Interpreting HOL in the calculus of constructions

Abstract

The purpose of this paper is to consider a representation of the HOL theorem-prover in the calculus of constructions with the property that consistency results from the calculus of constructions imply such results in HOL. This kind of representation is impossible using the propositions-as-types representation of logic and equality, but it is possible if a different representation is used.Since the basic formalism of higher-order logic (HOL) is weaker than that of the calculus of constructions, it seems natural to interpret the HOL theorem prover [1] in the latter. However, there are some problems in carrying out such an interpretation. The HOL theorem prover is based on classical logic, includes an -choice operator, and includes a principle of extensionality in the formThe calculus of constructions is based on constructive logic, and although it can be consistently extended to classical logic by postulating excluded middle, it has been known for some time that excluded middle together with an -choice operator leads to proof irrele-✩ This paper represents a presentation entitled "Logic and Type Theory in Theorem Provers" which was presented to the conference "Thirty-Five Years of AUTOMATH" held at