Mechanized Reasoning about Functional Programs

Abstract

This report describes a tool for reasoning about lazy functional programs written in a subset of Haskell. A first-order logic language for formulating theorems is defined. The syntax of the logic language is included in the functional language syntax; the semantics is classical. Structural induction and case analysis axioms are generated internally for suitable user-defined data types.

The logic language is given an operational semantics, which is implemented by the theorem proving tool. The operational semantics involves depth-first search, with limited backtracking. Proof can be goal-directed (backwards) or forward. Special care has been taken to restrict the search space. For example, induction is applied only where requested by the user. Pattern-matching is used instead of unification.

The theorem prover is in a sense ‘automatic’, as there is no user involvement in the proof of a goal. The approach taken can be called declarative theorem proving, since the user cannot manipulate a proof state using a sequence of tactics. The theorem prover can be regarded as an interpreter for a declarative language.