Lemma Sub.1

If x FV(E) then E[x:=F] E

Proof

We proceed by induction on n, the height of E.

Base Case n=0

Inductive Step

Suppose for a given n, for all i n we have

Consider an expression E of height n+1 We must show our result holds for E.

  • Sub-case 3: x FV(G) , u FV(G) ,

    In this sub-case, by the inductive hypothesis,

    By FV1. Thus x FV(G[u:=w]) or x=w.