4 - 4-1Substitution andα-equivalenceI. Substitution in...

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Unformatted text preview: 4-1Substitution andα-equivalenceI. Substitution in Termsr(x/t): replace all free occurrences ofxinrbyt.Definition,by structural induction onr:(a)x(x/t) ≡ty(x/t) ≡y(ynegationslash≡x)(b)c(x/t) ≡c(c)f(r1, . . . , rn)(x/t) ≡f(r1(x/t), . . . , rn(x/t))Q: Isr(x1/t1)(x2/t2) ≡r(x2/t2)(x1/t1)?Lemmas.(a)r(x/t) ≡rifx /∈FV(r)(b)r(x/y)(y/x) ≡rify /∈FV(r)(c)r(x1/t1)(x2/t2) ≡r(x2/t2)(x1/t1)ifx1/∈FV(t2)andx2/∈FV(t1)andx1negationslash≡x2(d)r(x1/t1)(x2/t2) ≡r(x2/t2)(bigx1/t1(x2/t2))bigifx1/∈FV(t2)andx1negationslash≡x2(e)r(x/t1)(x/t2) ≡r(bigx/t1(x/t2))bigProof:Structural induction onr.squareSimultaneous substitutionin terms:r(x1/t1, . . . , xn/tn)whereinegationslash=j=⇒xinegationslash≡xjExDefine this by structural induction onr.JZ CAS701 F094-2II. Substitution in FormulasTrickier than substitution in terms, because of bound variables!First define:Definition(α-equivalence).ϕisα-equivalenttoψ, denotedϕ≡αψif they differ only in the naming of bound variables.if they differ only in the naming of bound variables....
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This note was uploaded on 10/26/2009 for the course CAS 701 taught by Professor Zucker during the Fall '09 term at McMaster University.

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4 - 4-1Substitution andα-equivalenceI. Substitution in...

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