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\begin{document}

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\section*{proof assistant}

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

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

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

[[!include type theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{proof assistant} or \emph{proof management system} is a kind of software designed to help with [[proofs]] in formalized [[mathematics]]. Many proof assistants resemble and/or include a [[programming language]].

There are two threads of current development in proof systems: foundational and coverage.

The foundational work tries to find the best meta-theory to formalize mathematics (see also at \emph{[[foundations of mathematics]]}). Out of that work first came [[dependent type theory|dependent types]] ([[Automath]], in the late 60s), then the [[calculus of constructions]] (early [[Coq]]), and the [[calculus of inductive constructions]] (current [[Coq]]). More recently a new wave of such work is being done in [[homotopy type theory]] as another step in this direction. Coq's library is not that large, except in the area of [[group theory]] where the results of the work on [[Feit-Thompson theorem]] has produce something larger.

The much larger work has happened for decades building [[Mizar]]`s library (Mizar is based on [[Tarski–Grothendieck set theory]] rather than [[type theory]]). Its library is a couple of orders of magnitude larger than anyone else's. On the other hand, despite this quantity, it remains an issue to attack problems of contemporary research interest in these systems, see also at \emph{\href{Mizar#ProblemOfPertinence}{Mizar -- problem of pertinence}}.

Similar to Mizar is [[NuPRL]], [[HOL light]] and [[Isabelle]], which all have decently sized libraries. (Isabelle can be used with either [[material set theory]], like Mizar, or [[higher-order type theory]], like the others.)

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

[[!include proof assistants and formalization projects -- list]]

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

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Proof_assistant}{Proof assistant}}


\item Oscar Lanford III., \emph{Computer assisted proofs in analysis}, Proceedings of the International Congress of Mathematicians, 1986 (\href{http://www.mathunion.org/ICM/ICM1986.2/Main/icm1986.2.1385.1394.ocr.pdf}{pdf})


\item [[Freek Wiedijk]], \emph{Digital math by alphabet} (\href{http://www.cs.ru.nl/~freek/digimath/index.html}{web})


\item [[Carlos Simpson]], \emph{\href{http://math.unice.fr/~carlos/themes/verif.html}{Verification and creation of proofs by computer}}


\item [[Jeremy Avigad]], \emph{Interactive Theorem Proving, Automated Reasoning, and Mathematical Computation}, 2012 \href{http://www.andrew.cmu.edu/user/avigad/Talks/icerm.pdf}{pdf slides}


\item \emph{\href{http://www.cicm-conference.org/}{Conference Series on Intelligent Computer Mathematics}}

(\href{http://cicm-conference.org/2014/cicm.php}{2014})


\item Conference series on Interactive theorem proving

(\href{http://www.cs.uwyo.edu/~ruben/itp-2014/}{2014})



\end{itemize}
Parts of the above text are taken from \href{http://mathoverflow.net/questions/133572/at-which-level-is-it-currently-possible-to-write-formal-proofs/134009#134009}{this MO comment} by [[Jacques Carette]].

See also

\begin{itemize}%
\item [[Kevin Buzzard]], \emph{\href{https://xenaproject.wordpress.com/2018/10/07/what-is-the-xena-project/}{Xena project}}

on [[formal proof]] and proof assistants in undergraduate mathematics.



\end{itemize}
[[!redirects proof assistant]] [[!redirects proof assistants]]

[[!redirects proof management system]] [[!redirects proof management systems]]

[[!redirects computer proof assistant]] [[!redirects computer proof assistants]]

[[!redirects theorem prover]]



\end{document}