Lemma PT5

If E,F are terms of the -calculus, and v is a variable of the aforementioned, and p is a path which properly addresses E and not seen(v,E,p) then there is an E' which is -congruent to E for which

Proof

Let E' be formed from E by -converting any -abstraction found on p whose bound variable is a member of FV(F) to an abstraction whose bound variable is not a member of FV(F).

We now proceed by induction on visible(v,p)

Base Case

In the base-case visible(v,p)=0, so that E = \v.H. Hence E'[v:=F][p] = E'[p] by S5.

Inductive Step

Suppose for some natural number n that visible(v,p) n implies that Consider a path p for which length(p) = n+1 Then p = s::p', where p'=tl(p), and length(p')=n.

Lemma PT6

If E,F,G are terms of the -calculus, and v is a variable of the aforementioned, and p is a path which properly addresses E and seen(v,E,p) then there is an E' which is -congruent to E for which

Proof

Let E' be formed from E by -converting any -abstraction found on p whose bound variable is a member of FV(F) to an abstraction whose bound variable is not a member of FV(F).

We now proceed by induction on the length of p.

Base Case

In the base-case length(p)=0, so that p = (). In this case, since by UP1 E'[p:=G]=G and v is in scope, we have While

Inductive Step

Suppose for some natural number n that length(p) n implies that Consider a path p for which length(p) = n+1 Then p = s::p', where p'=tl(p), and length(p')=n

Figure for Proposition LC3

Proposition LC3

Let (E = ( u.C) G) be a redex of the -calculus.

Let p be a path for which E[p] = ( u.D) H) where hd(p)=0 and hd(tl(p))=1. Let p' = tl(tl(p)). Then

Proof

Let C' be formed from C by -converting any -abstraction whose bound variable is a member of FV(G) FV(H) to an abstraction whose bound variable is not a member of FV(G)) FV(H).

Let (E' = ( u'.C') G) and let u',D', H' be the converted forms of u,D,H.

Consider that by Lemma PT3.

using the definition of path-addressing, and expanding E. This is a redex. So, we can work out that is using lemma PT6, as being On the other hand, we can immediately apply the definition of to obtaining Expanding E' and using (twice) the definition of path-addressing, we obtain which, by the definition of is Now, we can use lemma PT4 to distribute the outer-substitution across the path-update: Whence we can apply Lemma Sub.4 because we have chosen v' FV(G) FV(H). Thus we have shown that local-confluence holds for E', which is -congruent to E. Local confluence for Eitself follows from -conversion.