▁Tra ug ott ' s ▁rule ▁is ▁generalized ▁to ▁allow ▁several ▁pair wise ▁distinct ▁sub form ulas ▁ ▁of ▁ ▁and ▁ ▁of ▁, ▁as ▁long ▁as ▁ ▁have ▁a ▁common ▁most ▁general ▁un ifier , ▁say ▁. ▁The ▁generalized ▁resol vent ▁is ▁obtained ▁after ▁applying ▁ ▁to ▁the ▁parent ▁formulas , ▁thus ▁making ▁the ▁propos itional ▁version ▁applicable . ▁Tra ug ott ' s ▁comple ten ess ▁proof ▁re lies ▁on ▁the ▁assumption ▁that ▁this ▁fully ▁general ▁rule ▁is ▁used ; ▁it ▁is ▁not ▁clear ▁whether ▁his ▁rule ▁would ▁remain ▁complete ▁if ▁restricted ▁to ▁ ▁and ▁. ▁ ▁Param od ulation ▁▁ ▁Param od ulation ▁is ▁a ▁related ▁technique ▁for ▁reasoning ▁on ▁sets ▁of ▁cla uses ▁where ▁the ▁predicate ▁symbol ▁is ▁equality . ▁It ▁generates ▁all ▁" equal " ▁versions ▁of ▁cla uses , ▁except ▁ref lex ive ▁ident ities . ▁The ▁param od ulation ▁operation ▁takes ▁a ▁positive ▁from ▁clause , ▁which ▁must ▁contain ▁an ▁equality ▁literal . ▁It ▁then ▁searches ▁an ▁into ▁clause ▁with ▁a ▁sub term ▁that ▁un ifies ▁with ▁one ▁side ▁of ▁the ▁equality . ▁The ▁sub term ▁is ▁then ▁replaced ▁by ▁the ▁other ▁side ▁of ▁the ▁equality . ▁The ▁general ▁aim ▁of ▁param od ulation ▁is ▁to ▁reduce ▁the ▁system ▁to ▁atoms , ▁reducing ▁the ▁size ▁of ▁the ▁terms ▁when ▁subst itut ing . ▁ ▁Im plement ations ▁▁▁ ▁C AR INE ▁ ▁Ot ter ▁ ▁Pro ver 9 ▁ ▁SN AR K ▁ ▁S PASS ▁ ▁V amp ire ▁ ▁See ▁also ▁▁ ▁Cond ensed ▁det