BT00XN

central_binom_legendre_valuation_balance

Alpha body-checked · checked-use disabled

Factorial Legendre equality exposes the central carry balance.

Exact expanded PA statement

forall p n c e A B. ((~(p = 1) /\ forall frm_prime_left_b5cvlb_prime frm_prime_right_b5cvlb_prime. p = frm_prime_left_b5cvlb_prime * frm_prime_right_b5cvlb_prime -> frm_prime_left_b5cvlb_prime = 1 \/ frm_prime_right_b5cvlb_prime = 1)) -> (((exists bcf_lt_gap_b5cvlb_central_out_of_range. bcf_lt_gap_b5cvlb_central_out_of_range + S (n + n) = n) /\ c = 0) \/ ((exists bcf_le_gap_b5cvlb_central_in_range. bcf_le_gap_b5cvlb_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_b5cvlb_central bcf_row_code_scale_b5cvlb_central bcf_row_scale_code_b5cvlb_central bcf_row_scale_scale_b5cvlb_central bcf_row_code_b5cvlb_central bcf_row_scale_b5cvlb_central. ((forall bcf_row_index_b5cvlb_central_table. (exists bcf_lt_gap_b5cvlb_central_table_row_bound. bcf_lt_gap_b5cvlb_central_table_row_bound + S (bcf_row_index_b5cvlb_central_table) = S (n + n)) -> exists bcf_row_code_b5cvlb_central_table bcf_row_scale_b5cvlb_central_table. ((((exists bcf_height_b5cvlb_central_table_decoded_row_code. bcf_height_b5cvlb_central_table_decoded_row_code + S (bcf_row_code_b5cvlb_central_table) = S ((S (bcf_row_index_b5cvlb_central_table)) * bcf_row_code_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_table_decoded_row_code. bcf_row_code_code_b5cvlb_central = bcf_quotient_b5cvlb_central_table_decoded_row_code * S ((S (bcf_row_index_b5cvlb_central_table)) * bcf_row_code_scale_b5cvlb_central) + (bcf_row_code_b5cvlb_central_table))) /\ ((((exists bcf_height_b5cvlb_central_table_decoded_row_scale. bcf_height_b5cvlb_central_table_decoded_row_scale + S (bcf_row_scale_b5cvlb_central_table) = S ((S (bcf_row_index_b5cvlb_central_table)) * bcf_row_scale_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_table_decoded_row_scale. bcf_row_scale_code_b5cvlb_central = bcf_quotient_b5cvlb_central_table_decoded_row_scale * S ((S (bcf_row_index_b5cvlb_central_table)) * bcf_row_scale_scale_b5cvlb_central) + (bcf_row_scale_b5cvlb_central_table))) /\ ((bcf_row_index_b5cvlb_central_table = 0 /\ (forall bcf_index_b5cvlb_central_table_zero_row. (exists bcf_lt_gap_b5cvlb_central_table_zero_row_bound. bcf_lt_gap_b5cvlb_central_table_zero_row_bound + S (bcf_index_b5cvlb_central_table_zero_row) = S (n + n)) -> exists bcf_value_b5cvlb_central_table_zero_row. ((((exists bcf_height_b5cvlb_central_table_zero_row_entry. bcf_height_b5cvlb_central_table_zero_row_entry + S (bcf_value_b5cvlb_central_table_zero_row) = S ((S (bcf_index_b5cvlb_central_table_zero_row)) * bcf_row_scale_b5cvlb_central_table)) /\ exists bcf_quotient_b5cvlb_central_table_zero_row_entry. bcf_row_code_b5cvlb_central_table = bcf_quotient_b5cvlb_central_table_zero_row_entry * S ((S (bcf_index_b5cvlb_central_table_zero_row)) * bcf_row_scale_b5cvlb_central_table) + (bcf_value_b5cvlb_central_table_zero_row))) /\ ((bcf_index_b5cvlb_central_table_zero_row = 0 /\ bcf_value_b5cvlb_central_table_zero_row = 1) \/ exists bcf_predecessor_b5cvlb_central_table_zero_row. bcf_index_b5cvlb_central_table_zero_row = S bcf_predecessor_b5cvlb_central_table_zero_row /\ bcf_value_b5cvlb_central_table_zero_row = 0)))) \/ exists bcf_predecessor_b5cvlb_central_table bcf_previous_code_b5cvlb_central_table bcf_previous_scale_b5cvlb_central_table. bcf_row_index_b5cvlb_central_table = S bcf_predecessor_b5cvlb_central_table /\ ((((exists bcf_height_b5cvlb_central_table_decoded_previous_code. bcf_height_b5cvlb_central_table_decoded_previous_code + S (bcf_previous_code_b5cvlb_central_table) = S ((S (bcf_predecessor_b5cvlb_central_table)) * bcf_row_code_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_table_decoded_previous_code. bcf_row_code_code_b5cvlb_central = bcf_quotient_b5cvlb_central_table_decoded_previous_code * S ((S (bcf_predecessor_b5cvlb_central_table)) * bcf_row_code_scale_b5cvlb_central) + (bcf_previous_code_b5cvlb_central_table))) /\ ((((exists bcf_height_b5cvlb_central_table_decoded_previous_scale. bcf_height_b5cvlb_central_table_decoded_previous_scale + S (bcf_previous_scale_b5cvlb_central_table) = S ((S (bcf_predecessor_b5cvlb_central_table)) * bcf_row_scale_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_table_decoded_previous_scale. bcf_row_scale_code_b5cvlb_central = bcf_quotient_b5cvlb_central_table_decoded_previous_scale * S ((S (bcf_predecessor_b5cvlb_central_table)) * bcf_row_scale_scale_b5cvlb_central) + (bcf_previous_scale_b5cvlb_central_table))) /\ (forall bcf_index_b5cvlb_central_table_row_step. (exists bcf_lt_gap_b5cvlb_central_table_row_step_bound. bcf_lt_gap_b5cvlb_central_table_row_step_bound + S (bcf_index_b5cvlb_central_table_row_step) = S (n + n)) -> exists bcf_value_b5cvlb_central_table_row_step. ((((exists bcf_height_b5cvlb_central_table_row_step_entry. bcf_height_b5cvlb_central_table_row_step_entry + S (bcf_value_b5cvlb_central_table_row_step) = S ((S (bcf_index_b5cvlb_central_table_row_step)) * bcf_row_scale_b5cvlb_central_table)) /\ exists bcf_quotient_b5cvlb_central_table_row_step_entry. bcf_row_code_b5cvlb_central_table = bcf_quotient_b5cvlb_central_table_row_step_entry * S ((S (bcf_index_b5cvlb_central_table_row_step)) * bcf_row_scale_b5cvlb_central_table) + (bcf_value_b5cvlb_central_table_row_step))) /\ ((bcf_index_b5cvlb_central_table_row_step = 0 /\ bcf_value_b5cvlb_central_table_row_step = 1) \/ exists bcf_predecessor_b5cvlb_central_table_row_step bcf_left_b5cvlb_central_table_row_step bcf_right_b5cvlb_central_table_row_step. bcf_index_b5cvlb_central_table_row_step = S bcf_predecessor_b5cvlb_central_table_row_step /\ ((((exists bcf_height_b5cvlb_central_table_row_step_previous_left. bcf_height_b5cvlb_central_table_row_step_previous_left + S (bcf_left_b5cvlb_central_table_row_step) = S ((S (bcf_predecessor_b5cvlb_central_table_row_step)) * bcf_previous_scale_b5cvlb_central_table)) /\ exists bcf_quotient_b5cvlb_central_table_row_step_previous_left. bcf_previous_code_b5cvlb_central_table = bcf_quotient_b5cvlb_central_table_row_step_previous_left * S ((S (bcf_predecessor_b5cvlb_central_table_row_step)) * bcf_previous_scale_b5cvlb_central_table) + (bcf_left_b5cvlb_central_table_row_step))) /\ ((((exists bcf_height_b5cvlb_central_table_row_step_previous_right. bcf_height_b5cvlb_central_table_row_step_previous_right + S (bcf_right_b5cvlb_central_table_row_step) = S ((S (S (bcf_predecessor_b5cvlb_central_table_row_step))) * bcf_previous_scale_b5cvlb_central_table)) /\ exists bcf_quotient_b5cvlb_central_table_row_step_previous_right. bcf_previous_code_b5cvlb_central_table = bcf_quotient_b5cvlb_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_b5cvlb_central_table_row_step))) * bcf_previous_scale_b5cvlb_central_table) + (bcf_right_b5cvlb_central_table_row_step))) /\ bcf_value_b5cvlb_central_table_row_step = bcf_left_b5cvlb_central_table_row_step + bcf_right_b5cvlb_central_table_row_step))))))))))) /\ ((((exists bcf_height_b5cvlb_central_decoded_row_code. bcf_height_b5cvlb_central_decoded_row_code + S (bcf_row_code_b5cvlb_central) = S ((S (n + n)) * bcf_row_code_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_decoded_row_code. bcf_row_code_code_b5cvlb_central = bcf_quotient_b5cvlb_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_b5cvlb_central) + (bcf_row_code_b5cvlb_central))) /\ ((((exists bcf_height_b5cvlb_central_decoded_row_scale. bcf_height_b5cvlb_central_decoded_row_scale + S (bcf_row_scale_b5cvlb_central) = S ((S (n + n)) * bcf_row_scale_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_decoded_row_scale. bcf_row_scale_code_b5cvlb_central = bcf_quotient_b5cvlb_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_b5cvlb_central) + (bcf_row_scale_b5cvlb_central))) /\ (((exists bcf_height_b5cvlb_central_decoded_value. bcf_height_b5cvlb_central_decoded_value + S (c) = S ((S (n)) * bcf_row_scale_b5cvlb_central)) /\ exists bcf_quotient_b5cvlb_central_decoded_value. bcf_row_code_b5cvlb_central = bcf_quotient_b5cvlb_central_decoded_value * S ((S (n)) * bcf_row_scale_b5cvlb_central) + (c))))))))) -> (((exists bpv_gap_b5cvlb_value_exponent_bound. bpv_gap_b5cvlb_value_exponent_bound + e = c) /\ (exists bpv_result_b5cvlb_value_selected. ((exists ff_b_b5cvlb_value_selected_power ff_c_b5cvlb_value_selected_power. ((forall ff_i_b5cvlb_value_selected_power_repeat. (exists ff_lt_b5cvlb_value_selected_power_repeat_bound. ff_lt_b5cvlb_value_selected_power_repeat_bound + S ff_i_b5cvlb_value_selected_power_repeat = e) -> (((exists ff_h_b5cvlb_value_selected_power_repeat_decoded. ff_h_b5cvlb_value_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvlb_value_selected_power_repeat)) * ff_c_b5cvlb_value_selected_power)) /\ exists ff_q_b5cvlb_value_selected_power_repeat_decoded. ff_b_b5cvlb_value_selected_power = ff_q_b5cvlb_value_selected_power_repeat_decoded * S ((S (ff_i_b5cvlb_value_selected_power_repeat)) * ff_c_b5cvlb_value_selected_power) + (p)))) /\ (exists ff_u_b5cvlb_value_selected_power_product ff_v_b5cvlb_value_selected_power_product. ((((exists ff_h_b5cvlb_value_selected_power_product_start. ff_h_b5cvlb_value_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_value_selected_power_product)) /\ exists ff_q_b5cvlb_value_selected_power_product_start. ff_u_b5cvlb_value_selected_power_product = ff_q_b5cvlb_value_selected_power_product_start * S ((S (0)) * ff_v_b5cvlb_value_selected_power_product) + (1))) /\ ((((exists ff_h_b5cvlb_value_selected_power_product_terminal. ff_h_b5cvlb_value_selected_power_product_terminal + S (bpv_result_b5cvlb_value_selected) = S ((S (e)) * ff_v_b5cvlb_value_selected_power_product)) /\ exists ff_q_b5cvlb_value_selected_power_product_terminal. ff_u_b5cvlb_value_selected_power_product = ff_q_b5cvlb_value_selected_power_product_terminal * S ((S (e)) * ff_v_b5cvlb_value_selected_power_product) + (bpv_result_b5cvlb_value_selected))) /\ forall ff_i_b5cvlb_value_selected_power_product. (exists ff_lt_b5cvlb_value_selected_power_product_bound. ff_lt_b5cvlb_value_selected_power_product_bound + S ff_i_b5cvlb_value_selected_power_product = e) -> exists ff_p_b5cvlb_value_selected_power_product ff_r_b5cvlb_value_selected_power_product ff_s_b5cvlb_value_selected_power_product. ((((exists ff_h_b5cvlb_value_selected_power_product_factor. ff_h_b5cvlb_value_selected_power_product_factor + S (ff_p_b5cvlb_value_selected_power_product) = S ((S (ff_i_b5cvlb_value_selected_power_product)) * ff_c_b5cvlb_value_selected_power)) /\ exists ff_q_b5cvlb_value_selected_power_product_factor. ff_b_b5cvlb_value_selected_power = ff_q_b5cvlb_value_selected_power_product_factor * S ((S (ff_i_b5cvlb_value_selected_power_product)) * ff_c_b5cvlb_value_selected_power) + (ff_p_b5cvlb_value_selected_power_product))) /\ ((((exists ff_h_b5cvlb_value_selected_power_product_partial. ff_h_b5cvlb_value_selected_power_product_partial + S (ff_r_b5cvlb_value_selected_power_product) = S ((S (ff_i_b5cvlb_value_selected_power_product)) * ff_v_b5cvlb_value_selected_power_product)) /\ exists ff_q_b5cvlb_value_selected_power_product_partial. ff_u_b5cvlb_value_selected_power_product = ff_q_b5cvlb_value_selected_power_product_partial * S ((S (ff_i_b5cvlb_value_selected_power_product)) * ff_v_b5cvlb_value_selected_power_product) + (ff_r_b5cvlb_value_selected_power_product))) /\ ((((exists ff_h_b5cvlb_value_selected_power_product_successor. ff_h_b5cvlb_value_selected_power_product_successor + S (ff_s_b5cvlb_value_selected_power_product) = S ((S (S ff_i_b5cvlb_value_selected_power_product)) * ff_v_b5cvlb_value_selected_power_product)) /\ exists ff_q_b5cvlb_value_selected_power_product_successor. ff_u_b5cvlb_value_selected_power_product = ff_q_b5cvlb_value_selected_power_product_successor * S ((S (S ff_i_b5cvlb_value_selected_power_product)) * ff_v_b5cvlb_value_selected_power_product) + (ff_s_b5cvlb_value_selected_power_product))) /\ ff_s_b5cvlb_value_selected_power_product = ff_r_b5cvlb_value_selected_power_product * ff_p_b5cvlb_value_selected_power_product)))))))) /\ (exists bpv_factor_b5cvlb_value_selected_divides. c = bpv_result_b5cvlb_value_selected * bpv_factor_b5cvlb_value_selected_divides)))) /\ forall bpv_candidate_b5cvlb_value. (exists bpv_gap_b5cvlb_value_candidate_bound. bpv_gap_b5cvlb_value_candidate_bound + bpv_candidate_b5cvlb_value = c) -> (exists bpv_result_b5cvlb_value_candidate. ((exists ff_b_b5cvlb_value_candidate_power ff_c_b5cvlb_value_candidate_power. ((forall ff_i_b5cvlb_value_candidate_power_repeat. (exists ff_lt_b5cvlb_value_candidate_power_repeat_bound. ff_lt_b5cvlb_value_candidate_power_repeat_bound + S ff_i_b5cvlb_value_candidate_power_repeat = bpv_candidate_b5cvlb_value) -> (((exists ff_h_b5cvlb_value_candidate_power_repeat_decoded. ff_h_b5cvlb_value_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvlb_value_candidate_power_repeat)) * ff_c_b5cvlb_value_candidate_power)) /\ exists ff_q_b5cvlb_value_candidate_power_repeat_decoded. ff_b_b5cvlb_value_candidate_power = ff_q_b5cvlb_value_candidate_power_repeat_decoded * S ((S (ff_i_b5cvlb_value_candidate_power_repeat)) * ff_c_b5cvlb_value_candidate_power) + (p)))) /\ (exists ff_u_b5cvlb_value_candidate_power_product ff_v_b5cvlb_value_candidate_power_product. ((((exists ff_h_b5cvlb_value_candidate_power_product_start. ff_h_b5cvlb_value_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_value_candidate_power_product)) /\ exists ff_q_b5cvlb_value_candidate_power_product_start. ff_u_b5cvlb_value_candidate_power_product = ff_q_b5cvlb_value_candidate_power_product_start * S ((S (0)) * ff_v_b5cvlb_value_candidate_power_product) + (1))) /\ ((((exists ff_h_b5cvlb_value_candidate_power_product_terminal. ff_h_b5cvlb_value_candidate_power_product_terminal + S (bpv_result_b5cvlb_value_candidate) = S ((S (bpv_candidate_b5cvlb_value)) * ff_v_b5cvlb_value_candidate_power_product)) /\ exists ff_q_b5cvlb_value_candidate_power_product_terminal. ff_u_b5cvlb_value_candidate_power_product = ff_q_b5cvlb_value_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvlb_value)) * ff_v_b5cvlb_value_candidate_power_product) + (bpv_result_b5cvlb_value_candidate))) /\ forall ff_i_b5cvlb_value_candidate_power_product. (exists ff_lt_b5cvlb_value_candidate_power_product_bound. ff_lt_b5cvlb_value_candidate_power_product_bound + S ff_i_b5cvlb_value_candidate_power_product = bpv_candidate_b5cvlb_value) -> exists ff_p_b5cvlb_value_candidate_power_product ff_r_b5cvlb_value_candidate_power_product ff_s_b5cvlb_value_candidate_power_product. ((((exists ff_h_b5cvlb_value_candidate_power_product_factor. ff_h_b5cvlb_value_candidate_power_product_factor + S (ff_p_b5cvlb_value_candidate_power_product) = S ((S (ff_i_b5cvlb_value_candidate_power_product)) * ff_c_b5cvlb_value_candidate_power)) /\ exists ff_q_b5cvlb_value_candidate_power_product_factor. ff_b_b5cvlb_value_candidate_power = ff_q_b5cvlb_value_candidate_power_product_factor * S ((S (ff_i_b5cvlb_value_candidate_power_product)) * ff_c_b5cvlb_value_candidate_power) + (ff_p_b5cvlb_value_candidate_power_product))) /\ ((((exists ff_h_b5cvlb_value_candidate_power_product_partial. ff_h_b5cvlb_value_candidate_power_product_partial + S (ff_r_b5cvlb_value_candidate_power_product) = S ((S (ff_i_b5cvlb_value_candidate_power_product)) * ff_v_b5cvlb_value_candidate_power_product)) /\ exists ff_q_b5cvlb_value_candidate_power_product_partial. ff_u_b5cvlb_value_candidate_power_product = ff_q_b5cvlb_value_candidate_power_product_partial * S ((S (ff_i_b5cvlb_value_candidate_power_product)) * ff_v_b5cvlb_value_candidate_power_product) + (ff_r_b5cvlb_value_candidate_power_product))) /\ ((((exists ff_h_b5cvlb_value_candidate_power_product_successor. ff_h_b5cvlb_value_candidate_power_product_successor + S (ff_s_b5cvlb_value_candidate_power_product) = S ((S (S ff_i_b5cvlb_value_candidate_power_product)) * ff_v_b5cvlb_value_candidate_power_product)) /\ exists ff_q_b5cvlb_value_candidate_power_product_successor. ff_u_b5cvlb_value_candidate_power_product = ff_q_b5cvlb_value_candidate_power_product_successor * S ((S (S ff_i_b5cvlb_value_candidate_power_product)) * ff_v_b5cvlb_value_candidate_power_product) + (ff_s_b5cvlb_value_candidate_power_product))) /\ ff_s_b5cvlb_value_candidate_power_product = ff_r_b5cvlb_value_candidate_power_product * ff_p_b5cvlb_value_candidate_power_product)))))))) /\ (exists bpv_factor_b5cvlb_value_candidate_divides. c = bpv_result_b5cvlb_value_candidate * bpv_factor_b5cvlb_value_candidate_divides))) -> (exists bpv_gap_b5cvlb_value_maximal. bpv_gap_b5cvlb_value_maximal + bpv_candidate_b5cvlb_value = e)) -> (exists bls_code_b5cvlb_total bls_scale_b5cvlb_total. ((forall bls_index_b5cvlb_total_prefix. (exists bls_gap_b5cvlb_total_prefix_bound. bls_gap_b5cvlb_total_prefix_bound + S (bls_index_b5cvlb_total_prefix) = ((n + n))) -> exists bls_power_b5cvlb_total_prefix bls_quotient_b5cvlb_total_prefix bls_remainder_b5cvlb_total_prefix. ((exists bpvi_b_bls_b5cvlb_total_prefix_power bpvi_c_bls_b5cvlb_total_prefix_power. ((forall bpvi_i_bls_b5cvlb_total_prefix_power. (exists bpvi_repeat_gap_bls_b5cvlb_total_prefix_power. bpvi_repeat_gap_bls_b5cvlb_total_prefix_power + S bpvi_i_bls_b5cvlb_total_prefix_power = S bls_index_b5cvlb_total_prefix) -> (((exists bpvi_h_bls_b5cvlb_total_prefix_power_repeat. bpvi_h_bls_b5cvlb_total_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_b5cvlb_total_prefix_power)) * bpvi_c_bls_b5cvlb_total_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_total_prefix_power_repeat. bpvi_b_bls_b5cvlb_total_prefix_power = bpvi_q_bls_b5cvlb_total_prefix_power_repeat * S ((S (bpvi_i_bls_b5cvlb_total_prefix_power)) * bpvi_c_bls_b5cvlb_total_prefix_power) + (p)))) /\ (exists bpvi_u_bls_b5cvlb_total_prefix_power bpvi_v_bls_b5cvlb_total_prefix_power. ((((exists bpvi_h_bls_b5cvlb_total_prefix_power_start. bpvi_h_bls_b5cvlb_total_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_b5cvlb_total_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_total_prefix_power_start. bpvi_u_bls_b5cvlb_total_prefix_power = bpvi_q_bls_b5cvlb_total_prefix_power_start * S ((S (0)) * bpvi_v_bls_b5cvlb_total_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_b5cvlb_total_prefix_power_terminal. bpvi_h_bls_b5cvlb_total_prefix_power_terminal + S (bls_power_b5cvlb_total_prefix) = S ((S (S bls_index_b5cvlb_total_prefix)) * bpvi_v_bls_b5cvlb_total_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_total_prefix_power_terminal. bpvi_u_bls_b5cvlb_total_prefix_power = bpvi_q_bls_b5cvlb_total_prefix_power_terminal * S ((S (S bls_index_b5cvlb_total_prefix)) * bpvi_v_bls_b5cvlb_total_prefix_power) + (bls_power_b5cvlb_total_prefix))) /\ forall bpvi_j_bls_b5cvlb_total_prefix_power. (exists bpvi_product_gap_bls_b5cvlb_total_prefix_power. bpvi_product_gap_bls_b5cvlb_total_prefix_power + S bpvi_j_bls_b5cvlb_total_prefix_power = S bls_index_b5cvlb_total_prefix) -> exists bpvi_factor_bls_b5cvlb_total_prefix_power bpvi_partial_bls_b5cvlb_total_prefix_power bpvi_successor_bls_b5cvlb_total_prefix_power. ((((exists bpvi_h_bls_b5cvlb_total_prefix_power_factor. bpvi_h_bls_b5cvlb_total_prefix_power_factor + S (bpvi_factor_bls_b5cvlb_total_prefix_power) = S ((S (bpvi_j_bls_b5cvlb_total_prefix_power)) * bpvi_c_bls_b5cvlb_total_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_total_prefix_power_factor. bpvi_b_bls_b5cvlb_total_prefix_power = bpvi_q_bls_b5cvlb_total_prefix_power_factor * S ((S (bpvi_j_bls_b5cvlb_total_prefix_power)) * bpvi_c_bls_b5cvlb_total_prefix_power) + (bpvi_factor_bls_b5cvlb_total_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlb_total_prefix_power_partial. bpvi_h_bls_b5cvlb_total_prefix_power_partial + S (bpvi_partial_bls_b5cvlb_total_prefix_power) = S ((S (bpvi_j_bls_b5cvlb_total_prefix_power)) * bpvi_v_bls_b5cvlb_total_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_total_prefix_power_partial. bpvi_u_bls_b5cvlb_total_prefix_power = bpvi_q_bls_b5cvlb_total_prefix_power_partial * S ((S (bpvi_j_bls_b5cvlb_total_prefix_power)) * bpvi_v_bls_b5cvlb_total_prefix_power) + (bpvi_partial_bls_b5cvlb_total_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlb_total_prefix_power_successor. bpvi_h_bls_b5cvlb_total_prefix_power_successor + S (bpvi_successor_bls_b5cvlb_total_prefix_power) = S ((S (S bpvi_j_bls_b5cvlb_total_prefix_power)) * bpvi_v_bls_b5cvlb_total_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_total_prefix_power_successor. bpvi_u_bls_b5cvlb_total_prefix_power = bpvi_q_bls_b5cvlb_total_prefix_power_successor * S ((S (S bpvi_j_bls_b5cvlb_total_prefix_power)) * bpvi_v_bls_b5cvlb_total_prefix_power) + (bpvi_successor_bls_b5cvlb_total_prefix_power))) /\ bpvi_successor_bls_b5cvlb_total_prefix_power = bpvi_partial_bls_b5cvlb_total_prefix_power * bpvi_factor_bls_b5cvlb_total_prefix_power)))))))) /\ ((((exists ff_h_bls_b5cvlb_total_prefix_quotient_entry. ff_h_bls_b5cvlb_total_prefix_quotient_entry + S (bls_quotient_b5cvlb_total_prefix) = S ((S (bls_index_b5cvlb_total_prefix)) * bls_scale_b5cvlb_total)) /\ exists ff_q_bls_b5cvlb_total_prefix_quotient_entry. bls_code_b5cvlb_total = ff_q_bls_b5cvlb_total_prefix_quotient_entry * S ((S (bls_index_b5cvlb_total_prefix)) * bls_scale_b5cvlb_total) + (bls_quotient_b5cvlb_total_prefix))) /\ (((n + n) = bls_power_b5cvlb_total_prefix * bls_quotient_b5cvlb_total_prefix + bls_remainder_b5cvlb_total_prefix /\ exists bls_remainder_gap_b5cvlb_total_prefix_division. bls_remainder_gap_b5cvlb_total_prefix_division + S (bls_remainder_b5cvlb_total_prefix) = bls_power_b5cvlb_total_prefix))))) /\ (exists ff_u_bls_b5cvlb_total_sum ff_v_bls_b5cvlb_total_sum. ((((exists ff_h_bls_b5cvlb_total_sum_start. ff_h_bls_b5cvlb_total_sum_start + S (0) = S ((S (0)) * ff_v_bls_b5cvlb_total_sum)) /\ exists ff_q_bls_b5cvlb_total_sum_start. ff_u_bls_b5cvlb_total_sum = ff_q_bls_b5cvlb_total_sum_start * S ((S (0)) * ff_v_bls_b5cvlb_total_sum) + (0))) /\ ((((exists ff_h_bls_b5cvlb_total_sum_terminal. ff_h_bls_b5cvlb_total_sum_terminal + S (A) = S ((S ((n + n))) * ff_v_bls_b5cvlb_total_sum)) /\ exists ff_q_bls_b5cvlb_total_sum_terminal. ff_u_bls_b5cvlb_total_sum = ff_q_bls_b5cvlb_total_sum_terminal * S ((S ((n + n))) * ff_v_bls_b5cvlb_total_sum) + (A))) /\ forall ff_i_bls_b5cvlb_total_sum. (exists ff_lt_bls_b5cvlb_total_sum_bound. ff_lt_bls_b5cvlb_total_sum_bound + S ff_i_bls_b5cvlb_total_sum = (n + n)) -> exists ff_a_bls_b5cvlb_total_sum ff_r_bls_b5cvlb_total_sum ff_s_bls_b5cvlb_total_sum. ((((exists ff_h_bls_b5cvlb_total_sum_summand. ff_h_bls_b5cvlb_total_sum_summand + S (ff_a_bls_b5cvlb_total_sum) = S ((S (ff_i_bls_b5cvlb_total_sum)) * bls_scale_b5cvlb_total)) /\ exists ff_q_bls_b5cvlb_total_sum_summand. bls_code_b5cvlb_total = ff_q_bls_b5cvlb_total_sum_summand * S ((S (ff_i_bls_b5cvlb_total_sum)) * bls_scale_b5cvlb_total) + (ff_a_bls_b5cvlb_total_sum))) /\ ((((exists ff_h_bls_b5cvlb_total_sum_partial. ff_h_bls_b5cvlb_total_sum_partial + S (ff_r_bls_b5cvlb_total_sum) = S ((S (ff_i_bls_b5cvlb_total_sum)) * ff_v_bls_b5cvlb_total_sum)) /\ exists ff_q_bls_b5cvlb_total_sum_partial. ff_u_bls_b5cvlb_total_sum = ff_q_bls_b5cvlb_total_sum_partial * S ((S (ff_i_bls_b5cvlb_total_sum)) * ff_v_bls_b5cvlb_total_sum) + (ff_r_bls_b5cvlb_total_sum))) /\ ((((exists ff_h_bls_b5cvlb_total_sum_successor. ff_h_bls_b5cvlb_total_sum_successor + S (ff_s_bls_b5cvlb_total_sum) = S ((S (S ff_i_bls_b5cvlb_total_sum)) * ff_v_bls_b5cvlb_total_sum)) /\ exists ff_q_bls_b5cvlb_total_sum_successor. ff_u_bls_b5cvlb_total_sum = ff_q_bls_b5cvlb_total_sum_successor * S ((S (S ff_i_bls_b5cvlb_total_sum)) * ff_v_bls_b5cvlb_total_sum) + (ff_s_bls_b5cvlb_total_sum))) /\ ff_s_bls_b5cvlb_total_sum = ff_r_bls_b5cvlb_total_sum + ff_a_bls_b5cvlb_total_sum)))))))) -> (exists bls_code_b5cvlb_column bls_scale_b5cvlb_column. ((forall bls_index_b5cvlb_column_prefix. (exists bls_gap_b5cvlb_column_prefix_bound. bls_gap_b5cvlb_column_prefix_bound + S (bls_index_b5cvlb_column_prefix) = (n)) -> exists bls_power_b5cvlb_column_prefix bls_quotient_b5cvlb_column_prefix bls_remainder_b5cvlb_column_prefix. ((exists bpvi_b_bls_b5cvlb_column_prefix_power bpvi_c_bls_b5cvlb_column_prefix_power. ((forall bpvi_i_bls_b5cvlb_column_prefix_power. (exists bpvi_repeat_gap_bls_b5cvlb_column_prefix_power. bpvi_repeat_gap_bls_b5cvlb_column_prefix_power + S bpvi_i_bls_b5cvlb_column_prefix_power = S bls_index_b5cvlb_column_prefix) -> (((exists bpvi_h_bls_b5cvlb_column_prefix_power_repeat. bpvi_h_bls_b5cvlb_column_prefix_power_repeat + S (p) = S ((S (bpvi_i_bls_b5cvlb_column_prefix_power)) * bpvi_c_bls_b5cvlb_column_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_column_prefix_power_repeat. bpvi_b_bls_b5cvlb_column_prefix_power = bpvi_q_bls_b5cvlb_column_prefix_power_repeat * S ((S (bpvi_i_bls_b5cvlb_column_prefix_power)) * bpvi_c_bls_b5cvlb_column_prefix_power) + (p)))) /\ (exists bpvi_u_bls_b5cvlb_column_prefix_power bpvi_v_bls_b5cvlb_column_prefix_power. ((((exists bpvi_h_bls_b5cvlb_column_prefix_power_start. bpvi_h_bls_b5cvlb_column_prefix_power_start + S (1) = S ((S (0)) * bpvi_v_bls_b5cvlb_column_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_column_prefix_power_start. bpvi_u_bls_b5cvlb_column_prefix_power = bpvi_q_bls_b5cvlb_column_prefix_power_start * S ((S (0)) * bpvi_v_bls_b5cvlb_column_prefix_power) + (1))) /\ ((((exists bpvi_h_bls_b5cvlb_column_prefix_power_terminal. bpvi_h_bls_b5cvlb_column_prefix_power_terminal + S (bls_power_b5cvlb_column_prefix) = S ((S (S bls_index_b5cvlb_column_prefix)) * bpvi_v_bls_b5cvlb_column_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_column_prefix_power_terminal. bpvi_u_bls_b5cvlb_column_prefix_power = bpvi_q_bls_b5cvlb_column_prefix_power_terminal * S ((S (S bls_index_b5cvlb_column_prefix)) * bpvi_v_bls_b5cvlb_column_prefix_power) + (bls_power_b5cvlb_column_prefix))) /\ forall bpvi_j_bls_b5cvlb_column_prefix_power. (exists bpvi_product_gap_bls_b5cvlb_column_prefix_power. bpvi_product_gap_bls_b5cvlb_column_prefix_power + S bpvi_j_bls_b5cvlb_column_prefix_power = S bls_index_b5cvlb_column_prefix) -> exists bpvi_factor_bls_b5cvlb_column_prefix_power bpvi_partial_bls_b5cvlb_column_prefix_power bpvi_successor_bls_b5cvlb_column_prefix_power. ((((exists bpvi_h_bls_b5cvlb_column_prefix_power_factor. bpvi_h_bls_b5cvlb_column_prefix_power_factor + S (bpvi_factor_bls_b5cvlb_column_prefix_power) = S ((S (bpvi_j_bls_b5cvlb_column_prefix_power)) * bpvi_c_bls_b5cvlb_column_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_column_prefix_power_factor. bpvi_b_bls_b5cvlb_column_prefix_power = bpvi_q_bls_b5cvlb_column_prefix_power_factor * S ((S (bpvi_j_bls_b5cvlb_column_prefix_power)) * bpvi_c_bls_b5cvlb_column_prefix_power) + (bpvi_factor_bls_b5cvlb_column_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlb_column_prefix_power_partial. bpvi_h_bls_b5cvlb_column_prefix_power_partial + S (bpvi_partial_bls_b5cvlb_column_prefix_power) = S ((S (bpvi_j_bls_b5cvlb_column_prefix_power)) * bpvi_v_bls_b5cvlb_column_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_column_prefix_power_partial. bpvi_u_bls_b5cvlb_column_prefix_power = bpvi_q_bls_b5cvlb_column_prefix_power_partial * S ((S (bpvi_j_bls_b5cvlb_column_prefix_power)) * bpvi_v_bls_b5cvlb_column_prefix_power) + (bpvi_partial_bls_b5cvlb_column_prefix_power))) /\ ((((exists bpvi_h_bls_b5cvlb_column_prefix_power_successor. bpvi_h_bls_b5cvlb_column_prefix_power_successor + S (bpvi_successor_bls_b5cvlb_column_prefix_power) = S ((S (S bpvi_j_bls_b5cvlb_column_prefix_power)) * bpvi_v_bls_b5cvlb_column_prefix_power)) /\ exists bpvi_q_bls_b5cvlb_column_prefix_power_successor. bpvi_u_bls_b5cvlb_column_prefix_power = bpvi_q_bls_b5cvlb_column_prefix_power_successor * S ((S (S bpvi_j_bls_b5cvlb_column_prefix_power)) * bpvi_v_bls_b5cvlb_column_prefix_power) + (bpvi_successor_bls_b5cvlb_column_prefix_power))) /\ bpvi_successor_bls_b5cvlb_column_prefix_power = bpvi_partial_bls_b5cvlb_column_prefix_power * bpvi_factor_bls_b5cvlb_column_prefix_power)))))))) /\ ((((exists ff_h_bls_b5cvlb_column_prefix_quotient_entry. ff_h_bls_b5cvlb_column_prefix_quotient_entry + S (bls_quotient_b5cvlb_column_prefix) = S ((S (bls_index_b5cvlb_column_prefix)) * bls_scale_b5cvlb_column)) /\ exists ff_q_bls_b5cvlb_column_prefix_quotient_entry. bls_code_b5cvlb_column = ff_q_bls_b5cvlb_column_prefix_quotient_entry * S ((S (bls_index_b5cvlb_column_prefix)) * bls_scale_b5cvlb_column) + (bls_quotient_b5cvlb_column_prefix))) /\ ((n = bls_power_b5cvlb_column_prefix * bls_quotient_b5cvlb_column_prefix + bls_remainder_b5cvlb_column_prefix /\ exists bls_remainder_gap_b5cvlb_column_prefix_division. bls_remainder_gap_b5cvlb_column_prefix_division + S (bls_remainder_b5cvlb_column_prefix) = bls_power_b5cvlb_column_prefix))))) /\ (exists ff_u_bls_b5cvlb_column_sum ff_v_bls_b5cvlb_column_sum. ((((exists ff_h_bls_b5cvlb_column_sum_start. ff_h_bls_b5cvlb_column_sum_start + S (0) = S ((S (0)) * ff_v_bls_b5cvlb_column_sum)) /\ exists ff_q_bls_b5cvlb_column_sum_start. ff_u_bls_b5cvlb_column_sum = ff_q_bls_b5cvlb_column_sum_start * S ((S (0)) * ff_v_bls_b5cvlb_column_sum) + (0))) /\ ((((exists ff_h_bls_b5cvlb_column_sum_terminal. ff_h_bls_b5cvlb_column_sum_terminal + S (B) = S ((S (n)) * ff_v_bls_b5cvlb_column_sum)) /\ exists ff_q_bls_b5cvlb_column_sum_terminal. ff_u_bls_b5cvlb_column_sum = ff_q_bls_b5cvlb_column_sum_terminal * S ((S (n)) * ff_v_bls_b5cvlb_column_sum) + (B))) /\ forall ff_i_bls_b5cvlb_column_sum. (exists ff_lt_bls_b5cvlb_column_sum_bound. ff_lt_bls_b5cvlb_column_sum_bound + S ff_i_bls_b5cvlb_column_sum = n) -> exists ff_a_bls_b5cvlb_column_sum ff_r_bls_b5cvlb_column_sum ff_s_bls_b5cvlb_column_sum. ((((exists ff_h_bls_b5cvlb_column_sum_summand. ff_h_bls_b5cvlb_column_sum_summand + S (ff_a_bls_b5cvlb_column_sum) = S ((S (ff_i_bls_b5cvlb_column_sum)) * bls_scale_b5cvlb_column)) /\ exists ff_q_bls_b5cvlb_column_sum_summand. bls_code_b5cvlb_column = ff_q_bls_b5cvlb_column_sum_summand * S ((S (ff_i_bls_b5cvlb_column_sum)) * bls_scale_b5cvlb_column) + (ff_a_bls_b5cvlb_column_sum))) /\ ((((exists ff_h_bls_b5cvlb_column_sum_partial. ff_h_bls_b5cvlb_column_sum_partial + S (ff_r_bls_b5cvlb_column_sum) = S ((S (ff_i_bls_b5cvlb_column_sum)) * ff_v_bls_b5cvlb_column_sum)) /\ exists ff_q_bls_b5cvlb_column_sum_partial. ff_u_bls_b5cvlb_column_sum = ff_q_bls_b5cvlb_column_sum_partial * S ((S (ff_i_bls_b5cvlb_column_sum)) * ff_v_bls_b5cvlb_column_sum) + (ff_r_bls_b5cvlb_column_sum))) /\ ((((exists ff_h_bls_b5cvlb_column_sum_successor. ff_h_bls_b5cvlb_column_sum_successor + S (ff_s_bls_b5cvlb_column_sum) = S ((S (S ff_i_bls_b5cvlb_column_sum)) * ff_v_bls_b5cvlb_column_sum)) /\ exists ff_q_bls_b5cvlb_column_sum_successor. ff_u_bls_b5cvlb_column_sum = ff_q_bls_b5cvlb_column_sum_successor * S ((S (S ff_i_bls_b5cvlb_column_sum)) * ff_v_bls_b5cvlb_column_sum) + (ff_s_bls_b5cvlb_column_sum))) /\ ff_s_bls_b5cvlb_column_sum = ff_r_bls_b5cvlb_column_sum + ff_a_bls_b5cvlb_column_sum)))))))) -> A = (B + B) + e

Structural proof guide

Factorial Legendre equality exposes the central carry balance.

Direct prerequisites: factorial_valuation_exists, prime_factorial_valuation_eq_legendre_sum, central_binom_factorial_valuation_balance. The authored body proceeds by case analysis (2), intermediate claims (5), equality transport (2).

Proof neighborhood

Direct dependencies

BT00RM factorial_valuation_exists BT00T1 prime_factorial_valuation_eq_legendre_sum BT00XM central_binom_factorial_valuation_balance

Direct dependents

BT00Y4 central_binom_carry_bit_count

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. 0002intro n
  3. 0003intro c
  4. 0004intro e
  5. 0005intro A
  6. 0006intro B
  7. 0007intro hp
  8. 0008intro hcentral
  9. 0009intro hvalue
  10. 0010intro htotal_legendre
  11. 0011intro hcolumn_legendre
  12. 0012have htotal : exists a. exists b5cv_factorial_b5cvlb_total_factorial. ((exists ff_b_b5cvlb_total_factorial_factorial ff_c_b5cvlb_total_factorial_factorial. ((forall ff_i_b5cvlb_total_factorial_factorial_range. (exists ff_lt_b5cvlb_total_factorial_factorial_range_bound. ff_lt_b5cvlb_total_factorial_factorial_range_bound + S ff_i_b5cvlb_total_factorial_factorial_range = (n + n)) -> (((exists ff_h_b5cvlb_total_factorial_factorial_range_decoded. ff_h_b5cvlb_total_factorial_factorial_range_decoded + S (1 + ff_i_b5cvlb_total_factorial_factorial_range) = S ((S (ff_i_b5cvlb_total_factorial_factorial_range)) * ff_c_b5cvlb_total_factorial_factorial)) /\ exists ff_q_b5cvlb_total_factorial_factorial_range_decoded. ff_b_b5cvlb_total_factorial_factorial = ff_q_b5cvlb_total_factorial_factorial_range_decoded * S ((S (ff_i_b5cvlb_total_factorial_factorial_range)) * ff_c_b5cvlb_total_factorial_factorial) + (1 + ff_i_b5cvlb_total_factorial_factorial_range)))) /\ (exists ff_u_b5cvlb_total_factorial_factorial_product ff_v_b5cvlb_total_factorial_factorial_product. ((((exists ff_h_b5cvlb_total_factorial_factorial_product_start. ff_h_b5cvlb_total_factorial_factorial_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_total_factorial_factorial_product)) /\ exists ff_q_b5cvlb_total_factorial_factorial_product_start. ff_u_b5cvlb_total_factorial_factorial_product = ff_q_b5cvlb_total_factorial_factorial_product_start * S ((S (0)) * ff_v_b5cvlb_total_factorial_factorial_product) + (1))) /\ ((((exists ff_h_b5cvlb_total_factorial_factorial_product_terminal. ff_h_b5cvlb_total_factorial_factorial_product_terminal + S (b5cv_factorial_b5cvlb_total_factorial) = S ((S ((n + n))) * ff_v_b5cvlb_total_factorial_factorial_product)) /\ exists ff_q_b5cvlb_total_factorial_factorial_product_terminal. ff_u_b5cvlb_total_factorial_factorial_product = ff_q_b5cvlb_total_factorial_factorial_product_terminal * S ((S ((n + n))) * ff_v_b5cvlb_total_factorial_factorial_product) + (b5cv_factorial_b5cvlb_total_factorial))) /\ forall ff_i_b5cvlb_total_factorial_factorial_product. (exists ff_lt_b5cvlb_total_factorial_factorial_product_bound. ff_lt_b5cvlb_total_factorial_factorial_product_bound + S ff_i_b5cvlb_total_factorial_factorial_product = (n + n)) -> exists ff_p_b5cvlb_total_factorial_factorial_product ff_r_b5cvlb_total_factorial_factorial_product ff_s_b5cvlb_total_factorial_factorial_product. ((((exists ff_h_b5cvlb_total_factorial_factorial_product_factor. ff_h_b5cvlb_total_factorial_factorial_product_factor + S (ff_p_b5cvlb_total_factorial_factorial_product) = S ((S (ff_i_b5cvlb_total_factorial_factorial_product)) * ff_c_b5cvlb_total_factorial_factorial)) /\ exists ff_q_b5cvlb_total_factorial_factorial_product_factor. ff_b_b5cvlb_total_factorial_factorial = ff_q_b5cvlb_total_factorial_factorial_product_factor * S ((S (ff_i_b5cvlb_total_factorial_factorial_product)) * ff_c_b5cvlb_total_factorial_factorial) + (ff_p_b5cvlb_total_factorial_factorial_product))) /\ ((((exists ff_h_b5cvlb_total_factorial_factorial_product_partial. ff_h_b5cvlb_total_factorial_factorial_product_partial + S (ff_r_b5cvlb_total_factorial_factorial_product) = S ((S (ff_i_b5cvlb_total_factorial_factorial_product)) * ff_v_b5cvlb_total_factorial_factorial_product)) /\ exists ff_q_b5cvlb_total_factorial_factorial_product_partial. ff_u_b5cvlb_total_factorial_factorial_product = ff_q_b5cvlb_total_factorial_factorial_product_partial * S ((S (ff_i_b5cvlb_total_factorial_factorial_product)) * ff_v_b5cvlb_total_factorial_factorial_product) + (ff_r_b5cvlb_total_factorial_factorial_product))) /\ ((((exists ff_h_b5cvlb_total_factorial_factorial_product_successor. ff_h_b5cvlb_total_factorial_factorial_product_successor + S (ff_s_b5cvlb_total_factorial_factorial_product) = S ((S (S ff_i_b5cvlb_total_factorial_factorial_product)) * ff_v_b5cvlb_total_factorial_factorial_product)) /\ exists ff_q_b5cvlb_total_factorial_factorial_product_successor. ff_u_b5cvlb_total_factorial_factorial_product = ff_q_b5cvlb_total_factorial_factorial_product_successor * S ((S (S ff_i_b5cvlb_total_factorial_factorial_product)) * ff_v_b5cvlb_total_factorial_factorial_product) + (ff_s_b5cvlb_total_factorial_factorial_product))) /\ ff_s_b5cvlb_total_factorial_factorial_product = ff_r_b5cvlb_total_factorial_factorial_product * ff_p_b5cvlb_total_factorial_factorial_product)))))))) /\ (((exists bpv_gap_b5cvlb_total_factorial_valuation_exponent_bound. bpv_gap_b5cvlb_total_factorial_valuation_exponent_bound + a = b5cv_factorial_b5cvlb_total_factorial) /\ (exists bpv_result_b5cvlb_total_factorial_valuation_selected. ((exists ff_b_b5cvlb_total_factorial_valuation_selected_power ff_c_b5cvlb_total_factorial_valuation_selected_power. ((forall ff_i_b5cvlb_total_factorial_valuation_selected_power_repeat. (exists ff_lt_b5cvlb_total_factorial_valuation_selected_power_repeat_bound. ff_lt_b5cvlb_total_factorial_valuation_selected_power_repeat_bound + S ff_i_b5cvlb_total_factorial_valuation_selected_power_repeat = a) -> (((exists ff_h_b5cvlb_total_factorial_valuation_selected_power_repeat_decoded. ff_h_b5cvlb_total_factorial_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvlb_total_factorial_valuation_selected_power_repeat)) * ff_c_b5cvlb_total_factorial_valuation_selected_power)) /\ exists ff_q_b5cvlb_total_factorial_valuation_selected_power_repeat_decoded. ff_b_b5cvlb_total_factorial_valuation_selected_power = ff_q_b5cvlb_total_factorial_valuation_selected_power_repeat_decoded * S ((S (ff_i_b5cvlb_total_factorial_valuation_selected_power_repeat)) * ff_c_b5cvlb_total_factorial_valuation_selected_power) + (p)))) /\ (exists ff_u_b5cvlb_total_factorial_valuation_selected_power_product ff_v_b5cvlb_total_factorial_valuation_selected_power_product. ((((exists ff_h_b5cvlb_total_factorial_valuation_selected_power_product_start. ff_h_b5cvlb_total_factorial_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_selected_power_product_start. ff_u_b5cvlb_total_factorial_valuation_selected_power_product = ff_q_b5cvlb_total_factorial_valuation_selected_power_product_start * S ((S (0)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_b5cvlb_total_factorial_valuation_selected_power_product_terminal. ff_h_b5cvlb_total_factorial_valuation_selected_power_product_terminal + S (bpv_result_b5cvlb_total_factorial_valuation_selected) = S ((S (a)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_selected_power_product_terminal. ff_u_b5cvlb_total_factorial_valuation_selected_power_product = ff_q_b5cvlb_total_factorial_valuation_selected_power_product_terminal * S ((S (a)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product) + (bpv_result_b5cvlb_total_factorial_valuation_selected))) /\ forall ff_i_b5cvlb_total_factorial_valuation_selected_power_product. (exists ff_lt_b5cvlb_total_factorial_valuation_selected_power_product_bound. ff_lt_b5cvlb_total_factorial_valuation_selected_power_product_bound + S ff_i_b5cvlb_total_factorial_valuation_selected_power_product = a) -> exists ff_p_b5cvlb_total_factorial_valuation_selected_power_product ff_r_b5cvlb_total_factorial_valuation_selected_power_product ff_s_b5cvlb_total_factorial_valuation_selected_power_product. ((((exists ff_h_b5cvlb_total_factorial_valuation_selected_power_product_factor. ff_h_b5cvlb_total_factorial_valuation_selected_power_product_factor + S (ff_p_b5cvlb_total_factorial_valuation_selected_power_product) = S ((S (ff_i_b5cvlb_total_factorial_valuation_selected_power_product)) * ff_c_b5cvlb_total_factorial_valuation_selected_power)) /\ exists ff_q_b5cvlb_total_factorial_valuation_selected_power_product_factor. ff_b_b5cvlb_total_factorial_valuation_selected_power = ff_q_b5cvlb_total_factorial_valuation_selected_power_product_factor * S ((S (ff_i_b5cvlb_total_factorial_valuation_selected_power_product)) * ff_c_b5cvlb_total_factorial_valuation_selected_power) + (ff_p_b5cvlb_total_factorial_valuation_selected_power_product))) /\ ((((exists ff_h_b5cvlb_total_factorial_valuation_selected_power_product_partial. ff_h_b5cvlb_total_factorial_valuation_selected_power_product_partial + S (ff_r_b5cvlb_total_factorial_valuation_selected_power_product) = S ((S (ff_i_b5cvlb_total_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_selected_power_product_partial. ff_u_b5cvlb_total_factorial_valuation_selected_power_product = ff_q_b5cvlb_total_factorial_valuation_selected_power_product_partial * S ((S (ff_i_b5cvlb_total_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product) + (ff_r_b5cvlb_total_factorial_valuation_selected_power_product))) /\ ((((exists ff_h_b5cvlb_total_factorial_valuation_selected_power_product_successor. ff_h_b5cvlb_total_factorial_valuation_selected_power_product_successor + S (ff_s_b5cvlb_total_factorial_valuation_selected_power_product) = S ((S (S ff_i_b5cvlb_total_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_selected_power_product_successor. ff_u_b5cvlb_total_factorial_valuation_selected_power_product = ff_q_b5cvlb_total_factorial_valuation_selected_power_product_successor * S ((S (S ff_i_b5cvlb_total_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_total_factorial_valuation_selected_power_product) + (ff_s_b5cvlb_total_factorial_valuation_selected_power_product))) /\ ff_s_b5cvlb_total_factorial_valuation_selected_power_product = ff_r_b5cvlb_total_factorial_valuation_selected_power_product * ff_p_b5cvlb_total_factorial_valuation_selected_power_product)))))))) /\ (exists bpv_factor_b5cvlb_total_factorial_valuation_selected_divides. b5cv_factorial_b5cvlb_total_factorial = bpv_result_b5cvlb_total_factorial_valuation_selected * bpv_factor_b5cvlb_total_factorial_valuation_selected_divides)))) /\ forall bpv_candidate_b5cvlb_total_factorial_valuation. (exists bpv_gap_b5cvlb_total_factorial_valuation_candidate_bound. bpv_gap_b5cvlb_total_factorial_valuation_candidate_bound + bpv_candidate_b5cvlb_total_factorial_valuation = b5cv_factorial_b5cvlb_total_factorial) -> (exists bpv_result_b5cvlb_total_factorial_valuation_candidate. ((exists ff_b_b5cvlb_total_factorial_valuation_candidate_power ff_c_b5cvlb_total_factorial_valuation_candidate_power. ((forall ff_i_b5cvlb_total_factorial_valuation_candidate_power_repeat. (exists ff_lt_b5cvlb_total_factorial_valuation_candidate_power_repeat_bound. ff_lt_b5cvlb_total_factorial_valuation_candidate_power_repeat_bound + S ff_i_b5cvlb_total_factorial_valuation_candidate_power_repeat = bpv_candidate_b5cvlb_total_factorial_valuation) -> (((exists ff_h_b5cvlb_total_factorial_valuation_candidate_power_repeat_decoded. ff_h_b5cvlb_total_factorial_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvlb_total_factorial_valuation_candidate_power_repeat)) * ff_c_b5cvlb_total_factorial_valuation_candidate_power)) /\ exists ff_q_b5cvlb_total_factorial_valuation_candidate_power_repeat_decoded. ff_b_b5cvlb_total_factorial_valuation_candidate_power = ff_q_b5cvlb_total_factorial_valuation_candidate_power_repeat_decoded * S ((S (ff_i_b5cvlb_total_factorial_valuation_candidate_power_repeat)) * ff_c_b5cvlb_total_factorial_valuation_candidate_power) + (p)))) /\ (exists ff_u_b5cvlb_total_factorial_valuation_candidate_power_product ff_v_b5cvlb_total_factorial_valuation_candidate_power_product. ((((exists ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_start. ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_start. ff_u_b5cvlb_total_factorial_valuation_candidate_power_product = ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_start * S ((S (0)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_terminal. ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_terminal + S (bpv_result_b5cvlb_total_factorial_valuation_candidate) = S ((S (bpv_candidate_b5cvlb_total_factorial_valuation)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_terminal. ff_u_b5cvlb_total_factorial_valuation_candidate_power_product = ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvlb_total_factorial_valuation)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product) + (bpv_result_b5cvlb_total_factorial_valuation_candidate))) /\ forall ff_i_b5cvlb_total_factorial_valuation_candidate_power_product. (exists ff_lt_b5cvlb_total_factorial_valuation_candidate_power_product_bound. ff_lt_b5cvlb_total_factorial_valuation_candidate_power_product_bound + S ff_i_b5cvlb_total_factorial_valuation_candidate_power_product = bpv_candidate_b5cvlb_total_factorial_valuation) -> exists ff_p_b5cvlb_total_factorial_valuation_candidate_power_product ff_r_b5cvlb_total_factorial_valuation_candidate_power_product ff_s_b5cvlb_total_factorial_valuation_candidate_power_product. ((((exists ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_factor. ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_factor + S (ff_p_b5cvlb_total_factorial_valuation_candidate_power_product) = S ((S (ff_i_b5cvlb_total_factorial_valuation_candidate_power_product)) * ff_c_b5cvlb_total_factorial_valuation_candidate_power)) /\ exists ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_factor. ff_b_b5cvlb_total_factorial_valuation_candidate_power = ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_factor * S ((S (ff_i_b5cvlb_total_factorial_valuation_candidate_power_product)) * ff_c_b5cvlb_total_factorial_valuation_candidate_power) + (ff_p_b5cvlb_total_factorial_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_partial. ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_partial + S (ff_r_b5cvlb_total_factorial_valuation_candidate_power_product) = S ((S (ff_i_b5cvlb_total_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_partial. ff_u_b5cvlb_total_factorial_valuation_candidate_power_product = ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_partial * S ((S (ff_i_b5cvlb_total_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product) + (ff_r_b5cvlb_total_factorial_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_successor. ff_h_b5cvlb_total_factorial_valuation_candidate_power_product_successor + S (ff_s_b5cvlb_total_factorial_valuation_candidate_power_product) = S ((S (S ff_i_b5cvlb_total_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_successor. ff_u_b5cvlb_total_factorial_valuation_candidate_power_product = ff_q_b5cvlb_total_factorial_valuation_candidate_power_product_successor * S ((S (S ff_i_b5cvlb_total_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_total_factorial_valuation_candidate_power_product) + (ff_s_b5cvlb_total_factorial_valuation_candidate_power_product))) /\ ff_s_b5cvlb_total_factorial_valuation_candidate_power_product = ff_r_b5cvlb_total_factorial_valuation_candidate_power_product * ff_p_b5cvlb_total_factorial_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_b5cvlb_total_factorial_valuation_candidate_divides. b5cv_factorial_b5cvlb_total_factorial = bpv_result_b5cvlb_total_factorial_valuation_candidate * bpv_factor_b5cvlb_total_factorial_valuation_candidate_divides))) -> (exists bpv_gap_b5cvlb_total_factorial_valuation_maximal. bpv_gap_b5cvlb_total_factorial_valuation_maximal + bpv_candidate_b5cvlb_total_factorial_valuation = a)))
  13. 0013specialize factorial_valuation_exists p
  14. 0014specialize factorial_valuation_exists (n + n)
  15. 0015exact factorial_valuation_exists
  16. 0016cases htotal
  17. 0017have hcolumn : exists b. exists bfv_factorial_b5cvlb_column_factorial. ((exists ff_b_b5cvlb_column_factorial_factorial ff_c_b5cvlb_column_factorial_factorial. ((forall ff_i_b5cvlb_column_factorial_factorial_range. (exists ff_lt_b5cvlb_column_factorial_factorial_range_bound. ff_lt_b5cvlb_column_factorial_factorial_range_bound + S ff_i_b5cvlb_column_factorial_factorial_range = n) -> (((exists ff_h_b5cvlb_column_factorial_factorial_range_decoded. ff_h_b5cvlb_column_factorial_factorial_range_decoded + S (1 + ff_i_b5cvlb_column_factorial_factorial_range) = S ((S (ff_i_b5cvlb_column_factorial_factorial_range)) * ff_c_b5cvlb_column_factorial_factorial)) /\ exists ff_q_b5cvlb_column_factorial_factorial_range_decoded. ff_b_b5cvlb_column_factorial_factorial = ff_q_b5cvlb_column_factorial_factorial_range_decoded * S ((S (ff_i_b5cvlb_column_factorial_factorial_range)) * ff_c_b5cvlb_column_factorial_factorial) + (1 + ff_i_b5cvlb_column_factorial_factorial_range)))) /\ (exists ff_u_b5cvlb_column_factorial_factorial_product ff_v_b5cvlb_column_factorial_factorial_product. ((((exists ff_h_b5cvlb_column_factorial_factorial_product_start. ff_h_b5cvlb_column_factorial_factorial_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_column_factorial_factorial_product)) /\ exists ff_q_b5cvlb_column_factorial_factorial_product_start. ff_u_b5cvlb_column_factorial_factorial_product = ff_q_b5cvlb_column_factorial_factorial_product_start * S ((S (0)) * ff_v_b5cvlb_column_factorial_factorial_product) + (1))) /\ ((((exists ff_h_b5cvlb_column_factorial_factorial_product_terminal. ff_h_b5cvlb_column_factorial_factorial_product_terminal + S (bfv_factorial_b5cvlb_column_factorial) = S ((S (n)) * ff_v_b5cvlb_column_factorial_factorial_product)) /\ exists ff_q_b5cvlb_column_factorial_factorial_product_terminal. ff_u_b5cvlb_column_factorial_factorial_product = ff_q_b5cvlb_column_factorial_factorial_product_terminal * S ((S (n)) * ff_v_b5cvlb_column_factorial_factorial_product) + (bfv_factorial_b5cvlb_column_factorial))) /\ forall ff_i_b5cvlb_column_factorial_factorial_product. (exists ff_lt_b5cvlb_column_factorial_factorial_product_bound. ff_lt_b5cvlb_column_factorial_factorial_product_bound + S ff_i_b5cvlb_column_factorial_factorial_product = n) -> exists ff_p_b5cvlb_column_factorial_factorial_product ff_r_b5cvlb_column_factorial_factorial_product ff_s_b5cvlb_column_factorial_factorial_product. ((((exists ff_h_b5cvlb_column_factorial_factorial_product_factor. ff_h_b5cvlb_column_factorial_factorial_product_factor + S (ff_p_b5cvlb_column_factorial_factorial_product) = S ((S (ff_i_b5cvlb_column_factorial_factorial_product)) * ff_c_b5cvlb_column_factorial_factorial)) /\ exists ff_q_b5cvlb_column_factorial_factorial_product_factor. ff_b_b5cvlb_column_factorial_factorial = ff_q_b5cvlb_column_factorial_factorial_product_factor * S ((S (ff_i_b5cvlb_column_factorial_factorial_product)) * ff_c_b5cvlb_column_factorial_factorial) + (ff_p_b5cvlb_column_factorial_factorial_product))) /\ ((((exists ff_h_b5cvlb_column_factorial_factorial_product_partial. ff_h_b5cvlb_column_factorial_factorial_product_partial + S (ff_r_b5cvlb_column_factorial_factorial_product) = S ((S (ff_i_b5cvlb_column_factorial_factorial_product)) * ff_v_b5cvlb_column_factorial_factorial_product)) /\ exists ff_q_b5cvlb_column_factorial_factorial_product_partial. ff_u_b5cvlb_column_factorial_factorial_product = ff_q_b5cvlb_column_factorial_factorial_product_partial * S ((S (ff_i_b5cvlb_column_factorial_factorial_product)) * ff_v_b5cvlb_column_factorial_factorial_product) + (ff_r_b5cvlb_column_factorial_factorial_product))) /\ ((((exists ff_h_b5cvlb_column_factorial_factorial_product_successor. ff_h_b5cvlb_column_factorial_factorial_product_successor + S (ff_s_b5cvlb_column_factorial_factorial_product) = S ((S (S ff_i_b5cvlb_column_factorial_factorial_product)) * ff_v_b5cvlb_column_factorial_factorial_product)) /\ exists ff_q_b5cvlb_column_factorial_factorial_product_successor. ff_u_b5cvlb_column_factorial_factorial_product = ff_q_b5cvlb_column_factorial_factorial_product_successor * S ((S (S ff_i_b5cvlb_column_factorial_factorial_product)) * ff_v_b5cvlb_column_factorial_factorial_product) + (ff_s_b5cvlb_column_factorial_factorial_product))) /\ ff_s_b5cvlb_column_factorial_factorial_product = ff_r_b5cvlb_column_factorial_factorial_product * ff_p_b5cvlb_column_factorial_factorial_product)))))))) /\ (((exists bpv_gap_b5cvlb_column_factorial_valuation_exponent_bound. bpv_gap_b5cvlb_column_factorial_valuation_exponent_bound + b = bfv_factorial_b5cvlb_column_factorial) /\ (exists bpv_result_b5cvlb_column_factorial_valuation_selected. ((exists ff_b_b5cvlb_column_factorial_valuation_selected_power ff_c_b5cvlb_column_factorial_valuation_selected_power. ((forall ff_i_b5cvlb_column_factorial_valuation_selected_power_repeat. (exists ff_lt_b5cvlb_column_factorial_valuation_selected_power_repeat_bound. ff_lt_b5cvlb_column_factorial_valuation_selected_power_repeat_bound + S ff_i_b5cvlb_column_factorial_valuation_selected_power_repeat = b) -> (((exists ff_h_b5cvlb_column_factorial_valuation_selected_power_repeat_decoded. ff_h_b5cvlb_column_factorial_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvlb_column_factorial_valuation_selected_power_repeat)) * ff_c_b5cvlb_column_factorial_valuation_selected_power)) /\ exists ff_q_b5cvlb_column_factorial_valuation_selected_power_repeat_decoded. ff_b_b5cvlb_column_factorial_valuation_selected_power = ff_q_b5cvlb_column_factorial_valuation_selected_power_repeat_decoded * S ((S (ff_i_b5cvlb_column_factorial_valuation_selected_power_repeat)) * ff_c_b5cvlb_column_factorial_valuation_selected_power) + (p)))) /\ (exists ff_u_b5cvlb_column_factorial_valuation_selected_power_product ff_v_b5cvlb_column_factorial_valuation_selected_power_product. ((((exists ff_h_b5cvlb_column_factorial_valuation_selected_power_product_start. ff_h_b5cvlb_column_factorial_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_selected_power_product_start. ff_u_b5cvlb_column_factorial_valuation_selected_power_product = ff_q_b5cvlb_column_factorial_valuation_selected_power_product_start * S ((S (0)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_b5cvlb_column_factorial_valuation_selected_power_product_terminal. ff_h_b5cvlb_column_factorial_valuation_selected_power_product_terminal + S (bpv_result_b5cvlb_column_factorial_valuation_selected) = S ((S (b)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_selected_power_product_terminal. ff_u_b5cvlb_column_factorial_valuation_selected_power_product = ff_q_b5cvlb_column_factorial_valuation_selected_power_product_terminal * S ((S (b)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product) + (bpv_result_b5cvlb_column_factorial_valuation_selected))) /\ forall ff_i_b5cvlb_column_factorial_valuation_selected_power_product. (exists ff_lt_b5cvlb_column_factorial_valuation_selected_power_product_bound. ff_lt_b5cvlb_column_factorial_valuation_selected_power_product_bound + S ff_i_b5cvlb_column_factorial_valuation_selected_power_product = b) -> exists ff_p_b5cvlb_column_factorial_valuation_selected_power_product ff_r_b5cvlb_column_factorial_valuation_selected_power_product ff_s_b5cvlb_column_factorial_valuation_selected_power_product. ((((exists ff_h_b5cvlb_column_factorial_valuation_selected_power_product_factor. ff_h_b5cvlb_column_factorial_valuation_selected_power_product_factor + S (ff_p_b5cvlb_column_factorial_valuation_selected_power_product) = S ((S (ff_i_b5cvlb_column_factorial_valuation_selected_power_product)) * ff_c_b5cvlb_column_factorial_valuation_selected_power)) /\ exists ff_q_b5cvlb_column_factorial_valuation_selected_power_product_factor. ff_b_b5cvlb_column_factorial_valuation_selected_power = ff_q_b5cvlb_column_factorial_valuation_selected_power_product_factor * S ((S (ff_i_b5cvlb_column_factorial_valuation_selected_power_product)) * ff_c_b5cvlb_column_factorial_valuation_selected_power) + (ff_p_b5cvlb_column_factorial_valuation_selected_power_product))) /\ ((((exists ff_h_b5cvlb_column_factorial_valuation_selected_power_product_partial. ff_h_b5cvlb_column_factorial_valuation_selected_power_product_partial + S (ff_r_b5cvlb_column_factorial_valuation_selected_power_product) = S ((S (ff_i_b5cvlb_column_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_selected_power_product_partial. ff_u_b5cvlb_column_factorial_valuation_selected_power_product = ff_q_b5cvlb_column_factorial_valuation_selected_power_product_partial * S ((S (ff_i_b5cvlb_column_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product) + (ff_r_b5cvlb_column_factorial_valuation_selected_power_product))) /\ ((((exists ff_h_b5cvlb_column_factorial_valuation_selected_power_product_successor. ff_h_b5cvlb_column_factorial_valuation_selected_power_product_successor + S (ff_s_b5cvlb_column_factorial_valuation_selected_power_product) = S ((S (S ff_i_b5cvlb_column_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_selected_power_product_successor. ff_u_b5cvlb_column_factorial_valuation_selected_power_product = ff_q_b5cvlb_column_factorial_valuation_selected_power_product_successor * S ((S (S ff_i_b5cvlb_column_factorial_valuation_selected_power_product)) * ff_v_b5cvlb_column_factorial_valuation_selected_power_product) + (ff_s_b5cvlb_column_factorial_valuation_selected_power_product))) /\ ff_s_b5cvlb_column_factorial_valuation_selected_power_product = ff_r_b5cvlb_column_factorial_valuation_selected_power_product * ff_p_b5cvlb_column_factorial_valuation_selected_power_product)))))))) /\ (exists bpv_factor_b5cvlb_column_factorial_valuation_selected_divides. bfv_factorial_b5cvlb_column_factorial = bpv_result_b5cvlb_column_factorial_valuation_selected * bpv_factor_b5cvlb_column_factorial_valuation_selected_divides)))) /\ forall bpv_candidate_b5cvlb_column_factorial_valuation. (exists bpv_gap_b5cvlb_column_factorial_valuation_candidate_bound. bpv_gap_b5cvlb_column_factorial_valuation_candidate_bound + bpv_candidate_b5cvlb_column_factorial_valuation = bfv_factorial_b5cvlb_column_factorial) -> (exists bpv_result_b5cvlb_column_factorial_valuation_candidate. ((exists ff_b_b5cvlb_column_factorial_valuation_candidate_power ff_c_b5cvlb_column_factorial_valuation_candidate_power. ((forall ff_i_b5cvlb_column_factorial_valuation_candidate_power_repeat. (exists ff_lt_b5cvlb_column_factorial_valuation_candidate_power_repeat_bound. ff_lt_b5cvlb_column_factorial_valuation_candidate_power_repeat_bound + S ff_i_b5cvlb_column_factorial_valuation_candidate_power_repeat = bpv_candidate_b5cvlb_column_factorial_valuation) -> (((exists ff_h_b5cvlb_column_factorial_valuation_candidate_power_repeat_decoded. ff_h_b5cvlb_column_factorial_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvlb_column_factorial_valuation_candidate_power_repeat)) * ff_c_b5cvlb_column_factorial_valuation_candidate_power)) /\ exists ff_q_b5cvlb_column_factorial_valuation_candidate_power_repeat_decoded. ff_b_b5cvlb_column_factorial_valuation_candidate_power = ff_q_b5cvlb_column_factorial_valuation_candidate_power_repeat_decoded * S ((S (ff_i_b5cvlb_column_factorial_valuation_candidate_power_repeat)) * ff_c_b5cvlb_column_factorial_valuation_candidate_power) + (p)))) /\ (exists ff_u_b5cvlb_column_factorial_valuation_candidate_power_product ff_v_b5cvlb_column_factorial_valuation_candidate_power_product. ((((exists ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_start. ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_start. ff_u_b5cvlb_column_factorial_valuation_candidate_power_product = ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_start * S ((S (0)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_terminal. ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_terminal + S (bpv_result_b5cvlb_column_factorial_valuation_candidate) = S ((S (bpv_candidate_b5cvlb_column_factorial_valuation)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_terminal. ff_u_b5cvlb_column_factorial_valuation_candidate_power_product = ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvlb_column_factorial_valuation)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product) + (bpv_result_b5cvlb_column_factorial_valuation_candidate))) /\ forall ff_i_b5cvlb_column_factorial_valuation_candidate_power_product. (exists ff_lt_b5cvlb_column_factorial_valuation_candidate_power_product_bound. ff_lt_b5cvlb_column_factorial_valuation_candidate_power_product_bound + S ff_i_b5cvlb_column_factorial_valuation_candidate_power_product = bpv_candidate_b5cvlb_column_factorial_valuation) -> exists ff_p_b5cvlb_column_factorial_valuation_candidate_power_product ff_r_b5cvlb_column_factorial_valuation_candidate_power_product ff_s_b5cvlb_column_factorial_valuation_candidate_power_product. ((((exists ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_factor. ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_factor + S (ff_p_b5cvlb_column_factorial_valuation_candidate_power_product) = S ((S (ff_i_b5cvlb_column_factorial_valuation_candidate_power_product)) * ff_c_b5cvlb_column_factorial_valuation_candidate_power)) /\ exists ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_factor. ff_b_b5cvlb_column_factorial_valuation_candidate_power = ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_factor * S ((S (ff_i_b5cvlb_column_factorial_valuation_candidate_power_product)) * ff_c_b5cvlb_column_factorial_valuation_candidate_power) + (ff_p_b5cvlb_column_factorial_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_partial. ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_partial + S (ff_r_b5cvlb_column_factorial_valuation_candidate_power_product) = S ((S (ff_i_b5cvlb_column_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_partial. ff_u_b5cvlb_column_factorial_valuation_candidate_power_product = ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_partial * S ((S (ff_i_b5cvlb_column_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product) + (ff_r_b5cvlb_column_factorial_valuation_candidate_power_product))) /\ ((((exists ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_successor. ff_h_b5cvlb_column_factorial_valuation_candidate_power_product_successor + S (ff_s_b5cvlb_column_factorial_valuation_candidate_power_product) = S ((S (S ff_i_b5cvlb_column_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product)) /\ exists ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_successor. ff_u_b5cvlb_column_factorial_valuation_candidate_power_product = ff_q_b5cvlb_column_factorial_valuation_candidate_power_product_successor * S ((S (S ff_i_b5cvlb_column_factorial_valuation_candidate_power_product)) * ff_v_b5cvlb_column_factorial_valuation_candidate_power_product) + (ff_s_b5cvlb_column_factorial_valuation_candidate_power_product))) /\ ff_s_b5cvlb_column_factorial_valuation_candidate_power_product = ff_r_b5cvlb_column_factorial_valuation_candidate_power_product * ff_p_b5cvlb_column_factorial_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_b5cvlb_column_factorial_valuation_candidate_divides. bfv_factorial_b5cvlb_column_factorial = bpv_result_b5cvlb_column_factorial_valuation_candidate * bpv_factor_b5cvlb_column_factorial_valuation_candidate_divides))) -> (exists bpv_gap_b5cvlb_column_factorial_valuation_maximal. bpv_gap_b5cvlb_column_factorial_valuation_maximal + bpv_candidate_b5cvlb_column_factorial_valuation = b)))
  18. 0018specialize factorial_valuation_exists p
  19. 0019specialize factorial_valuation_exists n
  20. 0020exact factorial_valuation_exists
  21. 0021cases hcolumn
  22. 0022have hbalance : x = (x1 + x1) + e
  23. 0023specialize central_binom_factorial_valuation_balance p
  24. 0024specialize central_binom_factorial_valuation_balance n
  25. 0025specialize central_binom_factorial_valuation_balance c
  26. 0026specialize central_binom_factorial_valuation_balance e
  27. 0027specialize central_binom_factorial_valuation_balance x
  28. 0028specialize central_binom_factorial_valuation_balance x1
  29. 0029apply central_binom_factorial_valuation_balance
  30. 0030exact hp
  31. 0031exact hcentral
  32. 0032exact hvalue
  33. 0033exact htotal_witness
  34. 0034exact hcolumn_witness
  35. 0035have htotal_eq : x = A
  36. 0036specialize prime_factorial_valuation_eq_legendre_sum p
  37. 0037specialize prime_factorial_valuation_eq_legendre_sum (n + n)
  38. 0038specialize prime_factorial_valuation_eq_legendre_sum x
  39. 0039specialize prime_factorial_valuation_eq_legendre_sum A
  40. 0040apply prime_factorial_valuation_eq_legendre_sum
  41. 0041exact hp
  42. 0042exact htotal_witness
  43. 0043exact htotal_legendre
  44. 0044have hcolumn_eq : x1 = B
  45. 0045specialize prime_factorial_valuation_eq_legendre_sum p
  46. 0046specialize prime_factorial_valuation_eq_legendre_sum n
  47. 0047specialize prime_factorial_valuation_eq_legendre_sum x1
  48. 0048specialize prime_factorial_valuation_eq_legendre_sum B
  49. 0049apply prime_factorial_valuation_eq_legendre_sum
  50. 0050exact hp
  51. 0051exact hcolumn_witness
  52. 0052exact hcolumn_legendre
  53. 0053trans x
  54. 0054symm
  55. 0055exact htotal_eq
  56. 0056trans (x1 + x1) + e
  57. 0057exact hbalance
  58. 0058rewrite hcolumn_eq
  59. 0059rewrite hcolumn_eq
  60. 0060refl