úÎ$ú!~B      !"#$%&'()*+,-./0123456789:;<=>?@A'(c) Gershom Bazerman, Jeff Polakow 2011 BSD 3 Clausegershomb@gmail.com experimentalNone!"23468<=FKM|We can wrap any underying atom dype in an Ion to give it a "polarity" and add handling of "not" to our simplification tools.smart constructors for QueryRep [QueryReps can be queried for clauses within them, and clauses within them can be extracted.pretty printer class0A formula is either an atom (of some type, e.g. String).cA non-atomic formula (which is either a disjunction or a conjunction) is n-ary and consists of a Set of atoms and a set of non-atomic subformulas of dual connective, i.e. the non-atomic subformulas of a disjunction must all be conjunctions. The type system enforces this since there is no QFlipTyp instance for QAtomTyp.5disjunction is the dual of conjunction and vice-versaOWe'll start with three types of formulas: disjunctions, conjunctions, and atoms0convenience constructors, not particularly smartsmart constructor for QOpT does following optimization: a /\ (a \/ b) <-> a, or dually: a \/ (a /\ b) <-> aconvenience functionsG(a /\ b) \/ (a /\ c) \/ d <-> (a /\ (b \/ c)) \/ d (and also the dual) —Given a set of QueryReps, extracts a common clause if possible, returning the clause, the terms from which the clause has been extracted, and the rest."wTakes any given simplifier and repeatedly applies it until it ceases to reduce the size of the query reprepresentation.& • a /\ (b \/ ~b) /\ (c \/ d) <-> a /\ (c \/ d) a /\ ~a /\ (b \/ c) <-> False (a \/ ~a) /\ (b \/ ~b) <-> True (*) and duals +N.B. 0-ary \/ is False and 0-ary /\ is TrueD  !"#$%&'()*+,-./0123456789:;<=>?BC@A(  !"#$%&'BA@?>=< ;: 987654321 !"#$%0/.-,+*)(&'8   !"#$%&'()*+,-./0123456789:;<=>?BC@AD      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFboolsimplifier-0.1.8Data.BoolSimplifierQNotQNegqNot PPConstQR ppConstQRIonPosNegCombineQandqorq HasClause hasClause stripClause PPQueryRep ppQueryRepQueryRepQOpQAtomQFlipTypQAtomTypQAndTypQOrTyp extractAs extractCsqOrqAndqop extractAtomCsandqsorqssimplifyQueryRepgetCommonClauseAsgetCommonClauseCsfixSimplifyQueryRepqAtom isEmptyQR isConstQR simplifyIons maximumByNote $fQNotQAndTyp $fQNotQOrTyp$fQNotQAtomTyp $fPPConstQRa$fPPConstQRQOrTyp$fPPConstQRQAndTyp$fPPQueryRepQueryRep$fPPQueryRepQueryRep0$fPPQueryRepQueryRep1$fCombineQaQAtomTypQAtomTyp$fCombineQaQAtomTypQOrTyp$fCombineQaQAtomTypQAndTyp$fCombineQaQOrTypQAtomTyp$fCombineQaQOrTypQOrTyp$fCombineQaQOrTypQAndTyp$fCombineQaQAndTypQAtomTyp$fCombineQaQAndTypQOrTyp$fCombineQaQAndTypQAndTyp$fHasClausefifeqtyp$fHasClausefifeQAtomTyp$fPPQueryRepQueryRep2$fShowQueryRep $fOrdQueryRep $fEqQueryRep $fShowQAndTyp $fShowQOrTypTFCo:R:QFlipTypQAndTypTFCo:R:QFlipTypQOrTyp