6 - 6-1Semantics of Predicate Logic (contd)I: Evaluation of...

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Unformatted text preview: 6-1Semantics of Predicate Logic (contd)I: Evaluation of TermsWe will define avalue functionValA:Tm()State(A)AWe also write[[t]]A=dfValA(t, )Aso that[[t]]A:State(A)A.The def. of [[t]]Ais bystructural induction ont:[[x]]A=(x)[[c]]A=cA[[f(t1, . . . , tn)]]A=fA([[t1]]A, . . . ,[[tn]]A)(n >0)Example. Fortsuc(suc(0) +suc(x)),[[t]]N1=Note. S&A uses the notation valA,(t) forValA(t, ).Definition. ForMVar,(relM)orM(agrees withonM) if and only ifM=Mi.e.,for allxM,(x) =(x).Lemma.For fixedM,Mis an equivalence relation onState(A).JZ CAS701 F096-2Theorem 1 (Coincidence Lemma for terms).(relFV(t)) =[[t]]A= [[t]]A.Proof:Structural induction ont.squareCorollary.Iftis closed then[[t]]Ais independent of.Notation. Hence fortclosed, write [[t]]AastA,calledvalA(t) on p.5-4.More generally: supposeFV(t) {x1, . . . , xn}Puttt(x1, . . . , xn).Then [[t]]Adepends only on the values ofatx1, . . . , xn.So for anya1, . . . , anA, writetA(a1, . . . , an)to mean: [[t]]Afor anysuch that(xi) =aifori= 1, . . . , n.Theorem 2 (Substitution Lemma for terms).[[r(x/t)]]A= [[r]]A{x/a}wherea= [[t]]A.Proof:Structural induction onr.squareExGive the proof....
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6 - 6-1Semantics of Predicate Logic (contd)I: Evaluation of...

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