We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecursive proofs. Thus we obtain reliable, corecursive algorithms for computing on real numbers.

This paper explains the setting of an extensive formalisation of the theory of sequences (finite and infinite lists of elements of some data type) in the Prototype Verification System pvs. This formalisation is based on the characterisation of sequences as a final coalgebra, which is used as an axiom. The resulting theories comprise standard operations on sequences like composition (or concatenation), filtering, flattening, and their properties. They also involve the prefix ordering and proofs that sequences form an algebraic complete partial order. The finality axiom gives rise to various reasoning principles, like bisimulation, simulation, invariance, and induction for admissible predicates. Most of the proofs of equality statements are based on bisimulations, and most of the proofs of prefix order statements use simulations. Some significant aspects of these theories are described in detail. This coalgebraic formalisation of sequences is presented as a concrete example that shows t...

We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called ‘productive’ if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit. Whereas productivity is undecidable for stream definitions in general, we show that it can be decided for ‘pure’ stream definitions. For every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements, called ‘pebbles’, in a finite ‘pebbleflow net(work)’. And the production of a pebbleflow net associated with a pure stream definition, that is, the amount of pebbles the net is able to produce at its output port, can be calculated by reducing nets to trivial nets.

A type theoretic programming language is introduced that is based on lambda calculus with coproducts, products and inductive types, and additionally allows the definition of recursive functions in the way that is common in most functional programming languages. A formal system is presented that checks whether such a definition is structurally recursive and a soundness theorem is shown for this system. Thus all functions passing this check are ensured to terminate on all inputs. For the moment only non-mutual recursive functions are considered. 1

The concept of recursive coalgebra of a functor was introduced in the 1970s by Osius in his work on categorical set theory to discuss the relationship between wellfounded induction and recursively specified functions. In this paper, we motivate the use of recursive coalgebras as a paradigm of structured recursion in programming semantics, list some basic facts about recursive coalgebras and, centrally, give new conditions for the recursiveness of a coalgebra based on comonads, comonadcoalgebras and distributive laws of functors over comonads. We also present an alternative construction using countable products instead of cofree comonads.

Coiterative functions can be explained categorically as final coalgebraic morphisms, once coinductive types are viewed as final coalgebras. However, the coiteration schema which arises in this way is too rigid to accommodate directly many interesting classes of circular specifications. In this paper, building on the notion of T -coiteration introduced by the third author and capitalizing on recent work on bialgebras by Turi-Plotkin and Bartels, we introduce and illustrate various generalized coiteration patterns. First we show that, by choosing the appropriate monad T , T -coiteration captures naturally a wide range of coiteration schemata, such as the duals of primitive recursion and course-of-value iteration, and mutual coiteration. Then we show that, in the more structured categorical setting of bialgebras, T -coiteration captures guarded coiterations schemata, i.e. specifications where recursive calls appear guarded by predefined algebraic operations.

Abstract. This paper gives a reduction-preserving translation from Coquand’s dependent pattern matching [4] into a traditional type theory [11] with universes, inductive types and relations and the axiom K [22]. This translation serves as a proof of termination for structurally recursive pattern matching programs, provides an implementable compilation technique in the style of functional programming languages, and demonstrates the equivalence with a more easily understood type theory. Dedicated to Professor Joseph Goguen on the occasion of his 65th birthday. 1

Abstract. We present four constructions for standard equipment which can be generated for every inductive datatype: case analysis, structural recursion, no confusion, acyclicity. Our constructions follow a two-level approach—they require less work than the standard techniques which inspired them [11, 8]. Moreover, given a suitably heterogeneous notion of equality, they extend without difficulty to inductive families of datatypes. These constructions are vital components of the translation from dependently typed programs in pattern matching style [7] to the equivalent programs expressed in terms of induction principles [21] and as such play a crucial behind-the-scenes rôle in Epigram [25]. 1

C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e

The open calculus of constructions integrates key features of Martin-Löf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higher-order style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a first-order semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.