\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{function type}

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

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

[[!include type theory - contents]]

\hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{as_a_negative_type}{As a negative type}\dotfill \pageref*{as_a_negative_type} \linebreak
\noindent\hyperlink{as_a_positive_type}{As a positive type}\dotfill \pageref*{as_a_positive_type} \linebreak
\noindent\hyperlink{positive_versus_negative}{Positive versus negative}\dotfill \pageref*{positive_versus_negative} \linebreak
\noindent\hyperlink{as_a_special_case_of_the_dependent_product}{As a special case of the dependent product}\dotfill \pageref*{as_a_special_case_of_the_dependent_product} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_dependent_product_types}{Relation to dependent product types}\dotfill \pageref*{relation_to_dependent_product_types} \linebreak
\noindent\hyperlink{application_in_logic}{Application in logic}\dotfill \pageref*{application_in_logic} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[type theory]] a \emph{function type} $X \to Y$ for two [[types]] $X,Y$ is the [[type]] of [[functions]] from $X$ to $Y$.

In a [[model]] of the type theory in [[categorical semantics]], this is an [[exponential object]]. In [[set theory]], it is a [[function set]]. In [[dependent type theory]], it is a special case of a [[dependent product type]].

\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

[[!include function type natural deduction - table]]

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Like any type constructor in type theory, a function type is specified by rules saying when we can introduce it as a type, how to [[term introduction|construct terms]] of that type, how to use or ``[[term elimination|eliminate]]'' terms of that type, and how to compute when we combine the constructors with the eliminators.

The [[type formation]] rule to build a function type is easy:

\begin{displaymath}
\frac{A\colon Type \qquad B \colon Type}{A\to B\colon Type}
\end{displaymath}
\hypertarget{as_a_negative_type}{}\subsubsection*{{As a negative type}}\label{as_a_negative_type}

Function types are almost always defined as a [[negative type]]. In this presentation, primacy is given to the eliminators. The natural eliminator of a function type says that we can [[function application|apply]] it to any input:

\begin{displaymath}
\frac{f\colon A\to B \qquad a\colon A}{f(a) \colon B}
\end{displaymath}
The constructor is then determined as usual for a negative type: to construct a term of $A\to B$, we have to specify how it behaves when applied to any input. In other words, we should have a term of type $B$ containing a free variable of type $A$. This yields the usual ``$\lambda$-abstraction'' constructor:

\begin{displaymath}
\frac{x\colon A\vdash b\colon B}{\lambda x.b\colon A\to B}
\end{displaymath}
The [[∞-reduction]] rule is the obvious one (the ur-example of all $\beta$-reductions), saying that when we evaluate a $\lambda$-abstraction, we do it by substituting for the bound variable.

\begin{displaymath}
(\lambda x.b)(a) \;\to_\beta\; b[a/x]
\end{displaymath}
If we want an [[∞-conversion]] rule, the natural one says that every function is a $\lambda$-abstraction:

\begin{displaymath}
\lambda x.f(x) \;\to_\eta\; f
\end{displaymath}
\hypertarget{as_a_positive_type}{}\subsubsection*{{As a positive type}}\label{as_a_positive_type}

It is also possible to present function types as a [[positive type]]. However, this requires a stronger metatheory, such as a [[logical framework]]. We use the same constructor ($\lambda$-abstraction), but now the eliminator says that to define an operation using a function, it suffices to say what to do in the case that that function is a lambda abstraction.

\begin{displaymath}
\frac{(x\colon A \vdash b\colon B) \vdash c\colon C \qquad f\colon A\to B}{funsplit(c,f)\colon C}
\end{displaymath}
This rule cannot be formulated in the usual presentation of type theory, since it involves a ``higher-order judgment'': the context of the term $c\colon C$ involves a ``term of type $B$ containing a free variable of type $A$''. However, it is possible to make sense of it. In [[dependent type theory]], we need additionally to allow $C$ to depend on $A\to B$.

The natural $\beta$-reduction rule for this eliminator is

\begin{displaymath}
funsplit(c, \lambda x.g) \;\to_\beta c[g/b]
\end{displaymath}
and its $\eta$-conversion rule looks something like

\begin{displaymath}
funsplit(c[\lambda x.b / g], f) \;\to_\eta\; c[f/g].
\end{displaymath}
Here $g\colon A\to B \vdash c\colon C$ is a term containing a free variable of type $A\to B$. By substituting $\lambda x.b$ for $g$, we obtain a term of type $C$ which depends on ``a term $b\colon B$ containing a free variable $x\colon A$''. We then apply the positive eliminator at $f\colon A\to B$, and the $\eta$-rule says that this can be computed by just substituting $f$ for $g$ in $c$.

\hypertarget{positive_versus_negative}{}\subsubsection*{{Positive versus negative}}\label{positive_versus_negative}

As usual, the positive and negative formulations are equivalent in a suitable sense. They have the same constructor, while we can formulate the eliminators in terms of each other:

\begin{displaymath}
\begin{aligned}
  f(a) &\coloneqq funsplit(b[a/x], f)\\
  funsplit(c, f) &\coloneqq c[f(x)/b]
\end{aligned}
\end{displaymath}
The conversion rules also correspond.

In dependent type theory, this definition of $funsplit$ only gives us a properly typed dependent eliminator if the negative function type satisfies $\eta$-conversion. As usual, if it satisfies \href{/nlab/show/eta-conversion#Propositional}{propositional eta-conversion} then we can transport along that instead---and conversely, the dependent eliminator allows us to prove propositional $\eta$-conversion. This is the content of Propositions 3.5, 3.6, and 3.7 in \hyperlink{GarnerSDP}{(Garner)}.

\hypertarget{as_a_special_case_of_the_dependent_product}{}\subsubsection*{{As a special case of the dependent product}}\label{as_a_special_case_of_the_dependent_product}

In [[dependent type theory]] a function type $A \to B$ is the special case the [[dependent product]] over $a : A$ for the special case that $B$ is regarded as an $A$-[[dependent type]] that actually happens to be $A$-independent

\begin{displaymath}
A \to B =_{def} \prod_{a : A} B
  \,.
\end{displaymath}
In [[categorical semantics]] this is the statement that a [[section]] of a product [[projection]] $A \times B \to A$ is equivalently just a morphism $A \to B$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_dependent_product_types}{}\subsubsection*{{Relation to dependent product types}}\label{relation_to_dependent_product_types}

A function type is the special case of a [[dependent product type]] for the case where the [[dependent type]] does not actually depend.

\begin{displaymath}
(X \to A)
  = 
  \prod_{x \colon X} A
  \,.
\end{displaymath}
See also at \emph{[[function monad]]}.

\hypertarget{application_in_logic}{}\subsubsection*{{Application in logic}}\label{application_in_logic}

In [[logic]], functions types express [[implication]]. More precisely, for $\phi, \psi$ two [[propositions]], under [[propositions as types]] the [[implication]] $\phi \Rightarrow \psi$ is the function type $\phi \to \psi$ (or rather the [[bracket type]] of that if one wishes to force this to be of type $Prop$ again ).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[function application]]


\item [[function monad]]


\item [[dependent product type]]


\item [[lambda calculus]]


\item [[implication]], [[linear implication]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A textbook account in the context of [[programming languages]] is in section III of

\begin{itemize}%
\item [[Robert Harper]], \emph{[[Practical Foundations for Programming Languages]]}

\end{itemize}
See also

\begin{itemize}%
\item [[Richard Garner]], \emph{On the strength of dependent products in the type theory of Martin-L\"o{}f}.

\end{itemize}
[[!redirects function type]] [[!redirects function types]]



\end{document}