Inductive-Inductive Definitions

This article presents a new extension of inductive definitions, namely inductive-inductive definitions

Elaborating Inductive Definitions

We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a syntactic artifact -- to its code -- its semantics -- we obtain an internalized account of inductives inside the type theory itself: we claim that reasoning about inductive definitions could be carried in the type theory, not in the meta-theory as it is usually the case. Besides, we give a formal specification of that elaboration process. It is therefore amenable to formal reasoning too. We prove the soundness of our translation and hint at its correctness with respect to Coq's Inductive definitions. The practical benefits of this approach are numerous. For the type theorist, this is a small step toward bootstrapping, ie. implementing the inductive fragment in the type theory itself. For the programmer, this means better support for generic programming: we shall present a lightweight deriving mechanism, entirely definable by the programmer and therefore not requiring any extension to the type theory.Comment: 32 pages, technical repor

Quotient inductive-inductive definitions

In this thesis we present a theory of quotient inductive-inductive definitions, which are inductive-inductive definitions extended with constructors for equations. The resulting theory is an improvement over previous treatments of inductive-inductive and indexed inductive definitions in that it unifies and generalises these into a single framework. The framework can also be seen as a first approximation towards a theory of higher inductive types, but done in a set truncated setting. We give the type of specifications of quotient inductive-inductive definitions mutually with its interpretation as categories of algebras. A categorical characterisation of the induction principle is given and is shown to coincide with the property of being an initial object in the categories of algebras. From the categorical characterisation of induction, we derive a more type theoretic induction principle for our quotient inductive-inductive definitions that looks like the usual induction principles. The existence of initial objects in the categories of algebras associated to quotient inductive-inductive definitions is established for a class of definitions. This is done by a colimit construction that can be carried out in type theory itself in the presence of natural numbers, sum types and quotients or equivalently, coequalisers

A Categorical Semantics for Inductive-Inductive Definitions

Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A → Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules

Quotient inductive-inductive definitions

In this thesis we present a theory of quotient inductive-inductive definitions, which are inductive-inductive definitions extended with constructors for equations. The resulting theory is an improvement over previous treatments of inductive-inductive and indexed inductive definitions in that it unifies and generalises these into a single framework. The framework can also be seen as a first approximation towards a theory of higher inductive types, but done in a set truncated setting. We give the type of specifications of quotient inductive-inductive definitions mutually with its interpretation as categories of algebras. A categorical characterisation of the induction principle is given and is shown to coincide with the property of being an initial object in the categories of algebras. From the categorical characterisation of induction, we derive a more type theoretic induction principle for our quotient inductive-inductive definitions that looks like the usual induction principles. The existence of initial objects in the categories of algebras associated to quotient inductive-inductive definitions is established for a class of definitions. This is done by a colimit construction that can be carried out in type theory itself in the presence of natural numbers, sum types and quotients or equivalently, coequalisers