Reasoning about Inductive Definitions: Inversion, Induction, and Recursion

Previously, we defined the small Vapid programming language. Since the language has a finite number of programs, its syntax was very easy to define: just list all the programs! In turn it was straightforward to define its evaluation function by cases, literally enumerating the results for each individual program. Finally, since the evaluator was defined by listing out the individual cases (program-result pairs), we could prove some (not particularly interesting) properties of the language and its programs.1 In an effort to move toward a more realistic language, we have introduced the syntax of a language of Boolean expressions, which was more complex than Vapid in that there are an infinite number of Boolean expressions. We did this using inductive definitions, which are much more expressive and sophisticated than just listing out programs. However, we must now answer the question: how do we define an evaluator for this infinite-program language, and how can we prove properties of all programs in the language and the results of evaluating them? To answer this question, we introduce three new reasoning principles: inversion lemmas, proofs by induction, and definitions of functions by recursion.