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\begin{document}

\selectlanguage{english}

\title{\includegraphics[scale=0.5]{isabelle_sledgehammer} \\[4ex]
Hammering Away \\[\smallskipamount]
\Large A User's Guide to Sledgehammer for Isabelle/HOL}
\author{\hbox{} \\
Jasmin Christian Blanchette \\
{\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\
\hbox{}}

\maketitle

\tableofcontents

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\section{Introduction}
\label{introduction}

Sledgehammer is a tool that applies first-order automatic theorem provers (ATPs)
and satisfiability-modulo-theory (SMT) solvers on the current goal. The
supported ATPs are E \cite{schulz-2002}, SPASS \cite{weidenbach-et-al-2009},
Vampire \cite{riazanov-voronkov-2002}, SInE-E \cite{sine}, and SNARK
\cite{snark}. The ATPs are run either locally or remotely via the
System\-On\-TPTP web service \cite{sutcliffe-2000}. In addition to the ATPs, the
\textit{smt} proof method (which typically relies on the Z3 SMT solver
\cite{z3}) is tried as well.

The problem passed to the automatic provers consists of your current goal
together with a heuristic selection of hundreds of facts (theorems) from the
current theory context, filtered by relevance. Because jobs are run in the
background, you can continue to work on your proof by other means. Provers can
be run in parallel. Any reply (which may arrive half a minute later) will appear
in the Proof General response buffer.

The result of a successful proof search is some source text that usually (but
not always) reconstructs the proof within Isabelle. For ATPs, the reconstructed
proof relies on the general-purpose Metis prover \cite{metis}, which is fully
integrated into Isabelle/HOL, with explicit inferences going through the kernel.
Thus its results are correct by construction.

In this manual, we will explicitly invoke the \textbf{sledgehammer} command.
Sledgehammer also provides an automatic mode that can be enabled via the
``Auto Sledgehammer'' option from the ``Isabelle'' menu in Proof General. In
this mode, Sledgehammer is run on every newly entered theorem. The time limit
for Auto Sledgehammer and other automatic tools can be set using the ``Auto
Tools Time Limit'' option.

\newbox\boxA
\setbox\boxA=\hbox{\texttt{nospam}}

Examples of Sledgehammer use can be found in Isabelle's
\texttt{src/HOL/Metis\_Examples} directory.
Comments and bug reports concerning Sledgehammer or this manual should be
directed to
\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
in.\allowbreak tum.\allowbreak de}.

\vskip2.5\smallskipamount

%\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
%suggesting several textual improvements.

\section{Installation}
\label{installation}

Sledgehammer is part of Isabelle, so you don't need to install it. However, it
relies on third-party automatic theorem provers (ATPs) and SAT solvers.
Currently, E, SPASS, and Vampire can be run locally; in addition, E, Vampire,
SInE-E, and SNARK are available remotely via SystemOnTPTP \cite{sutcliffe-2000}.
If you want better performance, you should install E and SPASS locally.

There are three main ways to install ATPs on your machine:

\begin{enum}
\item[$\bullet$] If you installed an official Isabelle package with everything
inside, it should already include properly setup executables for E and SPASS,
ready to use.%
\footnote{Vampire's license prevents us from doing the same for this otherwise
wonderful tool.}

\item[$\bullet$] Alternatively, you can download the Isabelle-aware E and SPASS
binary packages from Isabelle's download page. Extract the archives, then add a
line to your \texttt{\char`\~/.isabelle/etc/components} file with the absolute
path to E or SPASS. For example, if the \texttt{components} does not exist yet
and you extracted SPASS to \texttt{/usr/local/spass-3.7}, create the
\texttt{components} file with the single line

\prew
\texttt{/usr/local/spass-3.7}
\postw

in it.

\item[$\bullet$] If you prefer to build E or SPASS yourself, or obtained a
Vampire executable from somewhere (e.g., \url{http://www.vprover.org/}),
set the environment variable \texttt{E\_HOME}, \texttt{SPASS\_HOME}, or
\texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{eproof},
\texttt{SPASS}, or \texttt{vampire} executable. Sledgehammer has been tested
with E 1.0 and 1.2, SPASS 3.5 and 3.7, and Vampire 1.0%
\footnote{Following the rewrite of Vampire, the counter for version numbers was
reset to 0; hence the new Vampire 1.0 is more recent than Vampire 11.5.}%
. Since the ATPs' output formats are neither documented nor stable, other
versions of the ATPs might or might not work well with Sledgehammer.
\end{enum}

To check whether E and SPASS are installed, follow the example in
\S\ref{first-steps}.

Remote ATP invocation via the SystemOnTPTP web service requires Perl with the
World Wide Web Library (\texttt{libwww-perl}) installed. If you must use a proxy
server to access the Internet, set the \texttt{http\_proxy} environment variable
to the proxy, either in the environment in which Isabelle is launched or in your
\texttt{\char`\~/.isabelle/etc/settings} file. Here are a few examples:

\prew
\texttt{http\_proxy=http://proxy.example.org} \\
\texttt{http\_proxy=http://proxy.example.org:8080} \\
\texttt{http\_proxy=http://joeblow:[e-mail]}
\postw

\section{First Steps}
\label{first-steps}

To illustrate Sledgehammer in context, let us start a theory file and
attempt to prove a simple lemma:

\prew
\textbf{theory}~\textit{Scratch} \\
\textbf{imports}~\textit{Main} \\
\textbf{begin} \\[2\smallskipamount]
%
\textbf{lemma} ``$[a] = [b] \,\longleftrightarrow\, a = b$'' \\
\textbf{sledgehammer}
\postw

Instead of issuing the \textbf{sledgehammer} command, you can also find
Sledgehammer in the ``Commands'' submenu of the ``Isabelle'' menu in Proof
General or press the Emacs key sequence C-c C-a C-s.
Either way, Sledgehammer produces the following output after a few seconds:

\prew
\slshape
Sledgehammer: ``\textit{e}'' for subgoal 1: \\
$([a] = [b]) = (a = b)$ \\
Try this command: \textbf{by} (\textit{metis hd.simps}). \\
To minimize the number of lemmas, try this: \\
\textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{e}] (\textit{hd.simps}). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{spass}'' for subgoal 1: \\
$([a] = [b]) = (a = b)$ \\
Try this command: \textbf{by} (\textit{metis insert\_Nil last\_ConsL}). \\
To minimize the number of lemmas, try this: \\
\textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{spass}] (\textit{insert\_Nil last\_ConsL}). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{vampire}'' for subgoal 1: \\
$([a] = [b]) = (a = b)$ \\
Try this command: \textbf{by} (\textit{metis eq\_commute last\_snoc}) \\
To minimize the number of lemmas, try this: \\
\textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{vampire}]~(\textit{eq\_commute last\_snoc}). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{remote\_sine\_e}'' for subgoal 1: \\
$([a] = [b]) = (a = b)$ \\
Try this command: \textbf{by} (\textit{metis hd.simps}) \\
To minimize the number of lemmas, try this: \\
\textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{remote\_sine\_e}]~(\textit{hd.simps}).
\postw

Sledgehammer ran E, SPASS, Vampire, SInE-E, and the \textit{smt} proof method in
parallel. Depending on which provers are installed and how many processor cores
are available, some of the provers might be missing or present with a
\textit{remote\_} prefix.

For each successful prover, Sledgehammer gives a one-liner proof that uses the
\textit{metis} or \textit{smt} method. You can click the proof to insert it into
the theory text. You can click the ``\textbf{sledgehammer} \textit{minimize}''
command if you want to look for a shorter (and probably faster) proof. But here
the proof found by E looks perfect, so click it to finish the proof.

You can ask Sledgehammer for an Isar text proof by passing the
\textit{isar\_proof} option:

\prew
\textbf{sledgehammer} [\textit{isar\_proof}]
\postw

When Isar proof construction is successful, it can yield proofs that are more
readable and also faster than the \textit{metis} one-liners. This feature is
experimental and is only available for ATPs.

\section{Hints}
\label{hints}

For best results, first simplify your problem by calling \textit{auto} or at
least \textit{safe} followed by \textit{simp\_all}. None of the ATPs contain
arithmetic decision procedures. They are not especially good at heavy rewriting,
but because they regard equations as undirected, they often prove theorems that
require the reverse orientation of a \textit{simp} rule. Higher-order problems
can be tackled, but the success rate is better for first-order problems. Hence,
you may get better results if you first simplify the problem to remove
higher-order features.

Note that problems can be easy for \textit{auto} and difficult for ATPs, but the
reverse is also true, so don't be discouraged if your first attempts fail.
Because the system refers to all theorems known to Isabelle, it is particularly
suitable when your goal has a short proof from lemmas that you don't know about.

\section{Command Syntax}
\label{command-syntax}

Sledgehammer can be invoked at any point when there is an open goal by entering
the \textbf{sledgehammer} command in the theory file. Its general syntax is as
follows:

\prew
\textbf{sledgehammer} \textit{subcommand\/$^?$ options\/$^?$ facts\_override\/$^?$ num\/$^?$}
\postw

For convenience, Sledgehammer is also available in the ``Commands'' submenu of
the ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c
C-a C-s. This is equivalent to entering the \textbf{sledgehammer} command with
no arguments in the theory text.

In the general syntax, the \textit{subcommand} may be any of the following:

\begin{enum}
\item[$\bullet$] \textbf{\textit{run} (the default):} Runs Sledgehammer on
subgoal number \textit{num} (1 by default), with the given options and facts.

\item[$\bullet$] \textbf{\textit{minimize}:} Attempts to minimize the provided facts
(specified in the \textit{facts\_override} argument) to obtain a simpler proof
involving fewer facts. The options and goal number are as for \textit{run}.

\item[$\bullet$] \textbf{\textit{messages}:} Redisplays recent messages issued
by Sledgehammer. This allows you to examine results that might have been lost
due to Sledgehammer's asynchronous nature. The \textit{num} argument specifies a
limit on the number of messages to display (5 by default).

\item[$\bullet$] \textbf{\textit{available\_provers}:} Prints the list of
installed provers. See \S\ref{installation} and \S\ref{mode-of-operation} for
more information on how to install automatic provers.

\item[$\bullet$] \textbf{\textit{running\_provers}:} Prints information about
currently running automatic provers, including elapsed runtime and remaining
time until timeout.

\item[$\bullet$] \textbf{\textit{kill\_provers}:} Terminates all running
automatic provers.

\item[$\bullet$] \textbf{\textit{refresh\_tptp}:} Refreshes the list of remote
ATPs available at System\-On\-TPTP \cite{sutcliffe-2000}.
\end{enum}

Sledgehammer's behavior can be influenced by various \textit{options}, which can
be specified in brackets after the \textbf{sledgehammer} command. The
\textit{options} are a list of key--value pairs of the form ``[$k_1 = v_1,
\ldots, k_n = v_n$]''. For Boolean options, ``= \textit{true}'' is optional. For
example:

\prew
\textbf{sledgehammer} [\textit{isar\_proof}, \,\textit{timeout} = 120$\,s$]
\postw

Default values can be set using \textbf{sledgehammer\_\allowbreak params}:

\prew
\textbf{sledgehammer\_params} \textit{options}
\postw

The supported options are described in \S\ref{option-reference}.

The \textit{facts\_override} argument lets you alter the set of facts that go
through the relevance filter. It may be of the form ``(\textit{facts})'', where
\textit{facts} is a space-separated list of Isabelle facts (theorems, local
assumptions, etc.), in which case the relevance filter is bypassed and the given
facts are used. It may also be of the form ``(\textit{add}:\ \textit{facts}$_1$)'',
``(\textit{del}:\ \textit{facts}$_2$)'', or ``(\textit{add}:\ \textit{facts}$_1$\
\textit{del}:\ \textit{facts}$_2$)'', where the relevance filter is instructed to
proceed as usual except that it should consider \textit{facts}$_1$
highly-relevant and \textit{facts}$_2$ fully irrelevant.

You can instruct Sledgehammer to run automatically on newly entered theorems by
enabling the ``Auto Sledgehammer'' option from the ``Isabelle'' menu in Proof
General. For automatic runs, only the first prover set using \textit{provers}
(\S\ref{mode-of-operation}) is considered, \textit{verbose}
(\S\ref{output-format}) and \textit{debug} (\S\ref{output-format}) are disabled,
fewer facts are passed to the prover, and \textit{timeout}
(\S\ref{mode-of-operation}) is superseded by the ``Auto Tools Time Limit'' in
Proof General's ``Isabelle'' menu. Sledgehammer's output is also more concise.

\section{Option Reference}
\label{option-reference}

\def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
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\def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$}
\def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opsmartx#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill\\\hbox{}\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
\def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]}
\def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
\def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
\def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}

Sledgehammer's options are categorized as follows:\ mode of operation
(\S\ref{mode-of-operation}), problem encoding (\S\ref{problem-encoding}),
relevance filter (\S\ref{relevance-filter}), output format
(\S\ref{output-format}), and authentication (\S\ref{authentication}).

The descriptions below refer to the following syntactic quantities:

\begin{enum}
\item[$\bullet$] \qtybf{string}: A string.
\item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
\item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or
\textit{smart}.
\item[$\bullet$] \qtybf{int\/}: An integer.
\item[$\bullet$] \qtybf{float\_pair\/}: A pair of floating-point numbers
(e.g., 0.6 0.95).
\item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}.
\item[$\bullet$] \qtybf{float\_or\_none\/}: An integer (e.g., 60) or
floating-point number (e.g., 0.5) expressing a number of seconds, or the keyword
\textit{none} ($\infty$ seconds).
\end{enum}

Default values are indicated in square brackets. Boolean options have a negated
counterpart (e.g., \textit{blocking} vs.\ \textit{non\_blocking}). When setting
Boolean options, ``= \textit{true}'' may be omitted.

\subsection{Mode of Operation}
\label{mode-of-operation}

\begin{enum}
\opnodefault{provers}{string}
Specifies the automatic provers to use as a space-separated list (e.g.,
``\textit{e}~\textit{spass}''). The following provers are supported:

\begin{enum}
\item[$\bullet$] \textbf{\textit{e}:} E is an ATP developed by Stephan Schulz
\cite{schulz-2002}. To use E, set the environment variable
\texttt{E\_HOME} to the directory that contains the \texttt{eproof} executable,
or install the prebuilt E package from Isabelle's download page. See
\S\ref{installation} for details.

\item[$\bullet$] \textbf{\textit{spass}:} SPASS is an ATP developed by Christoph
Weidenbach et al.\ \cite{weidenbach-et-al-2009}. To use SPASS, set the
environment variable \texttt{SPASS\_HOME} to the directory that contains the
\texttt{SPASS} executable, or install the prebuilt SPASS package from Isabelle's
download page. Sledgehammer requires version 3.5 or above. See
\S\ref{installation} for details.

\item[$\bullet$] \textbf{\textit{vampire}:} Vampire is an ATP developed by
Andrei Voronkov and his colleagues \cite{riazanov-voronkov-2002}. To use
Vampire, set the environment variable \texttt{VAMPIRE\_HOME} to the directory
that contains the \texttt{vampire} executable.

\item[$\bullet$] \textbf{\textit{smt}:} The \textit{smt} proof method invokes an
external SMT prover (usually Z3 \cite{z3}).

\item[$\bullet$] \textbf{\textit{remote\_e}:} The remote version of E runs
on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.

\item[$\bullet$] \textbf{\textit{remote\_vampire}:} The remote version of
Vampire runs on Geoff Sutcliffe's Miami servers. Version 9 is used.

\item[$\bullet$] \textbf{\textit{remote\_sine\_e}:} SInE-E is a metaprover
developed by Kry\v stof Hoder \cite{sine} based on E. The remote version of
SInE runs on Geoff Sutcliffe's Miami servers.

\item[$\bullet$] \textbf{\textit{remote\_snark}:} SNARK is a prover
developed by Stickel et al.\ \cite{snark}. The remote version of
SNARK runs on Geoff Sutcliffe's Miami servers.

\item[$\bullet$] \textbf{\textit{remote\_smt}:} The remote version of the
\textit{smt} proof method runs the SMT solver on servers at the TU M\"unchen (or
wherever \texttt{REMOTE\_SMT\_URL} is set to point).

\end{enum}

By default, Sledgehammer will run E, SPASS, Vampire, SInE-E, and \textit{smt} in
parallel---either locally or remotely, depending on the number of processor
cores available. For historical reasons, the default value of this option can be
overridden using the option ``Sledgehammer: Provers'' from the ``Isabelle'' menu
in Proof General.

It is a good idea to run several provers in parallel, although it could slow
down your machine. Running E, SPASS, Vampire, and SInE-E together for 5 seconds
yields a better success rate than running the most effective of these (Vampire)
for 120 seconds \cite{boehme-nipkow-2010}.

\opnodefault{prover}{string}
Alias for \textit{provers}.

\opnodefault{atps}{string}
Legacy alias for \textit{provers}.

\opnodefault{atp}{string}
Legacy alias for \textit{provers}.

\opdefault{timeout}{float\_or\_none}{\upshape 30}
Specifies the maximum number of seconds that the automatic provers should spend
searching for a proof. For historical reasons, the default value of this option
can be overridden using the option ``Sledgehammer: Time Limit'' from the
``Isabelle'' menu in Proof General.

\opfalse{blocking}{non\_blocking}
Specifies whether the \textbf{sledgehammer} command should operate
synchronously. The asynchronous (non-blocking) mode lets the user start proving
the putative theorem manually while Sledgehammer looks for a proof, but it can
also be more confusing.

\opfalse{overlord}{no\_overlord}
Specifies whether Sledgehammer should put its temporary files in
\texttt{\$ISA\-BELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
debugging Sledgehammer but also unsafe if several instances of the tool are run
simultaneously. The files are identified by the prefix \texttt{prob\_}; you may
safely remove them after Sledgehammer has run.

\nopagebreak
{\small See also \textit{debug} (\S\ref{output-format}).}
\end{enum}

\subsection{Problem Encoding}
\label{problem-encoding}

\begin{enum}
\opfalse{explicit\_apply}{implicit\_apply}
Specifies whether function application should be encoded as an explicit
``apply'' operator in ATP problems. If the option is set to \textit{false}, each
function will be directly applied to as many arguments as possible. Enabling
this option can sometimes help discover higher-order proofs that otherwise would
not be found.

\opfalse{full\_types}{partial\_types}
Specifies whether full-type information is encoded in ATP problems. Enabling
this option can prevent the discovery of type-incorrect proofs, but it also
tends to slow down the ATPs significantly. For historical reasons, the default
value of this option can be overridden using the option ``Sledgehammer: Full
Types'' from the ``Isabelle'' menu in Proof General.
\end{enum}

\subsection{Relevance Filter}
\label{relevance-filter}

\begin{enum}
\opdefault{relevance\_thresholds}{float\_pair}{\upshape 0.45~0.85}
Specifies the thresholds above which facts are considered relevant by the
relevance filter. The first threshold is used for the first iteration of the
relevance filter and the second threshold is used for the last iteration (if it
is reached). The effective threshold is quadratically interpolated for the other
iterations. Each threshold ranges from 0 to 1, where 0 means that all theorems
are relevant and 1 only theorems that refer to previously seen constants.

\opsmart{max\_relevant}{int\_or\_smart}
Specifies the maximum number of facts that may be returned by the relevance
filter. If the option is set to \textit{smart}, it is set to a value that was
empirically found to be appropriate for the prover. A typical value would be
300.
\end{enum}

\subsection{Output Format}
\label{output-format}

\begin{enum}

\opfalse{verbose}{quiet}
Specifies whether the \textbf{sledgehammer} command should explain what it does.

\opfalse{debug}{no\_debug}
Specifies whether Sledgehammer should display additional debugging information
beyond what \textit{verbose} already displays. Enabling \textit{debug} also
enables \textit{verbose} behind the scenes.

\nopagebreak
{\small See also \textit{overlord} (\S\ref{mode-of-operation}).}

\opfalse{isar\_proof}{no\_isar\_proof}
Specifies whether Isar proofs should be output in addition to one-liner
\textit{metis} proofs. Isar proof construction is still experimental and often
fails; however, they are usually faster and sometimes more robust than
\textit{metis} proofs.

\opdefault{isar\_shrink\_factor}{int}{\upshape 1}
Specifies the granularity of the Isar proof. A value of $n$ indicates that each
Isar proof step should correspond to a group of up to $n$ consecutive proof
steps in the ATP proof.

\end{enum}

\subsection{Authentication}
\label{authentication}

\begin{enum}
\opnodefault{expect}{string}
Specifies the expected outcome, which must be one of the following:

\begin{enum}
\item[$\bullet$] \textbf{\textit{some}:} Sledgehammer found a (potentially
unsound) proof.
\item[$\bullet$] \textbf{\textit{none}:} Sledgehammer found no proof.
\item[$\bullet$] \textbf{\textit{unknown}:} Sledgehammer encountered some
problem.
\end{enum}

Sledgehammer emits an error (if \textit{blocking} is enabled) or a warning
(otherwise) if the actual outcome differs from the expected outcome. This option
is useful for regression testing.

\nopagebreak
{\small See also \textit{blocking} (\S\ref{mode-of-operation}).}
\end{enum}

\let\em=\sl
\bibliography{../manual}{}
\bibliographystyle{abbrv}

\end{document}