primorial_interval_divides_choose_between

Exact expanded PA statement

forall a l n k j c z. k + j = n -> (((exists bcf_lt_gap_bpidcb_choose_out_of_range. bcf_lt_gap_bpidcb_choose_out_of_range + S (n) = k) /\ c = 0) \/ ((exists bcf_le_gap_bpidcb_choose_in_range. bcf_le_gap_bpidcb_choose_in_range + (k) = n) /\ (exists bcf_row_code_code_bpidcb_choose bcf_row_code_scale_bpidcb_choose bcf_row_scale_code_bpidcb_choose bcf_row_scale_scale_bpidcb_choose bcf_row_code_bpidcb_choose bcf_row_scale_bpidcb_choose. ((forall bcf_row_index_bpidcb_choose_table. (exists bcf_lt_gap_bpidcb_choose_table_row_bound. bcf_lt_gap_bpidcb_choose_table_row_bound + S (bcf_row_index_bpidcb_choose_table) = S (n)) -> exists bcf_row_code_bpidcb_choose_table bcf_row_scale_bpidcb_choose_table. ((((exists bcf_height_bpidcb_choose_table_decoded_row_code. bcf_height_bpidcb_choose_table_decoded_row_code + S (bcf_row_code_bpidcb_choose_table) = S ((S (bcf_row_index_bpidcb_choose_table)) * bcf_row_code_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_table_decoded_row_code. bcf_row_code_code_bpidcb_choose = bcf_quotient_bpidcb_choose_table_decoded_row_code * S ((S (bcf_row_index_bpidcb_choose_table)) * bcf_row_code_scale_bpidcb_choose) + (bcf_row_code_bpidcb_choose_table))) /\ ((((exists bcf_height_bpidcb_choose_table_decoded_row_scale. bcf_height_bpidcb_choose_table_decoded_row_scale + S (bcf_row_scale_bpidcb_choose_table) = S ((S (bcf_row_index_bpidcb_choose_table)) * bcf_row_scale_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_table_decoded_row_scale. bcf_row_scale_code_bpidcb_choose = bcf_quotient_bpidcb_choose_table_decoded_row_scale * S ((S (bcf_row_index_bpidcb_choose_table)) * bcf_row_scale_scale_bpidcb_choose) + (bcf_row_scale_bpidcb_choose_table))) /\ ((bcf_row_index_bpidcb_choose_table = 0 /\ (forall bcf_index_bpidcb_choose_table_zero_row. (exists bcf_lt_gap_bpidcb_choose_table_zero_row_bound. bcf_lt_gap_bpidcb_choose_table_zero_row_bound + S (bcf_index_bpidcb_choose_table_zero_row) = S (n)) -> exists bcf_value_bpidcb_choose_table_zero_row. ((((exists bcf_height_bpidcb_choose_table_zero_row_entry. bcf_height_bpidcb_choose_table_zero_row_entry + S (bcf_value_bpidcb_choose_table_zero_row) = S ((S (bcf_index_bpidcb_choose_table_zero_row)) * bcf_row_scale_bpidcb_choose_table)) /\ exists bcf_quotient_bpidcb_choose_table_zero_row_entry. bcf_row_code_bpidcb_choose_table = bcf_quotient_bpidcb_choose_table_zero_row_entry * S ((S (bcf_index_bpidcb_choose_table_zero_row)) * bcf_row_scale_bpidcb_choose_table) + (bcf_value_bpidcb_choose_table_zero_row))) /\ ((bcf_index_bpidcb_choose_table_zero_row = 0 /\ bcf_value_bpidcb_choose_table_zero_row = 1) \/ exists bcf_predecessor_bpidcb_choose_table_zero_row. bcf_index_bpidcb_choose_table_zero_row = S bcf_predecessor_bpidcb_choose_table_zero_row /\ bcf_value_bpidcb_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_bpidcb_choose_table bcf_previous_code_bpidcb_choose_table bcf_previous_scale_bpidcb_choose_table. bcf_row_index_bpidcb_choose_table = S bcf_predecessor_bpidcb_choose_table /\ ((((exists bcf_height_bpidcb_choose_table_decoded_previous_code. bcf_height_bpidcb_choose_table_decoded_previous_code + S (bcf_previous_code_bpidcb_choose_table) = S ((S (bcf_predecessor_bpidcb_choose_table)) * bcf_row_code_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_table_decoded_previous_code. bcf_row_code_code_bpidcb_choose = bcf_quotient_bpidcb_choose_table_decoded_previous_code * S ((S (bcf_predecessor_bpidcb_choose_table)) * bcf_row_code_scale_bpidcb_choose) + (bcf_previous_code_bpidcb_choose_table))) /\ ((((exists bcf_height_bpidcb_choose_table_decoded_previous_scale. bcf_height_bpidcb_choose_table_decoded_previous_scale + S (bcf_previous_scale_bpidcb_choose_table) = S ((S (bcf_predecessor_bpidcb_choose_table)) * bcf_row_scale_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_table_decoded_previous_scale. bcf_row_scale_code_bpidcb_choose = bcf_quotient_bpidcb_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_bpidcb_choose_table)) * bcf_row_scale_scale_bpidcb_choose) + (bcf_previous_scale_bpidcb_choose_table))) /\ (forall bcf_index_bpidcb_choose_table_row_step. (exists bcf_lt_gap_bpidcb_choose_table_row_step_bound. bcf_lt_gap_bpidcb_choose_table_row_step_bound + S (bcf_index_bpidcb_choose_table_row_step) = S (n)) -> exists bcf_value_bpidcb_choose_table_row_step. ((((exists bcf_height_bpidcb_choose_table_row_step_entry. bcf_height_bpidcb_choose_table_row_step_entry + S (bcf_value_bpidcb_choose_table_row_step) = S ((S (bcf_index_bpidcb_choose_table_row_step)) * bcf_row_scale_bpidcb_choose_table)) /\ exists bcf_quotient_bpidcb_choose_table_row_step_entry. bcf_row_code_bpidcb_choose_table = bcf_quotient_bpidcb_choose_table_row_step_entry * S ((S (bcf_index_bpidcb_choose_table_row_step)) * bcf_row_scale_bpidcb_choose_table) + (bcf_value_bpidcb_choose_table_row_step))) /\ ((bcf_index_bpidcb_choose_table_row_step = 0 /\ bcf_value_bpidcb_choose_table_row_step = 1) \/ exists bcf_predecessor_bpidcb_choose_table_row_step bcf_left_bpidcb_choose_table_row_step bcf_right_bpidcb_choose_table_row_step. bcf_index_bpidcb_choose_table_row_step = S bcf_predecessor_bpidcb_choose_table_row_step /\ ((((exists bcf_height_bpidcb_choose_table_row_step_previous_left. bcf_height_bpidcb_choose_table_row_step_previous_left + S (bcf_left_bpidcb_choose_table_row_step) = S ((S (bcf_predecessor_bpidcb_choose_table_row_step)) * bcf_previous_scale_bpidcb_choose_table)) /\ exists bcf_quotient_bpidcb_choose_table_row_step_previous_left. bcf_previous_code_bpidcb_choose_table = bcf_quotient_bpidcb_choose_table_row_step_previous_left * S ((S (bcf_predecessor_bpidcb_choose_table_row_step)) * bcf_previous_scale_bpidcb_choose_table) + (bcf_left_bpidcb_choose_table_row_step))) /\ ((((exists bcf_height_bpidcb_choose_table_row_step_previous_right. bcf_height_bpidcb_choose_table_row_step_previous_right + S (bcf_right_bpidcb_choose_table_row_step) = S ((S (S (bcf_predecessor_bpidcb_choose_table_row_step))) * bcf_previous_scale_bpidcb_choose_table)) /\ exists bcf_quotient_bpidcb_choose_table_row_step_previous_right. bcf_previous_code_bpidcb_choose_table = bcf_quotient_bpidcb_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_bpidcb_choose_table_row_step))) * bcf_previous_scale_bpidcb_choose_table) + (bcf_right_bpidcb_choose_table_row_step))) /\ bcf_value_bpidcb_choose_table_row_step = bcf_left_bpidcb_choose_table_row_step + bcf_right_bpidcb_choose_table_row_step))))))))))) /\ ((((exists bcf_height_bpidcb_choose_decoded_row_code. bcf_height_bpidcb_choose_decoded_row_code + S (bcf_row_code_bpidcb_choose) = S ((S (n)) * bcf_row_code_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_decoded_row_code. bcf_row_code_code_bpidcb_choose = bcf_quotient_bpidcb_choose_decoded_row_code * S ((S (n)) * bcf_row_code_scale_bpidcb_choose) + (bcf_row_code_bpidcb_choose))) /\ ((((exists bcf_height_bpidcb_choose_decoded_row_scale. bcf_height_bpidcb_choose_decoded_row_scale + S (bcf_row_scale_bpidcb_choose) = S ((S (n)) * bcf_row_scale_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_decoded_row_scale. bcf_row_scale_code_bpidcb_choose = bcf_quotient_bpidcb_choose_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_bpidcb_choose) + (bcf_row_scale_bpidcb_choose))) /\ (((exists bcf_height_bpidcb_choose_decoded_value. bcf_height_bpidcb_choose_decoded_value + S (c) = S ((S (k)) * bcf_row_scale_bpidcb_choose)) /\ exists bcf_quotient_bpidcb_choose_decoded_value. bcf_row_code_bpidcb_choose = bcf_quotient_bpidcb_choose_decoded_value * S ((S (k)) * bcf_row_scale_bpidcb_choose) + (c))))))))) -> (exists bpr_code_bpidcb_interval bpr_scale_bpidcb_interval. ((forall bpr_index_bpidcb_interval_mask. (exists bpr_gap_bpidcb_interval_mask_bound. bpr_gap_bpidcb_interval_mask_bound + S (bpr_index_bpidcb_interval_mask) = l) -> exists bpr_value_bpidcb_interval_mask. ((((exists bpr_height_bpidcb_interval_mask_decoded. bpr_height_bpidcb_interval_mask_decoded + S (bpr_value_bpidcb_interval_mask) = S ((S (bpr_index_bpidcb_interval_mask)) * bpr_scale_bpidcb_interval)) /\ exists bpr_quotient_bpidcb_interval_mask_decoded. bpr_code_bpidcb_interval = bpr_quotient_bpidcb_interval_mask_decoded * S ((S (bpr_index_bpidcb_interval_mask)) * bpr_scale_bpidcb_interval) + (bpr_value_bpidcb_interval_mask))) /\ (((((~(S (a + bpr_index_bpidcb_interval_mask) = 1) /\ forall bpr_left_bpidcb_interval_mask_choice_prime bpr_right_bpidcb_interval_mask_choice_prime. S (a + bpr_index_bpidcb_interval_mask) = bpr_left_bpidcb_interval_mask_choice_prime * bpr_right_bpidcb_interval_mask_choice_prime -> bpr_left_bpidcb_interval_mask_choice_prime = 1 \/ bpr_right_bpidcb_interval_mask_choice_prime = 1)) /\ bpr_value_bpidcb_interval_mask = S (a + bpr_index_bpidcb_interval_mask)) \/ (~((~(S (a + bpr_index_bpidcb_interval_mask) = 1) /\ forall bpr_left_bpidcb_interval_mask_choice_prime bpr_right_bpidcb_interval_mask_choice_prime. S (a + bpr_index_bpidcb_interval_mask) = bpr_left_bpidcb_interval_mask_choice_prime * bpr_right_bpidcb_interval_mask_choice_prime -> bpr_left_bpidcb_interval_mask_choice_prime = 1 \/ bpr_right_bpidcb_interval_mask_choice_prime = 1)) /\ bpr_value_bpidcb_interval_mask = 1))))) /\ (exists ff_u_bpidcb_interval_product ff_v_bpidcb_interval_product. ((((exists ff_h_bpidcb_interval_product_start. ff_h_bpidcb_interval_product_start + S (1) = S ((S (0)) * ff_v_bpidcb_interval_product)) /\ exists ff_q_bpidcb_interval_product_start. ff_u_bpidcb_interval_product = ff_q_bpidcb_interval_product_start * S ((S (0)) * ff_v_bpidcb_interval_product) + (1))) /\ ((((exists ff_h_bpidcb_interval_product_terminal. ff_h_bpidcb_interval_product_terminal + S (z) = S ((S (l)) * ff_v_bpidcb_interval_product)) /\ exists ff_q_bpidcb_interval_product_terminal. ff_u_bpidcb_interval_product = ff_q_bpidcb_interval_product_terminal * S ((S (l)) * ff_v_bpidcb_interval_product) + (z))) /\ forall ff_i_bpidcb_interval_product. (exists ff_lt_bpidcb_interval_product_bound. ff_lt_bpidcb_interval_product_bound + S ff_i_bpidcb_interval_product = l) -> exists ff_p_bpidcb_interval_product ff_r_bpidcb_interval_product ff_s_bpidcb_interval_product. ((((exists ff_h_bpidcb_interval_product_factor. ff_h_bpidcb_interval_product_factor + S (ff_p_bpidcb_interval_product) = S ((S (ff_i_bpidcb_interval_product)) * bpr_scale_bpidcb_interval)) /\ exists ff_q_bpidcb_interval_product_factor. bpr_code_bpidcb_interval = ff_q_bpidcb_interval_product_factor * S ((S (ff_i_bpidcb_interval_product)) * bpr_scale_bpidcb_interval) + (ff_p_bpidcb_interval_product))) /\ ((((exists ff_h_bpidcb_interval_product_partial. ff_h_bpidcb_interval_product_partial + S (ff_r_bpidcb_interval_product) = S ((S (ff_i_bpidcb_interval_product)) * ff_v_bpidcb_interval_product)) /\ exists ff_q_bpidcb_interval_product_partial. ff_u_bpidcb_interval_product = ff_q_bpidcb_interval_product_partial * S ((S (ff_i_bpidcb_interval_product)) * ff_v_bpidcb_interval_product) + (ff_r_bpidcb_interval_product))) /\ ((((exists ff_h_bpidcb_interval_product_successor. ff_h_bpidcb_interval_product_successor + S (ff_s_bpidcb_interval_product) = S ((S (S ff_i_bpidcb_interval_product)) * ff_v_bpidcb_interval_product)) /\ exists ff_q_bpidcb_interval_product_successor. ff_u_bpidcb_interval_product = ff_q_bpidcb_interval_product_successor * S ((S (S ff_i_bpidcb_interval_product)) * ff_v_bpidcb_interval_product) + (ff_s_bpidcb_interval_product))) /\ ff_s_bpidcb_interval_product = ff_r_bpidcb_interval_product * ff_p_bpidcb_interval_product)))))))) -> (forall bpr_index_bpidcb_bounds. (exists bpr_gap_bpidcb_bounds_index. bpr_gap_bpidcb_bounds_index + S (bpr_index_bpidcb_bounds) = l) -> ((exists bpr_gap_bpidcb_bounds_left. bpr_gap_bpidcb_bounds_left + S (k) = S (a + bpr_index_bpidcb_bounds)) /\ ((exists bpr_gap_bpidcb_bounds_right. bpr_gap_bpidcb_bounds_right + S (j) = S (a + bpr_index_bpidcb_bounds)) /\ (exists bpr_le_gap_bpidcb_bounds_upper. bpr_le_gap_bpidcb_bounds_upper + (S (a + bpr_index_bpidcb_bounds)) = (n))))) -> (exists bpr_quotient_bpidcb_result. c = (z) * bpr_quotient_bpidcb_result)