LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the meta-theory of LEGO's type systems leading to a machine-checked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...

Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.

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Formal proofs generated by mechanised theorem proving systems may consist of a large number of inferences. As these theorem proving systems are usually very complex, it is extremely difficult if not impossible to formally verify them. This calls for an independent means of ensuring the consistency of mechanically generated proofs. This paper describes a method of recording HOL proofs in terms of a sequence of applications of inference rules. The recorded proofs can then be checked by an independent proof checker. Also described in this paper is an efficient proof checker which is able to check a practical proof consisting of thousands of inference steps. 1 Introduction Formal methods have been used in the development of many safety-critical systems in the form of formal specification and formal proof of correctness. Formal proofs are usually carried out using theorem provers or proof assistants. These systems are based on well-founded formal logic, and provide a programming environmen...

We investigate the verification of a translation phase of the Multiway Decision Graphs (MDG) verification system using the Higher Order Logic (HOL) theorem prover. In this paper, we deeply embed the semantics of a subset of the MDG-HDL language and its Table subset into HOL. We define a set of functions which translate this subset MDG-HDL language to its Table subset. A correctness theorem for this translator, which quantifies over its syntactic structure, has been proved. This theorem states that the semantics of the MDG-HDL program is equivalent to the semantics of its Table subset.

What facilities should an interactive verification system provide? We take the pragmatic view that the particular logic underlying a proof system is not as important as the support that is provided. Although a plethora of logics have been implemented, we think that there is a common kernel of support that a proof system ought to provide. Towards this end, we give detailed suggestions for verification support in three major areas: formalization, proof, and interface. Although our perspective comes from experience with highly expressive logics such as set theory, higher order logic, and type theory, we think our analyses apply more generally. Introduction Currently, theorem provers are used in the verification of both hardware and software [GM93, ORS92, BM90, HRS90, FFMH92], the formalization of informal mathematical proofs [FGT90, CH85, Pau90b], the teaching of logic[AMC84], and as tools of mathematical and metamathematical research [WWM + 90, CAB + 86]. 1 In this paper we describ...

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