We describe a proof plan that characterises a family of proofs corresponding to the synthesis of recursive functional programs. This plan provides a significant degree of automation in the construction of recursive programs from specifications, together with correctness proofs. This plan makes use of meta-variables to allow successive refinement of the identity of unknowns, and so allows the program and the proof to be developed hand in hand. We illustrate the plan with parts of a substantial example --- the synthesis of a unification algorithm.

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

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This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and Martin-Lof's intuitionistic theory of types. This paper treats Mahlo's -numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of Martin-Lof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's -numbers and relating the history of universes in Martin-Lof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...

Abstract. A robust software component fulfills a contract: it expects data satisfying a certain property and promises to return data satisfying another property. The object-oriented community uses the design-bycontract approach extensively. Proposals for language extensions that add contracts to higher-order functional programming have appeared recently. In this paper we propose an embedded domain-specific language for typed, higher-order and first-class contracts, which is both more expressive than previous proposals, and allows for a more informative blame assignment. We take some first steps towards an algebra of contracts, and we show how to define a generic contract combinator for arbitrary algebraic data types. The contract language is implemented as a library in Haskell using the concept of generalised algebraic data types. 1

One objective of this paper is the determination of the proof-theoretic strength of Martin-Löf's type theory with a universe and the type of well--founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As Martin-Löf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman.

We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO. Keywords: Hilbert and Natural-Deduction proof systems for Modal Logics, Logical Frameworks, Typed -calculus, Proof Assistants. Introduction In this paper we address the issue of designing proof development environments (i.e. "proof editors" or, even better, "proof assistants") for Modal Logics, in the style of [11, 12]. To this end, we explore the possibility of using Logical Frameworks (LF's) based on Type Theory...

The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.

Abstract. An untyped algorithm to test βη-equality for Martin-Löf’s Logical Framework with strong Σ-types is presented and proven complete using a model of partial equivalence relations between untyped terms. 1

The continuum is here presented as a formal space by means of a finitary inductive definition. In this setting a constructive proof of the Heine-Borel covering theorem is given. 1 Introduction It is well known that the usual classical proofs of the Heine-Borel covering theorem are not acceptable from a constructive point of view (cf. [vS, F]). An intuitionistic alternative proof that relies on the fan theorem was given by Brouwer (cf. [B, H]). In view of the relevance of constructive mathematics for computer science, relying on the connection between constructive proofs and computations, it is natural to look for a completely constructive proof of the theorem in its most general form, namely for intervals with real-valued endpoints. By using formal topology the continuum, as well as the closed intervals of the real line, can be defined by means of finitary inductive definitions. This approach allows a proof of the Heine-Borel theorem that, besides being constructive, can also be compl...