Contents 1 Introduction 2 2 A first example 3 3 Description of the system 10 3.1 The two main windows : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.2 The mouse : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.3 The scratch area : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.3.1 The File-menu : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.3.2 The Define-menu : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.3.3 The Construct-menu : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.3.4 The Edit-menu : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3.3.5 The Goal-menu : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 3.3.6 The Context-menu : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 3.3.7 The View-menu

This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents Martin-Lof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.

General recursive algorithms are such that the recursive calls are performed on arguments satisfying no condition that guarantees termination. Hence, there is no direct way of formalising them in type theory. The standard way of handling general recursion in type theory uses a well-founded recursion principle. Unfortunately, this way of formalising general recursive algorithms often produces unnecessarily long and complicated codes. On the other hand, functional programming languages like Haskell impose no restrictions on recursive programs, and then writing general recursive algorithms is straightforward. In addition, functional programs are usually short and self-explanatory. However, the existing frameworks for reasoning about the correctness of Haskell-like programs are weaker than the framework provided by type theory. The goal of this work is to present a method that combines the advantages of both programming styles when writing simple general recursive algorithms....

The paper presents sound and complete translations of several fragments of Martin-Lof's monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover Gandalf and several experimental results are described. 1 Introduction The subject of this paper is the problem of automated theorem proving in Martin-Lof's monomorphic type theory [19, 8], which is the underlying logic of the interactive proof development system ALF [2, 14]. In the scope of our paper the task of automated theorem proving in type theory is understood as demonstrating that a certain type is inhabited by constructing a term of that type. The problem of inhabitedness of a type A is understood in the following way: given a set of judgements \Gamma (these may be constant declarations, explicit definitions and defining equalities), find a term a such that a2A is derivable from \Gam...

We will present a formalization of pointfree topology in Martin-Löf's type theory. A notion of point will be introduced and we will show that the points of a Scott topology form a Scott domain. This work follows closely the intuitionistic approach to pointfree topology and domain theory, developed mainly by Martin-Löf and Sambin. The important difference is that the definitions and proofs are machine checked by the proof assistant ALF.

Contents 1 Introduction 2 2 Type Theory as a Programming Language 3 2.1 Hello World in Type Theory . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Hiding and argument synthesis . . . . . . . . . . . . . . . . . . . . . 4 2.3 Using dependent types in programming . . . . . . . . . . . . . . . . 4 2.4 Higher-order sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Logic for free 8 3.1 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 Inductively defined relations . . . . . . . . . . . . . . . . . . . . . . . 13 4 ALF's Type Theory 14 4.1 Judgements of Type Theory . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Conventions

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 formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in Martin-Lof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs 4 2.1 An inductive formulation of almost-fullness (AF ID ) : : : : : : : : : : 5 2.1.1 Intuitionistic Ramsey Theorem (IRT ID ) : : : : : : : : : : : : 7 2.1.2 Higman's Lemma (HL ID ) : : : : : : : : : : : : : : : : : : : : 12 2.2 A negationless inductive formulation of almost-fullness (AF I ) : : : : : 17 2.2.1 Intuitionistic Ramsey Theorem (IRT I ) : : : : : : : : : : : : : 17 2.3 Equivalence between the various formulations of almost-fullness : : : 20 3 The programs 22 3.1 A higher order program : : : : : : : : : : : : : : : : : : : : : : : : : 24 3.2 A first order program : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4 Computational content of classical proofs 28 4.1 A cl...

Introduction ALF is the implementation of a logical framework. This implies that ALF is an open system, we can use it for different approaches to Type Theory or even to encode conventional logic. ALF will only check whether our theories typecheck. We are responsible to make sure that the theory is consistent --- i.e. it is no problem to encode an inconsistent theory like System U in ALF. We could restrict ourselves to the monomorphic set theory as it is described in [NPS90], chapter 19. It is straightforward to implement this theory in ALF and to construct proof-objects by explicit definitions. However, this does not reflect the current usage of ALF, i.e.: ffl We want to be able to introduce new sets by giving a sequence of constructors. ffl We want to define non-canonical constants by pattern matching. Peter Dybjer has developed a notion of schemes to capture inductively defined sets [Dyb91, Dyb94]. Thierry Coq

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...