It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

Most general purpose proof assistants support versions of typed higher order logic. Experience has shown that these logics are capable of representing most of the mathematical models needed in Computer Science. However, perhaps there exist applications where ZF-style set theory is more natural, or even necessary. Examples may include Scott's classical inverse-limit construction of a model of the untyped - calculus (D1 ) and the semantics of parts of the Z specification notation. This paper

Set theory is the standard foundation for mathematics, but the majority of general purpose mechanised proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, EHDM, HOL, IMPS, LAMBDA, LEGO, Nuprl, PVS and Veritas. For many applications type theory works well and provides, for specification, the benefits of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, whereas type theory may appear inaccessable and so be an obstacle to the uptake of proof assistants based on it. This paper describes some experiments (using HOL) in combining set theory and type theory; the aim is to get the best of both worlds in a single system. Three approaches have been tried, all based on an axiomatically specified type V of ZF-like sets: (i) HOL is used without any additions besides V; (ii) an emb...

The need to use partial functions arises frequently in formal descriptions of computer systems. However, most proof assistants are based on logics of total functions. One way to address this mismatch is to invent and mechanize a new logic. Another is to develop practical workarounds in existing settings. In this paper we take the latter course: we survey and compare methods used to support partiality in a mechanization of a higher order logic featuring only total functions. The techniques we discuss are generally applicable and are illustrated by relatively large examples. 1.

. The majority of general purpose mechanised proof assistants support versions of typed higher order logic, even though set theory is the standard foundation for mathematics. For many applications higher order logic works well and provides, for specification, the benefits of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, but not higher order logic. This paper discusses some approaches to getting the best of both worlds: the expressiveness and standardness of set theory with the efficient treatment of functions provided by typed higher order logic. 1 Introduction Higher order logic is a successful and popular formalism for computer assisted reasoning. Proof systems based on higher order logic include ALF [18], Automath [20], Coq [9], EHDM [19], HOL [13], IMPS [10], LAMBDA [11], LEGO [17], Nuprl [6], PVS [22]...

The use of higher order logic (simple type theory) is often limited by its restrictive type system. Set theory allows many constructions on sets that are not possible on types in higher order logic. This paper presents a comparison of two theorem provers supporting set theory, namely HOL-ST and Isabelle/ZF, based on a formalization of the inverse limit construction of domain theory � this construction cannot be formalized in higher order logic directly. We argue that whilst the combination of higher order logic and set theory in HOL-ST has advantages over the rst order set theory in Isabelle/ZF, the proof infrastructure of Isabelle/ZF has better support for set theory proofs than HOL-ST. Proofs in Isabelle/ZF are both considerably shorter and easier to write. 1

this paper is the product category, defined by

HOLCF is the definitional extension of Church's Higher-Order Logic with Scott's Logic for Computable Functions that has been implemented in the theorem prover Isabelle. This results in a flexible setup for reasoning about functional programs. HOLCF supports standard domain theory (in particular fixpoint reasoning and recursive domain equations) but also coinductive arguments about lazy datatypes. This paper describes in detail how domain theory is embedded in HOL and presents applications from functional programming, concurrency and denotational semantics. 1 Introduction HOLCF is a logic for reasoning about functional programs. It provides arbitrary forms of recursion (via a fixpoint operator) and a package for defining datatypes. The latter caters for infinite objects, induction and coinduction. HOLCF is a synthesis of two logical systems, HOL and LCF, combining the best of both worlds. Before we go into technicalities (of which there is no shortage), we sketch the historical and log...

Introduction The goal of this project was to demonstrate the feasibility of the independent and trusted validation of the proofs generated by existing theorem provers. Our intention was to design, implement and formally verify a proof checking program for HOL [5] generated proofs. A proof checker can be much simpler than a full theorem prover such as HOL as it is only concerned with checking existing proofs rather than searching for or generating them. Our work has clearly demonstrated the feasibility of this approach. In particular, the main achievements of the project are as follows. ffl We have developed a computer representation suitable for communicating large, formal, machine generated proofs. ffl We have modified the HOL system to allow primitive inference proofs to be recorded in the above format. ffl We have formalised, within the HOL theorem proving system, theories of higher-order logic, Hilb