The Nuprl system is a framework for reasoning about mathematics and programming. Over the years its design has been substantially improved to meet the demands of large-scale applications. Nuprl LPE, the newest release, features an open, distributed architecture centered around a flexible knowledge base and supports the cooperation of independent formal tools. This paper gives a brief overview of the system and the objectives that are addressed by its new architecture.

Real-world applications of automated theorem proving require modern software environments that enable modularisation, networked inter-operability, robustness, and scalability. These requirements are met by the Agent-Oriented Programming paradigm of Distributed Artificial Intelligence. We argue that a reasonable framework for automated theorem proving in the large regards typical mathematical services as autonomous agents that provide internal functionality to the outside and that, in turn, are able to access a variety of existing external services. This article describes...

This paper reports on the integration of the higher-order theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in Tps; in interactive mode, all features of the Tps-system are available to the user. If the subproblem which is passed to Tps contains concepts defined in Ωmega's database of mathematical theories, these definitions are not instantiated but are also passed to Tps. Using a special theory which contains proof tactics that model the ND-calculus variant of Tps within mega, any complete or partial proof generated in Tps can be translated one to one into an mega proof plan. Proof transformation is realised by proof plan expansion in Ωmega's 3-dimensional proof data structure, and remains transparent to the user.

Isabelle and NuPRL are two theorem proving environments that are written in different dialects of ML using different formula syntaxes and different logical foundations. In spite of this, they have similar sets of basic theories, representing the same mathematical concepts.

. 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...

We describe a low-level proof format, which can be used for independent proof checking and as an intermediate language for translating proofs between systems. The checker is presented as a virtual machine and the proof format as the bytecode. We compare HOL and Coq with a view to designing this pivot language, and describe a prototype which converts recorded HOL proofs into this intermediate format, and then translates them into Coq. 1 Communication between Proof Assistants There are several motives for wanting to enable communication between proof assistants. The most important is that users of one proof assistant might want to use proofs written with another. Another is using one system to check another. There is also an ecumenical interest in forging links between theorem proving communities. There has been some work in providing a general framework for different logics. The MathWeb [AHJ + 00] and Open Mechanized Reasoning Systems projects [GPT94] aim to provide such a ...

In type-theory based proof systems that provide inductive structures, computation tools are automatically associated to inductive de nitions. Choosing a particular representation for a given concept has a strong inuence on proof structure. We propose a method to make the change from one representation to another easier, by systematically translating proofs from one context to another. We show how this method works by using it on natural numbers, for which a unary representation (based on Peano axioms) and a binary representation are available. This method leads to an automatic translation tool that we have implemented in Coq and successfully applied to several arithmetical theorems.

. The CLaM proof planner has been interfaced to the HOL interactive theorem prover to provide the power of proof planning to people using HOL for formal verification, etc. The interface sends HOL goals to CLaM for planning and translates plans back into HOL tactics that solve the initial goals. The project homepage can be found at http://www.cl.cam.ac.uk/Research/HVG/Clam.HOL/intro.html. 1 Introduction CLaM [2] is a proof planning system for Oyster, a tactic-based implementation of the constructive type theory of Martin-Lof. CLaM works by using formalized pre- and post-conditions of Oyster tactics as the basis of plan search. These specifications of tactics are called methods. When a plan for a goal is found, the expectation is that the resulting tactic will solve the goal. Experience shows that the search space for plans is often tractable: CLaM has been able to automatically plan many proofs. A particular emphasis of research with CLaM has been the automation of inductive proo...

We report on the integration of Tps as an external reasoning component into the mathematical assistant system Ωmega. Thereby Tps can be used both as an automatic theorem prover for higher order logic as well as interactively employed from within the Ωmega environment. Tps proofs can be directly incorporated into Ωmega on a tactic level enabling their visualization and verbalization. Using an example we show how Tps proofs can be inserted into Ωmega's knowledge base by expanding them to calculus level using both Ωmega's tactic mechanism and the first order theorem prover Otter. Furthermore we demonstrate how the facts from Ωmega's knowledge base can be used to build a Tps library.

In previous work [ Boulton et al., 1998 ] , we have built a proof system that joins an interactive tactic-based theorem prover for a classical logic [ Gordon and Melham, 1993 ] with a fully automatic proof planner for a constructive logic [ Bundy et al., 1990 ] . This combined system significantly improves the automation of the interactive prover, but is unsatisfactory for several reasons. To overcome these drawbacks, we propose a general scheme whereby the combined system can pass through multiple rounds of interaction in searching for a proof. This scheme has been implemented and offers an important increase in the power and usability of the combined system. 1 Introduction Proof planning [ Bundy, 1991 ] uses artificial intelligence planning techniques to discover proofs of theorems in mathematics and logic. CLaM is a proof planner, implemented in Prolog, for Oyster, an implementation of MartinL of Type Theory. Its main area of application is in finding proofs by mathematical inducti...