Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we argue that a commonly advocated approach to partial setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1

We describe a construction of the real numbers carried out in the Coq proof assistant. The basis is a set of axioms for the constructive real numbers as used in the FTA (Fundamental Theorem of Algebra) project, carried out at Nijmegen University. The aim of this work is to show that these axioms can be satisfied, by constructing a model for them. Apart from that, we show the robustness of the set of axioms for constructive real numbers, by proving (in Coq) that any two models of it are isomorphic. Finally, we show that our axioms are equivalent to the set of axioms for constructive reals introduced by Bridges in [2]. The construction of the reals is done in the ‘classical way’: first the rational numbers are built and they are shown to be a (constructive) ordered field and then the constructive real numbers are introduced as the usual Cauchy completion of the rational numbers. 1

We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecursive proofs. Thus we obtain reliable, corecursive algorithms for computing on real numbers.

This paper describes a construction of the real numbers in the HOL theorem-prover by strictly definitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in verification and computer algebra. Keywords: Mathematical Logic; Deduction and Theorem Proving 1 The real numbers For some mathematical tasks, the natural numbers N = f0; 1; 2; : : :g are sufficient. However for many purposes it is convenient to use a more extensive system, such as the integers (Z) or the rational (Q ), real (R) or complex (C ) numbers. In particular the real numbers are normally used for the measurement of physical quantities which (at least in abstract models) are continuously variable, and are therefore ubiquitous in scientific applications. 1.1 Properties of the real numbers We can characterize the reals as the unique `complete ordered field'. More precisely, the reals are a set ...

The first paper in this thesis presents a machine checked formalisation, in Martin-Löf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the Heine-Borel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the Hahn-Banach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional Martin-Löf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...

ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semi-simple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...

We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fully-checked example based on our representation of naive set theory. Contents 1 Introduction 1 2 Fundamentals 3 2.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Discrete sets . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 The category of sets . . . . . . . . . . . . . . . . . . . . . 5 2.1.5 Multi-variate maps . . . . . . . . . . . . . . . . . . . . . . 6 2.1.6 Predicates and relations . . . . . . . . . . . . . . . . . . . 7 2.1.7 Subsets and powerset . . . . . . . . . . . . . . . . . . . . 7 2.1.8 Quotients . . . . . . . . . . . . . . . ...

formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group at the University of Munich (7). As a former member of this group I was mainly involved in the theoretical background steering the implementation of the system. The system also exploits the so-called proofs-as-programs paradigm as a logical approach to correct software development: from a formal proof that a certain specication has a solution one fully automatically extracts a program that provably meets the specication. We carried out a number of extended case studies extracting programs from proofs in areas such as arithmetic (6), graph theory (7), innitary combinatorics (7), and lambda calculus (1,2). Special emphasis has been put on an ecient implemen