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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{}}

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 \section{Introduction}

 \label{introduction}

 Sledgehammer is a tool that applies first-order automatic theorem provers (ATPs)

 and satisfiability-modulo-theories (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}, SNARK

 \cite{snark}, and ToFoF-E \cite{tofof}. 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 Yices \cite{yices} and CVC3 \cite{cvc3} as well; these

 are run either locally or on a server in Munich.

 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}}

 \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,

 SInE-E, SNARK, and ToFoF-E 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 and 1.2, 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., ``1.2'').

 \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:pAsSwRd@proxy.example.org}

 \postw

 \subsection{Installing SMT Solvers}

 CVC3, Yices, and Z3 can be run locally or remotely on a Munich 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} ``$[a] = [b] \,\longleftrightarrow\, a = b$'' \\

 \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}). \\[3\smallskipamount]

 %

 Sledgehammer: ``\textit{remote\_z3}'' 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 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.

 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 (\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 \textit{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]{{\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}. 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.

 \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}'').

 \item[$\bullet$] \textbf{\textit{timeout}} (\S\ref{mode-of-operation}) 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.

 \item[$\bullet$] \textbf{\textit{full\_types}} (\S\ref{problem-encoding})

 specifies whether type-sound encodings should be used. By default, Sledgehammer

 employs a mixture of type-sound and type-unsound encodings, occasionally

 yielding unsound ATP proofs. (SMT solver proofs should always be sound, although

 we occasionally find soundness bugs in the solvers.)

 \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}).

 \end{enum}

 Options can be set globally using \textbf{sledgehammer\_params}

 (\S\ref{command-syntax}). 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}

 \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 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 \textit{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

 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. This usually indicates that

 Sledgehammer found a type-incorrect proof. Sledgehammer erases some type

 information to speed up the search. Try Sledgehammer again with full type

 information: \textit{full\_types} (\S\ref{problem-encoding}), or choose a

 specific type encoding with \textit{type\_sys} (\S\ref{problem-encoding}). Older

 versions of Sledgehammer were frequent victims of this problem. Now this should

 very seldom be an issue, but if you notice many unsound proofs, contact the

 author at \authoremail.

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

 First, if \emph{metis} (or \emph{metisFT}) can reconstruct it, the proof is

 sound (modulo soundness of Isabelle's inference kernel). If it fails or runs

 seemingly forever, you can try

 \prew

 \textbf{apply}~\textbf{--} \\

 \textbf{sledgehammer} [\textit{type\_sys} = \textit{poly\_tags}] (\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 refind the proof very

 quickly if it is sound, and the \textit{type\_sys} $=$ \textit{poly\_tags}

 option (\S\ref{problem-encoding}) ensures that no unsound proofs are found.

 The \textit{full\_types} option (\S\ref{problem-encoding}) can also be used

 here, but it is unsound in extremely rare degenerate cases such as the

 following:

 \prew

 \textbf{lemma} ``$\forall x\> y\Colon{'}a.\ x = y \,\Longrightarrow \exists f\> g\Colon\mathit{nat} \Rightarrow {'}a.\ f \not= g$'' \\

 \textbf{sledgehammer} [\textit{full\_types}] (\textit{nat.distinct\/}(1))

 \postw

 \point{How does Sledgehammer select the facts that should be passed to the

 automatic provers?}

 Briefly, 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 quicker 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, 50 or 100.

 \point{Why are the Isar proofs generated by Sledgehammer so ugly?}

 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{Should I let Sledgehammer minimize the number of lemmas?}

 In general, minimization is a good idea, because proofs involving fewer lemmas

 tend to be shorter as well, and hence easier to re-find by Metis. But the

 opposite is sometimes the case.

 \point{Why does the minimizer sometimes starts of its own?}

 There are two scenarios in which this can happen. First, some provers (e.g.,

 CVC3 and Yices) do not provide proofs or 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. Second, if a

 prover returns a proof with lots of facts, the minimizer is invoked

 automatically since Metis is unlikely to refind the proof.

 \point{What is metisFT?}

 The \textit{metisFT} proof method is the fully-typed version of Metis. It is

 much 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.

 If you see the warning

 \prew

 \textsl

 Metis: Falling back on ``\textit{metisFT}''.

 \postw

 in a successful Metis proof, you can advantageously replace the \textit{metis}

 call with \textit{metisFT}.

 \point{I got a strange error from Sledgehammer---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}'', ``\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} \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{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 \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, fewer facts are passed to the prover,

 \textit{slicing} (\S\ref{mode-of-operation}) is disabled, \textit{timeout}

 (\S\ref{mode-of-operation}) is superseded by the ``Auto Tools Time Limit'' in

 Proof General's ``Isabelle'' menu, \textit{full\_types}

 (\S\ref{problem-encoding}) is enabled, and \textit{verbose}

 (\S\ref{output-format}) and \textit{debug} (\S\ref{output-format}) are disabled.

 Sledgehammer's output is also more concise.

 \section{Option Reference}

 \label{option-reference}

 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}

 \def\qty#1{$\left<\textit{#1}\right>$}

 \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\/}: 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{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. Sledgehammer has been tested with

 versions 11, 0.6, and 1.0.

 \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{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{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. 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{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\_tofof\_e}:} ToFoF-E is a metaprover

 developed by Geoff Sutcliffe \cite{tofof} based on E running on his Miami

 servers. This ATP supports a fragment of the TPTP many-typed first-order format

 (TFF). It is supported primarily for experimenting with the

 \textit{type\_sys} $=$ \textit{simple} option (\S\ref{problem-encoding}).

 \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\_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\_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, 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. 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.

 \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.

 \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}

 \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 prevents the discovery of type-incorrect proofs, but it can slow

 down the ATP slightly. This option is implicitly enabled for automatic runs. 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.

 \opdefault{type\_sys}{string}{smart}

 Specifies the type system to use in ATP problems. The option can take the

 following values:

 \begin{enum}

 \item[$\bullet$] \textbf{\textit{poly\_preds}:} Types are encoded using a predicate

 $\mathit{has\_type\/}(\tau, t)$ 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}:} Each term and subterm is tagged with

 its type using a function $\mathit{type\_info\/}(\tau, t)$.

 \item[$\bullet$] \textbf{\textit{poly\_args}:}

 Like for the other sound encodings, constants are annotated with their types to

 resolve overloading, but otherwise no type information is encoded.

 \item[$\bullet$] \textbf{\textit{erased}:} No type information is supplied to

 the ATP. Types are simply erased.

 \item[$\bullet$]

 \textbf{%

 \textit{mono\_preds},

 \textit{mono\_tags},

 \textit{mono\_args}:} \\

 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{simple}:} Use the prover's support for simply

 typed first-order logic if available; otherwise, fall back on

 \textit{mangled\_preds}. The problem is monomorphized.

 \item[$\bullet$]

 \textbf{%

 \textit{mangled\_preds},

 \textit{mangled\_tags},

 \textit{mangled\_args}:} \\

 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{mono\_preds}?, \textit{mono\_tags}?, \textit{simple}?, \\

 \textit{mangled\_preds}?, \textit{mangled\_tags}?:} \\

 The type systems \textit{mono\_preds}, \textit{mono\_tags}, \textit{simple},

 \textit{mangled\_preds}, and \textit{mangled\_tags} are fully typed and

 virtually sound---except for pathological cases, all found proofs are

 type-correct. For each of these, Sledgehammer also provides a lighter (but

 virtually sound) variant identified by a question mark (`{?}')\ that detects and

 erases monotonic types, notably infinite types. (For \textit{simple}, the types

 are not actually erased but rather replaced by a shared uniform type of

 individuals.)

 \item[$\bullet$]

 \textbf{%

 \textit{poly\_tags}!, \textit{mono\_preds}!, \textit{mono\_tags}!, \\

 \textit{simple}!, \textit{mangled\_preds}!, \textit{mangled\_tags}!:} \\

 The type systems \textit{poly\_preds}, \textit{poly\_tags},

 \textit{mono\_preds}, \textit{mono\_tags}, \textit{simple},

 \textit{mangled\_preds}, and \textit{mangled\_tags} also admit a somewhat

 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}, the types are not actually erased

 but rather replaced by a shared uniform type of individuals.)

 \item[$\bullet$] \textbf{\textit{smart}:} If \textit{full\_types} is enabled,

 uses a fully typed, virtually sound encoding; otherwise, uses any encoding. The

 actual encoding used depends on the ATP and should be the most efficient for

 that ATP.

 \end{enum}

 In addition, all the \textit{preds} and \textit{tags} type systems 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 and ToFoF-E, the type system is always \textit{simple},

 irrespective of the value of this option.

 \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.

 \opdefault{max\_new\_mono\_instances}{int}{\upshape 400}

 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 system used.

 \nopagebreak

 {\small See also \textit{type\_sys} (\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 system used.

 \nopagebreak

 {\small See also \textit{type\_sys} (\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{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}