Abstract

Abstract. We use a formally proven wellfounded recursion theorem as the basis upon which to build a function definition facility for Higher Order Logic. This approach offers flexibility in the choice of wellfounded relations used, the deferral of termination arguments, and automatic isolation of termination conditions. Building on this platform, we provide the ability to define recursive functions via pattern matching. The system is parameterized and has been instantiated to quite different theorem provers. 1 Introduction One use of higher-order logic theorem provers is to verify pure functional programs. By far the easiest way to model such programs is with the built-in functions of the logic. In a logic of total functions, such a naive modelling can capture only a proper subset of all programs, but this subset, although it excludes important examples such as interpreters, is still quite interesting. A genuine advantage is that no extra reasoning infrastructure need be built: the logic already provides it and hence there is no need for a preparatory, and often arduous, phase of "verification theory " construction. Therefore, a verification approach that represents functional programs by functions of higher order logic is attractive in spite of its limitations. The first step toward making the verification of such programs commonplace is to support their definition in the logic. Here we confront a shortcoming of many higher-order logic theorem-proving systems: often only primitive recursive definitions are supported. The functions definable by higher-order primitive recursion model a large class of programs, but the syntactic restrictions of primitive recursion are bothersome. To avoid this bother, we choose instead to define functions by appeal to the wellfounded recursion theorem. The freedom this gives us in writing right-hand sides of function definitions is complemented by extending the left-hand sides to allow ML-style pattern matching. Thus we arrive at a system wherein functions syntactically similar to ML or Haskell programs can be directly defined and reasoned about in the logic.

Citations

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