Abstract. Nuprl and HOL are both tactic-based interactive theorem provers for higher-order logic, and both have been used in many substantial applications over the last decade. However, the HOL community has accumulated a much larger collection of formalized mathematics of the kind useful for hardware and software veri cation. This collection would be of great bene t in applying Nuprl to veri cation problems of real practical interest. This paper describes a connection we have implemented between HOL and Nuprl that gives Nuprl e ective access to mathematics formalized in HOL. In designing this connection, we had to overcome a number of problems related to di erences in the logics, logical infrastructures and stylistic conventions of Nuprl and HOL. 1

We give a new semantics for Nuprl's constructive type theory that justifies a useful embedding of the logic of the HOL theorem prover inside Nuprl. The embedding gives Nuprl effective access to most of the large body of formalized mathematics that the HOL community has amassed over the last decade. The new semantics is dramatically simpler than the old, and gives a novel and general way of adding set-theoretic equivalence classes to untyped functional programming languages.

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The concept of locales for Isabelle enables local definition and assumption for interactive mechanical proofs. Furthermore, dependent types are constructed in Isabelle/HOL for first class representation of structure. These two concepts are introduced briefly. Although each of them has proved useful in itself, their real power lies in combination. This paper illustrates by examples from abstract algebra how this combination works and argues that it enables modular reasoning.

. We give a new semantic foundation for type theories in the lineage of Martin-Lof's "polymorphic extensional" type theory, and use it to give a model of the constructive type theory of the interactive theorem proving system Nuprl. These type theories are based on an operational semantics of an untyped programming language. We show how to integrate classical set-theoretic objects, such as functions-as-graphs and equivalence classes, into this operational framework. The new semantics is dramatically simpler than the previous ones, and enables direct reasoning about classical mathematics. A practical consequence is that it justifies a useful embedding of the logic of the HOL theorem prover that gives Nuprl effective access to most of the large body of formalized mathematics that the HOL community has amassed over the years. 1 Introduction The so-called "polymorphic extensional" type theory of Martin-Lof (Martin-Lof, 1982) has two features that set it apart from other constructive type t...

. This paper illustrates the differences between the style of theory mechanisation of Coq and of HOL. This comparative study is based on the mechanisation of fragments of the theory of computation in these systems. Examples from these implementations are given to support some of the arguments discussed in this paper. The mechanisms for specifying definitions and for theorem proving are discussed separately, building in parallel two pictures of the different approaches of mechanisation given by these systems. 1 Introduction This paper compares the different theorem proving approaches of the HOL [10] and Coq [5] proof assistants. This comparison is based on a case study involving the mechanisation of parts of the theory of computation in the two systems. This paper does not illustrate these mechanisations but rather discusses the differences between the two systems and backs up certain points by examples taken from the case studies. One motivation of this work is that many users of theo...

This paper describes an approach to capturing the relation between circuits and their behaviours within a formal theory. The method exploits dependent types to achieve a rigorous yet theoretically simple connection between circuits (treated as graphs) and their behavioural specifications (treated as predicates). An example is given of a behavioural extraction function and it is shown how a type for modules can be defined that is sufficiently fine to guarantee that the behaviour of a module will satisfy its behavioural specification. The method is discussed in relation to VHDL and in relation to formal synthesis, (a process whereby one starts with a behavioural specification and, using an interactive goal-directed approach, ends up with a circuit and a formal proof that it satisfies the given behavioural specification).

A number of difficulties in the formalism of Pure Type Systems (PTS's) is discussed and an alternative classification system for typed calculi is proposed. In the new approach the main novelty is that one first explicitly specifies the dependencies that may occur. This is especially useful in order to describe constants. But it also facilitates the description of other type theoretic features, like dependent sums. Capsule review The paper reexamines the classification of typed lambda calculi as pure type systems, put forward by Barendregt. A problematic aspect of the PTS-framework is the incorporation of constants: this can be done, but at the cost of ad hoc extensions with products. The paper proposes an alternative description of type systems, essentially by enriching PTS's, with settings plus features, obtaining a neat mechanism for handling constants. 1 Introduction The phrase cubism will be used for the school that advocates classification of typed -calculi in terms of Pure Typ...

Introduction Quotients are used throughout mathematics for constructing new objects from old, by collapsing part of the structure, see for example any textbook on algebra or topology. Here we give a completely general description of such quotients in a type theoretic language. We assume a simple type theory, together with a predicate logic to reason about types and terms. Then quotients can be described as a left adjoint to a certain equality-predicate functor. This gives us all the rules we need: formation, introduction, elimination and (fi)- and (j)-conversions for quotients. These will be described in the next section below. Subsequently, the new syntax is put to use in constructing Z from N, a poset from a preorder, the abelianization of a group, and tensor products\Omega and sums \Phi of abelian groups. All these constructions involve taking a suitable quotient. They will be de

This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists.