lucas_pascal_congruence_step

Exact expanded first-order arithmetic statement

forall p n k A B C X Y. (((exists bcf_lt_gap_lucas_convolution_pascal_left_out_of_range. bcf_lt_gap_lucas_convolution_pascal_left_out_of_range + S (n) = k) /\ A = 0) \/ ((exists bcf_le_gap_lucas_convolution_pascal_left_in_range. bcf_le_gap_lucas_convolution_pascal_left_in_range + (k) = n) /\ (exists bcf_row_code_code_lucas_convolution_pascal_left bcf_row_code_scale_lucas_convolution_pascal_left bcf_row_scale_code_lucas_convolution_pascal_left bcf_row_scale_scale_lucas_convolution_pascal_left bcf_row_code_lucas_convolution_pascal_left bcf_row_scale_lucas_convolution_pascal_left. ((forall bcf_row_index_lucas_convolution_pascal_left_table. (exists bcf_lt_gap_lucas_convolution_pascal_left_table_row_bound. bcf_lt_gap_lucas_convolution_pascal_left_table_row_bound + S (bcf_row_index_lucas_convolution_pascal_left_table) = S (n)) -> exists bcf_row_code_lucas_convolution_pascal_left_table bcf_row_scale_lucas_convolution_pascal_left_table. ((((exists bcf_height_lucas_convolution_pascal_left_table_decoded_row_code. bcf_height_lucas_convolution_pascal_left_table_decoded_row_code + S (bcf_row_code_lucas_convolution_pascal_left_table) = S ((S (bcf_row_index_lucas_convolution_pascal_left_table)) * bcf_row_code_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_decoded_row_code. bcf_row_code_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_table_decoded_row_code * S ((S (bcf_row_index_lucas_convolution_pascal_left_table)) * bcf_row_code_scale_lucas_convolution_pascal_left) + (bcf_row_code_lucas_convolution_pascal_left_table))) /\ ((((exists bcf_height_lucas_convolution_pascal_left_table_decoded_row_scale. bcf_height_lucas_convolution_pascal_left_table_decoded_row_scale + S (bcf_row_scale_lucas_convolution_pascal_left_table) = S ((S (bcf_row_index_lucas_convolution_pascal_left_table)) * bcf_row_scale_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_decoded_row_scale. bcf_row_scale_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_table_decoded_row_scale * S ((S (bcf_row_index_lucas_convolution_pascal_left_table)) * bcf_row_scale_scale_lucas_convolution_pascal_left) + (bcf_row_scale_lucas_convolution_pascal_left_table))) /\ ((bcf_row_index_lucas_convolution_pascal_left_table = 0 /\ (forall bcf_index_lucas_convolution_pascal_left_table_zero_row. (exists bcf_lt_gap_lucas_convolution_pascal_left_table_zero_row_bound. bcf_lt_gap_lucas_convolution_pascal_left_table_zero_row_bound + S (bcf_index_lucas_convolution_pascal_left_table_zero_row) = S (n)) -> exists bcf_value_lucas_convolution_pascal_left_table_zero_row. ((((exists bcf_height_lucas_convolution_pascal_left_table_zero_row_entry. bcf_height_lucas_convolution_pascal_left_table_zero_row_entry + S (bcf_value_lucas_convolution_pascal_left_table_zero_row) = S ((S (bcf_index_lucas_convolution_pascal_left_table_zero_row)) * bcf_row_scale_lucas_convolution_pascal_left_table)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_zero_row_entry. bcf_row_code_lucas_convolution_pascal_left_table = bcf_quotient_lucas_convolution_pascal_left_table_zero_row_entry * S ((S (bcf_index_lucas_convolution_pascal_left_table_zero_row)) * bcf_row_scale_lucas_convolution_pascal_left_table) + (bcf_value_lucas_convolution_pascal_left_table_zero_row))) /\ ((bcf_index_lucas_convolution_pascal_left_table_zero_row = 0 /\ bcf_value_lucas_convolution_pascal_left_table_zero_row = 1) \/ exists bcf_predecessor_lucas_convolution_pascal_left_table_zero_row. bcf_index_lucas_convolution_pascal_left_table_zero_row = S bcf_predecessor_lucas_convolution_pascal_left_table_zero_row /\ bcf_value_lucas_convolution_pascal_left_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_convolution_pascal_left_table bcf_previous_code_lucas_convolution_pascal_left_table bcf_previous_scale_lucas_convolution_pascal_left_table. bcf_row_index_lucas_convolution_pascal_left_table = S bcf_predecessor_lucas_convolution_pascal_left_table /\ ((((exists bcf_height_lucas_convolution_pascal_left_table_decoded_previous_code. bcf_height_lucas_convolution_pascal_left_table_decoded_previous_code + S (bcf_previous_code_lucas_convolution_pascal_left_table) = S ((S (bcf_predecessor_lucas_convolution_pascal_left_table)) * bcf_row_code_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_decoded_previous_code. bcf_row_code_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_convolution_pascal_left_table)) * bcf_row_code_scale_lucas_convolution_pascal_left) + (bcf_previous_code_lucas_convolution_pascal_left_table))) /\ ((((exists bcf_height_lucas_convolution_pascal_left_table_decoded_previous_scale. bcf_height_lucas_convolution_pascal_left_table_decoded_previous_scale + S (bcf_previous_scale_lucas_convolution_pascal_left_table) = S ((S (bcf_predecessor_lucas_convolution_pascal_left_table)) * bcf_row_scale_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_decoded_previous_scale. bcf_row_scale_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_convolution_pascal_left_table)) * bcf_row_scale_scale_lucas_convolution_pascal_left) + (bcf_previous_scale_lucas_convolution_pascal_left_table))) /\ (forall bcf_index_lucas_convolution_pascal_left_table_row_step. (exists bcf_lt_gap_lucas_convolution_pascal_left_table_row_step_bound. bcf_lt_gap_lucas_convolution_pascal_left_table_row_step_bound + S (bcf_index_lucas_convolution_pascal_left_table_row_step) = S (n)) -> exists bcf_value_lucas_convolution_pascal_left_table_row_step. ((((exists bcf_height_lucas_convolution_pascal_left_table_row_step_entry. bcf_height_lucas_convolution_pascal_left_table_row_step_entry + S (bcf_value_lucas_convolution_pascal_left_table_row_step) = S ((S (bcf_index_lucas_convolution_pascal_left_table_row_step)) * bcf_row_scale_lucas_convolution_pascal_left_table)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_row_step_entry. bcf_row_code_lucas_convolution_pascal_left_table = bcf_quotient_lucas_convolution_pascal_left_table_row_step_entry * S ((S (bcf_index_lucas_convolution_pascal_left_table_row_step)) * bcf_row_scale_lucas_convolution_pascal_left_table) + (bcf_value_lucas_convolution_pascal_left_table_row_step))) /\ ((bcf_index_lucas_convolution_pascal_left_table_row_step = 0 /\ bcf_value_lucas_convolution_pascal_left_table_row_step = 1) \/ exists bcf_predecessor_lucas_convolution_pascal_left_table_row_step bcf_left_lucas_convolution_pascal_left_table_row_step bcf_right_lucas_convolution_pascal_left_table_row_step. bcf_index_lucas_convolution_pascal_left_table_row_step = S bcf_predecessor_lucas_convolution_pascal_left_table_row_step /\ ((((exists bcf_height_lucas_convolution_pascal_left_table_row_step_previous_left. bcf_height_lucas_convolution_pascal_left_table_row_step_previous_left + S (bcf_left_lucas_convolution_pascal_left_table_row_step) = S ((S (bcf_predecessor_lucas_convolution_pascal_left_table_row_step)) * bcf_previous_scale_lucas_convolution_pascal_left_table)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_row_step_previous_left. bcf_previous_code_lucas_convolution_pascal_left_table = bcf_quotient_lucas_convolution_pascal_left_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_convolution_pascal_left_table_row_step)) * bcf_previous_scale_lucas_convolution_pascal_left_table) + (bcf_left_lucas_convolution_pascal_left_table_row_step))) /\ ((((exists bcf_height_lucas_convolution_pascal_left_table_row_step_previous_right. bcf_height_lucas_convolution_pascal_left_table_row_step_previous_right + S (bcf_right_lucas_convolution_pascal_left_table_row_step) = S ((S (S (bcf_predecessor_lucas_convolution_pascal_left_table_row_step))) * bcf_previous_scale_lucas_convolution_pascal_left_table)) /\ exists bcf_quotient_lucas_convolution_pascal_left_table_row_step_previous_right. bcf_previous_code_lucas_convolution_pascal_left_table = bcf_quotient_lucas_convolution_pascal_left_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_convolution_pascal_left_table_row_step))) * bcf_previous_scale_lucas_convolution_pascal_left_table) + (bcf_right_lucas_convolution_pascal_left_table_row_step))) /\ bcf_value_lucas_convolution_pascal_left_table_row_step = bcf_left_lucas_convolution_pascal_left_table_row_step + bcf_right_lucas_convolution_pascal_left_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_convolution_pascal_left_decoded_row_code. bcf_height_lucas_convolution_pascal_left_decoded_row_code + S (bcf_row_code_lucas_convolution_pascal_left) = S ((S (n)) * bcf_row_code_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_decoded_row_code. bcf_row_code_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_decoded_row_code * S ((S (n)) * bcf_row_code_scale_lucas_convolution_pascal_left) + (bcf_row_code_lucas_convolution_pascal_left))) /\ ((((exists bcf_height_lucas_convolution_pascal_left_decoded_row_scale. bcf_height_lucas_convolution_pascal_left_decoded_row_scale + S (bcf_row_scale_lucas_convolution_pascal_left) = S ((S (n)) * bcf_row_scale_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_decoded_row_scale. bcf_row_scale_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_lucas_convolution_pascal_left) + (bcf_row_scale_lucas_convolution_pascal_left))) /\ (((exists bcf_height_lucas_convolution_pascal_left_decoded_value. bcf_height_lucas_convolution_pascal_left_decoded_value + S (A) = S ((S (k)) * bcf_row_scale_lucas_convolution_pascal_left)) /\ exists bcf_quotient_lucas_convolution_pascal_left_decoded_value. bcf_row_code_lucas_convolution_pascal_left = bcf_quotient_lucas_convolution_pascal_left_decoded_value * S ((S (k)) * bcf_row_scale_lucas_convolution_pascal_left) + (A))))))))) -> (((exists bcf_lt_gap_lucas_convolution_pascal_right_out_of_range. bcf_lt_gap_lucas_convolution_pascal_right_out_of_range + S (n) = S k) /\ B = 0) \/ ((exists bcf_le_gap_lucas_convolution_pascal_right_in_range. bcf_le_gap_lucas_convolution_pascal_right_in_range + (S k) = n) /\ (exists bcf_row_code_code_lucas_convolution_pascal_right bcf_row_code_scale_lucas_convolution_pascal_right bcf_row_scale_code_lucas_convolution_pascal_right bcf_row_scale_scale_lucas_convolution_pascal_right bcf_row_code_lucas_convolution_pascal_right bcf_row_scale_lucas_convolution_pascal_right. ((forall bcf_row_index_lucas_convolution_pascal_right_table. (exists bcf_lt_gap_lucas_convolution_pascal_right_table_row_bound. bcf_lt_gap_lucas_convolution_pascal_right_table_row_bound + S (bcf_row_index_lucas_convolution_pascal_right_table) = S (n)) -> exists bcf_row_code_lucas_convolution_pascal_right_table bcf_row_scale_lucas_convolution_pascal_right_table. ((((exists bcf_height_lucas_convolution_pascal_right_table_decoded_row_code. bcf_height_lucas_convolution_pascal_right_table_decoded_row_code + S (bcf_row_code_lucas_convolution_pascal_right_table) = S ((S (bcf_row_index_lucas_convolution_pascal_right_table)) * bcf_row_code_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_decoded_row_code. bcf_row_code_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_table_decoded_row_code * S ((S (bcf_row_index_lucas_convolution_pascal_right_table)) * bcf_row_code_scale_lucas_convolution_pascal_right) + (bcf_row_code_lucas_convolution_pascal_right_table))) /\ ((((exists bcf_height_lucas_convolution_pascal_right_table_decoded_row_scale. bcf_height_lucas_convolution_pascal_right_table_decoded_row_scale + S (bcf_row_scale_lucas_convolution_pascal_right_table) = S ((S (bcf_row_index_lucas_convolution_pascal_right_table)) * bcf_row_scale_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_decoded_row_scale. bcf_row_scale_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_table_decoded_row_scale * S ((S (bcf_row_index_lucas_convolution_pascal_right_table)) * bcf_row_scale_scale_lucas_convolution_pascal_right) + (bcf_row_scale_lucas_convolution_pascal_right_table))) /\ ((bcf_row_index_lucas_convolution_pascal_right_table = 0 /\ (forall bcf_index_lucas_convolution_pascal_right_table_zero_row. (exists bcf_lt_gap_lucas_convolution_pascal_right_table_zero_row_bound. bcf_lt_gap_lucas_convolution_pascal_right_table_zero_row_bound + S (bcf_index_lucas_convolution_pascal_right_table_zero_row) = S (n)) -> exists bcf_value_lucas_convolution_pascal_right_table_zero_row. ((((exists bcf_height_lucas_convolution_pascal_right_table_zero_row_entry. bcf_height_lucas_convolution_pascal_right_table_zero_row_entry + S (bcf_value_lucas_convolution_pascal_right_table_zero_row) = S ((S (bcf_index_lucas_convolution_pascal_right_table_zero_row)) * bcf_row_scale_lucas_convolution_pascal_right_table)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_zero_row_entry. bcf_row_code_lucas_convolution_pascal_right_table = bcf_quotient_lucas_convolution_pascal_right_table_zero_row_entry * S ((S (bcf_index_lucas_convolution_pascal_right_table_zero_row)) * bcf_row_scale_lucas_convolution_pascal_right_table) + (bcf_value_lucas_convolution_pascal_right_table_zero_row))) /\ ((bcf_index_lucas_convolution_pascal_right_table_zero_row = 0 /\ bcf_value_lucas_convolution_pascal_right_table_zero_row = 1) \/ exists bcf_predecessor_lucas_convolution_pascal_right_table_zero_row. bcf_index_lucas_convolution_pascal_right_table_zero_row = S bcf_predecessor_lucas_convolution_pascal_right_table_zero_row /\ bcf_value_lucas_convolution_pascal_right_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_convolution_pascal_right_table bcf_previous_code_lucas_convolution_pascal_right_table bcf_previous_scale_lucas_convolution_pascal_right_table. bcf_row_index_lucas_convolution_pascal_right_table = S bcf_predecessor_lucas_convolution_pascal_right_table /\ ((((exists bcf_height_lucas_convolution_pascal_right_table_decoded_previous_code. bcf_height_lucas_convolution_pascal_right_table_decoded_previous_code + S (bcf_previous_code_lucas_convolution_pascal_right_table) = S ((S (bcf_predecessor_lucas_convolution_pascal_right_table)) * bcf_row_code_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_decoded_previous_code. bcf_row_code_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_convolution_pascal_right_table)) * bcf_row_code_scale_lucas_convolution_pascal_right) + (bcf_previous_code_lucas_convolution_pascal_right_table))) /\ ((((exists bcf_height_lucas_convolution_pascal_right_table_decoded_previous_scale. bcf_height_lucas_convolution_pascal_right_table_decoded_previous_scale + S (bcf_previous_scale_lucas_convolution_pascal_right_table) = S ((S (bcf_predecessor_lucas_convolution_pascal_right_table)) * bcf_row_scale_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_decoded_previous_scale. bcf_row_scale_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_convolution_pascal_right_table)) * bcf_row_scale_scale_lucas_convolution_pascal_right) + (bcf_previous_scale_lucas_convolution_pascal_right_table))) /\ (forall bcf_index_lucas_convolution_pascal_right_table_row_step. (exists bcf_lt_gap_lucas_convolution_pascal_right_table_row_step_bound. bcf_lt_gap_lucas_convolution_pascal_right_table_row_step_bound + S (bcf_index_lucas_convolution_pascal_right_table_row_step) = S (n)) -> exists bcf_value_lucas_convolution_pascal_right_table_row_step. ((((exists bcf_height_lucas_convolution_pascal_right_table_row_step_entry. bcf_height_lucas_convolution_pascal_right_table_row_step_entry + S (bcf_value_lucas_convolution_pascal_right_table_row_step) = S ((S (bcf_index_lucas_convolution_pascal_right_table_row_step)) * bcf_row_scale_lucas_convolution_pascal_right_table)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_row_step_entry. bcf_row_code_lucas_convolution_pascal_right_table = bcf_quotient_lucas_convolution_pascal_right_table_row_step_entry * S ((S (bcf_index_lucas_convolution_pascal_right_table_row_step)) * bcf_row_scale_lucas_convolution_pascal_right_table) + (bcf_value_lucas_convolution_pascal_right_table_row_step))) /\ ((bcf_index_lucas_convolution_pascal_right_table_row_step = 0 /\ bcf_value_lucas_convolution_pascal_right_table_row_step = 1) \/ exists bcf_predecessor_lucas_convolution_pascal_right_table_row_step bcf_left_lucas_convolution_pascal_right_table_row_step bcf_right_lucas_convolution_pascal_right_table_row_step. bcf_index_lucas_convolution_pascal_right_table_row_step = S bcf_predecessor_lucas_convolution_pascal_right_table_row_step /\ ((((exists bcf_height_lucas_convolution_pascal_right_table_row_step_previous_left. bcf_height_lucas_convolution_pascal_right_table_row_step_previous_left + S (bcf_left_lucas_convolution_pascal_right_table_row_step) = S ((S (bcf_predecessor_lucas_convolution_pascal_right_table_row_step)) * bcf_previous_scale_lucas_convolution_pascal_right_table)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_row_step_previous_left. bcf_previous_code_lucas_convolution_pascal_right_table = bcf_quotient_lucas_convolution_pascal_right_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_convolution_pascal_right_table_row_step)) * bcf_previous_scale_lucas_convolution_pascal_right_table) + (bcf_left_lucas_convolution_pascal_right_table_row_step))) /\ ((((exists bcf_height_lucas_convolution_pascal_right_table_row_step_previous_right. bcf_height_lucas_convolution_pascal_right_table_row_step_previous_right + S (bcf_right_lucas_convolution_pascal_right_table_row_step) = S ((S (S (bcf_predecessor_lucas_convolution_pascal_right_table_row_step))) * bcf_previous_scale_lucas_convolution_pascal_right_table)) /\ exists bcf_quotient_lucas_convolution_pascal_right_table_row_step_previous_right. bcf_previous_code_lucas_convolution_pascal_right_table = bcf_quotient_lucas_convolution_pascal_right_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_convolution_pascal_right_table_row_step))) * bcf_previous_scale_lucas_convolution_pascal_right_table) + (bcf_right_lucas_convolution_pascal_right_table_row_step))) /\ bcf_value_lucas_convolution_pascal_right_table_row_step = bcf_left_lucas_convolution_pascal_right_table_row_step + bcf_right_lucas_convolution_pascal_right_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_convolution_pascal_right_decoded_row_code. bcf_height_lucas_convolution_pascal_right_decoded_row_code + S (bcf_row_code_lucas_convolution_pascal_right) = S ((S (n)) * bcf_row_code_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_decoded_row_code. bcf_row_code_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_decoded_row_code * S ((S (n)) * bcf_row_code_scale_lucas_convolution_pascal_right) + (bcf_row_code_lucas_convolution_pascal_right))) /\ ((((exists bcf_height_lucas_convolution_pascal_right_decoded_row_scale. bcf_height_lucas_convolution_pascal_right_decoded_row_scale + S (bcf_row_scale_lucas_convolution_pascal_right) = S ((S (n)) * bcf_row_scale_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_decoded_row_scale. bcf_row_scale_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_lucas_convolution_pascal_right) + (bcf_row_scale_lucas_convolution_pascal_right))) /\ (((exists bcf_height_lucas_convolution_pascal_right_decoded_value. bcf_height_lucas_convolution_pascal_right_decoded_value + S (B) = S ((S (S k)) * bcf_row_scale_lucas_convolution_pascal_right)) /\ exists bcf_quotient_lucas_convolution_pascal_right_decoded_value. bcf_row_code_lucas_convolution_pascal_right = bcf_quotient_lucas_convolution_pascal_right_decoded_value * S ((S (S k)) * bcf_row_scale_lucas_convolution_pascal_right) + (B))))))))) -> (((exists bcf_lt_gap_lucas_convolution_pascal_target_out_of_range. bcf_lt_gap_lucas_convolution_pascal_target_out_of_range + S (S n) = S k) /\ C = 0) \/ ((exists bcf_le_gap_lucas_convolution_pascal_target_in_range. bcf_le_gap_lucas_convolution_pascal_target_in_range + (S k) = S n) /\ (exists bcf_row_code_code_lucas_convolution_pascal_target bcf_row_code_scale_lucas_convolution_pascal_target bcf_row_scale_code_lucas_convolution_pascal_target bcf_row_scale_scale_lucas_convolution_pascal_target bcf_row_code_lucas_convolution_pascal_target bcf_row_scale_lucas_convolution_pascal_target. ((forall bcf_row_index_lucas_convolution_pascal_target_table. (exists bcf_lt_gap_lucas_convolution_pascal_target_table_row_bound. bcf_lt_gap_lucas_convolution_pascal_target_table_row_bound + S (bcf_row_index_lucas_convolution_pascal_target_table) = S (S n)) -> exists bcf_row_code_lucas_convolution_pascal_target_table bcf_row_scale_lucas_convolution_pascal_target_table. ((((exists bcf_height_lucas_convolution_pascal_target_table_decoded_row_code. bcf_height_lucas_convolution_pascal_target_table_decoded_row_code + S (bcf_row_code_lucas_convolution_pascal_target_table) = S ((S (bcf_row_index_lucas_convolution_pascal_target_table)) * bcf_row_code_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_decoded_row_code. bcf_row_code_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_table_decoded_row_code * S ((S (bcf_row_index_lucas_convolution_pascal_target_table)) * bcf_row_code_scale_lucas_convolution_pascal_target) + (bcf_row_code_lucas_convolution_pascal_target_table))) /\ ((((exists bcf_height_lucas_convolution_pascal_target_table_decoded_row_scale. bcf_height_lucas_convolution_pascal_target_table_decoded_row_scale + S (bcf_row_scale_lucas_convolution_pascal_target_table) = S ((S (bcf_row_index_lucas_convolution_pascal_target_table)) * bcf_row_scale_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_decoded_row_scale. bcf_row_scale_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_table_decoded_row_scale * S ((S (bcf_row_index_lucas_convolution_pascal_target_table)) * bcf_row_scale_scale_lucas_convolution_pascal_target) + (bcf_row_scale_lucas_convolution_pascal_target_table))) /\ ((bcf_row_index_lucas_convolution_pascal_target_table = 0 /\ (forall bcf_index_lucas_convolution_pascal_target_table_zero_row. (exists bcf_lt_gap_lucas_convolution_pascal_target_table_zero_row_bound. bcf_lt_gap_lucas_convolution_pascal_target_table_zero_row_bound + S (bcf_index_lucas_convolution_pascal_target_table_zero_row) = S (S n)) -> exists bcf_value_lucas_convolution_pascal_target_table_zero_row. ((((exists bcf_height_lucas_convolution_pascal_target_table_zero_row_entry. bcf_height_lucas_convolution_pascal_target_table_zero_row_entry + S (bcf_value_lucas_convolution_pascal_target_table_zero_row) = S ((S (bcf_index_lucas_convolution_pascal_target_table_zero_row)) * bcf_row_scale_lucas_convolution_pascal_target_table)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_zero_row_entry. bcf_row_code_lucas_convolution_pascal_target_table = bcf_quotient_lucas_convolution_pascal_target_table_zero_row_entry * S ((S (bcf_index_lucas_convolution_pascal_target_table_zero_row)) * bcf_row_scale_lucas_convolution_pascal_target_table) + (bcf_value_lucas_convolution_pascal_target_table_zero_row))) /\ ((bcf_index_lucas_convolution_pascal_target_table_zero_row = 0 /\ bcf_value_lucas_convolution_pascal_target_table_zero_row = 1) \/ exists bcf_predecessor_lucas_convolution_pascal_target_table_zero_row. bcf_index_lucas_convolution_pascal_target_table_zero_row = S bcf_predecessor_lucas_convolution_pascal_target_table_zero_row /\ bcf_value_lucas_convolution_pascal_target_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_convolution_pascal_target_table bcf_previous_code_lucas_convolution_pascal_target_table bcf_previous_scale_lucas_convolution_pascal_target_table. bcf_row_index_lucas_convolution_pascal_target_table = S bcf_predecessor_lucas_convolution_pascal_target_table /\ ((((exists bcf_height_lucas_convolution_pascal_target_table_decoded_previous_code. bcf_height_lucas_convolution_pascal_target_table_decoded_previous_code + S (bcf_previous_code_lucas_convolution_pascal_target_table) = S ((S (bcf_predecessor_lucas_convolution_pascal_target_table)) * bcf_row_code_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_decoded_previous_code. bcf_row_code_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_convolution_pascal_target_table)) * bcf_row_code_scale_lucas_convolution_pascal_target) + (bcf_previous_code_lucas_convolution_pascal_target_table))) /\ ((((exists bcf_height_lucas_convolution_pascal_target_table_decoded_previous_scale. bcf_height_lucas_convolution_pascal_target_table_decoded_previous_scale + S (bcf_previous_scale_lucas_convolution_pascal_target_table) = S ((S (bcf_predecessor_lucas_convolution_pascal_target_table)) * bcf_row_scale_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_decoded_previous_scale. bcf_row_scale_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_convolution_pascal_target_table)) * bcf_row_scale_scale_lucas_convolution_pascal_target) + (bcf_previous_scale_lucas_convolution_pascal_target_table))) /\ (forall bcf_index_lucas_convolution_pascal_target_table_row_step. (exists bcf_lt_gap_lucas_convolution_pascal_target_table_row_step_bound. bcf_lt_gap_lucas_convolution_pascal_target_table_row_step_bound + S (bcf_index_lucas_convolution_pascal_target_table_row_step) = S (S n)) -> exists bcf_value_lucas_convolution_pascal_target_table_row_step. ((((exists bcf_height_lucas_convolution_pascal_target_table_row_step_entry. bcf_height_lucas_convolution_pascal_target_table_row_step_entry + S (bcf_value_lucas_convolution_pascal_target_table_row_step) = S ((S (bcf_index_lucas_convolution_pascal_target_table_row_step)) * bcf_row_scale_lucas_convolution_pascal_target_table)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_row_step_entry. bcf_row_code_lucas_convolution_pascal_target_table = bcf_quotient_lucas_convolution_pascal_target_table_row_step_entry * S ((S (bcf_index_lucas_convolution_pascal_target_table_row_step)) * bcf_row_scale_lucas_convolution_pascal_target_table) + (bcf_value_lucas_convolution_pascal_target_table_row_step))) /\ ((bcf_index_lucas_convolution_pascal_target_table_row_step = 0 /\ bcf_value_lucas_convolution_pascal_target_table_row_step = 1) \/ exists bcf_predecessor_lucas_convolution_pascal_target_table_row_step bcf_left_lucas_convolution_pascal_target_table_row_step bcf_right_lucas_convolution_pascal_target_table_row_step. bcf_index_lucas_convolution_pascal_target_table_row_step = S bcf_predecessor_lucas_convolution_pascal_target_table_row_step /\ ((((exists bcf_height_lucas_convolution_pascal_target_table_row_step_previous_left. bcf_height_lucas_convolution_pascal_target_table_row_step_previous_left + S (bcf_left_lucas_convolution_pascal_target_table_row_step) = S ((S (bcf_predecessor_lucas_convolution_pascal_target_table_row_step)) * bcf_previous_scale_lucas_convolution_pascal_target_table)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_row_step_previous_left. bcf_previous_code_lucas_convolution_pascal_target_table = bcf_quotient_lucas_convolution_pascal_target_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_convolution_pascal_target_table_row_step)) * bcf_previous_scale_lucas_convolution_pascal_target_table) + (bcf_left_lucas_convolution_pascal_target_table_row_step))) /\ ((((exists bcf_height_lucas_convolution_pascal_target_table_row_step_previous_right. bcf_height_lucas_convolution_pascal_target_table_row_step_previous_right + S (bcf_right_lucas_convolution_pascal_target_table_row_step) = S ((S (S (bcf_predecessor_lucas_convolution_pascal_target_table_row_step))) * bcf_previous_scale_lucas_convolution_pascal_target_table)) /\ exists bcf_quotient_lucas_convolution_pascal_target_table_row_step_previous_right. bcf_previous_code_lucas_convolution_pascal_target_table = bcf_quotient_lucas_convolution_pascal_target_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_convolution_pascal_target_table_row_step))) * bcf_previous_scale_lucas_convolution_pascal_target_table) + (bcf_right_lucas_convolution_pascal_target_table_row_step))) /\ bcf_value_lucas_convolution_pascal_target_table_row_step = bcf_left_lucas_convolution_pascal_target_table_row_step + bcf_right_lucas_convolution_pascal_target_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_convolution_pascal_target_decoded_row_code. bcf_height_lucas_convolution_pascal_target_decoded_row_code + S (bcf_row_code_lucas_convolution_pascal_target) = S ((S (S n)) * bcf_row_code_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_decoded_row_code. bcf_row_code_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_decoded_row_code * S ((S (S n)) * bcf_row_code_scale_lucas_convolution_pascal_target) + (bcf_row_code_lucas_convolution_pascal_target))) /\ ((((exists bcf_height_lucas_convolution_pascal_target_decoded_row_scale. bcf_height_lucas_convolution_pascal_target_decoded_row_scale + S (bcf_row_scale_lucas_convolution_pascal_target) = S ((S (S n)) * bcf_row_scale_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_decoded_row_scale. bcf_row_scale_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_decoded_row_scale * S ((S (S n)) * bcf_row_scale_scale_lucas_convolution_pascal_target) + (bcf_row_scale_lucas_convolution_pascal_target))) /\ (((exists bcf_height_lucas_convolution_pascal_target_decoded_value. bcf_height_lucas_convolution_pascal_target_decoded_value + S (C) = S ((S (S k)) * bcf_row_scale_lucas_convolution_pascal_target)) /\ exists bcf_quotient_lucas_convolution_pascal_target_decoded_value. bcf_row_code_lucas_convolution_pascal_target = bcf_quotient_lucas_convolution_pascal_target_decoded_value * S ((S (S k)) * bcf_row_scale_lucas_convolution_pascal_target) + (C))))))))) -> (exists lcv_left_pascal_mod_left lcv_right_pascal_mod_left. (A) + (p) * lcv_left_pascal_mod_left = (X) + (p) * lcv_right_pascal_mod_left) -> (exists lcv_left_pascal_mod_right lcv_right_pascal_mod_right. (B) + (p) * lcv_left_pascal_mod_right = (Y) + (p) * lcv_right_pascal_mod_right) -> (exists lcv_left_pascal_mod_result lcv_right_pascal_mod_result. (C) + (p) * lcv_left_pascal_mod_result = (X + Y) + (p) * lcv_right_pascal_mod_result)