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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{}}

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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.%

 \footnote{The distinction between ATPs and SMT solvers is convenient but mostly

 historical. The two communities are converging, with more and more ATPs

 supporting typical SMT features such as arithmetic and sorts, and a few SMT

 solvers parsing ATP syntaxes. There is also a strong technological connection

 between instantiation-based ATPs (such as iProver and iProver-Eq) and SMT

 solvers.}

 %

 The supported ATPs are E \cite{schulz-2002}, E-SInE \cite{sine}, E-ToFoF

 \cite{tofof}, iProver \cite{korovin-2009}, iProver-Eq

 \cite{korovin-sticksel-2010}, LEO-II \cite{leo2}, Satallax \cite{satallax},

 SNARK \cite{snark}, SPASS \cite{weidenbach-et-al-2009}, 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 Alt-Ergo \cite{alt-ergo},

 CVC3 \cite{cvc3}, and Yices \cite{yices} as well; these are run either locally

 or (for CVC3 and Z3) 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 \textit{metis} proof method, which

 integrates the Metis ATP in 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 in Proof General's ``Isabelle'' menu. 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 provers (ATPs and SMT solvers).

 Among the ATPs, E, LEO-II, Satallax, SPASS, and Vampire can be run locally; in

 addition, E, E-SInE, E-ToFoF, iProver, iProver-Eq, LEO-II, Satallax, SNARK,

 Vampire, 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.

 Among the SMT solvers, Alt-Ergo, CVC3, Yices, and Z3 can be run locally, and

 CVC3 and Z3 can be run 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 without an Internet connection, you should at least install Z3

 locally.

 There are three main ways to install automatic provers on your machine:

 \begin{sloppy}

 \begin{enum}

 \item[\labelitemi] If you installed an official Isabelle package, it should

 already include properly setup executables for CVC3, E, SPASS, and Z3, ready to use.%

 \footnote{Vampire's and Yices's licenses prevent us from doing the same for

 these otherwise remarkable tools.}

 For Z3, you must additionally set the variable

 \texttt{Z3\_NON\_COMMERCIAL} to ``yes'' to confirm that you are a

 noncommercial user, either in the environment in which Isabelle is

 launched or in your

 \texttt{\$ISABELLE\_HOME\_USER/etc/settings} file.

 \item[\labelitemi] Alternatively, you can download the Isabelle-aware CVC3, E,

 SPASS, and Z3 binary packages from \download. Extract the archives, then add a

 line to your \texttt{\$ISABELLE\_HOME\_USER\slash etc\slash components}%

 \footnote{The variable \texttt{\$ISABELLE\_HOME\_USER} is set by Isabelle at

 startup. Its value can be retrieved by executing \texttt{isabelle}

 \texttt{getenv} \texttt{ISABELLE\_HOME\_USER} on the command line.}

 file with the absolute path to CVC3, E, SPASS, or Z3. For example, if the

 \texttt{components} file does not exist yet and you extracted SPASS to

 \texttt{/usr/local/spass-3.8ds}, create it with the single line

 \prew

 \texttt{/usr/local/spass-3.8ds}

 \postw

 in it.

 \item[\labelitemi] If you prefer to build E, LEO-II, Satallax, or SPASS

 manually, or found a Vampire executable somewhere (e.g.,

 \url{http://www.vprover.org/}), set the environment variable \texttt{E\_HOME},

 \texttt{LEO2\_HOME}, \texttt{SATALLAX\_HOME}, \texttt{SPASS\_HOME}, or

 \texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{eproof},

 \texttt{leo}, \texttt{satallax}, \texttt{SPASS}, or \texttt{vampire} executable.

 Sledgehammer has been tested with E 1.0 to 1.4, LEO-II 1.3.4, Satallax 2.2, 2.3,

 and 2.4, SPASS 3.8ds, and Vampire 0.6, 1.0, and 1.8.%

 \footnote{Following the rewrite of Vampire, the counter for version numbers was

 reset to 0; hence the (new) Vampire versions 0.6, 1.0, and 1.8 are more recent

 than 9.0 or 11.5.}%

 Since the ATPs' output formats are neither documented nor stable, other

 versions might not work well with Sledgehammer. Ideally,

 you should also set \texttt{E\_VERSION}, \texttt{LEO2\_VERSION},

 \texttt{SATALLAX\_VERSION}, \texttt{SPASS\_VERSION}, or

 \texttt{VAMPIRE\_VERSION} to the prover's version number (e.g., ``1.4'').

 Similarly, if you want to build Alt-Ergo or CVC3, or found a

 Yices or Z3 executable somewhere (e.g.,

 \url{http://yices.csl.sri.com/download.shtml} or

 \url{http://research.microsoft.com/en-us/um/redmond/projects/z3/download.html}),

 set the environment variable \texttt{CVC3\_\allowbreak SOLVER},

 \texttt{YICES\_SOLVER}, or \texttt{Z3\_SOLVER} to the complete path of

 the executable, \emph{including the file name}. Sledgehammer has been tested

 with Alt-Ergo 0.93, CVC3 2.2 and 2.4.1, Yices 1.0.28 and 1.0.33, and Z3 3.0,

 3.1, 3.2, and 4.0. Since the SMT solvers' output formats are somewhat unstable,

 other versions of the solvers might not work well with Sledgehammer. Ideally,

 also set \texttt{CVC3\_VERSION}, \texttt{YICES\_VERSION}, or

 \texttt{Z3\_VERSION} to the solver's version number (e.g., ``4.0'').

 \end{enum}

 \end{sloppy}

 To check whether E, SPASS, Vampire, and/or Z3 are successfully installed, try

 out the example in \S\ref{first-steps}. If the remote versions of any of these

 provers is used (identified by the prefix ``\emph{remote\_\/}''), or if the

 local versions fail to solve the easy goal presented there, something must be

 wrong with the installation.

 Remote prover invocation 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{\$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

 \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{z3\/}'' on goal \\

 $[a] = [b] \,\Longrightarrow\, a = b$ \\

 Try this: \textbf{by} (\textit{metis list.inject}) (20 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\_e\_sine\/}'' on goal \\

 $[a] = [b] \,\Longrightarrow\, a = b$ \\

 Try this: \textbf{by} (\textit{metis hd.simps}) (18 ms).

 \postw

 Sledgehammer ran E, E-SInE, 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

 the \textit{metis} or \textit{smt} proof method. 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 \textit{metis} or \textit{smt} 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. Frequently asked questions are answered in

 \S\ref{frequently-asked-questions}.

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

 \newcommand\point[1]{\subsection{\emph{#1}}}

 \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 E, SPASS, Vampire, and Z3 are locally 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[\labelitemi] \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[\labelitemi] \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[\labelitemi] \textbf{\textit{isar\_proof}} (\S\ref{output-format}) specifies

 that Isar proofs should be generated, instead of one-liner \textit{metis} or

 \textit{smt} proofs. The length of the Isar proofs can be controlled by setting

 \textit{isar\_shrink\_factor} (\S\ref{output-format}).

 \item[\labelitemi] \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{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 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[\labelitemi] Try the \textit{isar\_proof} option (\S\ref{output-format}) to

 obtain a step-by-step Isar proof where each step is justified by \textit{metis}.

 Since the steps are fairly small, \textit{metis} is more likely to be able to

 replay them.

 \item[\labelitemi] Try the \textit{smt} proof method instead of \textit{metis}. It

 is usually stronger, but you need to either 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[\labelitemi] 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, \textit{metis} fails fairly quickly, and you get the error

 message

 \prew

 \slshape

 One-line proof reconstruction failed.

 \postw

 This message indicates that Sledgehammer determined that the goal is provable,

 but the proof is, for technical reasons, beyond \textit{metis}'s power. You can

 then try again with the \textit{strict} option (\S\ref{problem-encoding}).

 If the goal is actually unprovable and you did not specify an unsound encoding

 using \textit{type\_enc} (\S\ref{problem-encoding}), this is a bug, and you are

 strongly encouraged to report this to the author at \authoremail.

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

 The current implementation of the Isar proof feature,

 enabled by the \textit{isar\_proof} option (\S\ref{output-format}),

 is highly experimental. 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{How can I tell whether a suggested proof is sound?}

 Earlier versions of Sledgehammer often suggested unsound proofs---either proofs

 of nontheorems or simply proofs that rely on type-unsound inferences. This

 is a thing of the past, unless you explicitly specify an unsound encoding

 using \textit{type\_enc} (\S\ref{problem-encoding}).

 %

 Officially, the only form of ``unsoundness'' that lurks in the sound

 encodings is related to missing characteristic theorems of datatypes. For

 example,

 \prew

 \textbf{lemma}~``$\exists \mathit{xs}.\; \mathit{xs} \neq []$'' \\

 \textbf{sledgehammer} ()

 \postw

 suggests an argumentless \textit{metis} call that fails. However, the conjecture

 does actually hold, and the \textit{metis} call can be repaired by adding

 \textit{list.distinct}.

 %

 We hope to address this problem in a future version of Isabelle. In the

 meantime, you can avoid it by passing the \textit{strict} option

 (\S\ref{problem-encoding}).

 \point{What are the \textit{full\_types}, \textit{no\_types}, and

 \textit{mono\_tags} arguments to Metis?}

 The \textit{metis}~(\textit{full\_types}) proof method

 and its cousin \textit{metis}~(\textit{mono\_tags}) are 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 ``$x = \const{True} \mathrel{\lor} x = \const{False}$'' 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 3 seconds each time to ensure that the generated

 one-line proofs actually work and to display timing information. This can be

 configured using the \textit{preplay\_timeout} and \textit{dont\_preplay}

 options (\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.

 %

 See the \textit{type\_enc} option (\S\ref{problem-encoding}) for details.

 Incidentally, if you ever see warnings such as

 \prew

 \slshape

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

 \postw

 for a successful \textit{metis} proof, you can advantageously pass the

 \textit{full\_types} option to \textit{metis} directly.

 \point{And what are the \textit{lifting} and \textit{hide\_lams} arguments

 to Metis?}

 Orthogonally to the encoding of types, it is important to choose an appropriate

 translation of $\lambda$-abstractions. Metis supports three translation schemes,

 in decreasing order of power: Curry combinators (the default),

 $\lambda$-lifting, and a ``hiding'' scheme that disables all reasoning under

 $\lambda$-abstractions. The more powerful schemes also give the automatic

 provers more rope to hang themselves. See the \textit{lam\_trans} option (\S\ref{problem-encoding}) for details.

 \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, \textit{metis}, or

 \textit{smt} 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 or preplay it).

 In addition, some provers (e.g., Yices) do not provide proofs or sometimes

 produce incomplete proofs. The minimizer is then invoked to find out which facts

 are actually needed from the (large) set of facts that was initially 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.

 %

 Automatic minimization can be forced or disabled using the \textit{minimize}

 option (\S\ref{mode-of-operation}).

 \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}

 \subsection{Sledgehammer}

 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[\labelitemi] \textbf{\textit{run} (the default):} Runs Sledgehammer on

 subgoal number \qty{num} (1 by default), with the given options and facts.

 \item[\labelitemi] \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[\labelitemi] \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 (10 by default).

 \item[\labelitemi] \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[\labelitemi] \textbf{\textit{running\_provers}:} Prints information about

 currently running automatic provers, including elapsed runtime and remaining

 time until timeout.

 \item[\labelitemi] \textbf{\textit{kill\_provers}:} Terminates all running

 automatic provers.

 \item[\labelitemi] \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 in Proof General's ``Isabelle'' menu.

 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{slice} (\S\ref{mode-of-operation}) is disabled, \textit{strict}

 (\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.

 \subsection{Metis}

 The \textit{metis} proof method has the syntax

 \prew

 \textbf{\textit{metis}}~(\qty{options})${}^?$~\qty{facts}${}^?$

 \postw

 where \qty{facts} is a list of arbitrary facts and \qty{options} is a

 comma-separated list consisting of at most one $\lambda$ translation scheme

 specification with the same semantics as Sledgehammer's \textit{lam\_trans}

 option (\S\ref{problem-encoding}) and at most one type encoding specification

 with the same semantics as Sledgehammer's \textit{type\_enc} option

 (\S\ref{problem-encoding}).

 %

 The supported $\lambda$ translation schemes are \textit{hide\_lams},

 \textit{lifting}, and \textit{combs} (the default).

 %

 All the untyped type encodings listed in \S\ref{problem-encoding} are supported.

 For convenience, the following aliases are provided:

 \begin{enum}

 \item[\labelitemi] \textbf{\textit{full\_types}:} Synonym for \textit{poly\_guards\_query}.

 \item[\labelitemi] \textbf{\textit{partial\_types}:} Synonym for \textit{poly\_args}.

 \item[\labelitemi] \textbf{\textit{no\_types}:} Synonym for \textit{erased}.

 \end{enum}

 \section{Option Reference}

 \label{option-reference}

 \def\defl{\{}

 \def\defr{\}}

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

 \def\optrueonly#1{\flushitem{\textit{#1} $\bigl[$= \textit{true}$\bigr]$\enskip}\nopagebreak\\[\parskip]}

 \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\opsmartx#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\enskip \defl\textit{smart}\defr\\\hbox{}\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[\labelitemi] \qtybf{string}: A string.

 \item[\labelitemi] \qtybf{bool\/}: \textit{true} or \textit{false}.

 \item[\labelitemi] \qtybf{smart\_bool\/}: \textit{true}, \textit{false}, or

 \textit{smart}.

 \item[\labelitemi] \qtybf{int\/}: An integer.

 %\item[\labelitemi] \qtybf{float\/}: A floating-point number (e.g., 2.5).

 \item[\labelitemi] \qtybf{float\_pair\/}: A pair of floating-point numbers

 (e.g., 0.6 0.95).

 \item[\labelitemi] \qtybf{smart\_int\/}: An integer or \textit{smart}.

 \item[\labelitemi] \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 negative counterpart (e.g., \textit{blocking} vs.\

 \textit{non\_blocking}). When setting Boolean options or their negative

 counterparts, ``= \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\/}'').

 Provers can be run locally or remotely; see \S\ref{installation} for

 installation instructions.

 The following local provers are supported:

 \begin{enum}

 \item[\labelitemi] \textbf{\textit{alt\_ergo}:} Alt-Ergo is a polymorphic

 SMT solver developed by Bobot et al.\ \cite{alt-ergo}.

 It supports the TPTP polymorphic typed first-order format (TFF1) via Why3

 \cite{why3}. It is included for experimental purposes. To use Alt-Ergo, set the

 environment variable \texttt{WHY3\_HOME} to the directory that contains the

 \texttt{why3} executable. Sledgehammer has been tested with Alt-Ergo 0.93 and an

 unidentified development version of Why3.

 \item[\labelitemi] \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, or install the prebuilt CVC3 package from

 \download. Sledgehammer has been tested with version 2.2 and 2.4.1.

 \item[\labelitemi] \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 and \texttt{E\_VERSION} to the version number (e.g., ``1.4''), or

 install the prebuilt E package from \download. Sledgehammer has been tested with

 versions 1.0 to 1.4.

 \item[\labelitemi] \textbf{\textit{leo2}:} LEO-II is an automatic

 higher-order prover developed by Christoph Benzm\"uller et al.\ \cite{leo2},

 with support for the TPTP typed higher-order syntax (THF0). To use LEO-II, set

 the environment variable \texttt{LEO2\_HOME} to the directory that contains the

 \texttt{leo} executable. Sledgehammer requires version 1.2.9 or above.

 \item[\labelitemi] \textbf{\textit{metis}:} Although it is much less powerful than

 the external provers, Metis itself can be used for proof search.

 \item[\labelitemi] \textbf{\textit{satallax}:} Satallax is an automatic

 higher-order prover developed by Chad Brown et al.\ \cite{satallax}, with

 support for the TPTP typed higher-order syntax (THF0). To use Satallax, set the

 environment variable \texttt{SATALLAX\_HOME} to the directory that contains the

 \texttt{satallax} executable. Sledgehammer requires version 2.2 or above.

 \item[\labelitemi] \textbf{\textit{smt}:} The \textit{smt} proof method with the

 current settings (usually:\ Z3 with proof reconstruction).

 \item[\labelitemi] \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 and \texttt{SPASS\_VERSION} to the

 version number (e.g., ``3.8ds''), or install the prebuilt SPASS package from

 \download. Sledgehammer requires version 3.8ds or above.

 \item[\labelitemi] \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 and \texttt{VAMPIRE\_VERSION} to the version number (e.g.,

 ``1.8rev1435''). Sledgehammer has been tested with versions 0.6, 1.0, and 1.8.

 Versions strictly above 1.8 (e.g., ``1.8rev1435'') support the TPTP typed

 first-order format (TFF0).

 \item[\labelitemi] \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 1.0.28.

 \item[\labelitemi] \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} to ``yes'' to confirm that you are a

 noncommercial user. Sledgehammer has been tested with versions 3.0, 3.1, 3.2,

 and 4.0.

 \item[\labelitemi] \textbf{\textit{z3\_tptp}:} This version of Z3 pretends to be

 an ATP, exploiting Z3's support for the TPTP untyped and typed first-order

 formats (FOF and TFF0). It is included for experimental purposes. It

 requires version 3.0 or above. To use it, set the environment variable

 \texttt{Z3\_HOME} to the directory that contains the \texttt{z3}

 executable.

 \end{enum}

 The following remote provers are supported:

 \begin{enum}

 \item[\labelitemi] \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[\labelitemi] \textbf{\textit{remote\_e}:} The remote version of E runs

 on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.

 \item[\labelitemi] \textbf{\textit{remote\_e\_sine}:} E-SInE is a metaprover

 developed by Kry\v stof Hoder \cite{sine} based on E. It runs on Geoff

 Sutcliffe's Miami servers.

 \item[\labelitemi] \textbf{\textit{remote\_e\_tofof}:} E-ToFoF is a metaprover

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

 servers. This ATP supports the TPTP typed first-order format (TFF0). The

 remote version of E-ToFoF runs on Geoff Sutcliffe's Miami servers.

 \item[\labelitemi] \textbf{\textit{remote\_iprover}:} iProver is a pure

 instantiation-based prover developed by Konstantin Korovin \cite{korovin-2009}. The

 remote version of iProver runs on Geoff Sutcliffe's Miami servers

 \cite{sutcliffe-2000}.

 \item[\labelitemi] \textbf{\textit{remote\_iprover\_eq}:} iProver-Eq is an

 instantiation-based prover with native support for equality developed by

 Konstantin Korovin and Christoph Sticksel \cite{korovin-sticksel-2010}. The

 remote version of iProver-Eq runs on Geoff Sutcliffe's Miami servers

 \cite{sutcliffe-2000}.

 \item[\labelitemi] \textbf{\textit{remote\_leo2}:} The remote version of LEO-II

 runs on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.

 \item[\labelitemi] \textbf{\textit{remote\_satallax}:} The remote version of

 Satallax runs on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.

 \item[\labelitemi] \textbf{\textit{remote\_snark}:} SNARK is a first-order

 resolution prover developed by Stickel et al.\ \cite{snark}. It supports the

 TPTP typed first-order format (TFF0). The remote version of SNARK runs on

 Geoff Sutcliffe's Miami servers.

 \item[\labelitemi] \textbf{\textit{remote\_vampire}:} The remote version of

 Vampire runs on Geoff Sutcliffe's Miami servers.

 \item[\labelitemi] \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[\labelitemi] \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[\labelitemi] \textbf{\textit{remote\_z3\_tptp}:} The remote version of ``Z3

 with TPTP syntax'' runs on Geoff Sutcliffe's Miami servers.

 \end{enum}

 By default, Sledgehammer runs E, E-SInE, SPASS, Vampire, Z3 (or whatever

 the SMT module's \textit{smt\_solver} configuration option is set to), and (if

 appropriate) Waldmeister 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'' in Proof General's ``Isabelle'' menu.

 It is generally a good idea to run several provers in parallel. 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}.

 \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{slice}{dont\_slice}

 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}).}

 \opsmart{minimize}{dont\_minimize}

 Specifies whether the minimization tool should be invoked automatically after

 proof search. By default, automatic minimization takes place only if

 it can be done in a reasonable amount of time (as determined by

 the number of facts in the original proof and the time it took to find or

 preplay it) or the proof involves an unreasonably large number of facts.

 \nopagebreak

 {\small See also \textit{preplay\_timeout} (\S\ref{timeouts})

 and \textit{dont\_preplay} (\S\ref{timeouts}).}

 \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}

 \newcommand\comb[1]{\const{#1}}

 \begin{enum}

 \opdefault{lam\_trans}{string}{smart}

 Specifies the $\lambda$ translation scheme to use in ATP problems. The supported

 translation schemes are listed below:

 \begin{enum}

 \item[\labelitemi] \textbf{\textit{hide\_lams}:} Hide the $\lambda$-abstractions

 by replacing them by unspecified fresh constants, effectively disabling all

 reasoning under $\lambda$-abstractions.

 \item[\labelitemi] \textbf{\textit{lifting}:} Introduce a new

 supercombinator \const{c} for each cluster of $n$~$\lambda$-abstractions,

 defined using an equation $\const{c}~x_1~\ldots~x_n = t$ ($\lambda$-lifting).

 \item[\labelitemi] \textbf{\textit{combs}:} Rewrite lambdas to the Curry

 combinators (\comb{I}, \comb{K}, \comb{S}, \comb{B}, \comb{C}). Combinators

 enable the ATPs to synthesize $\lambda$-terms but tend to yield bulkier formulas

 than $\lambda$-lifting: The translation is quadratic in the worst case, and the

 equational definitions of the combinators are very prolific in the context of

 resolution.

 \item[\labelitemi] \textbf{\textit{combs\_and\_lifting}:} Introduce a new

 supercombinator \const{c} for each cluster of $\lambda$-abstractions and characterize it both using a

 lifted equation $\const{c}~x_1~\ldots~x_n = t$ and via Curry combinators.

 \item[\labelitemi] \textbf{\textit{combs\_or\_lifting}:} For each cluster of

 $\lambda$-abstractions, heuristically choose between $\lambda$-lifting and Curry

 combinators.

 \item[\labelitemi] \textbf{\textit{keep\_lams}:}

 Keep the $\lambda$-abstractions in the generated problems. This is available

 only with provers that support the THF0 syntax.

 \item[\labelitemi] \textbf{\textit{smart}:} The actual translation scheme used

 depends on the ATP and should be the most efficient scheme for that ATP.

 \end{enum}

 For SMT solvers, the $\lambda$ translation scheme is always \textit{lifting},

 irrespective of the value of this option.

 \opsmartx{uncurried\_aliases}{no\_uncurried\_aliases}

 Specifies whether fresh function symbols should be generated as aliases for

 applications of curried functions in ATP problems.

 \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 \textit{metis}). The supported type encodings are

 listed below, with an indication of their soundness in parentheses.

 An asterisk (*) means that the encoding is slightly incomplete for

 reconstruction with \textit{metis}, unless the \emph{strict} option (described

 below) is enabled.

 \begin{enum}

 \item[\labelitemi] \textbf{\textit{erased} (very unsound):} No type information is

 supplied to the ATP, not even to resolve overloading. Types are simply erased.

 \item[\labelitemi] \textbf{\textit{poly\_guards} (sound):} Types are encoded using

 a predicate \const{g}$(\tau, t)$ that guards bound

 variables. Constants are annotated with their types, supplied as additional

 arguments, to resolve overloading.

 \item[\labelitemi] \textbf{\textit{poly\_tags} (sound):} Each term and subterm is

 tagged with its type using a function $\const{t\/}(\tau, t)$.

 \item[\labelitemi] \textbf{\textit{poly\_args} (unsound):}

 Like for \textit{poly\_guards} 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[\labelitemi]

 \textbf{%

 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags} (sound); \\

 \textit{raw\_mono\_args} (unsound):} \\

 Similar to \textit{poly\_guards}, \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[\labelitemi]

 \textbf{%

 \textit{mono\_guards}, \textit{mono\_tags} (sound);

 \textit{mono\_args} (unsound):} \\

 Similar to

 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, and

 \textit{raw\_mono\_args}, respectively but types are mangled in constant names

 instead of being supplied as ground term arguments. The binary predicate

 $\const{g}(\tau, t)$ becomes a unary predicate

 $\const{g\_}\tau(t)$, and the binary function

 $\const{t}(\tau, t)$ becomes a unary function

 $\const{t\_}\tau(t)$.

 \item[\labelitemi] \textbf{\textit{mono\_native} (sound):} Exploits native

 first-order types if the prover supports the TFF0, TFF1, or THF0 syntax;

 otherwise, falls back on \textit{mono\_guards}. The problem is monomorphized.

 \item[\labelitemi] \textbf{\textit{mono\_native\_higher} (sound):} Exploits

 native higher-order types if the prover supports the THF0 syntax; otherwise,

 falls back on \textit{mono\_native} or \textit{mono\_guards}. The problem is

 monomorphized.

 \item[\labelitemi] \textbf{\textit{poly\_native} (sound):} Exploits native

 polymorphic first-order types if the prover supports the TFF1 syntax; otherwise,

 falls back on \textit{mono\_native}.

 \item[\labelitemi]

 \textbf{%

 \textit{poly\_guards}?, \textit{poly\_tags}?, \textit{raw\_mono\_guards}?, \\

 \textit{raw\_mono\_tags}?, \textit{mono\_guards}?, \textit{mono\_tags}?, \\

 \textit{mono\_native}? (sound*):} \\

 The type encodings \textit{poly\_guards}, \textit{poly\_tags},

 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, \textit{mono\_guards},

 \textit{mono\_tags}, and \textit{mono\_native} are fully typed and sound. For

 each of these, Sledgehammer also provides a lighter variant identified by a

 question mark (`\hbox{?}')\ that detects and erases monotonic types, notably

 infinite types. (For \textit{mono\_native}, 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.

 \item[\labelitemi]

 \textbf{%

 \textit{poly\_guards}??, \textit{poly\_tags}??, \textit{raw\_mono\_guards}??, \\

 \textit{raw\_mono\_tags}??, \textit{mono\_guards}??, \textit{mono\_tags}?? \\

 (sound*):} \\

 Even lighter versions of the `\hbox{?}' encodings. As argument to the

 \textit{metis} proof method, the `\hbox{??}' suffix is replaced by

 \hbox{``\textit{\_query\_query\/}''}.

 \item[\labelitemi]

 \textbf{%

 \textit{poly\_guards}@?, \textit{raw\_mono\_guards}@? (sound*):} \\

 Alternative versions of the `\hbox{??}' encodings. As argument to the

 \textit{metis} proof method, the `\hbox{@?}' suffix is replaced by

 \hbox{``\textit{\_at\_query\/}''}.

 \item[\labelitemi]

 \textbf{%

 \textit{poly\_guards}!, \textit{poly\_tags}!, \textit{raw\_mono\_guards}!, \\

 \textit{raw\_mono\_tags}!, \textit{mono\_guards}!, \textit{mono\_tags}!, \\

 \textit{mono\_native}!, \textit{mono\_native\_higher}! (mildly unsound):} \\

 The type encodings \textit{poly\_guards}, \textit{poly\_tags},

 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, \textit{mono\_guards},

 \textit{mono\_tags}, \textit{mono\_native}, and \textit{mono\_native\_higher}

 also admit a mildly unsound (but very efficient) variant identified by an

 exclamation mark (`\hbox{!}') that detects and erases erases all types except

 those that are clearly finite (e.g., \textit{bool}). (For \textit{mono\_native}

 and \textit{mono\_native\_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 the suffix

 \hbox{``\textit{\_bang\/}''}.

 \item[\labelitemi]

 \textbf{%

 \textit{poly\_guards}!!, \textit{poly\_tags}!!, \textit{raw\_mono\_guards}!!, \\

 \textit{raw\_mono\_tags}!!, \textit{mono\_guards}!!, \textit{mono\_tags}!! \\

 (mildly unsound):} \\

 Even lighter versions of the `\hbox{!}' encodings. As argument to the

 \textit{metis} proof method, the `\hbox{!!}' suffix is replaced by

 \hbox{``\textit{\_bang\_bang\/}''}.

 \item[\labelitemi]

 \textbf{%

 \textit{poly\_guards}@!, \textit{raw\_mono\_guards}@! (mildly unsound):} \\

 Alternative versions of the `\hbox{!!}' encodings. As argument to the

 \textit{metis} proof method, the `\hbox{@!}' suffix is replaced by

 \hbox{``\textit{\_at\_bang\/}''}.

 \item[\labelitemi] \textbf{\textit{smart}:} The actual encoding used depends on

 the ATP and should be the most efficient sound encoding for that ATP.

 \end{enum}

 For SMT solvers, the type encoding is always \textit{mono\_native}, 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{strict}{non\_strict}

 Specifies whether Sledgehammer should run in its strict mode. In that mode,

 sound type encodings marked with an asterisk (*) above are made complete

 for reconstruction with \textit{metis}, at the cost of some clutter in the

 generated problems. This option has no effect if \textit{type\_enc} is

 deliberately 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. Typical values range between

 50 and 1000.

 \opdefault{max\_new\_mono\_instances}{int}{smart}

 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. If the option is set to \textit{smart}, it is set to a value

 that was empirically found to be appropriate for the prover. For most provers,

 this value is 200.

 \nopagebreak

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

 \opdefault{max\_mono\_iters}{int}{smart}

 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. If the option is set to \textit{smart}, it is set to a value

 that was empirically found to be appropriate for the prover. For most provers,

 this value is 3.

 \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[\labelitemi] \textbf{\textit{some}:} Sledgehammer found a proof.

 \item[\labelitemi] \textbf{\textit{none}:} Sledgehammer found no proof.

 \item[\labelitemi] \textbf{\textit{timeout}:} Sledgehammer timed out.

 \item[\labelitemi] \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'' in Proof General's ``Isabelle'' menu.

 \opdefault{preplay\_timeout}{float\_or\_none}{\upshape 3}

 Specifies the maximum number of seconds that \textit{metis} or \textit{smt}

 should spend 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 \textit{metis} calls.

 \nopagebreak

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

 \optrueonly{dont\_preplay}

 Alias for ``\textit{preplay\_timeout} = 0''.

 \end{enum}

 \let\em=\sl

 \bibliography{../manual}{}

 \bibliographystyle{abbrv}

 \end{document}