▁ ▁Care ▁must ▁be ▁taken ▁with ▁cases ▁when ▁a ▁unit ▁node ▁ ▁occurs ▁above ▁in ▁the ▁sub proof ▁that ▁der ives ▁another ▁unit ▁node ▁. ▁In ▁such ▁cases , ▁ ▁depends ▁on ▁. ▁Let ▁ ▁be ▁the ▁single ▁literal ▁of ▁the ▁unit ▁clause ▁of ▁. ▁Then ▁any ▁occurrence ▁of ▁ ▁in ▁the ▁sub proof ▁above ▁ ▁will ▁not ▁be ▁cancel led ▁by ▁resolution ▁infer ences ▁with ▁ ▁anymore . ▁Con sequently , ▁ ▁will ▁be ▁propag ated ▁down wards ▁when ▁the ▁proof ▁is ▁fixed ▁and ▁will ▁appear ▁in ▁the ▁clause ▁of ▁. ▁D iff icult ies ▁with ▁such ▁dependencies ▁can ▁be ▁easily ▁avoided ▁if ▁we ▁re insert ▁the ▁upper ▁unit ▁node ▁ ▁after ▁rein ser ting ▁the ▁unit ▁node ▁ ▁( i . e . ▁after ▁re insert ion , ▁ ▁must ▁appear ▁below ▁, ▁to ▁cancel ▁the ▁extra ▁literal ▁ ▁from ▁’ s ▁clause ). ▁This ▁can ▁be ▁ens ured ▁by ▁collect ing ▁the ▁unit ▁nodes ▁in ▁a ▁queue ▁during ▁a ▁bottom - up ▁travers al ▁of ▁the ▁proof ▁and ▁rein ser ting ▁them ▁in ▁the ▁order ▁they ▁were ▁que ued . ▁ ▁The ▁algorithm ▁for ▁fixing ▁a ▁proof ▁containing ▁many ▁roots ▁performs ▁a ▁top - down ▁travers al ▁of ▁the ▁proof , ▁re comput ing ▁the ▁resol vent s ▁and ▁replacing ▁broken ▁nodes ▁( e . g . ▁nodes ▁having ▁deleted Node Marker ▁as ▁one ▁of ▁their ▁parents ) ▁by ▁their ▁surv iving ▁parents ▁( e . g . ▁the ▁other ▁parent , ▁in ▁case ▁one ▁parent ▁was ▁deleted Node Marker