Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable declarations without variance information. Hence, the second rule is too liberal; we must insist on p = ◦. The corrected set of rules is then: ⋄ cxt ∆ cxt ∆, X ◦κ cxt ∆ cxt ∆ ⊢ A: ∗ ∆, x:A cxt Definition of welltyped terms (immediately following). Since wellformed typing contexts ∆ contain no variance information, hence ◦ ∆ = ∆, we might drop the “◦ ” in the instantiation rule (fifth rule). The new set of rules is consequently, (x:A) ∈ ∆ ∆ cxt ∆ ⊢ x: A ∆, X ◦κ ⊢ t: A ∆ ⊢ t: ∀X κ. A ∆, x:A ⊢ t: B ∆ ⊢ λx.t: A → B ∆ ⊢ t: ∀X κ. A ∆ ⊢ F: κ

Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using proof-theoretic methods for obtaining certified algorithms in exact real arithmetic. 1

We show that cyclic structures, i.e., finite or possibly infinite structures with backpointers, unwindable into possibly infinite structures, can be elegantly represented as nested datatypes. This representation is free of the various deficiencies characterizing the more naive representation as mixed-variant datatypes. It is inspired by the representation of lambda-terms as a nested datatype via the de Bruijn notation. 1

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. Folds and unfolds are at the heart of the algebra of programming. They allow the cognoscenti to derive and manipulate programs rigorously and effectively. Fundamental laws such as fusion codify basic optimisation principles. However, most, if not all, programs require some tweaking to be given the form of an (un-) fold, and thus make them amenable to formal manipulation. In this paper, we remedy the situation by introducing adjoint folds and unfolds. We demonstrate that most programs are already of the required form and thus are directly amenable to manipulation. Central to the development is the categorical notion of an adjunction, which links adjoint (un-) folds to standard (un-) folds. We discuss a number of adjunctions and show that they are directly relevant to programming.