BT00T1

prime_factorial_valuation_eq_legendre_sum

Alpha body-checked · checked-use disabled

At every prime, the factorial valuation exponent equals the finite Legendre sum.

Exact expanded PA statement

forall p n a b. ((~(p = 1) /\ forall frm_prime_left_blfg_equality_prime frm_prime_right_blfg_equality_prime. p = frm_prime_left_blfg_equality_prime * frm_prime_right_blfg_equality_prime -> frm_prime_left_blfg_equality_prime = 1 \/ frm_prime_right_blfg_equality_prime = 1)) -> (exists bfv_factorial_blfg_equality_factorial. ((exists ff_b_blfg_equality_factorial_factorial ff_c_blfg_equality_factorial_factorial. ((forall ff_i_blfg_equality_factorial_factorial_range. (exists ff_lt_blfg_equality_factorial_factorial_range_bound. ff_lt_blfg_equality_factorial_factorial_range_bound + S ff_i_blfg_equality_factorial_factorial_range = n) -> (((exists ff_h_blfg_equality_factorial_factorial_range_decoded. ff_h_blfg_equality_factorial_factorial_range_decoded + S (1 + ff_i_blfg_equality_factorial_factorial_range) = S ((S (ff_i_blfg_equality_factorial_factorial_range)) * ff_c_blfg_equality_factorial_factorial)) /\ exists ff_q_blfg_equality_factorial_factorial_range_decoded. ff_b_blfg_equality_factorial_factorial = ff_q_blfg_equality_factorial_factorial_range_decoded * S ((S (ff_i_blfg_equality_factorial_factorial_range)) * ff_c_blfg_equality_factorial_factorial) + (1 + ff_i_blfg_equality_factorial_factorial_range)))) /\ (exists ff_u_blfg_equality_factorial_factorial_product ff_v_blfg_equality_factorial_factorial_product. ((((exists ff_h_blfg_equality_factorial_factorial_product_start. ff_h_blfg_equality_factorial_factorial_product_start + S (1) = S ((S (0)) * ff_v_blfg_equality_factorial_factorial_product)) /\ exists ff_q_blfg_equality_factorial_factorial_product_start. ff_u_blfg_equality_factorial_factorial_product = ff_q_blfg_equality_factorial_factorial_product_start * S ((S (0)) * ff_v_blfg_equality_factorial_factorial_product) + (1))) /\ ((((exists ff_h_blfg_equality_factorial_factorial_product_terminal. ff_h_blfg_equality_factorial_factorial_product_terminal + S (bfv_factorial_blfg_equality_factorial) = S ((S (n)) * ff_v_blfg_equality_factorial_factorial_product)) /\ exists ff_q_blfg_equality_factorial_factorial_product_terminal. ff_u_blfg_equality_factorial_factorial_product = ff_q_blfg_equality_factorial_factorial_product_terminal * S ((S (n)) * ff_v_blfg_equality_factorial_factorial_product) + (bfv_factorial_blfg_equality_factorial))) /\ forall ff_i_blfg_equality_factorial_factorial_product. (exists ff_lt_blfg_equality_factorial_factorial_product_bound. ff_lt_blfg_equality_factorial_factorial_product_bound + S ff_i_blfg_equality_factorial_factorial_product = n) -> exists ff_p_blfg_equality_factorial_factorial_product ff_r_blfg_equality_factorial_factorial_product ff_s_blfg_equality_factorial_factorial_product. ((((exists ff_h_blfg_equality_factorial_factorial_product_factor. ff_h_blfg_equality_factorial_factorial_product_factor + S (ff_p_blfg_equality_factorial_factorial_product) = S ((S (ff_i_blfg_equality_factorial_factorial_product)) * ff_c_blfg_equality_factorial_factorial)) /\ exists ff_q_blfg_equality_factorial_factorial_product_factor. ff_b_blfg_equality_factorial_factorial = ff_q_blfg_equality_factorial_factorial_product_factor * S ((S (ff_i_blfg_equality_factorial_factorial_product)) * ff_c_blfg_equality_factorial_factorial) + (ff_p_blfg_equality_factorial_factorial_product))) /\ ((((exists ff_h_blfg_equality_factorial_factorial_product_partial. ff_h_blfg_equality_factorial_factorial_product_partial + S (ff_r_blfg_equality_factorial_factorial_product) = S ((S (ff_i_blfg_equality_factorial_factorial_product)) * ff_v_blfg_equality_factorial_factorial_product)) /\ exists ff_q_blfg_equality_factorial_factorial_product_partial. ff_u_blfg_equality_factorial_factorial_product = ff_q_blfg_equality_factorial_factorial_product_partial * S ((S (ff_i_blfg_equality_factorial_factorial_product)) * ff_v_blfg_equality_factorial_factorial_product) + (ff_r_blfg_equality_factorial_factorial_product))) /\ ((((exists ff_h_blfg_equality_factorial_factorial_product_successor. ff_h_blfg_equality_factorial_factorial_product_successor + S (ff_s_blfg_equality_factorial_factorial_product) = S ((S (S ff_i_blfg_equality_factorial_factorial_product)) * ff_v_blfg_equality_factorial_factorial_product)) /\ exists ff_q_blfg_equality_factorial_factorial_product_successor. ff_u_blfg_equality_factorial_factorial_product = ff_q_blfg_equality_factorial_factorial_product_successor * S ((S (S ff_i_blfg_equality_factorial_factorial_product)) * ff_v_blfg_equality_factorial_factorial_product) + (ff_s_blfg_equality_factorial_factorial_product))) /\ ff_s_blfg_equality_factorial_factorial_product = ff_r_blfg_equality_factorial_factorial_product * ff_p_blfg_equality_factorial_factorial_product)))))))) /\ (((exists bpv_gap_blfg_equality_factorial_valuation_exponent_bound. bpv_gap_blfg_equality_factorial_valuation_exponent_bound + a = bfv_factorial_blfg_equality_factorial) /\ (exists bpv_result_blfg_equality_factorial_valuation_selected. ((exists ff_b_blfg_equality_factorial_valuation_selected_power ff_c_blfg_equality_factorial_valuation_selected_power. ((forall ff_i_blfg_equality_factorial_valuation_selected_power_repeat. (exists ff_lt_blfg_equality_factorial_valuation_selected_power_repeat_bound. ff_lt_blfg_equality_factorial_valuation_selected_power_repeat_bound + S ff_i_blfg_equality_factorial_valuation_selected_power_repeat = a) -> (((exists ff_h_blfg_equality_factorial_valuation_selected_power_repeat_decoded. ff_h_blfg_equality_factorial_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_blfg_equality_factorial_valuation_selected_power_repeat)) * ff_c_blfg_equality_factorial_valuation_selected_power)) /\ exists ff_q_blfg_equality_factorial_valuation_selected_power_repeat_decoded. ff_b_blfg_equality_factorial_valuation_selected_power = ff_q_blfg_equality_factorial_valuation_selected_power_repeat_decoded * S ((S (ff_i_blfg_equality_factorial_valuation_selected_power_repeat)) * ff_c_blfg_equality_factorial_valuation_selected_power) + (p)))) /\ (exists ff_u_blfg_equality_factorial_valuation_selected_power_product ff_v_blfg_equality_factorial_valuation_selected_power_product. ((((exists ff_h_blfg_equality_factorial_valuation_selected_power_product_start. ff_h_blfg_equality_factorial_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_blfg_equality_factorial_valuation_selected_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_selected_power_product_start. ff_u_blfg_equality_factorial_valuation_selected_power_product = ff_q_blfg_equality_factorial_valuation_selected_power_product_start * S ((S (0)) * ff_v_blfg_equality_factorial_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_blfg_equality_factorial_valuation_selected_power_product_terminal. ff_h_blfg_equality_factorial_valuation_selected_power_product_terminal + S (bpv_result_blfg_equality_factorial_valuation_selected) = S ((S (a)) * ff_v_blfg_equality_factorial_valuation_selected_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_selected_power_product_terminal. ff_u_blfg_equality_factorial_valuation_selected_power_product = ff_q_blfg_equality_factorial_valuation_selected_power_product_terminal * S ((S (a)) * ff_v_blfg_equality_factorial_valuation_selected_power_product) + (bpv_result_blfg_equality_factorial_valuation_selected))) /\ forall ff_i_blfg_equality_factorial_valuation_selected_power_product. (exists ff_lt_blfg_equality_factorial_valuation_selected_power_product_bound. ff_lt_blfg_equality_factorial_valuation_selected_power_product_bound + S ff_i_blfg_equality_factorial_valuation_selected_power_product = a) -> exists ff_p_blfg_equality_factorial_valuation_selected_power_product ff_r_blfg_equality_factorial_valuation_selected_power_product ff_s_blfg_equality_factorial_valuation_selected_power_product. ((((exists ff_h_blfg_equality_factorial_valuation_selected_power_product_factor. ff_h_blfg_equality_factorial_valuation_selected_power_product_factor + S (ff_p_blfg_equality_factorial_valuation_selected_power_product) = S ((S (ff_i_blfg_equality_factorial_valuation_selected_power_product)) * ff_c_blfg_equality_factorial_valuation_selected_power)) /\ exists ff_q_blfg_equality_factorial_valuation_selected_power_product_factor. ff_b_blfg_equality_factorial_valuation_selected_power = ff_q_blfg_equality_factorial_valuation_selected_power_product_factor * S ((S (ff_i_blfg_equality_factorial_valuation_selected_power_product)) * ff_c_blfg_equality_factorial_valuation_selected_power) + (ff_p_blfg_equality_factorial_valuation_selected_power_product))) /\ ((((exists ff_h_blfg_equality_factorial_valuation_selected_power_product_partial. ff_h_blfg_equality_factorial_valuation_selected_power_product_partial + S (ff_r_blfg_equality_factorial_valuation_selected_power_product) = S ((S (ff_i_blfg_equality_factorial_valuation_selected_power_product)) * ff_v_blfg_equality_factorial_valuation_selected_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_selected_power_product_partial. ff_u_blfg_equality_factorial_valuation_selected_power_product = ff_q_blfg_equality_factorial_valuation_selected_power_product_partial * S ((S (ff_i_blfg_equality_factorial_valuation_selected_power_product)) * ff_v_blfg_equality_factorial_valuation_selected_power_product) + (ff_r_blfg_equality_factorial_valuation_selected_power_product))) /\ ((((exists ff_h_blfg_equality_factorial_valuation_selected_power_product_successor. ff_h_blfg_equality_factorial_valuation_selected_power_product_successor + S (ff_s_blfg_equality_factorial_valuation_selected_power_product) = S ((S (S ff_i_blfg_equality_factorial_valuation_selected_power_product)) * ff_v_blfg_equality_factorial_valuation_selected_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_selected_power_product_successor. ff_u_blfg_equality_factorial_valuation_selected_power_product = ff_q_blfg_equality_factorial_valuation_selected_power_product_successor * S ((S (S ff_i_blfg_equality_factorial_valuation_selected_power_product)) * ff_v_blfg_equality_factorial_valuation_selected_power_product) + (ff_s_blfg_equality_factorial_valuation_selected_power_product))) /\ ff_s_blfg_equality_factorial_valuation_selected_power_product = ff_r_blfg_equality_factorial_valuation_selected_power_product * ff_p_blfg_equality_factorial_valuation_selected_power_product)))))))) /\ (exists bpv_factor_blfg_equality_factorial_valuation_selected_divides. bfv_factorial_blfg_equality_factorial = bpv_result_blfg_equality_factorial_valuation_selected * bpv_factor_blfg_equality_factorial_valuation_selected_divides)))) /\ forall bpv_candidate_blfg_equality_factorial_valuation. (exists bpv_gap_blfg_equality_factorial_valuation_candidate_bound. bpv_gap_blfg_equality_factorial_valuation_candidate_bound + bpv_candidate_blfg_equality_factorial_valuation = bfv_factorial_blfg_equality_factorial) -> (exists bpv_result_blfg_equality_factorial_valuation_candidate. ((exists ff_b_blfg_equality_factorial_valuation_candidate_power ff_c_blfg_equality_factorial_valuation_candidate_power. ((forall ff_i_blfg_equality_factorial_valuation_candidate_power_repeat. (exists ff_lt_blfg_equality_factorial_valuation_candidate_power_repeat_bound. ff_lt_blfg_equality_factorial_valuation_candidate_power_repeat_bound + S ff_i_blfg_equality_factorial_valuation_candidate_power_repeat = bpv_candidate_blfg_equality_factorial_valuation) -> (((exists ff_h_blfg_equality_factorial_valuation_candidate_power_repeat_decoded. ff_h_blfg_equality_factorial_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_blfg_equality_factorial_valuation_candidate_power_repeat)) * ff_c_blfg_equality_factorial_valuation_candidate_power)) /\ exists ff_q_blfg_equality_factorial_valuation_candidate_power_repeat_decoded. ff_b_blfg_equality_factorial_valuation_candidate_power = ff_q_blfg_equality_factorial_valuation_candidate_power_repeat_decoded * S ((S (ff_i_blfg_equality_factorial_valuation_candidate_power_repeat)) * ff_c_blfg_equality_factorial_valuation_candidate_power) + (p)))) /\ (exists ff_u_blfg_equality_factorial_valuation_candidate_power_product ff_v_blfg_equality_factorial_valuation_candidate_power_product. ((((exists ff_h_blfg_equality_factorial_valuation_candidate_power_product_start. ff_h_blfg_equality_factorial_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_candidate_power_product_start. ff_u_blfg_equality_factorial_valuation_candidate_power_product = ff_q_blfg_equality_factorial_valuation_candidate_power_product_start * S ((S (0)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_blfg_equality_factorial_valuation_candidate_power_product_terminal. ff_h_blfg_equality_factorial_valuation_candidate_power_product_terminal + S (bpv_result_blfg_equality_factorial_valuation_candidate) = S ((S (bpv_candidate_blfg_equality_factorial_valuation)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_candidate_power_product_terminal. ff_u_blfg_equality_factorial_valuation_candidate_power_product = ff_q_blfg_equality_factorial_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_blfg_equality_factorial_valuation)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product) + (bpv_result_blfg_equality_factorial_valuation_candidate))) /\ forall ff_i_blfg_equality_factorial_valuation_candidate_power_product. (exists ff_lt_blfg_equality_factorial_valuation_candidate_power_product_bound. ff_lt_blfg_equality_factorial_valuation_candidate_power_product_bound + S ff_i_blfg_equality_factorial_valuation_candidate_power_product = bpv_candidate_blfg_equality_factorial_valuation) -> exists ff_p_blfg_equality_factorial_valuation_candidate_power_product ff_r_blfg_equality_factorial_valuation_candidate_power_product ff_s_blfg_equality_factorial_valuation_candidate_power_product. ((((exists ff_h_blfg_equality_factorial_valuation_candidate_power_product_factor. ff_h_blfg_equality_factorial_valuation_candidate_power_product_factor + S (ff_p_blfg_equality_factorial_valuation_candidate_power_product) = S ((S (ff_i_blfg_equality_factorial_valuation_candidate_power_product)) * ff_c_blfg_equality_factorial_valuation_candidate_power)) /\ exists ff_q_blfg_equality_factorial_valuation_candidate_power_product_factor. ff_b_blfg_equality_factorial_valuation_candidate_power = ff_q_blfg_equality_factorial_valuation_candidate_power_product_factor * S ((S (ff_i_blfg_equality_factorial_valuation_candidate_power_product)) * ff_c_blfg_equality_factorial_valuation_candidate_power) + (ff_p_blfg_equality_factorial_valuation_candidate_power_product))) /\ ((((exists ff_h_blfg_equality_factorial_valuation_candidate_power_product_partial. ff_h_blfg_equality_factorial_valuation_candidate_power_product_partial + S (ff_r_blfg_equality_factorial_valuation_candidate_power_product) = S ((S (ff_i_blfg_equality_factorial_valuation_candidate_power_product)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_candidate_power_product_partial. ff_u_blfg_equality_factorial_valuation_candidate_power_product = ff_q_blfg_equality_factorial_valuation_candidate_power_product_partial * S ((S (ff_i_blfg_equality_factorial_valuation_candidate_power_product)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product) + (ff_r_blfg_equality_factorial_valuation_candidate_power_product))) /\ ((((exists ff_h_blfg_equality_factorial_valuation_candidate_power_product_successor. ff_h_blfg_equality_factorial_valuation_candidate_power_product_successor + S (ff_s_blfg_equality_factorial_valuation_candidate_power_product) = S ((S (S ff_i_blfg_equality_factorial_valuation_candidate_power_product)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product)) /\ exists ff_q_blfg_equality_factorial_valuation_candidate_power_product_successor. ff_u_blfg_equality_factorial_valuation_candidate_power_product = ff_q_blfg_equality_factorial_valuation_candidate_power_product_successor * S ((S (S ff_i_blfg_equality_factorial_valuation_candidate_power_product)) * ff_v_blfg_equality_factorial_valuation_candidate_power_product) + (ff_s_blfg_equality_factorial_valuation_candidate_power_product))) /\ ff_s_blfg_equality_factorial_valuation_candidate_power_product = ff_r_blfg_equality_factorial_valuation_candidate_power_product * ff_p_blfg_equality_factorial_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_blfg_equality_factorial_valuation_candidate_divides. bfv_factorial_blfg_equality_factorial = bpv_result_blfg_equality_factorial_valuation_candidate * bpv_factor_blfg_equality_factorial_valuation_candidate_divides))) -> (exists bpv_gap_blfg_equality_factorial_valuation_maximal. bpv_gap_blfg_equality_factorial_valuation_maximal + bpv_candidate_blfg_equality_factorial_valuation = a)))) -> (exists bls_code_blfg_equality_legendre bls_scale_blfg_equality_legendre. ((forall bls_index_blfg_equality_legendre_prefix. (exists bls_gap_blfg_equality_legendre_prefix_bound. bls_gap_blfg_equality_legendre_prefix_bound + S (bls_index_blfg_equality_legendre_prefix) = (n)) -> exists bls_power_blfg_equality_legendre_prefix bls_quotient_blfg_equality_legendre_prefix bls_remainder_blfg_equality_legendre_prefix. ((exists bpvi_b_bls_blfg_equality_legendre_prefix_power bpvi_c_bls_blfg_equality_legendre_prefix_power. ((forall bpvi_i_bls_blfg_equality_legendre_prefix_power. (exists bpvi_repeat_gap_bls_blfg_equality_legendre_prefix_power. bpvi_repeat_gap_bls_blfg_equality_legendre_prefix_power + S bpvi_i_bls_blfg_equality_legendre_prefix_power = S bls_index_blfg_equality_legendre_prefix) -> (((exists bpvi_h_bls_blfg_equality_legendre_prefix_power_repeat. bpvi_h_bls_blfg_equality_legendre_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_blfg_equality_legendre_prefix_power)) * bpvi_c_bls_blfg_equality_legendre_prefix_power)) /\ exists bpvi_q_bls_blfg_equality_legendre_prefix_power_repeat. bpvi_b_bls_blfg_equality_legendre_prefix_power = bpvi_q_bls_blfg_equality_legendre_prefix_power_repeat * S ((S (bpvi_i_bls_blfg_equality_legendre_prefix_power)) * bpvi_c_bls_blfg_equality_legendre_prefix_power) + (p)))) /\ (exists bpvi_u_bls_blfg_equality_legendre_prefix_power bpvi_v_bls_blfg_equality_legendre_prefix_power. ((((exists bpvi_h_bls_blfg_equality_legendre_prefix_power_start. bpvi_h_bls_blfg_equality_legendre_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_blfg_equality_legendre_prefix_power)) /\ exists bpvi_q_bls_blfg_equality_legendre_prefix_power_start. bpvi_u_bls_blfg_equality_legendre_prefix_power = bpvi_q_bls_blfg_equality_legendre_prefix_power_start * S ((S (0)) * bpvi_v_bls_blfg_equality_legendre_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_blfg_equality_legendre_prefix_power_terminal. bpvi_h_bls_blfg_equality_legendre_prefix_power_terminal + S (bls_power_blfg_equality_legendre_prefix) = S ((S (S bls_index_blfg_equality_legendre_prefix)) * bpvi_v_bls_blfg_equality_legendre_prefix_power)) /\ exists bpvi_q_bls_blfg_equality_legendre_prefix_power_terminal. bpvi_u_bls_blfg_equality_legendre_prefix_power = bpvi_q_bls_blfg_equality_legendre_prefix_power_terminal * S ((S (S bls_index_blfg_equality_legendre_prefix)) * bpvi_v_bls_blfg_equality_legendre_prefix_power) + (bls_power_blfg_equality_legendre_prefix))) /\ forall bpvi_j_bls_blfg_equality_legendre_prefix_power. (exists bpvi_product_gap_bls_blfg_equality_legendre_prefix_power. bpvi_product_gap_bls_blfg_equality_legendre_prefix_power + S bpvi_j_bls_blfg_equality_legendre_prefix_power = S bls_index_blfg_equality_legendre_prefix) -> exists bpvi_factor_bls_blfg_equality_legendre_prefix_power bpvi_partial_bls_blfg_equality_legendre_prefix_power bpvi_successor_bls_blfg_equality_legendre_prefix_power. ((((exists bpvi_h_bls_blfg_equality_legendre_prefix_power_factor. bpvi_h_bls_blfg_equality_legendre_prefix_power_factor + S (bpvi_factor_bls_blfg_equality_legendre_prefix_power) = S ((S (bpvi_j_bls_blfg_equality_legendre_prefix_power)) * bpvi_c_bls_blfg_equality_legendre_prefix_power)) /\ exists bpvi_q_bls_blfg_equality_legendre_prefix_power_factor. bpvi_b_bls_blfg_equality_legendre_prefix_power = bpvi_q_bls_blfg_equality_legendre_prefix_power_factor * S ((S (bpvi_j_bls_blfg_equality_legendre_prefix_power)) * bpvi_c_bls_blfg_equality_legendre_prefix_power) + (bpvi_factor_bls_blfg_equality_legendre_prefix_power))) /\ ((((exists bpvi_h_bls_blfg_equality_legendre_prefix_power_partial. bpvi_h_bls_blfg_equality_legendre_prefix_power_partial + S (bpvi_partial_bls_blfg_equality_legendre_prefix_power) = S ((S (bpvi_j_bls_blfg_equality_legendre_prefix_power)) * bpvi_v_bls_blfg_equality_legendre_prefix_power)) /\ exists bpvi_q_bls_blfg_equality_legendre_prefix_power_partial. bpvi_u_bls_blfg_equality_legendre_prefix_power = bpvi_q_bls_blfg_equality_legendre_prefix_power_partial * S ((S (bpvi_j_bls_blfg_equality_legendre_prefix_power)) * bpvi_v_bls_blfg_equality_legendre_prefix_power) + (bpvi_partial_bls_blfg_equality_legendre_prefix_power))) /\ ((((exists bpvi_h_bls_blfg_equality_legendre_prefix_power_successor. bpvi_h_bls_blfg_equality_legendre_prefix_power_successor + S (bpvi_successor_bls_blfg_equality_legendre_prefix_power) = S ((S (S bpvi_j_bls_blfg_equality_legendre_prefix_power)) * bpvi_v_bls_blfg_equality_legendre_prefix_power)) /\ exists bpvi_q_bls_blfg_equality_legendre_prefix_power_successor. bpvi_u_bls_blfg_equality_legendre_prefix_power = bpvi_q_bls_blfg_equality_legendre_prefix_power_successor * S ((S (S bpvi_j_bls_blfg_equality_legendre_prefix_power)) * bpvi_v_bls_blfg_equality_legendre_prefix_power) + (bpvi_successor_bls_blfg_equality_legendre_prefix_power))) /\ bpvi_successor_bls_blfg_equality_legendre_prefix_power = bpvi_partial_bls_blfg_equality_legendre_prefix_power * bpvi_factor_bls_blfg_equality_legendre_prefix_power)))))))) /\ ((((exists ff_h_bls_blfg_equality_legendre_prefix_quotient_entry. ff_h_bls_blfg_equality_legendre_prefix_quotient_entry + S (bls_quotient_blfg_equality_legendre_prefix) = S ((S (bls_index_blfg_equality_legendre_prefix)) * bls_scale_blfg_equality_legendre)) /\ exists ff_q_bls_blfg_equality_legendre_prefix_quotient_entry. bls_code_blfg_equality_legendre = ff_q_bls_blfg_equality_legendre_prefix_quotient_entry * S ((S (bls_index_blfg_equality_legendre_prefix)) * bls_scale_blfg_equality_legendre) + (bls_quotient_blfg_equality_legendre_prefix))) /\ ((n = bls_power_blfg_equality_legendre_prefix * bls_quotient_blfg_equality_legendre_prefix + bls_remainder_blfg_equality_legendre_prefix /\ exists bls_remainder_gap_blfg_equality_legendre_prefix_division. bls_remainder_gap_blfg_equality_legendre_prefix_division + S (bls_remainder_blfg_equality_legendre_prefix) = bls_power_blfg_equality_legendre_prefix))))) /\ (exists ff_u_bls_blfg_equality_legendre_sum ff_v_bls_blfg_equality_legendre_sum. ((((exists ff_h_bls_blfg_equality_legendre_sum_start. ff_h_bls_blfg_equality_legendre_sum_start + S (0) = S ((S (0)) * ff_v_bls_blfg_equality_legendre_sum)) /\ exists ff_q_bls_blfg_equality_legendre_sum_start. ff_u_bls_blfg_equality_legendre_sum = ff_q_bls_blfg_equality_legendre_sum_start * S ((S (0)) * ff_v_bls_blfg_equality_legendre_sum) + (0))) /\ ((((exists ff_h_bls_blfg_equality_legendre_sum_terminal. ff_h_bls_blfg_equality_legendre_sum_terminal + S (b) = S ((S (n)) * ff_v_bls_blfg_equality_legendre_sum)) /\ exists ff_q_bls_blfg_equality_legendre_sum_terminal. ff_u_bls_blfg_equality_legendre_sum = ff_q_bls_blfg_equality_legendre_sum_terminal * S ((S (n)) * ff_v_bls_blfg_equality_legendre_sum) + (b))) /\ forall ff_i_bls_blfg_equality_legendre_sum. (exists ff_lt_bls_blfg_equality_legendre_sum_bound. ff_lt_bls_blfg_equality_legendre_sum_bound + S ff_i_bls_blfg_equality_legendre_sum = n) -> exists ff_a_bls_blfg_equality_legendre_sum ff_r_bls_blfg_equality_legendre_sum ff_s_bls_blfg_equality_legendre_sum. ((((exists ff_h_bls_blfg_equality_legendre_sum_summand. ff_h_bls_blfg_equality_legendre_sum_summand + S (ff_a_bls_blfg_equality_legendre_sum) = S ((S (ff_i_bls_blfg_equality_legendre_sum)) * bls_scale_blfg_equality_legendre)) /\ exists ff_q_bls_blfg_equality_legendre_sum_summand. bls_code_blfg_equality_legendre = ff_q_bls_blfg_equality_legendre_sum_summand * S ((S (ff_i_bls_blfg_equality_legendre_sum)) * bls_scale_blfg_equality_legendre) + (ff_a_bls_blfg_equality_legendre_sum))) /\ ((((exists ff_h_bls_blfg_equality_legendre_sum_partial. ff_h_bls_blfg_equality_legendre_sum_partial + S (ff_r_bls_blfg_equality_legendre_sum) = S ((S (ff_i_bls_blfg_equality_legendre_sum)) * ff_v_bls_blfg_equality_legendre_sum)) /\ exists ff_q_bls_blfg_equality_legendre_sum_partial. ff_u_bls_blfg_equality_legendre_sum = ff_q_bls_blfg_equality_legendre_sum_partial * S ((S (ff_i_bls_blfg_equality_legendre_sum)) * ff_v_bls_blfg_equality_legendre_sum) + (ff_r_bls_blfg_equality_legendre_sum))) /\ ((((exists ff_h_bls_blfg_equality_legendre_sum_successor. ff_h_bls_blfg_equality_legendre_sum_successor + S (ff_s_bls_blfg_equality_legendre_sum) = S ((S (S ff_i_bls_blfg_equality_legendre_sum)) * ff_v_bls_blfg_equality_legendre_sum)) /\ exists ff_q_bls_blfg_equality_legendre_sum_successor. ff_u_bls_blfg_equality_legendre_sum = ff_q_bls_blfg_equality_legendre_sum_successor * S ((S (S ff_i_bls_blfg_equality_legendre_sum)) * ff_v_bls_blfg_equality_legendre_sum) + (ff_s_bls_blfg_equality_legendre_sum))) /\ ff_s_bls_blfg_equality_legendre_sum = ff_r_bls_blfg_equality_legendre_sum + ff_a_bls_blfg_equality_legendre_sum)))))))) -> a = b

Structural proof guide

At every prime, the factorial valuation exponent equals the finite Legendre sum.

Direct prerequisites: prime_factorial_valuation_zero, legendre_sum_zero, factorial_valuation_exists, prime_legendre_sum_exists, power_valuation_exists, factorial_legendre_successor_agreement. The authored body proceeds by structural induction (1), case analysis (3), intermediate claims (6).

Proof neighborhood

Direct dependencies

BT00RO prime_factorial_valuation_zero BT00S4 legendre_sum_zero BT00RM factorial_valuation_exists BT00S2 prime_legendre_sum_exists BT00Q7 power_valuation_exists BT00T0 factorial_legendre_successor_agreement

Direct dependents

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.

  1. 0001intro p
  2. 0002induction n
  3. 0003intro a
  4. 0004intro b
  5. 0005intro hp
  6. 0006intro hfactorial
  7. 0007intro hlegendre
  8. 0008have ha : a = 0
  9. 0009specialize prime_factorial_valuation_zero p
  10. 0010specialize prime_factorial_valuation_zero 0
  11. 0011specialize prime_factorial_valuation_zero a
  12. 0012apply prime_factorial_valuation_zero
  13. 0013refl
  14. 0014exact hp
  15. 0015exact hfactorial
  16. 0016have hb : b = 0
  17. 0017specialize legendre_sum_zero p
  18. 0018specialize legendre_sum_zero 0
  19. 0019specialize legendre_sum_zero b
  20. 0020apply legendre_sum_zero
  21. 0021refl
  22. 0022exact hlegendre
  23. 0023trans 0
  24. 0024exact ha
  25. 0025symm
  26. 0026exact hb
  27. 0027intro a
  28. 0028intro b
  29. 0029intro hp
  30. 0030intro hfactorial
  31. 0031intro hlegendre
  32. 0032have hfactorial_old : exists e. exists bfv_factorial_blfg_induction_factorial_old. ((exists ff_b_blfg_induction_factorial_old_factorial ff_c_blfg_induction_factorial_old_factorial. ((forall ff_i_blfg_induction_factorial_old_factorial_range. (exists ff_lt_blfg_induction_factorial_old_factorial_range_bound. ff_lt_blfg_induction_factorial_old_factorial_range_bound + S ff_i_blfg_induction_factorial_old_factorial_range = n) -> (((exists ff_h_blfg_induction_factorial_old_factorial_range_decoded. ff_h_blfg_induction_factorial_old_factorial_range_decoded + S (1 + ff_i_blfg_induction_factorial_old_factorial_range) = S ((S (ff_i_blfg_induction_factorial_old_factorial_range)) * ff_c_blfg_induction_factorial_old_factorial)) /\ exists ff_q_blfg_induction_factorial_old_factorial_range_decoded. ff_b_blfg_induction_factorial_old_factorial = ff_q_blfg_induction_factorial_old_factorial_range_decoded * S ((S (ff_i_blfg_induction_factorial_old_factorial_range)) * ff_c_blfg_induction_factorial_old_factorial) + (1 + ff_i_blfg_induction_factorial_old_factorial_range)))) /\ (exists ff_u_blfg_induction_factorial_old_factorial_product ff_v_blfg_induction_factorial_old_factorial_product. ((((exists ff_h_blfg_induction_factorial_old_factorial_product_start. ff_h_blfg_induction_factorial_old_factorial_product_start + S (1) = S ((S (0)) * ff_v_blfg_induction_factorial_old_factorial_product)) /\ exists ff_q_blfg_induction_factorial_old_factorial_product_start. ff_u_blfg_induction_factorial_old_factorial_product = ff_q_blfg_induction_factorial_old_factorial_product_start * S ((S (0)) * ff_v_blfg_induction_factorial_old_factorial_product) + (1))) /\ ((((exists ff_h_blfg_induction_factorial_old_factorial_product_terminal. ff_h_blfg_induction_factorial_old_factorial_product_terminal + S (bfv_factorial_blfg_induction_factorial_old) = S ((S (n)) * ff_v_blfg_induction_factorial_old_factorial_product)) /\ exists ff_q_blfg_induction_factorial_old_factorial_product_terminal. ff_u_blfg_induction_factorial_old_factorial_product = ff_q_blfg_induction_factorial_old_factorial_product_terminal * S ((S (n)) * ff_v_blfg_induction_factorial_old_factorial_product) + (bfv_factorial_blfg_induction_factorial_old))) /\ forall ff_i_blfg_induction_factorial_old_factorial_product. (exists ff_lt_blfg_induction_factorial_old_factorial_product_bound. ff_lt_blfg_induction_factorial_old_factorial_product_bound + S ff_i_blfg_induction_factorial_old_factorial_product = n) -> exists ff_p_blfg_induction_factorial_old_factorial_product ff_r_blfg_induction_factorial_old_factorial_product ff_s_blfg_induction_factorial_old_factorial_product. ((((exists ff_h_blfg_induction_factorial_old_factorial_product_factor. ff_h_blfg_induction_factorial_old_factorial_product_factor + S (ff_p_blfg_induction_factorial_old_factorial_product) = S ((S (ff_i_blfg_induction_factorial_old_factorial_product)) * ff_c_blfg_induction_factorial_old_factorial)) /\ exists ff_q_blfg_induction_factorial_old_factorial_product_factor. ff_b_blfg_induction_factorial_old_factorial = ff_q_blfg_induction_factorial_old_factorial_product_factor * S ((S (ff_i_blfg_induction_factorial_old_factorial_product)) * ff_c_blfg_induction_factorial_old_factorial) + (ff_p_blfg_induction_factorial_old_factorial_product))) /\ ((((exists ff_h_blfg_induction_factorial_old_factorial_product_partial. ff_h_blfg_induction_factorial_old_factorial_product_partial + S (ff_r_blfg_induction_factorial_old_factorial_product) = S ((S (ff_i_blfg_induction_factorial_old_factorial_product)) * ff_v_blfg_induction_factorial_old_factorial_product)) /\ exists ff_q_blfg_induction_factorial_old_factorial_product_partial. ff_u_blfg_induction_factorial_old_factorial_product = ff_q_blfg_induction_factorial_old_factorial_product_partial * S ((S (ff_i_blfg_induction_factorial_old_factorial_product)) * ff_v_blfg_induction_factorial_old_factorial_product) + (ff_r_blfg_induction_factorial_old_factorial_product))) /\ ((((exists ff_h_blfg_induction_factorial_old_factorial_product_successor. ff_h_blfg_induction_factorial_old_factorial_product_successor + S (ff_s_blfg_induction_factorial_old_factorial_product) = S ((S (S ff_i_blfg_induction_factorial_old_factorial_product)) * ff_v_blfg_induction_factorial_old_factorial_product)) /\ exists ff_q_blfg_induction_factorial_old_factorial_product_successor. ff_u_blfg_induction_factorial_old_factorial_product = ff_q_blfg_induction_factorial_old_factorial_product_successor * S ((S (S ff_i_blfg_induction_factorial_old_factorial_product)) * ff_v_blfg_induction_factorial_old_factorial_product) + (ff_s_blfg_induction_factorial_old_factorial_product))) /\ ff_s_blfg_induction_factorial_old_factorial_product = ff_r_blfg_induction_factorial_old_factorial_product * ff_p_blfg_induction_factorial_old_factorial_product)))))))) /\ (((exists bpv_gap_blfg_induction_factorial_old_valuation_exponent_bound. bpv_gap_blfg_induction_factorial_old_valuation_exponent_bound + e = bfv_factorial_blfg_induction_factorial_old) /\ (exists bpv_result_blfg_induction_factorial_old_valuation_selected. ((exists ff_b_blfg_induction_factorial_old_valuation_selected_power ff_c_blfg_induction_factorial_old_valuation_selected_power. ((forall ff_i_blfg_induction_factorial_old_valuation_selected_power_repeat. (exists ff_lt_blfg_induction_factorial_old_valuation_selected_power_repeat_bound. ff_lt_blfg_induction_factorial_old_valuation_selected_power_repeat_bound + S ff_i_blfg_induction_factorial_old_valuation_selected_power_repeat = e) -> (((exists ff_h_blfg_induction_factorial_old_valuation_selected_power_repeat_decoded. ff_h_blfg_induction_factorial_old_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_blfg_induction_factorial_old_valuation_selected_power_repeat)) * ff_c_blfg_induction_factorial_old_valuation_selected_power)) /\ exists ff_q_blfg_induction_factorial_old_valuation_selected_power_repeat_decoded. ff_b_blfg_induction_factorial_old_valuation_selected_power = ff_q_blfg_induction_factorial_old_valuation_selected_power_repeat_decoded * S ((S (ff_i_blfg_induction_factorial_old_valuation_selected_power_repeat)) * ff_c_blfg_induction_factorial_old_valuation_selected_power) + (p)))) /\ (exists ff_u_blfg_induction_factorial_old_valuation_selected_power_product ff_v_blfg_induction_factorial_old_valuation_selected_power_product. ((((exists ff_h_blfg_induction_factorial_old_valuation_selected_power_product_start. ff_h_blfg_induction_factorial_old_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_selected_power_product_start. ff_u_blfg_induction_factorial_old_valuation_selected_power_product = ff_q_blfg_induction_factorial_old_valuation_selected_power_product_start * S ((S (0)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_blfg_induction_factorial_old_valuation_selected_power_product_terminal. ff_h_blfg_induction_factorial_old_valuation_selected_power_product_terminal + S (bpv_result_blfg_induction_factorial_old_valuation_selected) = S ((S (e)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_selected_power_product_terminal. ff_u_blfg_induction_factorial_old_valuation_selected_power_product = ff_q_blfg_induction_factorial_old_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product) + (bpv_result_blfg_induction_factorial_old_valuation_selected))) /\ forall ff_i_blfg_induction_factorial_old_valuation_selected_power_product. (exists ff_lt_blfg_induction_factorial_old_valuation_selected_power_product_bound. ff_lt_blfg_induction_factorial_old_valuation_selected_power_product_bound + S ff_i_blfg_induction_factorial_old_valuation_selected_power_product = e) -> exists ff_p_blfg_induction_factorial_old_valuation_selected_power_product ff_r_blfg_induction_factorial_old_valuation_selected_power_product ff_s_blfg_induction_factorial_old_valuation_selected_power_product. ((((exists ff_h_blfg_induction_factorial_old_valuation_selected_power_product_factor. ff_h_blfg_induction_factorial_old_valuation_selected_power_product_factor + S (ff_p_blfg_induction_factorial_old_valuation_selected_power_product) = S ((S (ff_i_blfg_induction_factorial_old_valuation_selected_power_product)) * ff_c_blfg_induction_factorial_old_valuation_selected_power)) /\ exists ff_q_blfg_induction_factorial_old_valuation_selected_power_product_factor. ff_b_blfg_induction_factorial_old_valuation_selected_power = ff_q_blfg_induction_factorial_old_valuation_selected_power_product_factor * S ((S (ff_i_blfg_induction_factorial_old_valuation_selected_power_product)) * ff_c_blfg_induction_factorial_old_valuation_selected_power) + (ff_p_blfg_induction_factorial_old_valuation_selected_power_product))) /\ ((((exists ff_h_blfg_induction_factorial_old_valuation_selected_power_product_partial. ff_h_blfg_induction_factorial_old_valuation_selected_power_product_partial + S (ff_r_blfg_induction_factorial_old_valuation_selected_power_product) = S ((S (ff_i_blfg_induction_factorial_old_valuation_selected_power_product)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_selected_power_product_partial. ff_u_blfg_induction_factorial_old_valuation_selected_power_product = ff_q_blfg_induction_factorial_old_valuation_selected_power_product_partial * S ((S (ff_i_blfg_induction_factorial_old_valuation_selected_power_product)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product) + (ff_r_blfg_induction_factorial_old_valuation_selected_power_product))) /\ ((((exists ff_h_blfg_induction_factorial_old_valuation_selected_power_product_successor. ff_h_blfg_induction_factorial_old_valuation_selected_power_product_successor + S (ff_s_blfg_induction_factorial_old_valuation_selected_power_product) = S ((S (S ff_i_blfg_induction_factorial_old_valuation_selected_power_product)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_selected_power_product_successor. ff_u_blfg_induction_factorial_old_valuation_selected_power_product = ff_q_blfg_induction_factorial_old_valuation_selected_power_product_successor * S ((S (S ff_i_blfg_induction_factorial_old_valuation_selected_power_product)) * ff_v_blfg_induction_factorial_old_valuation_selected_power_product) + (ff_s_blfg_induction_factorial_old_valuation_selected_power_product))) /\ ff_s_blfg_induction_factorial_old_valuation_selected_power_product = ff_r_blfg_induction_factorial_old_valuation_selected_power_product * ff_p_blfg_induction_factorial_old_valuation_selected_power_product)))))))) /\ (exists bpv_factor_blfg_induction_factorial_old_valuation_selected_divides. bfv_factorial_blfg_induction_factorial_old = bpv_result_blfg_induction_factorial_old_valuation_selected * bpv_factor_blfg_induction_factorial_old_valuation_selected_divides)))) /\ forall bpv_candidate_blfg_induction_factorial_old_valuation. (exists bpv_gap_blfg_induction_factorial_old_valuation_candidate_bound. bpv_gap_blfg_induction_factorial_old_valuation_candidate_bound + bpv_candidate_blfg_induction_factorial_old_valuation = bfv_factorial_blfg_induction_factorial_old) -> (exists bpv_result_blfg_induction_factorial_old_valuation_candidate. ((exists ff_b_blfg_induction_factorial_old_valuation_candidate_power ff_c_blfg_induction_factorial_old_valuation_candidate_power. ((forall ff_i_blfg_induction_factorial_old_valuation_candidate_power_repeat. (exists ff_lt_blfg_induction_factorial_old_valuation_candidate_power_repeat_bound. ff_lt_blfg_induction_factorial_old_valuation_candidate_power_repeat_bound + S ff_i_blfg_induction_factorial_old_valuation_candidate_power_repeat = bpv_candidate_blfg_induction_factorial_old_valuation) -> (((exists ff_h_blfg_induction_factorial_old_valuation_candidate_power_repeat_decoded. ff_h_blfg_induction_factorial_old_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_blfg_induction_factorial_old_valuation_candidate_power_repeat)) * ff_c_blfg_induction_factorial_old_valuation_candidate_power)) /\ exists ff_q_blfg_induction_factorial_old_valuation_candidate_power_repeat_decoded. ff_b_blfg_induction_factorial_old_valuation_candidate_power = ff_q_blfg_induction_factorial_old_valuation_candidate_power_repeat_decoded * S ((S (ff_i_blfg_induction_factorial_old_valuation_candidate_power_repeat)) * ff_c_blfg_induction_factorial_old_valuation_candidate_power) + (p)))) /\ (exists ff_u_blfg_induction_factorial_old_valuation_candidate_power_product ff_v_blfg_induction_factorial_old_valuation_candidate_power_product. ((((exists ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_start. ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_start. ff_u_blfg_induction_factorial_old_valuation_candidate_power_product = ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_start * S ((S (0)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_terminal. ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_terminal + S (bpv_result_blfg_induction_factorial_old_valuation_candidate) = S ((S (bpv_candidate_blfg_induction_factorial_old_valuation)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_terminal. ff_u_blfg_induction_factorial_old_valuation_candidate_power_product = ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_blfg_induction_factorial_old_valuation)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product) + (bpv_result_blfg_induction_factorial_old_valuation_candidate))) /\ forall ff_i_blfg_induction_factorial_old_valuation_candidate_power_product. (exists ff_lt_blfg_induction_factorial_old_valuation_candidate_power_product_bound. ff_lt_blfg_induction_factorial_old_valuation_candidate_power_product_bound + S ff_i_blfg_induction_factorial_old_valuation_candidate_power_product = bpv_candidate_blfg_induction_factorial_old_valuation) -> exists ff_p_blfg_induction_factorial_old_valuation_candidate_power_product ff_r_blfg_induction_factorial_old_valuation_candidate_power_product ff_s_blfg_induction_factorial_old_valuation_candidate_power_product. ((((exists ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_factor. ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_factor + S (ff_p_blfg_induction_factorial_old_valuation_candidate_power_product) = S ((S (ff_i_blfg_induction_factorial_old_valuation_candidate_power_product)) * ff_c_blfg_induction_factorial_old_valuation_candidate_power)) /\ exists ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_factor. ff_b_blfg_induction_factorial_old_valuation_candidate_power = ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_factor * S ((S (ff_i_blfg_induction_factorial_old_valuation_candidate_power_product)) * ff_c_blfg_induction_factorial_old_valuation_candidate_power) + (ff_p_blfg_induction_factorial_old_valuation_candidate_power_product))) /\ ((((exists ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_partial. ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_partial + S (ff_r_blfg_induction_factorial_old_valuation_candidate_power_product) = S ((S (ff_i_blfg_induction_factorial_old_valuation_candidate_power_product)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_partial. ff_u_blfg_induction_factorial_old_valuation_candidate_power_product = ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_partial * S ((S (ff_i_blfg_induction_factorial_old_valuation_candidate_power_product)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product) + (ff_r_blfg_induction_factorial_old_valuation_candidate_power_product))) /\ ((((exists ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_successor. ff_h_blfg_induction_factorial_old_valuation_candidate_power_product_successor + S (ff_s_blfg_induction_factorial_old_valuation_candidate_power_product) = S ((S (S ff_i_blfg_induction_factorial_old_valuation_candidate_power_product)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product)) /\ exists ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_successor. ff_u_blfg_induction_factorial_old_valuation_candidate_power_product = ff_q_blfg_induction_factorial_old_valuation_candidate_power_product_successor * S ((S (S ff_i_blfg_induction_factorial_old_valuation_candidate_power_product)) * ff_v_blfg_induction_factorial_old_valuation_candidate_power_product) + (ff_s_blfg_induction_factorial_old_valuation_candidate_power_product))) /\ ff_s_blfg_induction_factorial_old_valuation_candidate_power_product = ff_r_blfg_induction_factorial_old_valuation_candidate_power_product * ff_p_blfg_induction_factorial_old_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_blfg_induction_factorial_old_valuation_candidate_divides. bfv_factorial_blfg_induction_factorial_old = bpv_result_blfg_induction_factorial_old_valuation_candidate * bpv_factor_blfg_induction_factorial_old_valuation_candidate_divides))) -> (exists bpv_gap_blfg_induction_factorial_old_valuation_maximal. bpv_gap_blfg_induction_factorial_old_valuation_maximal + bpv_candidate_blfg_induction_factorial_old_valuation = e)))
  33. 0033specialize factorial_valuation_exists p
  34. 0034specialize factorial_valuation_exists n
  35. 0035exact factorial_valuation_exists
  36. 0036cases hfactorial_old
  37. 0037have hlegendre_old : exists s. exists bls_code_blfg_induction_legendre_old bls_scale_blfg_induction_legendre_old. ((forall bls_index_blfg_induction_legendre_old_prefix. (exists bls_gap_blfg_induction_legendre_old_prefix_bound. bls_gap_blfg_induction_legendre_old_prefix_bound + S (bls_index_blfg_induction_legendre_old_prefix) = (n)) -> exists bls_power_blfg_induction_legendre_old_prefix bls_quotient_blfg_induction_legendre_old_prefix bls_remainder_blfg_induction_legendre_old_prefix. ((exists bpvi_b_bls_blfg_induction_legendre_old_prefix_power bpvi_c_bls_blfg_induction_legendre_old_prefix_power. ((forall bpvi_i_bls_blfg_induction_legendre_old_prefix_power. (exists bpvi_repeat_gap_bls_blfg_induction_legendre_old_prefix_power. bpvi_repeat_gap_bls_blfg_induction_legendre_old_prefix_power + S bpvi_i_bls_blfg_induction_legendre_old_prefix_power = S bls_index_blfg_induction_legendre_old_prefix) -> (((exists bpvi_h_bls_blfg_induction_legendre_old_prefix_power_repeat. bpvi_h_bls_blfg_induction_legendre_old_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_c_bls_blfg_induction_legendre_old_prefix_power)) /\ exists bpvi_q_bls_blfg_induction_legendre_old_prefix_power_repeat. bpvi_b_bls_blfg_induction_legendre_old_prefix_power = bpvi_q_bls_blfg_induction_legendre_old_prefix_power_repeat * S ((S (bpvi_i_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_c_bls_blfg_induction_legendre_old_prefix_power) + (p)))) /\ (exists bpvi_u_bls_blfg_induction_legendre_old_prefix_power bpvi_v_bls_blfg_induction_legendre_old_prefix_power. ((((exists bpvi_h_bls_blfg_induction_legendre_old_prefix_power_start. bpvi_h_bls_blfg_induction_legendre_old_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power)) /\ exists bpvi_q_bls_blfg_induction_legendre_old_prefix_power_start. bpvi_u_bls_blfg_induction_legendre_old_prefix_power = bpvi_q_bls_blfg_induction_legendre_old_prefix_power_start * S ((S (0)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_blfg_induction_legendre_old_prefix_power_terminal. bpvi_h_bls_blfg_induction_legendre_old_prefix_power_terminal + S (bls_power_blfg_induction_legendre_old_prefix) = S ((S (S bls_index_blfg_induction_legendre_old_prefix)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power)) /\ exists bpvi_q_bls_blfg_induction_legendre_old_prefix_power_terminal. bpvi_u_bls_blfg_induction_legendre_old_prefix_power = bpvi_q_bls_blfg_induction_legendre_old_prefix_power_terminal * S ((S (S bls_index_blfg_induction_legendre_old_prefix)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power) + (bls_power_blfg_induction_legendre_old_prefix))) /\ forall bpvi_j_bls_blfg_induction_legendre_old_prefix_power. (exists bpvi_product_gap_bls_blfg_induction_legendre_old_prefix_power. bpvi_product_gap_bls_blfg_induction_legendre_old_prefix_power + S bpvi_j_bls_blfg_induction_legendre_old_prefix_power = S bls_index_blfg_induction_legendre_old_prefix) -> exists bpvi_factor_bls_blfg_induction_legendre_old_prefix_power bpvi_partial_bls_blfg_induction_legendre_old_prefix_power bpvi_successor_bls_blfg_induction_legendre_old_prefix_power. ((((exists bpvi_h_bls_blfg_induction_legendre_old_prefix_power_factor. bpvi_h_bls_blfg_induction_legendre_old_prefix_power_factor + S (bpvi_factor_bls_blfg_induction_legendre_old_prefix_power) = S ((S (bpvi_j_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_c_bls_blfg_induction_legendre_old_prefix_power)) /\ exists bpvi_q_bls_blfg_induction_legendre_old_prefix_power_factor. bpvi_b_bls_blfg_induction_legendre_old_prefix_power = bpvi_q_bls_blfg_induction_legendre_old_prefix_power_factor * S ((S (bpvi_j_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_c_bls_blfg_induction_legendre_old_prefix_power) + (bpvi_factor_bls_blfg_induction_legendre_old_prefix_power))) /\ ((((exists bpvi_h_bls_blfg_induction_legendre_old_prefix_power_partial. bpvi_h_bls_blfg_induction_legendre_old_prefix_power_partial + S (bpvi_partial_bls_blfg_induction_legendre_old_prefix_power) = S ((S (bpvi_j_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power)) /\ exists bpvi_q_bls_blfg_induction_legendre_old_prefix_power_partial. bpvi_u_bls_blfg_induction_legendre_old_prefix_power = bpvi_q_bls_blfg_induction_legendre_old_prefix_power_partial * S ((S (bpvi_j_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power) + (bpvi_partial_bls_blfg_induction_legendre_old_prefix_power))) /\ ((((exists bpvi_h_bls_blfg_induction_legendre_old_prefix_power_successor. bpvi_h_bls_blfg_induction_legendre_old_prefix_power_successor + S (bpvi_successor_bls_blfg_induction_legendre_old_prefix_power) = S ((S (S bpvi_j_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power)) /\ exists bpvi_q_bls_blfg_induction_legendre_old_prefix_power_successor. bpvi_u_bls_blfg_induction_legendre_old_prefix_power = bpvi_q_bls_blfg_induction_legendre_old_prefix_power_successor * S ((S (S bpvi_j_bls_blfg_induction_legendre_old_prefix_power)) * bpvi_v_bls_blfg_induction_legendre_old_prefix_power) + (bpvi_successor_bls_blfg_induction_legendre_old_prefix_power))) /\ bpvi_successor_bls_blfg_induction_legendre_old_prefix_power = bpvi_partial_bls_blfg_induction_legendre_old_prefix_power * bpvi_factor_bls_blfg_induction_legendre_old_prefix_power)))))))) /\ ((((exists ff_h_bls_blfg_induction_legendre_old_prefix_quotient_entry. ff_h_bls_blfg_induction_legendre_old_prefix_quotient_entry + S (bls_quotient_blfg_induction_legendre_old_prefix) = S ((S (bls_index_blfg_induction_legendre_old_prefix)) * bls_scale_blfg_induction_legendre_old)) /\ exists ff_q_bls_blfg_induction_legendre_old_prefix_quotient_entry. bls_code_blfg_induction_legendre_old = ff_q_bls_blfg_induction_legendre_old_prefix_quotient_entry * S ((S (bls_index_blfg_induction_legendre_old_prefix)) * bls_scale_blfg_induction_legendre_old) + (bls_quotient_blfg_induction_legendre_old_prefix))) /\ ((n = bls_power_blfg_induction_legendre_old_prefix * bls_quotient_blfg_induction_legendre_old_prefix + bls_remainder_blfg_induction_legendre_old_prefix /\ exists bls_remainder_gap_blfg_induction_legendre_old_prefix_division. bls_remainder_gap_blfg_induction_legendre_old_prefix_division + S (bls_remainder_blfg_induction_legendre_old_prefix) = bls_power_blfg_induction_legendre_old_prefix))))) /\ (exists ff_u_bls_blfg_induction_legendre_old_sum ff_v_bls_blfg_induction_legendre_old_sum. ((((exists ff_h_bls_blfg_induction_legendre_old_sum_start. ff_h_bls_blfg_induction_legendre_old_sum_start + S (0) = S ((S (0)) * ff_v_bls_blfg_induction_legendre_old_sum)) /\ exists ff_q_bls_blfg_induction_legendre_old_sum_start. ff_u_bls_blfg_induction_legendre_old_sum = ff_q_bls_blfg_induction_legendre_old_sum_start * S ((S (0)) * ff_v_bls_blfg_induction_legendre_old_sum) + (0))) /\ ((((exists ff_h_bls_blfg_induction_legendre_old_sum_terminal. ff_h_bls_blfg_induction_legendre_old_sum_terminal + S (s) = S ((S (n)) * ff_v_bls_blfg_induction_legendre_old_sum)) /\ exists ff_q_bls_blfg_induction_legendre_old_sum_terminal. ff_u_bls_blfg_induction_legendre_old_sum = ff_q_bls_blfg_induction_legendre_old_sum_terminal * S ((S (n)) * ff_v_bls_blfg_induction_legendre_old_sum) + (s))) /\ forall ff_i_bls_blfg_induction_legendre_old_sum. (exists ff_lt_bls_blfg_induction_legendre_old_sum_bound. ff_lt_bls_blfg_induction_legendre_old_sum_bound + S ff_i_bls_blfg_induction_legendre_old_sum = n) -> exists ff_a_bls_blfg_induction_legendre_old_sum ff_r_bls_blfg_induction_legendre_old_sum ff_s_bls_blfg_induction_legendre_old_sum. ((((exists ff_h_bls_blfg_induction_legendre_old_sum_summand. ff_h_bls_blfg_induction_legendre_old_sum_summand + S (ff_a_bls_blfg_induction_legendre_old_sum) = S ((S (ff_i_bls_blfg_induction_legendre_old_sum)) * bls_scale_blfg_induction_legendre_old)) /\ exists ff_q_bls_blfg_induction_legendre_old_sum_summand. bls_code_blfg_induction_legendre_old = ff_q_bls_blfg_induction_legendre_old_sum_summand * S ((S (ff_i_bls_blfg_induction_legendre_old_sum)) * bls_scale_blfg_induction_legendre_old) + (ff_a_bls_blfg_induction_legendre_old_sum))) /\ ((((exists ff_h_bls_blfg_induction_legendre_old_sum_partial. ff_h_bls_blfg_induction_legendre_old_sum_partial + S (ff_r_bls_blfg_induction_legendre_old_sum) = S ((S (ff_i_bls_blfg_induction_legendre_old_sum)) * ff_v_bls_blfg_induction_legendre_old_sum)) /\ exists ff_q_bls_blfg_induction_legendre_old_sum_partial. ff_u_bls_blfg_induction_legendre_old_sum = ff_q_bls_blfg_induction_legendre_old_sum_partial * S ((S (ff_i_bls_blfg_induction_legendre_old_sum)) * ff_v_bls_blfg_induction_legendre_old_sum) + (ff_r_bls_blfg_induction_legendre_old_sum))) /\ ((((exists ff_h_bls_blfg_induction_legendre_old_sum_successor. ff_h_bls_blfg_induction_legendre_old_sum_successor + S (ff_s_bls_blfg_induction_legendre_old_sum) = S ((S (S ff_i_bls_blfg_induction_legendre_old_sum)) * ff_v_bls_blfg_induction_legendre_old_sum)) /\ exists ff_q_bls_blfg_induction_legendre_old_sum_successor. ff_u_bls_blfg_induction_legendre_old_sum = ff_q_bls_blfg_induction_legendre_old_sum_successor * S ((S (S ff_i_bls_blfg_induction_legendre_old_sum)) * ff_v_bls_blfg_induction_legendre_old_sum) + (ff_s_bls_blfg_induction_legendre_old_sum))) /\ ff_s_bls_blfg_induction_legendre_old_sum = ff_r_bls_blfg_induction_legendre_old_sum + ff_a_bls_blfg_induction_legendre_old_sum)))))))
  38. 0038specialize prime_legendre_sum_exists p
  39. 0039specialize prime_legendre_sum_exists n
  40. 0040apply prime_legendre_sum_exists
  41. 0041exact hp
  42. 0042cases hlegendre_old
  43. 0043have hcontribution : exists f. (((exists blsr_le_gap_blfg_induction_contribution_exponent_bound. blsr_le_gap_blfg_induction_contribution_exponent_bound + (f) = (S n)) /\ (exists bpvi_result_blfg_induction_contribution_selected. ((exists bpvi_b_blfg_induction_contribution_selected_power bpvi_c_blfg_induction_contribution_selected_power. ((forall bpvi_i_blfg_induction_contribution_selected_power. (exists bpvi_repeat_gap_blfg_induction_contribution_selected_power. bpvi_repeat_gap_blfg_induction_contribution_selected_power + S bpvi_i_blfg_induction_contribution_selected_power = f) -> (((exists bpvi_h_blfg_induction_contribution_selected_power_repeat. bpvi_h_blfg_induction_contribution_selected_power_repeat + S (p) = S ((S (bpvi_i_blfg_induction_contribution_selected_power)) * bpvi_c_blfg_induction_contribution_selected_power)) /\ exists bpvi_q_blfg_induction_contribution_selected_power_repeat. bpvi_b_blfg_induction_contribution_selected_power = bpvi_q_blfg_induction_contribution_selected_power_repeat * S ((S (bpvi_i_blfg_induction_contribution_selected_power)) * bpvi_c_blfg_induction_contribution_selected_power) + (p)))) /\ (exists bpvi_u_blfg_induction_contribution_selected_power bpvi_v_blfg_induction_contribution_selected_power. ((((exists bpvi_h_blfg_induction_contribution_selected_power_start. bpvi_h_blfg_induction_contribution_selected_power_start + S (1) = S ((S (0)) * bpvi_v_blfg_induction_contribution_selected_power)) /\ exists bpvi_q_blfg_induction_contribution_selected_power_start. bpvi_u_blfg_induction_contribution_selected_power = bpvi_q_blfg_induction_contribution_selected_power_start * S ((S (0)) * bpvi_v_blfg_induction_contribution_selected_power) + (1))) /\ ((((exists bpvi_h_blfg_induction_contribution_selected_power_terminal. bpvi_h_blfg_induction_contribution_selected_power_terminal + S (bpvi_result_blfg_induction_contribution_selected) = S ((S (f)) * bpvi_v_blfg_induction_contribution_selected_power)) /\ exists bpvi_q_blfg_induction_contribution_selected_power_terminal. bpvi_u_blfg_induction_contribution_selected_power = bpvi_q_blfg_induction_contribution_selected_power_terminal * S ((S (f)) * bpvi_v_blfg_induction_contribution_selected_power) + (bpvi_result_blfg_induction_contribution_selected))) /\ forall bpvi_j_blfg_induction_contribution_selected_power. (exists bpvi_product_gap_blfg_induction_contribution_selected_power. bpvi_product_gap_blfg_induction_contribution_selected_power + S bpvi_j_blfg_induction_contribution_selected_power = f) -> exists bpvi_factor_blfg_induction_contribution_selected_power bpvi_partial_blfg_induction_contribution_selected_power bpvi_successor_blfg_induction_contribution_selected_power. ((((exists bpvi_h_blfg_induction_contribution_selected_power_factor. bpvi_h_blfg_induction_contribution_selected_power_factor + S (bpvi_factor_blfg_induction_contribution_selected_power) = S ((S (bpvi_j_blfg_induction_contribution_selected_power)) * bpvi_c_blfg_induction_contribution_selected_power)) /\ exists bpvi_q_blfg_induction_contribution_selected_power_factor. bpvi_b_blfg_induction_contribution_selected_power = bpvi_q_blfg_induction_contribution_selected_power_factor * S ((S (bpvi_j_blfg_induction_contribution_selected_power)) * bpvi_c_blfg_induction_contribution_selected_power) + (bpvi_factor_blfg_induction_contribution_selected_power))) /\ ((((exists bpvi_h_blfg_induction_contribution_selected_power_partial. bpvi_h_blfg_induction_contribution_selected_power_partial + S (bpvi_partial_blfg_induction_contribution_selected_power) = S ((S (bpvi_j_blfg_induction_contribution_selected_power)) * bpvi_v_blfg_induction_contribution_selected_power)) /\ exists bpvi_q_blfg_induction_contribution_selected_power_partial. bpvi_u_blfg_induction_contribution_selected_power = bpvi_q_blfg_induction_contribution_selected_power_partial * S ((S (bpvi_j_blfg_induction_contribution_selected_power)) * bpvi_v_blfg_induction_contribution_selected_power) + (bpvi_partial_blfg_induction_contribution_selected_power))) /\ ((((exists bpvi_h_blfg_induction_contribution_selected_power_successor. bpvi_h_blfg_induction_contribution_selected_power_successor + S (bpvi_successor_blfg_induction_contribution_selected_power) = S ((S (S bpvi_j_blfg_induction_contribution_selected_power)) * bpvi_v_blfg_induction_contribution_selected_power)) /\ exists bpvi_q_blfg_induction_contribution_selected_power_successor. bpvi_u_blfg_induction_contribution_selected_power = bpvi_q_blfg_induction_contribution_selected_power_successor * S ((S (S bpvi_j_blfg_induction_contribution_selected_power)) * bpvi_v_blfg_induction_contribution_selected_power) + (bpvi_successor_blfg_induction_contribution_selected_power))) /\ bpvi_successor_blfg_induction_contribution_selected_power = bpvi_partial_blfg_induction_contribution_selected_power * bpvi_factor_blfg_induction_contribution_selected_power)))))))) /\ exists bpvi_divisor_factor_blfg_induction_contribution_selected. S n = bpvi_result_blfg_induction_contribution_selected * bpvi_divisor_factor_blfg_induction_contribution_selected))) /\ forall blsr_candidate_blfg_induction_contribution. (exists blsr_le_gap_blfg_induction_contribution_candidate_bound. blsr_le_gap_blfg_induction_contribution_candidate_bound + (blsr_candidate_blfg_induction_contribution) = (S n)) -> (exists bpvi_result_blfg_induction_contribution_candidate. ((exists bpvi_b_blfg_induction_contribution_candidate_power bpvi_c_blfg_induction_contribution_candidate_power. ((forall bpvi_i_blfg_induction_contribution_candidate_power. (exists bpvi_repeat_gap_blfg_induction_contribution_candidate_power. bpvi_repeat_gap_blfg_induction_contribution_candidate_power + S bpvi_i_blfg_induction_contribution_candidate_power = blsr_candidate_blfg_induction_contribution) -> (((exists bpvi_h_blfg_induction_contribution_candidate_power_repeat. bpvi_h_blfg_induction_contribution_candidate_power_repeat + S (p) = S ((S (bpvi_i_blfg_induction_contribution_candidate_power)) * bpvi_c_blfg_induction_contribution_candidate_power)) /\ exists bpvi_q_blfg_induction_contribution_candidate_power_repeat. bpvi_b_blfg_induction_contribution_candidate_power = bpvi_q_blfg_induction_contribution_candidate_power_repeat * S ((S (bpvi_i_blfg_induction_contribution_candidate_power)) * bpvi_c_blfg_induction_contribution_candidate_power) + (p)))) /\ (exists bpvi_u_blfg_induction_contribution_candidate_power bpvi_v_blfg_induction_contribution_candidate_power. ((((exists bpvi_h_blfg_induction_contribution_candidate_power_start. bpvi_h_blfg_induction_contribution_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_blfg_induction_contribution_candidate_power)) /\ exists bpvi_q_blfg_induction_contribution_candidate_power_start. bpvi_u_blfg_induction_contribution_candidate_power = bpvi_q_blfg_induction_contribution_candidate_power_start * S ((S (0)) * bpvi_v_blfg_induction_contribution_candidate_power) + (1))) /\ ((((exists bpvi_h_blfg_induction_contribution_candidate_power_terminal. bpvi_h_blfg_induction_contribution_candidate_power_terminal + S (bpvi_result_blfg_induction_contribution_candidate) = S ((S (blsr_candidate_blfg_induction_contribution)) * bpvi_v_blfg_induction_contribution_candidate_power)) /\ exists bpvi_q_blfg_induction_contribution_candidate_power_terminal. bpvi_u_blfg_induction_contribution_candidate_power = bpvi_q_blfg_induction_contribution_candidate_power_terminal * S ((S (blsr_candidate_blfg_induction_contribution)) * bpvi_v_blfg_induction_contribution_candidate_power) + (bpvi_result_blfg_induction_contribution_candidate))) /\ forall bpvi_j_blfg_induction_contribution_candidate_power. (exists bpvi_product_gap_blfg_induction_contribution_candidate_power. bpvi_product_gap_blfg_induction_contribution_candidate_power + S bpvi_j_blfg_induction_contribution_candidate_power = blsr_candidate_blfg_induction_contribution) -> exists bpvi_factor_blfg_induction_contribution_candidate_power bpvi_partial_blfg_induction_contribution_candidate_power bpvi_successor_blfg_induction_contribution_candidate_power. ((((exists bpvi_h_blfg_induction_contribution_candidate_power_factor. bpvi_h_blfg_induction_contribution_candidate_power_factor + S (bpvi_factor_blfg_induction_contribution_candidate_power) = S ((S (bpvi_j_blfg_induction_contribution_candidate_power)) * bpvi_c_blfg_induction_contribution_candidate_power)) /\ exists bpvi_q_blfg_induction_contribution_candidate_power_factor. bpvi_b_blfg_induction_contribution_candidate_power = bpvi_q_blfg_induction_contribution_candidate_power_factor * S ((S (bpvi_j_blfg_induction_contribution_candidate_power)) * bpvi_c_blfg_induction_contribution_candidate_power) + (bpvi_factor_blfg_induction_contribution_candidate_power))) /\ ((((exists bpvi_h_blfg_induction_contribution_candidate_power_partial. bpvi_h_blfg_induction_contribution_candidate_power_partial + S (bpvi_partial_blfg_induction_contribution_candidate_power) = S ((S (bpvi_j_blfg_induction_contribution_candidate_power)) * bpvi_v_blfg_induction_contribution_candidate_power)) /\ exists bpvi_q_blfg_induction_contribution_candidate_power_partial. bpvi_u_blfg_induction_contribution_candidate_power = bpvi_q_blfg_induction_contribution_candidate_power_partial * S ((S (bpvi_j_blfg_induction_contribution_candidate_power)) * bpvi_v_blfg_induction_contribution_candidate_power) + (bpvi_partial_blfg_induction_contribution_candidate_power))) /\ ((((exists bpvi_h_blfg_induction_contribution_candidate_power_successor. bpvi_h_blfg_induction_contribution_candidate_power_successor + S (bpvi_successor_blfg_induction_contribution_candidate_power) = S ((S (S bpvi_j_blfg_induction_contribution_candidate_power)) * bpvi_v_blfg_induction_contribution_candidate_power)) /\ exists bpvi_q_blfg_induction_contribution_candidate_power_successor. bpvi_u_blfg_induction_contribution_candidate_power = bpvi_q_blfg_induction_contribution_candidate_power_successor * S ((S (S bpvi_j_blfg_induction_contribution_candidate_power)) * bpvi_v_blfg_induction_contribution_candidate_power) + (bpvi_successor_blfg_induction_contribution_candidate_power))) /\ bpvi_successor_blfg_induction_contribution_candidate_power = bpvi_partial_blfg_induction_contribution_candidate_power * bpvi_factor_blfg_induction_contribution_candidate_power)))))))) /\ exists bpvi_divisor_factor_blfg_induction_contribution_candidate. S n = bpvi_result_blfg_induction_contribution_candidate * bpvi_divisor_factor_blfg_induction_contribution_candidate)) -> (exists blsr_le_gap_blfg_induction_contribution_maximal. blsr_le_gap_blfg_induction_contribution_maximal + (blsr_candidate_blfg_induction_contribution) = (f)))
  44. 0044specialize power_valuation_exists p
  45. 0045specialize power_valuation_exists (S n)
  46. 0046exact power_valuation_exists
  47. 0047cases hcontribution
  48. 0048have hpredecessor_agreement : x = x1
  49. 0049specialize IH x
  50. 0050specialize IH x1
  51. 0051apply IH
  52. 0052exact hp
  53. 0053exact hfactorial_old_witness
  54. 0054exact hlegendre_old_witness
  55. 0055specialize factorial_legendre_successor_agreement p
  56. 0056specialize factorial_legendre_successor_agreement n
  57. 0057specialize factorial_legendre_successor_agreement x
  58. 0058specialize factorial_legendre_successor_agreement a
  59. 0059specialize factorial_legendre_successor_agreement x1
  60. 0060specialize factorial_legendre_successor_agreement b
  61. 0061specialize factorial_legendre_successor_agreement x2
  62. 0062apply factorial_legendre_successor_agreement
  63. 0063exact hp
  64. 0064exact hfactorial_old_witness
  65. 0065exact hfactorial
  66. 0066exact hcontribution_witness
  67. 0067exact hlegendre_old_witness
  68. 0068exact hlegendre
  69. 0069exact hpredecessor_agreement