BT00RM — factorial_valuation

Exact expanded PA statement

forall p n. exists e. (exists bfv_factorial_bfv_total. ((exists ff_b_bfv_total_factorial ff_c_bfv_total_factorial. ((forall ff_i_bfv_total_factorial_range. (exists ff_lt_bfv_total_factorial_range_bound. ff_lt_bfv_total_factorial_range_bound + S ff_i_bfv_total_factorial_range = n) -> (((exists ff_h_bfv_total_factorial_range_decoded. ff_h_bfv_total_factorial_range_decoded + S (1 + ff_i_bfv_total_factorial_range) = S ((S (ff_i_bfv_total_factorial_range)) * ff_c_bfv_total_factorial)) /\ exists ff_q_bfv_total_factorial_range_decoded. ff_b_bfv_total_factorial = ff_q_bfv_total_factorial_range_decoded * S ((S (ff_i_bfv_total_factorial_range)) * ff_c_bfv_total_factorial) + (1 + ff_i_bfv_total_factorial_range)))) /\ (exists ff_u_bfv_total_factorial_product ff_v_bfv_total_factorial_product. ((((exists ff_h_bfv_total_factorial_product_start. ff_h_bfv_total_factorial_product_start + S (1) = S ((S (0)) * ff_v_bfv_total_factorial_product)) /\ exists ff_q_bfv_total_factorial_product_start. ff_u_bfv_total_factorial_product = ff_q_bfv_total_factorial_product_start * S ((S (0)) * ff_v_bfv_total_factorial_product) + (1))) /\ ((((exists ff_h_bfv_total_factorial_product_terminal. ff_h_bfv_total_factorial_product_terminal + S (bfv_factorial_bfv_total) = S ((S (n)) * ff_v_bfv_total_factorial_product)) /\ exists ff_q_bfv_total_factorial_product_terminal. ff_u_bfv_total_factorial_product = ff_q_bfv_total_factorial_product_terminal * S ((S (n)) * ff_v_bfv_total_factorial_product) + (bfv_factorial_bfv_total))) /\ forall ff_i_bfv_total_factorial_product. (exists ff_lt_bfv_total_factorial_product_bound. ff_lt_bfv_total_factorial_product_bound + S ff_i_bfv_total_factorial_product = n) -> exists ff_p_bfv_total_factorial_product ff_r_bfv_total_factorial_product ff_s_bfv_total_factorial_product. ((((exists ff_h_bfv_total_factorial_product_factor. ff_h_bfv_total_factorial_product_factor + S (ff_p_bfv_total_factorial_product) = S ((S (ff_i_bfv_total_factorial_product)) * ff_c_bfv_total_factorial)) /\ exists ff_q_bfv_total_factorial_product_factor. ff_b_bfv_total_factorial = ff_q_bfv_total_factorial_product_factor * S ((S (ff_i_bfv_total_factorial_product)) * ff_c_bfv_total_factorial) + (ff_p_bfv_total_factorial_product))) /\ ((((exists ff_h_bfv_total_factorial_product_partial. ff_h_bfv_total_factorial_product_partial + S (ff_r_bfv_total_factorial_product) = S ((S (ff_i_bfv_total_factorial_product)) * ff_v_bfv_total_factorial_product)) /\ exists ff_q_bfv_total_factorial_product_partial. ff_u_bfv_total_factorial_product = ff_q_bfv_total_factorial_product_partial * S ((S (ff_i_bfv_total_factorial_product)) * ff_v_bfv_total_factorial_product) + (ff_r_bfv_total_factorial_product))) /\ ((((exists ff_h_bfv_total_factorial_product_successor. ff_h_bfv_total_factorial_product_successor + S (ff_s_bfv_total_factorial_product) = S ((S (S ff_i_bfv_total_factorial_product)) * ff_v_bfv_total_factorial_product)) /\ exists ff_q_bfv_total_factorial_product_successor. ff_u_bfv_total_factorial_product = ff_q_bfv_total_factorial_product_successor * S ((S (S ff_i_bfv_total_factorial_product)) * ff_v_bfv_total_factorial_product) + (ff_s_bfv_total_factorial_product))) /\ ff_s_bfv_total_factorial_product = ff_r_bfv_total_factorial_product * ff_p_bfv_total_factorial_product)))))))) /\ (((exists bpv_gap_bfv_total_valuation_exponent_bound. bpv_gap_bfv_total_valuation_exponent_bound + e = bfv_factorial_bfv_total) /\ (exists bpv_result_bfv_total_valuation_selected. ((exists ff_b_bfv_total_valuation_selected_power ff_c_bfv_total_valuation_selected_power. ((forall ff_i_bfv_total_valuation_selected_power_repeat. (exists ff_lt_bfv_total_valuation_selected_power_repeat_bound. ff_lt_bfv_total_valuation_selected_power_repeat_bound + S ff_i_bfv_total_valuation_selected_power_repeat = e) -> (((exists ff_h_bfv_total_valuation_selected_power_repeat_decoded. ff_h_bfv_total_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_total_valuation_selected_power_repeat)) * ff_c_bfv_total_valuation_selected_power)) /\ exists ff_q_bfv_total_valuation_selected_power_repeat_decoded. ff_b_bfv_total_valuation_selected_power = ff_q_bfv_total_valuation_selected_power_repeat_decoded * S ((S (ff_i_bfv_total_valuation_selected_power_repeat)) * ff_c_bfv_total_valuation_selected_power) + (p)))) /\ (exists ff_u_bfv_total_valuation_selected_power_product ff_v_bfv_total_valuation_selected_power_product. ((((exists ff_h_bfv_total_valuation_selected_power_product_start. ff_h_bfv_total_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_total_valuation_selected_power_product)) /\ exists ff_q_bfv_total_valuation_selected_power_product_start. ff_u_bfv_total_valuation_selected_power_product = ff_q_bfv_total_valuation_selected_power_product_start * S ((S (0)) * ff_v_bfv_total_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bfv_total_valuation_selected_power_product_terminal. ff_h_bfv_total_valuation_selected_power_product_terminal + S (bpv_result_bfv_total_valuation_selected) = S ((S (e)) * ff_v_bfv_total_valuation_selected_power_product)) /\ exists ff_q_bfv_total_valuation_selected_power_product_terminal. ff_u_bfv_total_valuation_selected_power_product = ff_q_bfv_total_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_total_valuation_selected_power_product) + (bpv_result_bfv_total_valuation_selected))) /\ forall ff_i_bfv_total_valuation_selected_power_product. (exists ff_lt_bfv_total_valuation_selected_power_product_bound. ff_lt_bfv_total_valuation_selected_power_product_bound + S ff_i_bfv_total_valuation_selected_power_product = e) -> exists ff_p_bfv_total_valuation_selected_power_product ff_r_bfv_total_valuation_selected_power_product ff_s_bfv_total_valuation_selected_power_product. ((((exists ff_h_bfv_total_valuation_selected_power_product_factor. ff_h_bfv_total_valuation_selected_power_product_factor + S (ff_p_bfv_total_valuation_selected_power_product) = S ((S (ff_i_bfv_total_valuation_selected_power_product)) * ff_c_bfv_total_valuation_selected_power)) /\ exists ff_q_bfv_total_valuation_selected_power_product_factor. ff_b_bfv_total_valuation_selected_power = ff_q_bfv_total_valuation_selected_power_product_factor * S ((S (ff_i_bfv_total_valuation_selected_power_product)) * ff_c_bfv_total_valuation_selected_power) + (ff_p_bfv_total_valuation_selected_power_product))) /\ ((((exists ff_h_bfv_total_valuation_selected_power_product_partial. ff_h_bfv_total_valuation_selected_power_product_partial + S (ff_r_bfv_total_valuation_selected_power_product) = S ((S (ff_i_bfv_total_valuation_selected_power_product)) * ff_v_bfv_total_valuation_selected_power_product)) /\ exists ff_q_bfv_total_valuation_selected_power_product_partial. ff_u_bfv_total_valuation_selected_power_product = ff_q_bfv_total_valuation_selected_power_product_partial * S ((S (ff_i_bfv_total_valuation_selected_power_product)) * ff_v_bfv_total_valuation_selected_power_product) + (ff_r_bfv_total_valuation_selected_power_product))) /\ ((((exists ff_h_bfv_total_valuation_selected_power_product_successor. ff_h_bfv_total_valuation_selected_power_product_successor + S (ff_s_bfv_total_valuation_selected_power_product) = S ((S (S ff_i_bfv_total_valuation_selected_power_product)) * ff_v_bfv_total_valuation_selected_power_product)) /\ exists ff_q_bfv_total_valuation_selected_power_product_successor. ff_u_bfv_total_valuation_selected_power_product = ff_q_bfv_total_valuation_selected_power_product_successor * S ((S (S ff_i_bfv_total_valuation_selected_power_product)) * ff_v_bfv_total_valuation_selected_power_product) + (ff_s_bfv_total_valuation_selected_power_product))) /\ ff_s_bfv_total_valuation_selected_power_product = ff_r_bfv_total_valuation_selected_power_product * ff_p_bfv_total_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bfv_total_valuation_selected_divides. bfv_factorial_bfv_total = bpv_result_bfv_total_valuation_selected * bpv_factor_bfv_total_valuation_selected_divides)))) /\ forall bpv_candidate_bfv_total_valuation. (exists bpv_gap_bfv_total_valuation_candidate_bound. bpv_gap_bfv_total_valuation_candidate_bound + bpv_candidate_bfv_total_valuation = bfv_factorial_bfv_total) -> (exists bpv_result_bfv_total_valuation_candidate. ((exists ff_b_bfv_total_valuation_candidate_power ff_c_bfv_total_valuation_candidate_power. ((forall ff_i_bfv_total_valuation_candidate_power_repeat. (exists ff_lt_bfv_total_valuation_candidate_power_repeat_bound. ff_lt_bfv_total_valuation_candidate_power_repeat_bound + S ff_i_bfv_total_valuation_candidate_power_repeat = bpv_candidate_bfv_total_valuation) -> (((exists ff_h_bfv_total_valuation_candidate_power_repeat_decoded. ff_h_bfv_total_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_total_valuation_candidate_power_repeat)) * ff_c_bfv_total_valuation_candidate_power)) /\ exists ff_q_bfv_total_valuation_candidate_power_repeat_decoded. ff_b_bfv_total_valuation_candidate_power = ff_q_bfv_total_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bfv_total_valuation_candidate_power_repeat)) * ff_c_bfv_total_valuation_candidate_power) + (p)))) /\ (exists ff_u_bfv_total_valuation_candidate_power_product ff_v_bfv_total_valuation_candidate_power_product. ((((exists ff_h_bfv_total_valuation_candidate_power_product_start. ff_h_bfv_total_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_total_valuation_candidate_power_product)) /\ exists ff_q_bfv_total_valuation_candidate_power_product_start. ff_u_bfv_total_valuation_candidate_power_product = ff_q_bfv_total_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bfv_total_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bfv_total_valuation_candidate_power_product_terminal. ff_h_bfv_total_valuation_candidate_power_product_terminal + S (bpv_result_bfv_total_valuation_candidate) = S ((S (bpv_candidate_bfv_total_valuation)) * ff_v_bfv_total_valuation_candidate_power_product)) /\ exists ff_q_bfv_total_valuation_candidate_power_product_terminal. ff_u_bfv_total_valuation_candidate_power_product = ff_q_bfv_total_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_total_valuation)) * ff_v_bfv_total_valuation_candidate_power_product) + (bpv_result_bfv_total_valuation_candidate))) /\ forall ff_i_bfv_total_valuation_candidate_power_product. (exists ff_lt_bfv_total_valuation_candidate_power_product_bound. ff_lt_bfv_total_valuation_candidate_power_product_bound + S ff_i_bfv_total_valuation_candidate_power_product = bpv_candidate_bfv_total_valuation) -> exists ff_p_bfv_total_valuation_candidate_power_product ff_r_bfv_total_valuation_candidate_power_product ff_s_bfv_total_valuation_candidate_power_product. ((((exists ff_h_bfv_total_valuation_candidate_power_product_factor. ff_h_bfv_total_valuation_candidate_power_product_factor + S (ff_p_bfv_total_valuation_candidate_power_product) = S ((S (ff_i_bfv_total_valuation_candidate_power_product)) * ff_c_bfv_total_valuation_candidate_power)) /\ exists ff_q_bfv_total_valuation_candidate_power_product_factor. ff_b_bfv_total_valuation_candidate_power = ff_q_bfv_total_valuation_candidate_power_product_factor * S ((S (ff_i_bfv_total_valuation_candidate_power_product)) * ff_c_bfv_total_valuation_candidate_power) + (ff_p_bfv_total_valuation_candidate_power_product))) /\ ((((exists ff_h_bfv_total_valuation_candidate_power_product_partial. ff_h_bfv_total_valuation_candidate_power_product_partial + S (ff_r_bfv_total_valuation_candidate_power_product) = S ((S (ff_i_bfv_total_valuation_candidate_power_product)) * ff_v_bfv_total_valuation_candidate_power_product)) /\ exists ff_q_bfv_total_valuation_candidate_power_product_partial. ff_u_bfv_total_valuation_candidate_power_product = ff_q_bfv_total_valuation_candidate_power_product_partial * S ((S (ff_i_bfv_total_valuation_candidate_power_product)) * ff_v_bfv_total_valuation_candidate_power_product) + (ff_r_bfv_total_valuation_candidate_power_product))) /\ ((((exists ff_h_bfv_total_valuation_candidate_power_product_successor. ff_h_bfv_total_valuation_candidate_power_product_successor + S (ff_s_bfv_total_valuation_candidate_power_product) = S ((S (S ff_i_bfv_total_valuation_candidate_power_product)) * ff_v_bfv_total_valuation_candidate_power_product)) /\ exists ff_q_bfv_total_valuation_candidate_power_product_successor. ff_u_bfv_total_valuation_candidate_power_product = ff_q_bfv_total_valuation_candidate_power_product_successor * S ((S (S ff_i_bfv_total_valuation_candidate_power_product)) * ff_v_bfv_total_valuation_candidate_power_product) + (ff_s_bfv_total_valuation_candidate_power_product))) /\ ff_s_bfv_total_valuation_candidate_power_product = ff_r_bfv_total_valuation_candidate_power_product * ff_p_bfv_total_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bfv_total_valuation_candidate_divides. bfv_factorial_bfv_total = bpv_result_bfv_total_valuation_candidate * bpv_factor_bfv_total_valuation_candidate_divides))) -> (exists bpv_gap_bfv_total_valuation_maximal. bpv_gap_bfv_total_valuation_maximal + bpv_candidate_bfv_total_valuation = e))))

Structural proof guide

Every factorial has a canonical bounded valuation at every base.

Direct prerequisites: factorial_exists, power_valuation_exists. The authored body proceeds by case analysis (2), intermediate claims (2).

Proof neighborhood

Direct dependencies

BT008Y factorial_exists BT00Q7 power_valuation_exists

Direct dependents

BT00T1 prime_factorial_valuation_eq_legendre_sum BT00XN central_binom_legendre_valuation_balance

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 n
0003have hfactorial : exists F. (exists ff_b_bfv_total_witness ff_c_bfv_total_witness. ((forall ff_i_bfv_total_witness_range. (exists ff_lt_bfv_total_witness_range_bound. ff_lt_bfv_total_witness_range_bound + S ff_i_bfv_total_witness_range = n) -> (((exists ff_h_bfv_total_witness_range_decoded. ff_h_bfv_total_witness_range_decoded + S (1 + ff_i_bfv_total_witness_range) = S ((S (ff_i_bfv_total_witness_range)) * ff_c_bfv_total_witness)) /\ exists ff_q_bfv_total_witness_range_decoded. ff_b_bfv_total_witness = ff_q_bfv_total_witness_range_decoded * S ((S (ff_i_bfv_total_witness_range)) * ff_c_bfv_total_witness) + (1 + ff_i_bfv_total_witness_range)))) /\ (exists ff_u_bfv_total_witness_product ff_v_bfv_total_witness_product. ((((exists ff_h_bfv_total_witness_product_start. ff_h_bfv_total_witness_product_start + S (1) = S ((S (0)) * ff_v_bfv_total_witness_product)) /\ exists ff_q_bfv_total_witness_product_start. ff_u_bfv_total_witness_product = ff_q_bfv_total_witness_product_start * S ((S (0)) * ff_v_bfv_total_witness_product) + (1))) /\ ((((exists ff_h_bfv_total_witness_product_terminal. ff_h_bfv_total_witness_product_terminal + S (F) = S ((S (n)) * ff_v_bfv_total_witness_product)) /\ exists ff_q_bfv_total_witness_product_terminal. ff_u_bfv_total_witness_product = ff_q_bfv_total_witness_product_terminal * S ((S (n)) * ff_v_bfv_total_witness_product) + (F))) /\ forall ff_i_bfv_total_witness_product. (exists ff_lt_bfv_total_witness_product_bound. ff_lt_bfv_total_witness_product_bound + S ff_i_bfv_total_witness_product = n) -> exists ff_p_bfv_total_witness_product ff_r_bfv_total_witness_product ff_s_bfv_total_witness_product. ((((exists ff_h_bfv_total_witness_product_factor. ff_h_bfv_total_witness_product_factor + S (ff_p_bfv_total_witness_product) = S ((S (ff_i_bfv_total_witness_product)) * ff_c_bfv_total_witness)) /\ exists ff_q_bfv_total_witness_product_factor. ff_b_bfv_total_witness = ff_q_bfv_total_witness_product_factor * S ((S (ff_i_bfv_total_witness_product)) * ff_c_bfv_total_witness) + (ff_p_bfv_total_witness_product))) /\ ((((exists ff_h_bfv_total_witness_product_partial. ff_h_bfv_total_witness_product_partial + S (ff_r_bfv_total_witness_product) = S ((S (ff_i_bfv_total_witness_product)) * ff_v_bfv_total_witness_product)) /\ exists ff_q_bfv_total_witness_product_partial. ff_u_bfv_total_witness_product = ff_q_bfv_total_witness_product_partial * S ((S (ff_i_bfv_total_witness_product)) * ff_v_bfv_total_witness_product) + (ff_r_bfv_total_witness_product))) /\ ((((exists ff_h_bfv_total_witness_product_successor. ff_h_bfv_total_witness_product_successor + S (ff_s_bfv_total_witness_product) = S ((S (S ff_i_bfv_total_witness_product)) * ff_v_bfv_total_witness_product)) /\ exists ff_q_bfv_total_witness_product_successor. ff_u_bfv_total_witness_product = ff_q_bfv_total_witness_product_successor * S ((S (S ff_i_bfv_total_witness_product)) * ff_v_bfv_total_witness_product) + (ff_s_bfv_total_witness_product))) /\ ff_s_bfv_total_witness_product = ff_r_bfv_total_witness_product * ff_p_bfv_total_witness_product))))))))
0004specialize factorial_exists n
0005exact factorial_exists
0006cases hfactorial
0007have hvaluation : exists e. (((exists bpv_gap_bfv_total_graph_exponent_bound. bpv_gap_bfv_total_graph_exponent_bound + e = x) /\ (exists bpv_result_bfv_total_graph_selected. ((exists ff_b_bfv_total_graph_selected_power ff_c_bfv_total_graph_selected_power. ((forall ff_i_bfv_total_graph_selected_power_repeat. (exists ff_lt_bfv_total_graph_selected_power_repeat_bound. ff_lt_bfv_total_graph_selected_power_repeat_bound + S ff_i_bfv_total_graph_selected_power_repeat = e) -> (((exists ff_h_bfv_total_graph_selected_power_repeat_decoded. ff_h_bfv_total_graph_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_total_graph_selected_power_repeat)) * ff_c_bfv_total_graph_selected_power)) /\ exists ff_q_bfv_total_graph_selected_power_repeat_decoded. ff_b_bfv_total_graph_selected_power = ff_q_bfv_total_graph_selected_power_repeat_decoded * S ((S (ff_i_bfv_total_graph_selected_power_repeat)) * ff_c_bfv_total_graph_selected_power) + (p)))) /\ (exists ff_u_bfv_total_graph_selected_power_product ff_v_bfv_total_graph_selected_power_product. ((((exists ff_h_bfv_total_graph_selected_power_product_start. ff_h_bfv_total_graph_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_total_graph_selected_power_product)) /\ exists ff_q_bfv_total_graph_selected_power_product_start. ff_u_bfv_total_graph_selected_power_product = ff_q_bfv_total_graph_selected_power_product_start * S ((S (0)) * ff_v_bfv_total_graph_selected_power_product) + (1))) /\ ((((exists ff_h_bfv_total_graph_selected_power_product_terminal. ff_h_bfv_total_graph_selected_power_product_terminal + S (bpv_result_bfv_total_graph_selected) = S ((S (e)) * ff_v_bfv_total_graph_selected_power_product)) /\ exists ff_q_bfv_total_graph_selected_power_product_terminal. ff_u_bfv_total_graph_selected_power_product = ff_q_bfv_total_graph_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_total_graph_selected_power_product) + (bpv_result_bfv_total_graph_selected))) /\ forall ff_i_bfv_total_graph_selected_power_product. (exists ff_lt_bfv_total_graph_selected_power_product_bound. ff_lt_bfv_total_graph_selected_power_product_bound + S ff_i_bfv_total_graph_selected_power_product = e) -> exists ff_p_bfv_total_graph_selected_power_product ff_r_bfv_total_graph_selected_power_product ff_s_bfv_total_graph_selected_power_product. ((((exists ff_h_bfv_total_graph_selected_power_product_factor. ff_h_bfv_total_graph_selected_power_product_factor + S (ff_p_bfv_total_graph_selected_power_product) = S ((S (ff_i_bfv_total_graph_selected_power_product)) * ff_c_bfv_total_graph_selected_power)) /\ exists ff_q_bfv_total_graph_selected_power_product_factor. ff_b_bfv_total_graph_selected_power = ff_q_bfv_total_graph_selected_power_product_factor * S ((S (ff_i_bfv_total_graph_selected_power_product)) * ff_c_bfv_total_graph_selected_power) + (ff_p_bfv_total_graph_selected_power_product))) /\ ((((exists ff_h_bfv_total_graph_selected_power_product_partial. ff_h_bfv_total_graph_selected_power_product_partial + S (ff_r_bfv_total_graph_selected_power_product) = S ((S (ff_i_bfv_total_graph_selected_power_product)) * ff_v_bfv_total_graph_selected_power_product)) /\ exists ff_q_bfv_total_graph_selected_power_product_partial. ff_u_bfv_total_graph_selected_power_product = ff_q_bfv_total_graph_selected_power_product_partial * S ((S (ff_i_bfv_total_graph_selected_power_product)) * ff_v_bfv_total_graph_selected_power_product) + (ff_r_bfv_total_graph_selected_power_product))) /\ ((((exists ff_h_bfv_total_graph_selected_power_product_successor. ff_h_bfv_total_graph_selected_power_product_successor + S (ff_s_bfv_total_graph_selected_power_product) = S ((S (S ff_i_bfv_total_graph_selected_power_product)) * ff_v_bfv_total_graph_selected_power_product)) /\ exists ff_q_bfv_total_graph_selected_power_product_successor. ff_u_bfv_total_graph_selected_power_product = ff_q_bfv_total_graph_selected_power_product_successor * S ((S (S ff_i_bfv_total_graph_selected_power_product)) * ff_v_bfv_total_graph_selected_power_product) + (ff_s_bfv_total_graph_selected_power_product))) /\ ff_s_bfv_total_graph_selected_power_product = ff_r_bfv_total_graph_selected_power_product * ff_p_bfv_total_graph_selected_power_product)))))))) /\ (exists bpv_factor_bfv_total_graph_selected_divides. x = bpv_result_bfv_total_graph_selected * bpv_factor_bfv_total_graph_selected_divides)))) /\ forall bpv_candidate_bfv_total_graph. (exists bpv_gap_bfv_total_graph_candidate_bound. bpv_gap_bfv_total_graph_candidate_bound + bpv_candidate_bfv_total_graph = x) -> (exists bpv_result_bfv_total_graph_candidate. ((exists ff_b_bfv_total_graph_candidate_power ff_c_bfv_total_graph_candidate_power. ((forall ff_i_bfv_total_graph_candidate_power_repeat. (exists ff_lt_bfv_total_graph_candidate_power_repeat_bound. ff_lt_bfv_total_graph_candidate_power_repeat_bound + S ff_i_bfv_total_graph_candidate_power_repeat = bpv_candidate_bfv_total_graph) -> (((exists ff_h_bfv_total_graph_candidate_power_repeat_decoded. ff_h_bfv_total_graph_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_total_graph_candidate_power_repeat)) * ff_c_bfv_total_graph_candidate_power)) /\ exists ff_q_bfv_total_graph_candidate_power_repeat_decoded. ff_b_bfv_total_graph_candidate_power = ff_q_bfv_total_graph_candidate_power_repeat_decoded * S ((S (ff_i_bfv_total_graph_candidate_power_repeat)) * ff_c_bfv_total_graph_candidate_power) + (p)))) /\ (exists ff_u_bfv_total_graph_candidate_power_product ff_v_bfv_total_graph_candidate_power_product. ((((exists ff_h_bfv_total_graph_candidate_power_product_start. ff_h_bfv_total_graph_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_total_graph_candidate_power_product)) /\ exists ff_q_bfv_total_graph_candidate_power_product_start. ff_u_bfv_total_graph_candidate_power_product = ff_q_bfv_total_graph_candidate_power_product_start * S ((S (0)) * ff_v_bfv_total_graph_candidate_power_product) + (1))) /\ ((((exists ff_h_bfv_total_graph_candidate_power_product_terminal. ff_h_bfv_total_graph_candidate_power_product_terminal + S (bpv_result_bfv_total_graph_candidate) = S ((S (bpv_candidate_bfv_total_graph)) * ff_v_bfv_total_graph_candidate_power_product)) /\ exists ff_q_bfv_total_graph_candidate_power_product_terminal. ff_u_bfv_total_graph_candidate_power_product = ff_q_bfv_total_graph_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_total_graph)) * ff_v_bfv_total_graph_candidate_power_product) + (bpv_result_bfv_total_graph_candidate))) /\ forall ff_i_bfv_total_graph_candidate_power_product. (exists ff_lt_bfv_total_graph_candidate_power_product_bound. ff_lt_bfv_total_graph_candidate_power_product_bound + S ff_i_bfv_total_graph_candidate_power_product = bpv_candidate_bfv_total_graph) -> exists ff_p_bfv_total_graph_candidate_power_product ff_r_bfv_total_graph_candidate_power_product ff_s_bfv_total_graph_candidate_power_product. ((((exists ff_h_bfv_total_graph_candidate_power_product_factor. ff_h_bfv_total_graph_candidate_power_product_factor + S (ff_p_bfv_total_graph_candidate_power_product) = S ((S (ff_i_bfv_total_graph_candidate_power_product)) * ff_c_bfv_total_graph_candidate_power)) /\ exists ff_q_bfv_total_graph_candidate_power_product_factor. ff_b_bfv_total_graph_candidate_power = ff_q_bfv_total_graph_candidate_power_product_factor * S ((S (ff_i_bfv_total_graph_candidate_power_product)) * ff_c_bfv_total_graph_candidate_power) + (ff_p_bfv_total_graph_candidate_power_product))) /\ ((((exists ff_h_bfv_total_graph_candidate_power_product_partial. ff_h_bfv_total_graph_candidate_power_product_partial + S (ff_r_bfv_total_graph_candidate_power_product) = S ((S (ff_i_bfv_total_graph_candidate_power_product)) * ff_v_bfv_total_graph_candidate_power_product)) /\ exists ff_q_bfv_total_graph_candidate_power_product_partial. ff_u_bfv_total_graph_candidate_power_product = ff_q_bfv_total_graph_candidate_power_product_partial * S ((S (ff_i_bfv_total_graph_candidate_power_product)) * ff_v_bfv_total_graph_candidate_power_product) + (ff_r_bfv_total_graph_candidate_power_product))) /\ ((((exists ff_h_bfv_total_graph_candidate_power_product_successor. ff_h_bfv_total_graph_candidate_power_product_successor + S (ff_s_bfv_total_graph_candidate_power_product) = S ((S (S ff_i_bfv_total_graph_candidate_power_product)) * ff_v_bfv_total_graph_candidate_power_product)) /\ exists ff_q_bfv_total_graph_candidate_power_product_successor. ff_u_bfv_total_graph_candidate_power_product = ff_q_bfv_total_graph_candidate_power_product_successor * S ((S (S ff_i_bfv_total_graph_candidate_power_product)) * ff_v_bfv_total_graph_candidate_power_product) + (ff_s_bfv_total_graph_candidate_power_product))) /\ ff_s_bfv_total_graph_candidate_power_product = ff_r_bfv_total_graph_candidate_power_product * ff_p_bfv_total_graph_candidate_power_product)))))))) /\ (exists bpv_factor_bfv_total_graph_candidate_divides. x = bpv_result_bfv_total_graph_candidate * bpv_factor_bfv_total_graph_candidate_divides))) -> (exists bpv_gap_bfv_total_graph_maximal. bpv_gap_bfv_total_graph_maximal + bpv_candidate_bfv_total_graph = e))
0008specialize power_valuation_exists p
0009specialize power_valuation_exists x
0010exact power_valuation_exists
0011cases hvaluation
0012exists x1
0013exists x
0014split
0015exact hfactorial_witness
0016exact hvaluation_witness