Abstract. We prove the correctness of a formalised realisability interpretation of extensions of first-order theories by inductive and coinductive definitions in an untyped λ-calculus with fixed-points. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation, and hint at further non-trivial applications in computable analysis. 1

Abstract. We study temporal properties over infinite binary red-blue trees in the setting of constructive type theory. We consider several familiar path-based properties, typical to linear-time and branching-time temporal logics like LTL and CTL ∗ , and the corresponding tree-based properties, in the spirit of the modal μ-calculus. We conduct a systematic study of the relationships of the path-based and tree-based versions of “eventually always blueness ” and mixed inductive-coinductive “almost always blueness ” and arrive at a diagram relating these properties to each other in terms of implications that hold either unconditionally or under specific assumptions (Weak Continuity for Numbers, the Fan Theorem, Lesser Principle of Omniscience, Bar Induction). We have fully formalized our development with the Coq proof assistant. 1

Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. to the binary signed digit representation. 1