15–212: Fundamental Structures of Computer Science II Some Notes on Structural Induction

These notes provide a brief introduction to structural induction for proving properties of ML programs. We assume that the reader is already familiar with ML and the notes on evaluation and natural number induction for pure ML programs. We write e k =⇒ e′ for a computation of k steps, e =⇒ e′ for a computation of any number of steps (including 0), e ↪→ v for a complete computation of e to a value v, and n = m or e = e′ for mathematical equality. We define e ∼= e′ (e is operationally equivalent to e′) to hold if for any value v, e ↪→ v iff e′ ↪→ v, that is, if e and e′ either both have values (in which case it must be the same), or neither has a value. This notion will have to be refined when the language is extended by effects. Structural inductions in ML arises as inductions over the structure of values defined by datatype declarations. Most datatype declarations give rise to an induction principle which may be used to prove properties of recursive functions with arguments of the given type.