Abstract. In this article, the Lorentzian manifolds isometrically embeddable in L N (for some large N , in the spirit of Nash's theorem) are characterized as a subclass of the set of all stably causal spacetimes; concretely, those which admit a smooth time function τ with |∇τ | > 1. Then, we prove that any globally hyperbolic spacetime (M, g) admits such a function, and, even more, a global orthogonal decomposition M = R × S, g = −βdt 2 + g t with bounded function β and Cauchy slices.In particular, a proof of a result stated by C.J.S. Clarke is obtained: any globally hyperbolic spacetime can be isometrically embedded in Minkowski spacetime L N . The role of the so-called "folk problems on smoothability" in Clarke's approach is also discussed.

a b s t r a c tThe aim of this article is to present the category of bounded Fréchet manifolds in respect to which we will review the geometry of Fréchet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some examples from Riemannian geometry are given.

In this paper, we describe the structure of Riemannian manifolds with a special kind of Codazzi spinors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy surface for any weakly irreducible holonomy representation with parallel spinors, t.m. with a holonomy group G R n−2 ⊂ SO(1, n − 1), where G ⊂ SO(n − 2) is trivial or a product of groups SU (k), Sp(l), G 2 or Spin(7).

HOLCF is the definitional extension of Church's Higher-Order Logic with Scott's Logic for Computable Functions that has been implemented in the theorem prover Isabelle. This results in a flexible setup for reasoning about functional programs. HOLCF supports standard domain theory (in particular fixpoint reasoning and recursive domain equations), but also coinductive arguments about lazy datatypes. This paper describes in detail how domain theory is embedded in HOL, and presents applications from functional programming, concurrency and denotational semantics.

We propose a combination of model checking and interactive theorem proving where the theorem prover is used to represent nite and in nite state systems, reason about them compositionally and reduce them to small nite systems by v eri ed abstractions. As an example we v erify a version of the Alternating Bit Protocol with unbounded lossy and duplicating channels: the channels are abstracted by i n teractive proof and the resulting nite state system is model checked.