A Type-Free Theory of Half-Monotone Inductive Definitions

This paper studies an extension of inductive de nitions in the context of a type-free theory. It is a kind of simultaneous inductive de nition of two predicates where the de ning formulas are monotone with respect to the rst predicate, but not monotone with respect to the second predicate. We call this inductive de nition half-monotone in analogy of Allen's term half-positive. We can regard this de nition as a variant of monotone inductive de nitions by introducing a re ned order between tuples of predicates. We give a general theory for halfmonotone inductive de nitions in a type-free rst-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta's technique. The mechanism of half-monotone inductive de nitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Lof's Type Theory. We can also formalize the provability relation \a term p is a proof of a proposition P" naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.