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\begin{document}

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\section*{inductive family}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{deduction_and_induction}{}\paragraph*{{Deduction and Induction}}\label{deduction_and_induction}

[[!include deduction and induction - contents]]

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{induction}{}\paragraph*{{Induction}}\label{induction}

[[!include induction - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{semantics}{Semantics}\dotfill \pageref*{semantics} \linebreak
\noindent\hyperlink{higher_categorical_version_homotopy_type_theory}{Higher categorical version/ homotopy type theory}\dotfill \pageref*{higher_categorical_version_homotopy_type_theory} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Inductive families} generalize [[inductive types]]. Instead of defining a single type inductively, one simultaneously defines a whole family of types. An alternative term is ``indexed inductive definition''.

A simple example of an inductive family is the type of vectors Vect n indexed by the dimension n. This is defined by two constructors: one for the empty vector of dimension 0, and another for the operation which constructs a vector of dimension n+1 by adding a component to a vector of dimension n. The family of finite types Fin n (indexed by n) can also be defined as an inductive family: the constructor 0 is in any Fin n, and there is a successor operation which constructs an element in Fin (n+1) from an element in Fin n.

By the identification of [[propositions as types]], inductive families correspond to inductively defined predicates. For example, the [[identity type]] on a type A can be defined inductively by the reflexivity rule stipulating that a is identical to a for any a : A, that is, identity is the least reflexive relation. The identity family of types in intuitionistic type theory results from the identification of this relation with a family of types.

\hypertarget{history}{}\subsection*{{History}}\label{history}

The inductively defined identity type was introduced by Martin-L\"o{}f 1973 in his first published paper on Intuitionistic Type Theory.

A general schema for inductive families in Intuitionistic Type Theory was defined in \hyperlink{Dybjer91}{Dybjer91}, \hyperlink{Dybjer94}{Dybjer94}. This general schema was based on Martin-L\"o{}f's 1971 schema for inductive definitions in predicate logic. Simultaneously, \hyperlink{CoquandPaulin}{Coquand} and \hyperlink{Paulin93}{Paulin} extended the Calculus of Constructions with a similar schema for inductive families. This resulted in the [[calculus of inductive constructions]].

\hyperlink{Dybjer2000}{Dybjer} and \hyperlink{DybjerSetzer2001}{Dybjer and Setzer} generalized this schema to inductive-recursive definitions, resulting in ``indexed induction-recursion''.

\hyperlink{DybjerSetzer}{Dybjer and Setzer} also distinguish two kinds of inductive (and inductive-recursive) families, restricted (due to Coquand) and general ones. The identity type is an example of the latter, but not of the former.

Inductive families are part of the axiomatic foundation in [[Coq]] and [[agda]]. Instead, [[Lean]] does not have fix-point expressions, match expressions, or a termination checker in the kernel. Instead, recursive definitions and pattern matching are compiled into eliminators outside of the kernel.

\hypertarget{semantics}{}\subsection*{{Semantics}}\label{semantics}

Standard inductive types, [[W-types]] can be interpreted in any topos with [[natural numbers object]] (Moerdijk-Palmgren). Gambino and Hyland construct initial algebras for dependent [[polynomial functors]]. \hyperlink{AGHMM}{Indexed containers} are the same as dependent polynomial functors. \hyperlink{AGHMM}{Indexed containers} are claimed to form a foundation for inductive families.

\hypertarget{higher_categorical_version_homotopy_type_theory}{}\subsection*{{Higher categorical version/ homotopy type theory}}\label{higher_categorical_version_homotopy_type_theory}

\hyperlink{vdBergMoerdijk13}{van den Berg and Moerdijk} show that (standard) W-types can be interpreted in certain [[model categories]]. In [[homotopy type theory]] with [[universes]], one can reduce indexed W-types to W-types. This has been formalized \href{https://github.com/pcapriotti/agda-base/blob/master/src/container/w/core.agda}{here}, \href{https://github.com/SkySkimmer/HoTTClasses/blob/inductives/theories/theory/inductives.v}{here} and \href{https://github.com/jashug/IWTypes}{here}. \hyperlink{Sattler}{Sattler} outlines a generalization of the reduction to [[homotopy type theory]] without the need of universes.

\hypertarget{References}{}\subsection*{{References}}\label{References}

\begin{itemize}%
\item Per Martin-L\"o{}f, \emph{Hauptsatz for the intuitionistic theory of iterated inductive definitions}, 1971, Studies in Logic and the Foundations of Mathematics - Elsevier
\item Per Martin-L\"o{}f, \emph{An Intuitionistic Theory of Types: Predicative Part}, 1975, in Logic Colloquium 1973.
\item Peter Dybjer, \emph{Inductive Families} Formal aspects of computing 6 (4), 440-465 \href{http://www.cse.chalmers.se/~peterd/papers/Inductive_Families.ps}{PS}
\item Peter Dybjer, \emph{Inductive sets and families in Martin-L\"o{}f's type theory and their set-theoretic semantics}, 1991 Logical frameworks 2, 6
\item Peter Dybjer, \emph{A general formulation of simultaneous inductive-recursive definitions in type theory}, 2000, The Journal of Symbolic Logic 65 (02), 525-549 \emph{ Peter Dybjer, Anton Setzer, \emph{Indexed induction-recursion}, 2001 Proof Theory in Computer Science, 93-113}
\item Christine Paulin-Mohring, \emph{Inductive definitions in the system Coq rules and properties}, 1993 Typed lambda calculi and applications, 328-345.
\item Thierry Coquand, Christine Paulin, \emph{Inductively defined types}, COLOG-88 Volume 417 of the series Lecture Notes in Computer Science pp 50-66 \href{http://link.springer.com/chapter/10.1007%2F3-540-52335-9_47}{Springer} \href{https://books.google.dk/books?id=o_f4UwiZtL0C&pg=PA50}{G books}
\item Thorsten Altenkirch, Neil Ghani, Peter Hancock, Conor McBride, and Peter Morris, \emph{Indexed containers} \href{http://strictlypositive.org/indexed-containers.pdf}{PDF}
\item Peter Dybjer and Anton Setzer, \emph{Indexed induction-recursion}, Journal of Logic and Algebraic Programming, volume 66, Issue 1, January 2006, Pages 1-49. \href{http://www.cse.chalmers.se/~peterd/papers/Indexed_IR.pdf}{PDF}
\item Nicola Gambino and Martin Hyland, Wellfounded Trees and Dependent Polynomial Functors. \href{http://www1.maths.leeds.ac.uk/~pmtng/Research/Papers/gambino-hyland.pdf}{PDF}
\item [[Benno van den Berg]], [[Ieke Moerdijk]], \emph{W-types in Homotopy Type Theory} (\href{http://arxiv.org/abs/1307.2765}{arXiv:1307.2765})
\item Christian Sattler, \emph{slides} \href{http://cs.ioc.ee/types15/slides/sattler-slides.pdf}{slides}

\end{itemize}
[[!redirects inductive families]] [[!redirects indexed inductive definition]] [[!redirects indexed inductive type]] [[!redirects indexed inductive types]]



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