Extending the Calculus of Constructions with Tarski's fix-point theorem

TL;DR

This paper introduces a method to incorporate Tarski's fix-point theorem into the calculus of inductive constructions, enabling the modeling and reasoning of potentially non-terminating recursive functions within type-theoretic proof systems.

Contribution

It extends the calculus of inductive constructions with Tarski's fix-point theorem and classical logic axioms, allowing for the representation of non-terminating functions.

Findings

Extended framework supports reasoning about terminating and non-terminating computations

Program extraction techniques adapted for new recursive functions

Potentially broadens the class of functions modeled in type-theory based proof tools

Abstract

We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to potentially non-terminating functions. This is only possible if we extend the logical framework by adding the axioms that correspond to classical logic. We claim that the extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities of the calculus of inductive construction, like program extraction can be extended to also handle the new functions.