Multi-stage programming languages provide a convenient notation for explicitly staging programs. Staging a definitional interpreter for a domain specific language is one way of deriving an implementation that is both readable and efficient. In an untyped setting, staging an interpreter "removes a complete layer of interpretive overhead", just like partial evaluation. In a typed setting however, Hindley-Milner type systems do not allow us to exploit typing information in the language being interpreted. In practice, this can have a slowdown cost factor of three or more times.

The nominal approach to abstract syntax deals with the issues of bound names and α-equivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax modulo α-equivalence. At the heart of this approach is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higher-order logic of principles of α-structural recursion and induction for α-equivalence classes from the ordinary versions of these principles for abstract syntax trees.

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.

Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal. 1

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.

1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and Martin-L"of [17, 18, 19]. The first occurrence of formal induction-recursion is Martin-L"of's definition of a universe `a la Tarski [19], which consists of a set U

Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols. 1.

We extend Bove's technique for formalising simple general recursive algorithms in constructive type theory to nested recursive algorithms. The method consists in defining an inductive special-purpose accessibility predicate, that characterises the inputs on which the algorithm terminates. As a result, the type-theoretic version of the algorithm can be defined by structural recursion on the proof that the input values satisfy this predicate. This technique results in definitions in which the computational and logical parts are clearly separated; hence, the type-theoretic version of the algorithm is given by its purely functional content, similarly to the corresponding program in a functional programming language. In the case of nested recursion, the special predicate and the type-theoretic algorithm must be defined simultaneously, because they depend on each other. This kind of definitions is not allowed in ordinary type theory, but it is provided in type theories extended wit...

We present a new approach to introducing an extensional propositional equality in Intensional Type Theory. Our construction is based on the observation that there is a sound, intensional setoid model in Intensional Type theory with a proof-irrelevant universe of propositions and -rules for - and -types. The Type Theory corresponding to this model is decidable, has no irreducible constants and permits large eliminations, which are essential for universes. Keywords. Type Theory, categorical models. 1. Introduction and Summary In Intensional Type Theory (see e.g. [11]) we differentiate between a decidable definitional equality (which we denote by =) and a propositional equality type (Id ( ; ) for any given type ) which requires proof. Typing only depends on definitional equality and hence is decidable. In Intensional Type Theory the type corresponding to the principle of extensionality Ext x2:(x) f;g2(x2:(x)) ( x2 Id (x) (f(x); g(x))) ! Id x2:(x) (f; g) is not...

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