CS 6110 Lecture 7 Well - Founded Induction 9 February 2009

Recall that some of the substitution rules mentioned the function FV : {λ-terms} → Var: FV(x) = {x} FV(e1 e2) = FV(e1) ∪ FV(e2) FV(λx. e) = FV(e)− {x}. Why does this definition uniquely determine the function FV? There are two issues here: • Existence: whether FV is defined on all λ-terms; • Uniqueness: whether the definition is unique. Of relevance here is the fact that there are three clauses in the definition of FV corresponding to the three clauses in the definition of λ-terms and that a λ-term can be formed in one and only one way by one of these three clauses. Note also that although the symbol FV occurs on the right-hand side in two of these three clauses, they are applied to proper (proper = strictly smaller) subterms. The idea underlying this definition is called structural induction. This is an instance of a general induction principle called induction on a well-founded relation.