bertrand_window_prime_divides_central_binom

Exact expanded first-order arithmetic statement

forall n p C. ((~(p = 1) /\ forall frm_prime_left_bpc_prime frm_prime_right_bpc_prime. p = frm_prime_left_bpc_prime * frm_prime_right_bpc_prime -> frm_prime_left_bpc_prime = 1 \/ frm_prime_right_bpc_prime = 1)) -> (exists bcf_lt_gap_bpc_lower. bcf_lt_gap_bpc_lower + S (n) = p) -> (exists bcf_lt_gap_bpc_upper. bcf_lt_gap_bpc_upper + S (p) = n + n) -> (((exists bcf_lt_gap_bpc_central_out_of_range. bcf_lt_gap_bpc_central_out_of_range + S (n + n) = n) /\ C = 0) \/ ((exists bcf_le_gap_bpc_central_in_range. bcf_le_gap_bpc_central_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bpc_central bcf_row_code_scale_bpc_central bcf_row_scale_code_bpc_central bcf_row_scale_scale_bpc_central bcf_row_code_bpc_central bcf_row_scale_bpc_central. ((forall bcf_row_index_bpc_central_table. (exists bcf_lt_gap_bpc_central_table_row_bound. bcf_lt_gap_bpc_central_table_row_bound + S (bcf_row_index_bpc_central_table) = S (n + n)) -> exists bcf_row_code_bpc_central_table bcf_row_scale_bpc_central_table. ((((exists bcf_height_bpc_central_table_decoded_row_code. bcf_height_bpc_central_table_decoded_row_code + S (bcf_row_code_bpc_central_table) = S ((S (bcf_row_index_bpc_central_table)) * bcf_row_code_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_table_decoded_row_code. bcf_row_code_code_bpc_central = bcf_quotient_bpc_central_table_decoded_row_code * S ((S (bcf_row_index_bpc_central_table)) * bcf_row_code_scale_bpc_central) + (bcf_row_code_bpc_central_table))) /\ ((((exists bcf_height_bpc_central_table_decoded_row_scale. bcf_height_bpc_central_table_decoded_row_scale + S (bcf_row_scale_bpc_central_table) = S ((S (bcf_row_index_bpc_central_table)) * bcf_row_scale_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_table_decoded_row_scale. bcf_row_scale_code_bpc_central = bcf_quotient_bpc_central_table_decoded_row_scale * S ((S (bcf_row_index_bpc_central_table)) * bcf_row_scale_scale_bpc_central) + (bcf_row_scale_bpc_central_table))) /\ ((bcf_row_index_bpc_central_table = 0 /\ (forall bcf_index_bpc_central_table_zero_row. (exists bcf_lt_gap_bpc_central_table_zero_row_bound. bcf_lt_gap_bpc_central_table_zero_row_bound + S (bcf_index_bpc_central_table_zero_row) = S (n + n)) -> exists bcf_value_bpc_central_table_zero_row. ((((exists bcf_height_bpc_central_table_zero_row_entry. bcf_height_bpc_central_table_zero_row_entry + S (bcf_value_bpc_central_table_zero_row) = S ((S (bcf_index_bpc_central_table_zero_row)) * bcf_row_scale_bpc_central_table)) /\ exists bcf_quotient_bpc_central_table_zero_row_entry. bcf_row_code_bpc_central_table = bcf_quotient_bpc_central_table_zero_row_entry * S ((S (bcf_index_bpc_central_table_zero_row)) * bcf_row_scale_bpc_central_table) + (bcf_value_bpc_central_table_zero_row))) /\ ((bcf_index_bpc_central_table_zero_row = 0 /\ bcf_value_bpc_central_table_zero_row = 1) \/ exists bcf_predecessor_bpc_central_table_zero_row. bcf_index_bpc_central_table_zero_row = S bcf_predecessor_bpc_central_table_zero_row /\ bcf_value_bpc_central_table_zero_row = 0)))) \/ exists bcf_predecessor_bpc_central_table bcf_previous_code_bpc_central_table bcf_previous_scale_bpc_central_table. bcf_row_index_bpc_central_table = S bcf_predecessor_bpc_central_table /\ ((((exists bcf_height_bpc_central_table_decoded_previous_code. bcf_height_bpc_central_table_decoded_previous_code + S (bcf_previous_code_bpc_central_table) = S ((S (bcf_predecessor_bpc_central_table)) * bcf_row_code_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_table_decoded_previous_code. bcf_row_code_code_bpc_central = bcf_quotient_bpc_central_table_decoded_previous_code * S ((S (bcf_predecessor_bpc_central_table)) * bcf_row_code_scale_bpc_central) + (bcf_previous_code_bpc_central_table))) /\ ((((exists bcf_height_bpc_central_table_decoded_previous_scale. bcf_height_bpc_central_table_decoded_previous_scale + S (bcf_previous_scale_bpc_central_table) = S ((S (bcf_predecessor_bpc_central_table)) * bcf_row_scale_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_table_decoded_previous_scale. bcf_row_scale_code_bpc_central = bcf_quotient_bpc_central_table_decoded_previous_scale * S ((S (bcf_predecessor_bpc_central_table)) * bcf_row_scale_scale_bpc_central) + (bcf_previous_scale_bpc_central_table))) /\ (forall bcf_index_bpc_central_table_row_step. (exists bcf_lt_gap_bpc_central_table_row_step_bound. bcf_lt_gap_bpc_central_table_row_step_bound + S (bcf_index_bpc_central_table_row_step) = S (n + n)) -> exists bcf_value_bpc_central_table_row_step. ((((exists bcf_height_bpc_central_table_row_step_entry. bcf_height_bpc_central_table_row_step_entry + S (bcf_value_bpc_central_table_row_step) = S ((S (bcf_index_bpc_central_table_row_step)) * bcf_row_scale_bpc_central_table)) /\ exists bcf_quotient_bpc_central_table_row_step_entry. bcf_row_code_bpc_central_table = bcf_quotient_bpc_central_table_row_step_entry * S ((S (bcf_index_bpc_central_table_row_step)) * bcf_row_scale_bpc_central_table) + (bcf_value_bpc_central_table_row_step))) /\ ((bcf_index_bpc_central_table_row_step = 0 /\ bcf_value_bpc_central_table_row_step = 1) \/ exists bcf_predecessor_bpc_central_table_row_step bcf_left_bpc_central_table_row_step bcf_right_bpc_central_table_row_step. bcf_index_bpc_central_table_row_step = S bcf_predecessor_bpc_central_table_row_step /\ ((((exists bcf_height_bpc_central_table_row_step_previous_left. bcf_height_bpc_central_table_row_step_previous_left + S (bcf_left_bpc_central_table_row_step) = S ((S (bcf_predecessor_bpc_central_table_row_step)) * bcf_previous_scale_bpc_central_table)) /\ exists bcf_quotient_bpc_central_table_row_step_previous_left. bcf_previous_code_bpc_central_table = bcf_quotient_bpc_central_table_row_step_previous_left * S ((S (bcf_predecessor_bpc_central_table_row_step)) * bcf_previous_scale_bpc_central_table) + (bcf_left_bpc_central_table_row_step))) /\ ((((exists bcf_height_bpc_central_table_row_step_previous_right. bcf_height_bpc_central_table_row_step_previous_right + S (bcf_right_bpc_central_table_row_step) = S ((S (S (bcf_predecessor_bpc_central_table_row_step))) * bcf_previous_scale_bpc_central_table)) /\ exists bcf_quotient_bpc_central_table_row_step_previous_right. bcf_previous_code_bpc_central_table = bcf_quotient_bpc_central_table_row_step_previous_right * S ((S (S (bcf_predecessor_bpc_central_table_row_step))) * bcf_previous_scale_bpc_central_table) + (bcf_right_bpc_central_table_row_step))) /\ bcf_value_bpc_central_table_row_step = bcf_left_bpc_central_table_row_step + bcf_right_bpc_central_table_row_step))))))))))) /\ ((((exists bcf_height_bpc_central_decoded_row_code. bcf_height_bpc_central_decoded_row_code + S (bcf_row_code_bpc_central) = S ((S (n + n)) * bcf_row_code_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_decoded_row_code. bcf_row_code_code_bpc_central = bcf_quotient_bpc_central_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bpc_central) + (bcf_row_code_bpc_central))) /\ ((((exists bcf_height_bpc_central_decoded_row_scale. bcf_height_bpc_central_decoded_row_scale + S (bcf_row_scale_bpc_central) = S ((S (n + n)) * bcf_row_scale_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_decoded_row_scale. bcf_row_scale_code_bpc_central = bcf_quotient_bpc_central_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bpc_central) + (bcf_row_scale_bpc_central))) /\ (((exists bcf_height_bpc_central_decoded_value. bcf_height_bpc_central_decoded_value + S (C) = S ((S (n)) * bcf_row_scale_bpc_central)) /\ exists bcf_quotient_bpc_central_decoded_value. bcf_row_code_bpc_central = bcf_quotient_bpc_central_decoded_value * S ((S (n)) * bcf_row_scale_bpc_central) + (C))))))))) -> (exists bcf_quotient_bpc_bpc_central_factor. C = p * bcf_quotient_bpc_bpc_central_factor)