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\begin{document}

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\section*{pure type system}

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

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

[[!include type theory - contents]]

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

[[!include deduction and induction - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{strongly_normalizing_systems}{Strongly normalizing systems}\dotfill \pageref*{strongly_normalizing_systems} \linebreak
\noindent\hyperlink{intuitionistic_higherorder_logic}{Intuitionistic (higher-order) logic}\dotfill \pageref*{intuitionistic_higherorder_logic} \linebreak
\noindent\hyperlink{lambda_cube}{Lambda cube}\dotfill \pageref*{lambda_cube} \linebreak
\noindent\hyperlink{CalculusOfConstructions}{Calculus of constructions}\dotfill \pageref*{CalculusOfConstructions} \linebreak
\noindent\hyperlink{inconsistent_systems}{Inconsistent systems}\dotfill \pageref*{inconsistent_systems} \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}

A pure type system is an explicitly typed [[lambda calculus]] using [[dependent product]] as the type of lambda expressions: the basic idea is that

\begin{displaymath}
\Gamma, x:A \vdash b:B
\end{displaymath}
implies

\begin{displaymath}
\Gamma \vdash (\lambda x:A . b) : (\prod x:A . B).
\end{displaymath}
In other words a \emph{pure type system} is

\begin{enumerate}%
\item a system of [[natural deduction]];


\item for [[dependent types]];


\item and with (only) the [[dependent product type]] [[type formation|formation]] rule specified.



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

A \emph{pure type system} is defined by

\begin{itemize}%
\item a set $S$ of \emph{sorts}, all of which are constants,


\item a set $A$ of \emph{axioms} of the form $c : s$ with $c$ a constant and $s$ a sort,


\item a set $R \hookrightarrow S \times S \times S$ of \emph{rules}: triples $(s_{1}, s_{2}, s_{3})$ of sorts.

Rules $(s_{1}, s_{2}, s_{2})$ are abbreviated as $(s_{1}, s_{2})$.



\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
These \emph{rules} will appear in the [[type formation]] rule for [[dependent product types]] below. They will say that for a type of sort $s_2$ [[dependent type|depending]] on a type of sort $s_1$ its dependent product is a type of sort $s_3$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
In fact all rules $(s_{1}, s_{2}, s_{2})$ appearing in the following have $s_2 = s_3$. So we can just as well regard $R$ as a binary relation $R \hookrightarrow S\times S$ (rather than a ternary one).

\end{remark}
From the above input data we derive the following

\begin{enumerate}%
\item The \textbf{terms} of a pure type system are the following (recursive definition):

\begin{itemize}%
\item variables and constants
\item $(\lambda x : A . B)$ (abstraction)
\item $(\prod x : A . B)$ (product)
\item $A B$ (application)

\end{itemize}
Here $x$ is a variable and $A$, $B$ are terms. The operators $\lambda$ and $\prod$ bind the variable $x$.


\item The \textbf{typing of terms} is inductively defined by the following rules.

A typing is of the form

\begin{displaymath}
x_{1} : A_{1}, \dots, x_{n} : A_{n} \vdash A : B
\end{displaymath}
meaning that the types of the variables declared on the left induces the term $A$ has type $B$. Note that types are also terms.

The order of variable declarations is significant: the declaration $x_{i} : A_{i}$ may depend on declarations to its left.



\end{enumerate}
The [[natural deduction]] rules are defined to be the following, for all $s \in S$ and where $x$ ranges over the set of variables.

\begin{tabular}{l|l|l}
name&rule&condition\\
\hline 
(axioms)&$\vdash c : s$&if $(c : s)$ is an axiom;\\
(start)&$\frac{\Gamma \vdash A:s}{\Gamma, x : A \vdash x : A}$&if $x \notin \Gamma$\\
(weakening)&$\frac{\Gamma \vdash A:B; \quad \Gamma \vdash C:s}{\Gamma, x : C \vdash A : B}$&if $x \notin \Gamma$\\
(product)&$\frac{\Gamma \vdash A : s_{1}; \quad \Gamma, x:A \vdash B : s_{2}}{\Gamma \vdash (\prod x:A . B) : s_{3}}$&if $(s_{1}, s_{2}, s_{3}) \in R$;\\
(application)&$\frac{\Gamma \vdash F : (\prod x:A . B); \quad \Gamma \vdash a : A}{\Gamma \vdash Fa : B [x := a]}$&\\
(abstraction)&$\frac{\Gamma, x:A \vdash b : B; \quad \Gamma \vdash (\prod x:A . B) : s}{\Gamma \vdash (\lambda x:A.b) : (\prod x:A.B)}$&\\
(conversion)&$\frac{\Gamma \vdash A : B; \quad \Gamma \vdash B' : s; \quad B =_{\beta} B'}{\Gamma \vdash A : B'}$&\\
\end{tabular}

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{strongly_normalizing_systems}{}\subsubsection*{{Strongly normalizing systems}}\label{strongly_normalizing_systems}

\hypertarget{intuitionistic_higherorder_logic}{}\paragraph*{{Intuitionistic (higher-order) logic}}\label{intuitionistic_higherorder_logic}

\hypertarget{lambda_cube}{}\paragraph*{{Lambda cube}}\label{lambda_cube}

The \emph{lambda cube} (\hyperlink{Barendregt91}{Barendregt 91}) consists of eight systems arranged in a cube. The most expressive is given by the following choice of sorts, axioms and rules:

\begin{tabular}{l|l}
symbol&actual value\\
\hline 
$S =$&$\{\ast, \square\}$\\
$A =$&$\{(\ast : \square)\}$\\
$R =$&$\{(\ast, \ast), (\ast, \square), (\square, \ast), (\square, \square)\}$\\
\end{tabular}

(Here $\{\ast, \Box\}$ denotes the 2-element set, see \hyperlink{Barendregt91}{Barendregt 91, 2.1})

The other systems omit some of the last three rules. Some specific systems are the following:

\begin{tabular}{l|l|l|l|l}
name&$(\ast, \ast)$&$(\ast, \square)$&$(\square, \ast)$&$(\square, \square)$\\
\hline 
$\lambda\rightarrow$ [[simply typed lambda calculus]]&yes&&&\\
${\lambda}P$ [[logical framework]]&yes&yes&&\\
$\lambda2$ [[system F]]&yes&&yes&\\
$\lambda\underline{\omega}$&yes&&&yes\\
${\lambda}C$ [[calculus of constructions]]&yes&yes&yes&yes\\
\end{tabular}

\hypertarget{CalculusOfConstructions}{}\paragraph*{{Calculus of constructions}}\label{CalculusOfConstructions}

For instance for the [[calculus of constructions]] we have

\begin{itemize}%
\item $* =$ [[Prop]], the \emph{[[type of propositions]]}


\item $\Box =$ [[Type]], the \emph{[[type of types]]}.



\end{itemize}
The single axiom $* \colon \Box$ hence says that $Prop \colon Type$, hence that [[Prop]] is a [[type]].

The rules express the usual rule for [[dependent product type]]:

\begin{itemize}%
\item a dependent product over a generic [[dependent type]] is itself some type;


\item but the dependent product of a dependent [[proposition]] is itself a proposition, namely the [[universal quantifier|universal quantification]].



\end{itemize}
\hypertarget{inconsistent_systems}{}\subsubsection*{{Inconsistent systems}}\label{inconsistent_systems}

The most permissive pure type system:

\begin{tabular}{l|l}
symbol&actual value\\
\hline 
$S =$&$\{\ast\}$\\
$A =$&$\{(\ast : \ast)\}$\\
$R =$&$\{(\ast, \ast)\}$\\
\end{tabular}

But there is an example with even non-circular system of axioms (System $\mathsf{U}^-$):

\begin{tabular}{l|l}
symbol&actual value\\
\hline 
$S =$&$\{ \ast, \square, \triangle\}$\\
$A =$&$\{(\ast : \square), (\square : \triangle)\}$\\
$R =$&$\{(\ast, \ast), (\square, \ast), (\square, \square), (\triangle, \square)\}$\\
\end{tabular}

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

\begin{itemize}%
\item Pure type systems can be augmented with a \emph{cumulativity} relation between sorts, so that if $s_1 \preceq s_2$, then any type in $s_1$ is also in $s_2$; see \hyperlink{BG05}{Barras-Gregoire}.


\item The \emph{[[calculus of inductive constructions]]} can be formulated as a particular pure type system (with a hierarchy of [[type of types]]) augmented by rules for introducing [[inductive types]].



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

\begin{itemize}%
\item Henk Barendregt, \emph{Introduction to generalized type systems}, J. Funct. Program. 1(2), 1991 (\href{http://patryshev.com/books/barendregt.pdf}{pdf})


\item Henk Barendregt (Catholic University Nijmegen), \emph{[[Lambda calculi with types]]}, To appear in [[Samson Abramsky]], D.M. Gabbay and T.S.E. Maibaum (eds.) \emph{Handbook of Logic in Computer Science}, Volume II, Oxford University Press. (\href{http://ttic.uchicago.edu/~dreyer/course/papers/barendregt.pdf}{preprint pdf})



\end{itemize}
Survey includes

\begin{itemize}%
\item Frade, \emph{Calculus of inductive constructions} (\href{http://www4.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf}{pdf})


\item [[Cody Roux]], \emph{Pure type systems: Dependents when you need them}, talk at Boston Haskell Meetup 2015 (\href{http://de.slideshare.net/imalsogreg/cody-roux-pure-type-systems-boston-haskell-meetup}{slides},\href{https://www.youtube.com/watch?v=ZGqKsalJi4s}{video})



\end{itemize}
A generalization to cumulativity can be found in

\begin{itemize}%
\item Bruno Barras and Benjamin Gregoire, \emph{On the role of type decorations in the Calculus of Inductive Constructions}, Lecture Notes in Computer Science Volume 3634, 2005, pp 151-166, \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.61.2666&rep=rep1&type=pdf}{PDF}

\end{itemize}
[[!redirects pure type systems]]



\end{document}