267 lines
7.0 KiB
TeX
267 lines
7.0 KiB
TeX
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\part{Other Interactive Theorem Provers}
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\frame[plain]{\partpage}
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\begin{frame}
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\frametitle{Other Interactive Theorem Provers}
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\begin{itemize}
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\item at the beginning we very briefly discussed other theorem provers
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\item now, with more knowledge about HOL 4 we can discuss other provers and their differences to HOL 4 in more detail
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\item HOL 4 is a good system
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\item it is very well suited for the tasks required by the PROSPER project
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\item however, as always \emph{choose the right tool for your task}
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\item you might find a different prover more suitable for your needs
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\item hopefully this course has enabled you to learn to use other provers on your own without much trouble
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\end{itemize}
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\end{frame}
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\section{HOL 4}
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\begin{frame}
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\frametitle{HOL 4}
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\begin{itemize}
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\item based on classical higher order logic
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\item logic is sweet spot between expressivity and automation
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\pro very trustworthy thanks to LCF approach
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\pro simple enough to understand easily
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\pro very easy to write custom proof tools, \ie own automation
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\pro reasonably fast and efficient
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\item decent automation
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\con no user-interface
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\con no special proof language
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\con no IDE, very little editor support
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\end{itemize}
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\end{frame}
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\section{HOL Omega}
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\begin{frame}
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\frametitle{HOL Omega}
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\begin{itemize}
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\item mainly developed by Peter Homeier \emph{\url{http://www.trustworthytools.com/}}
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\item extension of HOL 4
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\begin{itemize}
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\pro logic extended by kinds
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\pro allows type operator variables
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\pro allows quantification over type variables
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\end{itemize}
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\pro sometimes handy to \eg model category theory
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\con not very actively developed
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\con HOL 4 usually sufficient and better supported
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\end{itemize}
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\end{frame}
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\section{HOL Light}
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\begin{frame}
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\frametitle{HOL Light}
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\begin{itemize}
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\item mainly developed by John Harrison
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\item \emph{\url{https://github.com/jrh13/hol-light}}
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\item cleanup and reimplementation of HOL in OCaml
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\item little legacy code
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\item however, still very similar to HOL 4
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\pro much better automation for real analysis
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\pro cleaner
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\con OCaml introduces some minor issues with trustworthiness
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\con some other libs and tools of HOL 4 are missing
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\con HOL 4 has bigger community
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\end{itemize}
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\end{frame}
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\section{Isabelle}
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\begin{frame}
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\frametitle{Isabelle}
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\begin{itemize}
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\item Isabelle is also a descendant of LCF
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\item originally developed by Larry Paulson in Cambridge\\
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\emph{\url{https://www.cl.cam.ac.uk/research/hvg/Isabelle/}}
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\item meanwhile also developed at TU Munich by Tobias Nipkow
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\emph{\url{http://www21.in.tum.de}}
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\item huge contributions by Markarius Wenzel\\
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\emph{\url{http://sketis.net}}
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\item Isabelle is a generic theorem prover
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\item most used instantiation is Isabelle/HOL
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\item other important one is Isabelle/ZF
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Isabelle / HOL - Logic}
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\begin{itemize}
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\item logic of Isabelle / HOL very similar to HOL's logic
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\begin{itemize}
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\item meta logic leads to meta level quantification and object level quantification
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\pro type classes
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\pro powerful module system
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\pro existential variables
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\item \ldots
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\end{itemize}
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\item Isabelle is implemented using the LCF approach
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\item it uses SML (Poly/ML)
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\item many original tools (\eg simplifier) similar to HOL
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\item focused as HOL on equational reasoning
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\item many tools are exchanged between HOL 4 and Isabelle / HOL
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\begin{itemize}
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\item Metis
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\item Sledgehammer
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\item \ldots
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Isabelle / HOL - Engineering}
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\begin{itemize}
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\pro a lot of engineering went into Isabelle/HOL
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\pro it has a very nice GUI
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\begin{itemize}
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\item IDE based on JEdit
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\item special language for proofs (Isar)
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\item good error messages
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\item \ldots
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\end{itemize}
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\pro very good automation
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\pro efficient implementations
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\pro many libraries (Archive of Formal Proof)
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\pro excellent code extraction
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\pro good documentation
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\pro easy for new users
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Isabelle / HOL - Isar}
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\begin{itemize}
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\item special proof language Isar used
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\item this allows to write \emph{declarative proofs}
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\begin{itemize}
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\item very high level
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\item easy to read by humans
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\item very robust
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\item very good tool support
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\item \ldots
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\end{itemize}
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\con however, tactical proofs are not easily accessible any more
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\begin{itemize}
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\item many intermediate goals need to be stated (declared) explicitly
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\item this can be very tedious
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\item tools like verification condition generators are hard to use
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Isabelle / HOL - Drawbacks}
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\begin{itemize}
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\pro Isabelle/HOL provides excellent out of the box automation
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\pro it provides a very nice user interface
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\pro it is very nice for new users
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\con however, this comes at a price
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\begin{itemize}
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\item multiple layers added between kernel and user
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\item hard to understand all these layers
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\item a lot of knowledge is needed to write your own automation
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\end{itemize}
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\con hard to write own automation
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\con Isabelle/HOL due to focus on declarative proofs not well suited for \eg PROSPER
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\end{itemize}
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\end{frame}
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\section{Coq}
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\begin{frame}
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\frametitle{Coq}
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\begin{itemize}
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\item Coq is a proof assistant using the Calculus of Inductive Constructions
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\item inspired by HOL 88
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\item backward proofs as in HOL 4 used
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\item however, very big differences
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\begin{itemize}
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\item much more powerful logic
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\item dependent types
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\item constructive logic
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\item not exactly following LCF approach
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\end{itemize}
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\pro good user interface
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\pro very good community support
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Coq - Logic}
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\begin{itemize}
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\pro Coq's logic is very powerful
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\pro it is very natural for mathematicians
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\pro very natural for language theory
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\pro allows reasoning about proofs
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\item allows to add axioms as needed
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\item as a result, Coq is used often to
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\begin{itemize}
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\item formalise mathematics
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\item formalise programming language semantics
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\item reason about proof theory
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Coq - Drawbacks}
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\begin{itemize}
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\item Coq's power comes at a price
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\con there is not much automation
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\con proofs tend to be very long
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\begin{itemize}
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\item they are very simple though
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\pro comparably easy to maintain
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\end{itemize}
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\con Coq's proof checking can be very slow
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\con when verifying programs or hardware you notice that HOL was designed for this purpose
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\begin{itemize}
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\item need for \emph{obvious} termination is tedious
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\item missing automation
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\item very slow
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\end{itemize}
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\end{itemize}
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\end{frame}
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\section{Conclusion}
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\begin{frame}
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\frametitle{Summary}
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\begin{itemize}
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\item there are many good theorem provers out there
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\item \emph{pick the right tool for your purpose}
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\item the HOL theorem prover is a good system for many purposes
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\item for PROSPER it is a good choice
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\item I encourage you to continue learning about HOL and interactive theorem proving in general
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\item if you have any questions feel free to contact me (Thomas Tuerk, email \emph{thomas@tuerk-brechen.de})
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\end{itemize}
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\end{frame}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "current"
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%%% End:
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