hp
0013exact ha
0014exact hvaluation
0015cases hcharacterization
0016cases hcharacterization_left
0017cases hcharacterization_left_witness
0018cases hcharacterization_left_witness_right
0019exists x
0020exists x1
0021split
0022exact hcharacterization_left_witness_left
0023split
0024exact hcharacterization_left_witness_right_witness
0025split
0026intro hcofactor_zero
0027apply ha
0028trans x * x1
0029exact hcharacterization_left_witness_right_witness
0030rewrite hcofactor_zero
0031apply PA5
0032intro hcofactor_prime
0033apply hcharacterization_right
0034specialize power_divides_successor_of_cofactor p
0035specialize power_divides_successor_of_cofactor e
0036specialize power_divides_successor_of_cofactor a
0037specialize power_divides_successor_of_cofactor x
0038specialize power_divides_successor_of_cofactor x1
0039apply power_divides_successor_of_cofactor
0040exact hcharacterization_left_witness_left
0041exact hcharacterization_left_witness_right_witness
0042exact hcofactor_prime