doc-src/Sledgehammer/sledgehammer.tex

clarify fine soundness point

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\begin{document}

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\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} \\[4\smallskipamount]
{\normalsize with contributions from} \\[4\smallskipamount]
Lawrence C. Paulson \\
{\normalsize Computer Laboratory, University of Cambridge} \\
\hbox{}}

\maketitle

\tableofcontents

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\section{Introduction}
\label{introduction}

Sledgehammer is a tool that applies automatic theorem provers (ATPs)
and satisfiability-modulo-theories (SMT) solvers on the current goal. The
supported ATPs are E \cite{schulz-2002}, LEO-II \cite{leo2}, Satallax
\cite{satallax}, SInE-E \cite{sine}, SNARK \cite{snark}, SPASS
\cite{weidenbach-et-al-2009}, ToFoF-E \cite{tofof}, Vampire
\cite{riazanov-voronkov-2002}, and Waldmeister \cite{waldmeister}. The ATPs are
run either locally or remotely via the System\-On\-TPTP web service
\cite{sutcliffe-2000}. In addition to the ATPs, the SMT solvers Z3 \cite{z3} is
used by default, and you can tell Sledgehammer to try CVC3 \cite{cvc3} and Yices
\cite{yices} as well; these are run either locally or on a server at the TU
M\"unchen.

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, 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}}

\newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
in.\allowbreak tum.\allowbreak de}}

To run Sledgehammer, you must make sure that the theory \textit{Sledgehammer} is
imported---this is rarely a problem in practice since it is part of
\textit{Main}. 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 the author at \authoremail.

\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 SMT solvers.

\subsection{Installing ATPs}

Currently, E, SPASS, and Vampire can be run locally; in addition, E, Vampire,
LEO-II, Satallax, SInE-E, SNARK, ToFoF-E, and Waldmeister are available remotely
via System\-On\-TPTP \cite{sutcliffe-2000}. If you want better performance, you
should at least 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{\$ISABELLE\_HOME\_USER/etc/components}%
\footnote{The variable \texttt{\$ISABELLE\_HOME\_USER} is set by Isabelle at
startup. Its value can be retrieved by invoking \texttt{isabelle}
\texttt{getenv} \texttt{ISABELLE\_HOME\_USER} on the command line.}
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 to 1.3, SPASS 3.5 and 3.7, and Vampire 0.6 and 1.0%
\footnote{Following the rewrite of Vampire, the counter for version numbers was
reset to 0; hence the (new) Vampire versions 0.6 and 1.0 are more recent than,
say, 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. Ideally,
also set \texttt{E\_VERSION}, \texttt{SPASS\_VERSION}, or
\texttt{VAMPIRE\_VERSION} to the ATP's version number (e.g., ``1.3'').
\end{enum}

To check whether E and SPASS are successfully installed, follow the example in
\S\ref{first-steps}. If the remote versions of E and SPASS are used (identified
by the prefix ``\emph{remote\_}''), or if the local versions fail to solve the
easy goal presented there, this is a sign that something is wrong with your
installation.

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\_HOME\_USER/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

\subsection{Installing SMT Solvers}

CVC3, Yices, and Z3 can be run locally or (for CVC3 and Z3) remotely on a TU
M\"unchen server. If you want better performance and get the ability to replay
proofs that rely on the \emph{smt} proof method, you should at least install Z3
locally.

There are two main ways of installing SMT solvers locally.

\begin{enum}
\item[$\bullet$] If you installed an official Isabelle package with everything
inside, it should already include properly setup executables for CVC3 and Z3,
ready to use.%
\footnote{Yices's license prevents us from doing the same for this otherwise
wonderful tool.}
For Z3, you additionally need to set the environment variable
\texttt{Z3\_NON\_COMMERCIAL} to ``yes'' to confirm that you are a noncommercial
user.

\item[$\bullet$] Otherwise, follow the instructions documented in the \emph{SMT}
theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}).
\end{enum}

\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} ``$[a] = [b] \,\Longrightarrow\, a = b$'' \\
\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}'' on goal \\
$[a] = [b] \,\Longrightarrow\, a = b$ \\
Try this: \textbf{by} (\textit{metis last\_ConsL}) (64 ms). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{vampire}'' on goal \\
$[a] = [b] \,\Longrightarrow\, a = b$ \\
Try this: \textbf{by} (\textit{metis hd.simps}) (14 ms). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{spass}'' on goal \\
$[a] = [b] \,\Longrightarrow\, a = b$ \\
Try this: \textbf{by} (\textit{metis list.inject}) (17 ms). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{remote\_waldmeister}'' on goal \\
$[a] = [b] \,\Longrightarrow\, a = b$ \\
Try this: \textbf{by} (\textit{metis hd.simps}) (15 ms). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{remote\_sine\_e}'' on goal \\
$[a] = [b] \,\Longrightarrow\, a = b$ \\
Try this: \textbf{by} (\textit{metis hd.simps}) (18 ms). \\[3\smallskipamount]
%
Sledgehammer: ``\textit{remote\_z3}'' on goal \\
$[a] = [b] \,\Longrightarrow\, a = b$ \\
Try this: \textbf{by} (\textit{metis list.inject}) (20 ms).
\postw

Sledgehammer ran E, SInE-E, SPASS, Vampire, Waldmeister, and Z3 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. Waldmeister is run only for unit equational problems,
where the goal's conclusion is a (universally quantified) equation.

For each successful prover, Sledgehammer gives a one-liner proof that uses Metis
or the \textit{smt} proof method. For Metis, approximate timings are shown in
parentheses, indicating how fast the call is. You can click the proof to insert
it into the theory text.

In addition, you can ask Sledgehammer for an Isar text proof by passing the
\textit{isar\_proof} option (\S\ref{output-format}):

\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 Metis one-liners. This feature is experimental
and is only available for ATPs.

\section{Hints}
\label{hints}

This section presents a few hints that should help you get the most out of
Sledgehammer and Metis. Frequently (and infrequently) asked questions are
answered in \S\ref{frequently-asked-questions}.

\newcommand\point[1]{\medskip\par{\sl\bfseries#1}\par\nopagebreak}

\point{Presimplify the goal}

For best results, first simplify your problem by calling \textit{auto} or at
least \textit{safe} followed by \textit{simp\_all}. The SMT solvers provide
arithmetic decision procedures, but the ATPs typically do not (or if they do,
Sledgehammer does not use it yet). Apart from Waldmeister, 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.

\point{Make sure at least E, SPASS, Vampire, and Z3 are installed}

Locally installed provers are faster and more reliable than those running on
servers. See \S\ref{installation} for details on how to install them.

\point{Familiarize yourself with the most important options}

Sledgehammer's options are fully documented in \S\ref{command-syntax}. Many of
the options are very specialized, but serious users of the tool should at least
familiarize themselves with the following options:

\begin{enum}
\item[$\bullet$] \textbf{\textit{provers}} (\S\ref{mode-of-operation}) specifies
the automatic provers (ATPs and SMT solvers) that should be run whenever
Sledgehammer is invoked (e.g., ``\textit{provers}~= \textit{e spass
remote\_vampire}''). For convenience, you can omit ``\textit{provers}~=''
and simply write the prover names as a space-separated list (e.g., ``\textit{e
spass remote\_vampire}'').

\item[$\bullet$] \textbf{\textit{max\_relevant}} (\S\ref{relevance-filter})
specifies the maximum number of facts that should be passed to the provers. By
default, the value is prover-dependent but varies between about 150 and 1000. If
the provers time out, you can try lowering this value to, say, 100 or 50 and see
if that helps.

\item[$\bullet$] \textbf{\textit{isar\_proof}} (\S\ref{output-format}) specifies
that Isar proofs should be generated, instead of one-liner Metis proofs. The
length of the Isar proofs can be controlled by setting
\textit{isar\_shrink\_factor} (\S\ref{output-format}).

\item[$\bullet$] \textbf{\textit{timeout}} (\S\ref{timeouts}) controls the
provers' time limit. It is set to 30 seconds, but since Sledgehammer runs
asynchronously you should not hesitate to raise this limit to 60 or 120 seconds
if you are the kind of user who can think clearly while ATPs are active.
\end{enum}

Options can be set globally using \textbf{sledgehammer\_params}
(\S\ref{command-syntax}). The command also prints the list of all available
options with their current value. Fact selection can be influenced by specifying
``$(\textit{add}{:}~\textit{my\_facts})$'' after the \textbf{sledgehammer} call
to ensure that certain facts are included, or simply ``$(\textit{my\_facts})$''
to force Sledgehammer to run only with $\textit{my\_facts}$.

\section{Frequently Asked Questions}
\label{frequently-asked-questions}

This sections answers frequently (and infrequently) asked questions about
Sledgehammer. It is a good idea to skim over it now even if you don't have any
questions at this stage. And if you have any further questions not listed here,
send them to the author at \authoremail.

\point{Why does Metis fail to reconstruct the proof?}

There are many reasons. If Metis runs seemingly forever, that is a sign that the
proof is too difficult for it. Metis's search is complete, so it should
eventually find it, but that's little consolation. There are several possible
solutions:

\begin{enum}
\item[$\bullet$] Try the \textit{isar\_proof} option (\S\ref{output-format}) to
obtain a step-by-step Isar proof where each step is justified by Metis. Since
the steps are fairly small, Metis is more likely to be able to replay them.

\item[$\bullet$] Try the \textit{smt} proof method instead of Metis. It is
usually stronger, but you need to have Z3 available to replay the proofs, trust
the SMT solver, or use certificates. See the documentation in the \emph{SMT}
theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}) for details.

\item[$\bullet$] Try the \textit{blast} or \textit{auto} proof methods, passing
the necessary facts via \textbf{unfolding}, \textbf{using}, \textit{intro}{:},
\textit{elim}{:}, \textit{dest}{:}, or \textit{simp}{:}, as appropriate.
\end{enum}

In some rare cases, Metis fails fairly quickly, and you get the error message

\prew
\slshape
Proof reconstruction failed.
\postw

This message usually indicates that Sledgehammer found a type-incorrect proof.
This was a frequent issue with older versions of Sledgehammer, which did not
supply enough typing information to the ATPs by default. If you notice many
unsound proofs and are not using \textit{type\_enc} (\S\ref{problem-encoding}),
contact the author at \authoremail.

\point{How can I tell whether a generated proof is sound?}

First, if Metis can reconstruct it, the proof is sound (assuming Isabelle's
inference kernel is sound). If it fails or runs seemingly forever, you can try

\prew
\textbf{apply}~\textbf{--} \\
\textbf{sledgehammer} [\textit{sound}] (\textit{metis\_facts})
\postw

where \textit{metis\_facts} is the list of facts appearing in the suggested
Metis call. The automatic provers should be able to re-find the proof quickly if
it is sound, and the \textit{sound} option (\S\ref{problem-encoding}) ensures
that no unsound proofs are found.

\point{Which facts are passed to the automatic provers?}

The relevance filter assigns a score to every available fact (lemma, theorem,
definition, or axiom)\ based upon how many constants that fact shares with the
conjecture. This process iterates to include facts relevant to those just
accepted, but with a decay factor to ensure termination. The constants are
weighted to give unusual ones greater significance. The relevance filter copes
best when the conjecture contains some unusual constants; if all the constants
are common, it is unable to discriminate among the hundreds of facts that are
picked up. The relevance filter is also memoryless: It has no information about
how many times a particular fact has been used in a proof, and it cannot learn.

The number of facts included in a problem varies from prover to prover, since
some provers get overwhelmed more easily than others. You can show the number of
facts given using the \textit{verbose} option (\S\ref{output-format}) and the
actual facts using \textit{debug} (\S\ref{output-format}).

Sledgehammer is good at finding short proofs combining a handful of existing
lemmas. If you are looking for longer proofs, you must typically restrict the
number of facts, by setting the \textit{max\_relevant} option
(\S\ref{relevance-filter}) to, say, 25 or 50.

You can also influence which facts are actually selected in a number of ways. If
you simply want to ensure that a fact is included, you can specify it using the
``$(\textit{add}{:}~\textit{my\_facts})$'' syntax. For example:
%
\prew
\textbf{sledgehammer} (\textit{add}: \textit{hd.simps} \textit{tl.simps})
\postw
%
The specified facts then replace the least relevant facts that would otherwise be
included; the other selected facts remain the same.
If you want to direct the selection in a particular direction, you can specify
the facts via \textbf{using}:
%
\prew
\textbf{using} \textit{hd.simps} \textit{tl.simps} \\
\textbf{sledgehammer}
\postw
%
The facts are then more likely to be selected than otherwise, and if they are
selected at iteration $j$ they also influence which facts are selected at
iterations $j + 1$, $j + 2$, etc. To give them even more weight, try
%
\prew
\textbf{using} \textit{hd.simps} \textit{tl.simps} \\
\textbf{apply}~\textbf{--} \\
\textbf{sledgehammer}
\postw

\point{Why are the generated Isar proofs so ugly/detailed/broken?}

The current implementation is experimental and explodes exponentially in the
worst case. Work on a new implementation has begun. There is a large body of
research into transforming resolution proofs into natural deduction proofs (such
as Isar proofs), which we hope to leverage. In the meantime, a workaround is to
set the \textit{isar\_shrink\_factor} option (\S\ref{output-format}) to a larger
value or to try several provers and keep the nicest-looking proof.

\point{What are the \textit{full\_types} and \textit{no\_types} arguments to
Metis?}

The \textit{metis}~(\textit{full\_types}) proof method is the fully-typed
version of Metis. It is somewhat slower than \textit{metis}, but the proof
search is fully typed, and it also includes more powerful rules such as the
axiom ``$x = \mathit{True} \mathrel{\lor} x = \mathit{False}$'' for reasoning in
higher-order places (e.g., in set comprehensions). The method kicks in
automatically as a fallback when \textit{metis} fails, and it is sometimes
generated by Sledgehammer instead of \textit{metis} if the proof obviously
requires type information or if \textit{metis} failed when Sledgehammer
preplayed the proof. (By default, Sledgehammer tries to run \textit{metis} with
various options for up to 4 seconds to ensure that the generated one-line proofs
actually work and to display timing information. This can be configured using
the \textit{preplay\_timeout} option (\S\ref{timeouts}).)

At the other end of the soundness spectrum, \textit{metis} (\textit{no\_types})
uses no type information at all during the proof search, which is more efficient
but often fails. Calls to \textit{metis} (\textit{no\_types}) are occasionally
generated by Sledgehammer.

Incidentally, if you see the warning

\prew
\slshape
Metis: Falling back on ``\textit{metis} (\textit{full\_types})''.
\postw

in a successful Metis proof, you can advantageously pass the
\textit{full\_types} option to \textit{metis} directly.

\point{Are generated proofs minimal?}

Automatic provers frequently use many more facts than are necessary.
Sledgehammer inclues a minimization tool that takes a set of facts returned by a
given prover and repeatedly calls the same prover or Metis with subsets of those
axioms in order to find a minimal set. Reducing the number of axioms typically
improves Metis's speed and success rate, while also removing superfluous clutter
from the proof scripts.

In earlier versions of Sledgehammer, generated proofs were systematically
accompanied by a suggestion to invoke the minimization tool. This step is now
performed implicitly if it can be done in a reasonable amount of time (something
that can be guessed from the number of facts in the original proof and the time
it took to find it or replay it).

In addition, some provers (notably CVC3, Satallax, and Yices) do not provide
proofs or sometimes provide incomplete proofs. The minimizer is then invoked to
find out which facts are actually needed from the (large) set of facts that was
initinally given to the prover. Finally, if a prover returns a proof with lots
of facts, the minimizer is invoked automatically since Metis would be unlikely
to re-find the proof.

\point{A strange error occurred---what should I do?}

Sledgehammer tries to give informative error messages. Please report any strange
error to the author at \authoremail. This applies double if you get the message

\prew
\slshape
The prover found a type-unsound proof involving ``\textit{foo}'',
``\textit{bar}'', and ``\textit{baz}'' even though a supposedly type-sound
encoding was used (or, less likely, your axioms are inconsistent). You might
want to report this to the Isabelle developers.
\postw

\point{Auto can solve it---why not Sledgehammer?}

Problems can be easy for \textit{auto} and difficult for automatic provers, 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.

\point{Why are there so many options?}

Sledgehammer's philosophy should work out of the box, without user guidance.
Many of the options are meant to be used mostly by the Sledgehammer developers
for experimentation purposes. Of course, feel free to experiment with them if
you are so inclined.

\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} \qty{subcommand}$^?$ \qty{options}$^?$ \qty{facts\_override}$^?$ \qty{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 \qty{subcommand} may be any of the following:

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

\item[$\bullet$] \textbf{\textit{min}:} Attempts to minimize the facts
specified in the \qty{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 \qty{num} argument specifies a
limit on the number of messages to display (5 by default).

\item[$\bullet$] \textbf{\textit{supported\_provers}:} Prints the list of
automatic provers supported by Sledgehammer. 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 \qty{options}, which can be
specified in brackets after the \textbf{sledgehammer} command. The
\qty{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]
\postw

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

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

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

The \qty{facts\_override} argument lets you alter the set of facts that go
through the relevance filter. It may be of the form ``(\qty{facts})'', where
\qty{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}:\ \qty{facts\/_{\mathrm{1}}})'',
``(\textit{del}:\ \qty{facts\/_{\mathrm{2}}})'', or ``(\textit{add}:\ \qty{facts\/_{\mathrm{1}}}\
\textit{del}:\ \qty{facts\/_{\mathrm{2}}})'', where the relevance filter is instructed to
proceed as usual except that it should consider \qty{facts\/_{\mathrm{1}}}
highly-relevant and \qty{facts\/_{\mathrm{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, fewer facts are passed to the prover,
\textit{slicing} (\S\ref{mode-of-operation}) is disabled, \textit{sound}
(\S\ref{problem-encoding}) is enabled, \textit{verbose} (\S\ref{output-format})
and \textit{debug} (\S\ref{output-format}) are disabled, and \textit{timeout}
(\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in Proof
General's ``Isabelle'' menu. Sledgehammer's output is also more concise.

The \textit{metis} proof method has the syntax

\prew
\textbf{\textit{metis}}~(\qty{type\_enc})${}^?$~\qty{facts}${}^?$
\postw

where \qty{type\_enc} is a type encoding specification with the same semantics
as Sledgehammer's \textit{type\_enc} option (\S\ref{problem-encoding}) and
\qty{facts} is a list of arbitrary facts. In addition to the values listed in
\S\ref{problem-encoding}, \qty{type\_enc} may also be \textit{full\_types}, in
which case an appropriate type-sound encoding is chosen, \textit{partial\_types}
(the default type-unsound encoding), or \textit{no\_types}, a synonym for
\textit{erased}.

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

\def\defl{\{}
\def\defr{\}}

\def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
\def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{true}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{false}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\enskip \defl\textit{smart}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
\def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
\def\opnodefaultbrk#1#2{\flushitem{$\bigl[$\textit{#1} =$\bigr]$ \qtybf{#2}} \nopagebreak\\[\parskip]}
\def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\enskip \defl\textit{#3}\defr} \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{smart\_bool}$\bigr]$\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}), authentication (\S\ref{authentication}), and timeouts
(\S\ref{timeouts}).

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{smart\_bool\/}: \textit{true}, \textit{false}, or
\textit{smart}.
\item[$\bullet$] \qtybf{int\/}: An integer.
%\item[$\bullet$] \qtybf{float\/}: A floating-point number (e.g., 2.5).
\item[$\bullet$] \qtybf{float\_pair\/}: A pair of floating-point numbers
(e.g., 0.6 0.95).
\item[$\bullet$] \qtybf{smart\_int\/}: An integer or \textit{smart}.
\item[$\bullet$] \qtybf{float\_or\_none\/}: A floating-point number (e.g., 60 or
0.5) expressing a number of seconds, or the keyword \textit{none} ($\infty$
seconds).
\end{enum}

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

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

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

\begin{enum}
\item[$\bullet$] \textbf{\textit{cvc3}:} CVC3 is an SMT solver developed by
Clark Barrett, Cesare Tinelli, and their colleagues \cite{cvc3}. To use CVC3,
set the environment variable \texttt{CVC3\_SOLVER} to the complete path of the
executable, including the file name. Sledgehammer has been tested with version
2.2.

\item[$\bullet$] \textbf{\textit{e}:} E is a first-order resolution prover
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 a first-order resolution
prover 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{yices}:} Yices is an SMT solver developed at
SRI \cite{yices}. To use Yices, set the environment variable
\texttt{YICES\_SOLVER} to the complete path of the executable, including the
file name. Sledgehammer has been tested with version 1.0.

\item[$\bullet$] \textbf{\textit{vampire}:} Vampire is a first-order resolution
prover 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. Sledgehammer has been tested with versions 11, 0.6, and 1.0.

\item[$\bullet$] \textbf{\textit{z3}:} Z3 is an SMT solver developed at
Microsoft Research \cite{z3}. To use Z3, set the environment variable
\texttt{Z3\_SOLVER} to the complete path of the executable, including the file
name, and set \texttt{Z3\_NON\_COMMERCIAL=yes} to confirm that you are a
noncommercial user. Sledgehammer has been tested with versions 2.7 to 2.18.

\item[$\bullet$] \textbf{\textit{z3\_atp}:} This version of Z3 pretends to be an
ATP, exploiting Z3's undocumented support for the TPTP format. It is included
for experimental purposes. It requires version 2.18 or above.

\item[$\bullet$] \textbf{\textit{metis}:} Although it is much less powerful than
the external provers, Metis itself can be used for proof search.

\item[$\bullet$] \textbf{\textit{metis\_full\_types}:} Fully typed version of
Metis, corresponding to \textit{metis} (\textit{full\_types}).

\item[$\bullet$] \textbf{\textit{metis\_no\_types}:} Untyped version of Metis,
corresponding to \textit{metis} (\textit{no\_types}).
\end{enum}

In addition, the following remote provers are supported:

\begin{enum}
\item[$\bullet$] \textbf{\textit{remote\_cvc3}:} The remote version of CVC3 runs
on servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to
point).

\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\_leo2}:} LEO-II is an automatic
higher-order prover developed by Christoph Benzm\"uller et al.\ \cite{leo2},
with support for the TPTP higher-order syntax (THF). The remote version of
LEO-II runs on Geoff Sutcliffe's Miami servers. In the current setup, the
problems given to LEO-II are only mildly higher-order.

\item[$\bullet$] \textbf{\textit{remote\_satallax}:} Satallax is an automatic
higher-order prover developed by Chad Brown et al.\ \cite{satallax}, with
support for the TPTP higher-order syntax (THF). The remote version of Satallax
runs on Geoff Sutcliffe's Miami servers. In the current setup, the problems
given to Satallax are only mildly higher-order.

\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 first-order
resolution prover developed by Stickel et al.\ \cite{snark}. It supports the
TPTP many-typed first-order format (TFF). The remote version of SNARK runs on
Geoff Sutcliffe's Miami servers.

\item[$\bullet$] \textbf{\textit{remote\_tofof\_e}:} ToFoF-E is a metaprover
developed by Geoff Sutcliffe \cite{tofof} based on E running on his Miami
servers. This ATP supports the TPTP many-typed first-order format (TFF). The
remote version of ToFoF-E runs on Geoff Sutcliffe's Miami servers.

\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\_waldmeister}:} Waldmeister is a unit
equality prover developed by Hillenbrand et al.\ \cite{waldmeister}. It can be
used to prove universally quantified equations using unconditional equations,
corresponding to the TPTP CNF UEQ division. The remote version of Waldmeister
runs on Geoff Sutcliffe's Miami servers.

\item[$\bullet$] \textbf{\textit{remote\_z3}:} The remote version of Z3 runs on
servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to
point).

\item[$\bullet$] \textbf{\textit{remote\_z3\_atp}:} The remote version of ``Z3
as an ATP'' runs on Geoff Sutcliffe's Miami servers.
\end{enum}

By default, Sledgehammer will run E, SPASS, Vampire, SInE-E, and Z3 (or whatever
the SMT module's \textit{smt\_solver} configuration option is set to) 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, and Vampire for 5~seconds yields a similar
success rate to running the most effective of these for 120~seconds
\cite{boehme-nipkow-2010}.

For the \textit{min} subcommand, the default prover is \textit{metis}. If
several provers are set, the first one is used.

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

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

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

\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. Irrespective of the value of this option, Sledgehammer
is always run synchronously for the new jEdit-based user interface or if
\textit{debug} (\S\ref{output-format}) is enabled.

\optrue{slicing}{no\_slicing}
Specifies whether the time allocated to a prover should be sliced into several
segments, each of which has its own set of possibly prover-dependent options.
For SPASS and Vampire, the first slice tries the fast but incomplete
set-of-support (SOS) strategy, whereas the second slice runs without it. For E,
up to three slices are tried, with different weighted search strategies and
number of facts. For SMT solvers, several slices are tried with the same options
each time but fewer and fewer facts. According to benchmarks with a timeout of
30 seconds, slicing is a valuable optimization, and you should probably leave it
enabled unless you are conducting experiments. This option is implicitly
disabled for (short) automatic runs.

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

\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}
\opdefault{type\_enc}{string}{smart}
Specifies the type encoding to use in ATP problems. Some of the type encodings
are unsound, meaning that they can give rise to spurious proofs
(unreconstructible using Metis). The supported type encodings are listed below,
with an indication of their soundness in parentheses:

\begin{enum}
\item[$\bullet$] \textbf{\textit{erased} (very unsound):} No type information is
supplied to the ATP. Types are simply erased.

\item[$\bullet$] \textbf{\textit{poly\_preds} (sound):} Types are encoded using
a predicate \textit{has\_\allowbreak type\/}$(\tau, t)$ that restricts the range
of bound variables. Constants are annotated with their types, supplied as extra
arguments, to resolve overloading.

\item[$\bullet$] \textbf{\textit{poly\_tags} (sound):} Each term and subterm is
tagged with its type using a function $\mathit{type\_info\/}(\tau, t)$.

\item[$\bullet$] \textbf{\textit{poly\_args} (unsound):}
Like for \textit{poly\_preds} constants are annotated with their types to
resolve overloading, but otherwise no type information is encoded. This
coincides with the default encoding used by the \textit{metis} command.

\item[$\bullet$]
\textbf{%
\textit{mono\_preds}, \textit{mono\_tags} (sound);
\textit{mono\_args} (unsound):} \\
Similar to \textit{poly\_preds}, \textit{poly\_tags}, and \textit{poly\_args},
respectively, but the problem is additionally monomorphized, meaning that type
variables are instantiated with heuristically chosen ground types.
Monomorphization can simplify reasoning but also leads to larger fact bases,
which can slow down the ATPs.

\item[$\bullet$]
\textbf{%
\textit{mangled\_preds},
\textit{mangled\_tags} (sound); \\
\textit{mangled\_args} (unsound):} \\
Similar to
\textit{mono\_preds}, \textit{mono\_tags}, and \textit{mono\_args},
respectively but types are mangled in constant names instead of being supplied
as ground term arguments. The binary predicate $\mathit{has\_type\/}(\tau, t)$
becomes a unary predicate $\mathit{has\_type\_}\tau(t)$, and the binary function
$\mathit{type\_info\/}(\tau, t)$ becomes a unary function
$\mathit{type\_info\_}\tau(t)$.

\item[$\bullet$] \textbf{\textit{simple} (sound):} Exploit simple first-order
types if the prover supports the TFF or THF syntax; otherwise, fall back on
\textit{mangled\_preds}. The problem is monomorphized.

\item[$\bullet$] \textbf{\textit{simple\_higher} (sound):} Exploit simple
higher-order types if the prover supports the THF syntax; otherwise, fall back
on \textit{simple} or \textit{mangled\_preds\_heavy}. The problem is
monomorphized.

\item[$\bullet$]
\textbf{%
\textit{poly\_preds}?, \textit{poly\_tags}?, \textit{mono\_preds}?, \textit{mono\_tags}?, \\
\textit{mangled\_preds}?, \textit{mangled\_tags}?, \textit{simple}?, \textit{simple\_higher}? (quasi-sound):} \\
The type encodings \textit{poly\_preds}, \textit{poly\_tags},
\textit{mono\_preds}, \textit{mono\_tags}, \textit{mangled\_preds},
\textit{mangled\_tags}, \textit{simple}, and \textit{simple\_higher} are fully
typed and sound. For each of these, Sledgehammer also provides a lighter,
virtually sound variant identified by a question mark (`{?}')\ that detects and
erases monotonic types, notably infinite types. (For \textit{simple} and
\textit{simple\_higher}, the types are not actually erased but rather replaced
by a shared uniform type of individuals.) As argument to the \textit{metis}
proof method, the question mark is replaced by a \hbox{``\textit{\_query}''}
suffix. If the \emph{sound} option is enabled, these encodings are fully sound.

\item[$\bullet$]
\textbf{%
\textit{poly\_preds}!, \textit{poly\_tags}!, \textit{mono\_preds}!, \textit{mono\_tags}!, \\
\textit{mangled\_preds}!, \textit{mangled\_tags}!, \textit{simple}!, \textit{simple\_higher}! \\
(mildly unsound):} \\
The type encodings \textit{poly\_preds}, \textit{poly\_tags},
\textit{mono\_preds}, \textit{mono\_tags}, \textit{mangled\_preds},
\textit{mangled\_tags}, \textit{simple}, and \textit{simple\_higher} also admit
a mildly unsound (but very efficient) variant identified by an exclamation mark
(`{!}') that detects and erases erases all types except those that are clearly
finite (e.g., \textit{bool}). (For \textit{simple} and \textit{simple\_higher},
the types are not actually erased but rather replaced by a shared uniform type
of individuals.) As argument to the \textit{metis} proof method, the exclamation
mark is replaced by a \hbox{``\textit{\_bang}''} suffix.

\item[$\bullet$] \textbf{\textit{smart}:} The actual encoding used depends on
the ATP and should be the most efficient virtually sound encoding for that ATP.
\end{enum}

In addition, all the \textit{preds} and \textit{tags} type encodings are
available in two variants, a lightweight and a heavyweight variant. The
lightweight variants are generally more efficient and are the default; the
heavyweight variants are identified by a \textit{\_heavy} suffix (e.g.,
\textit{mangled\_preds\_heavy}{?}).

For SMT solvers, the type encoding is always \textit{simple}, irrespective of
the value of this option.

\nopagebreak
{\small See also \textit{max\_new\_mono\_instances} (\S\ref{relevance-filter})
and \textit{max\_mono\_iters} (\S\ref{relevance-filter}).}

\opfalse{sound}{unsound}
Specifies whether Sledgehammer should run in its fully sound mode. In that mode,
quasi-sound type encodings (which are the default) are made fully sound, at the
cost of some clutter in the generated problems. This option is ignored if
\textit{type\_enc} is explicitly set to an unsound encoding.
\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.

\opdefault{max\_relevant}{smart\_int}{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
250.

\opdefault{max\_new\_mono\_instances}{int}{\upshape 200}
Specifies the maximum number of monomorphic instances to generate beyond
\textit{max\_relevant}. The higher this limit is, the more monomorphic instances
are potentially generated. Whether monomorphization takes place depends on the
type encoding used.

\nopagebreak
{\small See also \textit{type\_enc} (\S\ref{problem-encoding}).}

\opdefault{max\_mono\_iters}{int}{\upshape 3}
Specifies the maximum number of iterations for the monomorphization fixpoint
construction. The higher this limit is, the more monomorphic instances are
potentially generated. Whether monomorphization takes place depends on the
type encoding used.

\nopagebreak
{\small See also \textit{type\_enc} (\S\ref{problem-encoding}).}
\end{enum}

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

\begin{enum}

\opfalse{verbose}{quiet}
Specifies whether the \textbf{sledgehammer} command should explain what it does.
This option is implicitly disabled for automatic runs.

\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} and \textit{blocking} (\S\ref{mode-of-operation})
behind the scenes. The \textit{debug} option is implicitly disabled for
automatic runs.

\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{timeout}:} Sledgehammer timed out.
\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}) and
\textit{timeout} (\S\ref{timeouts}).}
\end{enum}

\subsection{Timeouts}
\label{timeouts}

\begin{enum}
\opdefault{timeout}{float\_or\_none}{\upshape 30}
Specifies the maximum number of seconds that the automatic provers should spend
searching for a proof. This excludes problem preparation and is a soft limit.
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.

\opdefault{preplay\_timeout}{float\_or\_none}{\upshape 4}
Specifies the maximum number of seconds that Metis should be spent trying to
``preplay'' the found proof. If this option is set to 0, no preplaying takes
place, and no timing information is displayed next to the suggested Metis calls.
\end{enum}

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

\end{document}