Several proof-assistants rely on the very formal basis of Pure Type Systems (PTS) as their foundations. We are concerned with the issues involved in the development of large proofs in these provers such as namespace management, development of reusable proof libraries and separate verification. Although actual implementations offer many features to address them, few theoretical foundations have been laid for them up to now.

this paper appears in Theorem Proving in Higher Order Logics, TPHOLs 2000, c

This thesis is about exploring the possibilities of a limited version of Martin-Löf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type theory itself is studied and one in which it is used to prove Higman's lemma. In the first paper, A Lambda Calculus Model of Martin-Löf's Theory of Types with Explicit Substitution, we present the formal calculus in complete detail. It consists of Martin-Lof's logical framework with explicit substitution extended with some inductively defined sets, also given in complete detail. These inductively defined sets are precisely those we need in the second paper of this thesis for the formal proof of Higman's lemma. The limitations of the formalism come from the fact that we do not introduce universes. It is known that for other versions of type theory, the absence of universes implies the impossib...

. We present an example of formalization of systems of algebras using an extension of Martin-Lof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of Martin-Lof's theory of logical types [14] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of types, from now on referred to as the logical framework, has been presented in [15, 7]. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A...

This article presents a formulation of the fan theorem in Martin-Löf's type theory. Starting from one of the standard versions of the fan theorem we gradually introduce reformulations leading to a final version which is easy to interpret in type theory. Finally we describe a formal proof of that final version of the fan theorem.

The extension of Martin-Löf's theory of types with record types and subtyping has elsewhere been presented. We give a concise description of that theory and motivate its use for the formalization of systems of algebras. We also give a short account of a proof checker that has been implemented on machine. The logical heart of the checker is constituted by the procedures for the mechanical verification of the forms of judgement of a particular formulation of the extension. The case study that we put forward in this work has been developed and mechanically verified using the implemented system. We illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions.

This paper presents a proof-irrelevant model of Martin-Lof's theory of types with explicit substitution; that is, a model in the style of [Smi88], in which types are interpreted as truth values and objects (or proofs) are irrelevant. The fundamental difference here is the need to cope with a formal system which in addition to types has sets and substitutions. This difference leads us to a whole reformulation of the model which consists in defining an interpretation in terms of the untyped lambda calculus. From this interpretation the proof-irrelevant model is obtained as a particular instance. Finally, the paper outlines the definition of a realizability model which is also obtained as a particular instance. Keywords: type theory, explicit substitution, models of type theory, proof-irrelevant model, realizability model. Contents 1 Introduction 1 2 Type theory 2 2.1 Syntax : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 3 A lambda calculus model 8 3.1 Semantic...

ABSTRACT iii

Records and record types in semantic theory