Yarrow is an interactive proof assistant based on the theory of Pure Type Systems, a family of typed lambda calculi. Yarrow has been designed as a flexible environment for experimentation with various typed lambda calculi. It offers both graphical and textual interfaces. It has been coded entirely in Haskell, making use of the Fudget library for the graphical interface. In this paper we concentrate on the software architecture of Yarrow, in particular the use of monads, the coupling of user interface and proof engine, polymorphic output routines, and flexible representations of lambda terms. We also treat the presentation of proofs in the flag-style format. 1 1 Introduction In this paper we describe the system Yarrow, an interactive proof assistant based on the theory of Pure Type Systems, a family of typed lambda calculi. In typed lambda calculi, theorems and proofs can be represented as well-typed terms, proof checking amounts to type checking, and proof construction to the...

We give syntax and a PER-model semantics for a typed h-calculus with subtypes and singleton types. The calculus may be seen as a minimal calculus of subtyping with a simple form of dependent types. The aim is to study singleton types and to take a canny step towards more complex dependent subtyping systems. Singleton types have applications in the use of type systems for specification and program extraction: given a program P we can form the very tight specification {P} which is met uniquely by P. Singletons integrate abbreviational definitions into a type system: the hypothesis x : {M} asserts x = M. The addition of singleton types is a non- conservative extension of familiar subtyping theories. In our system, more terms are typable and previously typable terms have more (non-dependent) types.

We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples. WTT is a refinement of de Bruijn's Mathematical Vernacular (MV) and hence: WTT is faithful to the mathematician's language yet is formal and avoids ambiguities.

Abstract. The Curry-Howard correspondence connects Natural Deduction derivation with the lambda-calculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason ‘backwards ’ from the desired conclusion towards the assumptions. At intermediate stages we may have an ‘incomplete derivation’, with ‘holes’. This is natural in informal practice; the challenge is to formalise it. To this end we use a one-and-a-halfth order technique based on nominal terms, with two levels of variable. Predicates are types, derivations are terms — and the two levels of variable are respectively the assumptions and the ‘holes ’ of an incomplete derivation. 1