We give two nite axiomatizations of indexed inductive-recursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductive-recursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductive-recursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.

Constructive type theory is an expressive programming language where both algorithms and proofs can be represented. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination.

Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the Stern-Brocot representation for rational numbers. This formalisation uses advanced machinery of the Coq theorem prover and applies recent developments in formalising general recursive functions. This formalisation highlights the rôle of type theory both as a tool to verify hand-written programs and as a tool to generate verified programs. 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.

We present a Tait-style proof to show that a simple functional normaliser for a combinatory version of System T terminates. The main interest in our construction is methodological, it is an alternative to the usual small-step operational semantics on the one side and normalisation by evaluation on the other. Our work is motivated by our goal to verify implementations of Type Theory such as Epigram. Keywords: Normalisation,Strong Computability 1.

Abstract. We show that certain input-output relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. Inductive invariants can be used to enhance recursive function definition packages in higher-order logic mechanizations. We demonstrate the usefulness of inductive invariants on a large example of the BDD algorithm Apply. Finally, we introduce a related concept of inductive fixpoints with the property that for every functional in higher-order logic there exists a largest partial function that is such a fixpoint. 1

Traditionally, decidability of conversion for typed λ-calculi is established by showing that small-step reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simply-typed λ-calculus with explicit substitutions and βη-equality, a system which is not strongly normalising. We also show how the construction can be extended to System T with the usual β-rules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the Church-Rosser property of a term rewriting system.

In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.

In this work we present a generalisation of Dybjer's schema for simultaneous inductive-recursive definitions for the cases where we have several mutually recursive predicates defined simultaneously with several functions which, in turn, are defined by recursion on those predicates.

Abstract. Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. In both cases, we could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second one evaluates them, and the last one computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor: a partial recursive function is created by giving the first and third of the above components. As in one of our previous methods, every partial recursive function is associated with an inductive domain predicate and the evaluation of the function requires a proof that the predicate holds on the input values. We give a constructive justification for the new construct by means of an interpretation from the extended type theory into the base one. This shows that the extended theory is consistent and constructive. 1