doc-src/Sledgehammer/sledgehammer.tex
author haftmann
Wed Sep 07 23:38:52 2011 +0200 (2011-09-07)
changeset 44818 27ba81ad0890
parent 44816 efa1f532508b
child 45048 59ca831deef4
permissions -rw-r--r--
theory of saturated naturals contributed by Peter Gammie
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    13 \usepackage{../../lib/texinputs/isabelle,../iman,../pdfsetup}
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    32 \def\unk{{?}}
    33 \def\undef{(\lambda x.\; \unk)}
    34 %\def\unr{\textit{others}}
    35 \def\unr{\ldots}
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    37 \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}}
    38 
    39 \urlstyle{tt}
    40 
    41 \begin{document}
    42 
    43 \selectlanguage{english}
    44 
    45 \title{\includegraphics[scale=0.5]{isabelle_sledgehammer} \\[4ex]
    46 Hammering Away \\[\smallskipamount]
    47 \Large A User's Guide to Sledgehammer for Isabelle/HOL}
    48 \author{\hbox{} \\
    49 Jasmin Christian Blanchette \\
    50 {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\[4\smallskipamount]
    51 {\normalsize with contributions from} \\[4\smallskipamount]
    52 Lawrence C. Paulson \\
    53 {\normalsize Computer Laboratory, University of Cambridge} \\
    54 \hbox{}}
    55 
    56 \maketitle
    57 
    58 \tableofcontents
    59 
    60 \setlength{\parskip}{.7em plus .2em minus .1em}
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    66 
    67 % General-purpose enum environment with correct spacing
    68 \newenvironment{enum}%
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    75 
    76 \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin
    77 \advance\rightskip by\leftmargin}
    78 \def\post{\vskip0pt plus1ex\endgroup}
    79 
    80 \def\prew{\pre\advance\rightskip by-\leftmargin}
    81 \def\postw{\post}
    82 
    83 \section{Introduction}
    84 \label{introduction}
    85 
    86 Sledgehammer is a tool that applies automatic theorem provers (ATPs)
    87 and satisfiability-modulo-theories (SMT) solvers on the current goal. The
    88 supported ATPs are E \cite{schulz-2002}, E-SInE \cite{sine}, E-ToFoF
    89 \cite{tofof}, LEO-II \cite{leo2}, Satallax \cite{satallax}, SNARK \cite{snark},
    90 SPASS \cite{weidenbach-et-al-2009}, Vampire \cite{riazanov-voronkov-2002}, and
    91 Waldmeister \cite{waldmeister}. The ATPs are run either locally or remotely via
    92 the System\-On\-TPTP web service \cite{sutcliffe-2000}. In addition to the ATPs,
    93 the SMT solvers Z3 \cite{z3} is used by default, and you can tell Sledgehammer
    94 to try CVC3 \cite{cvc3} and Yices \cite{yices} as well; these are run either
    95 locally or on a server at the TU M\"unchen.
    96 
    97 The problem passed to the automatic provers consists of your current goal
    98 together with a heuristic selection of hundreds of facts (theorems) from the
    99 current theory context, filtered by relevance. Because jobs are run in the
   100 background, you can continue to work on your proof by other means. Provers can
   101 be run in parallel. Any reply (which may arrive half a minute later) will appear
   102 in the Proof General response buffer.
   103 
   104 The result of a successful proof search is some source text that usually (but
   105 not always) reconstructs the proof within Isabelle. For ATPs, the reconstructed
   106 proof relies on the general-purpose Metis prover, which is fully integrated into
   107 Isabelle/HOL, with explicit inferences going through the kernel. Thus its
   108 results are correct by construction.
   109 
   110 In this manual, we will explicitly invoke the \textbf{sledgehammer} command.
   111 Sledgehammer also provides an automatic mode that can be enabled via the ``Auto
   112 Sledgehammer'' option in Proof General's ``Isabelle'' menu. In this mode,
   113 Sledgehammer is run on every newly entered theorem. The time limit for Auto
   114 Sledgehammer and other automatic tools can be set using the ``Auto Tools Time
   115 Limit'' option.
   116 
   117 \newbox\boxA
   118 \setbox\boxA=\hbox{\texttt{nospam}}
   119 
   120 \newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak
   121 in.\allowbreak tum.\allowbreak de}}
   122 
   123 To run Sledgehammer, you must make sure that the theory \textit{Sledgehammer} is
   124 imported---this is rarely a problem in practice since it is part of
   125 \textit{Main}. Examples of Sledgehammer use can be found in Isabelle's
   126 \texttt{src/HOL/Metis\_Examples} directory.
   127 Comments and bug reports concerning Sledgehammer or this manual should be
   128 directed to the author at \authoremail.
   129 
   130 \vskip2.5\smallskipamount
   131 
   132 %\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for
   133 %suggesting several textual improvements.
   134 
   135 \section{Installation}
   136 \label{installation}
   137 
   138 Sledgehammer is part of Isabelle, so you don't need to install it. However, it
   139 relies on third-party automatic theorem provers (ATPs) and SMT solvers.
   140 
   141 \subsection{Installing ATPs}
   142 
   143 Currently, E, LEO-II, Satallax, SPASS, and Vampire can be run locally; in
   144 addition, E, E-SInE, E-ToFoF, LEO-II, Satallax, SNARK, Waldmeister, and Vampire
   145 are available remotely via System\-On\-TPTP \cite{sutcliffe-2000}. If you want
   146 better performance, you should at least install E and SPASS locally.
   147 
   148 There are three main ways to install ATPs on your machine:
   149 
   150 \begin{enum}
   151 \item[$\bullet$] If you installed an official Isabelle package with everything
   152 inside, it should already include properly setup executables for E and SPASS,
   153 ready to use.%
   154 \footnote{Vampire's license prevents us from doing the same for this otherwise
   155 wonderful tool.}
   156 
   157 \item[$\bullet$] Alternatively, you can download the Isabelle-aware E and SPASS
   158 binary packages from Isabelle's download page. Extract the archives, then add a
   159 line to your \texttt{\$ISABELLE\_HOME\_USER/etc/components}%
   160 \footnote{The variable \texttt{\$ISABELLE\_HOME\_USER} is set by Isabelle at
   161 startup. Its value can be retrieved by invoking \texttt{isabelle}
   162 \texttt{getenv} \texttt{ISABELLE\_HOME\_USER} on the command line.}
   163 file with the absolute
   164 path to E or SPASS. For example, if the \texttt{components} does not exist yet
   165 and you extracted SPASS to \texttt{/usr/local/spass-3.7}, create the
   166 \texttt{components} file with the single line
   167 
   168 \prew
   169 \texttt{/usr/local/spass-3.7}
   170 \postw
   171 
   172 in it.
   173 
   174 \item[$\bullet$] If you prefer to build E or SPASS yourself, or obtained a
   175 Vampire executable from somewhere (e.g., \url{http://www.vprover.org/}),
   176 set the environment variable \texttt{E\_HOME}, \texttt{SPASS\_HOME}, or
   177 \texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{eproof},
   178 \texttt{SPASS}, or \texttt{vampire} executable. Sledgehammer has been tested
   179 with E 1.0 to 1.3, SPASS 3.5 and 3.7, and Vampire 0.6, 1.0, and 1.8%
   180 \footnote{Following the rewrite of Vampire, the counter for version numbers was
   181 reset to 0; hence the (new) Vampire versions 0.6, 1.0, and 1.8 are more recent
   182 than, say, Vampire 9.0 or 11.5.}%
   183 . Since the ATPs' output formats are neither documented nor stable, other
   184 versions of the ATPs might or might not work well with Sledgehammer. Ideally,
   185 also set \texttt{E\_VERSION}, \texttt{SPASS\_VERSION}, or
   186 \texttt{VAMPIRE\_VERSION} to the ATP's version number (e.g., ``1.4'').
   187 \end{enum}
   188 
   189 To check whether E and SPASS are successfully installed, follow the example in
   190 \S\ref{first-steps}. If the remote versions of E and SPASS are used (identified
   191 by the prefix ``\emph{remote\_}''), or if the local versions fail to solve the
   192 easy goal presented there, this is a sign that something is wrong with your
   193 installation.
   194 
   195 Remote ATP invocation via the SystemOnTPTP web service requires Perl with the
   196 World Wide Web Library (\texttt{libwww-perl}) installed. If you must use a proxy
   197 server to access the Internet, set the \texttt{http\_proxy} environment variable
   198 to the proxy, either in the environment in which Isabelle is launched or in your
   199 \texttt{\char`\~/\$ISABELLE\_HOME\_USER/etc/settings} file. Here are a few examples:
   200 
   201 \prew
   202 \texttt{http\_proxy=http://proxy.example.org} \\
   203 \texttt{http\_proxy=http://proxy.example.org:8080} \\
   204 \texttt{http\_proxy=http://joeblow:pAsSwRd@proxy.example.org}
   205 \postw
   206 
   207 \subsection{Installing SMT Solvers}
   208 
   209 CVC3, Yices, and Z3 can be run locally or (for CVC3 and Z3) remotely on a TU
   210 M\"unchen server. If you want better performance and get the ability to replay
   211 proofs that rely on the \emph{smt} proof method, you should at least install Z3
   212 locally.
   213 
   214 There are two main ways of installing SMT solvers locally.
   215 
   216 \begin{enum}
   217 \item[$\bullet$] If you installed an official Isabelle package with everything
   218 inside, it should already include properly setup executables for CVC3 and Z3,
   219 ready to use.%
   220 \footnote{Yices's license prevents us from doing the same for this otherwise
   221 wonderful tool.}
   222 For Z3, you additionally need to set the environment variable
   223 \texttt{Z3\_NON\_COMMERCIAL} to ``yes'' to confirm that you are a noncommercial
   224 user.
   225 
   226 \item[$\bullet$] Otherwise, follow the instructions documented in the \emph{SMT}
   227 theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}).
   228 \end{enum}
   229 
   230 \section{First Steps}
   231 \label{first-steps}
   232 
   233 To illustrate Sledgehammer in context, let us start a theory file and
   234 attempt to prove a simple lemma:
   235 
   236 \prew
   237 \textbf{theory}~\textit{Scratch} \\
   238 \textbf{imports}~\textit{Main} \\
   239 \textbf{begin} \\[2\smallskipamount]
   240 %
   241 \textbf{lemma} ``$[a] = [b] \,\Longrightarrow\, a = b$'' \\
   242 \textbf{sledgehammer}
   243 \postw
   244 
   245 Instead of issuing the \textbf{sledgehammer} command, you can also find
   246 Sledgehammer in the ``Commands'' submenu of the ``Isabelle'' menu in Proof
   247 General or press the Emacs key sequence C-c C-a C-s.
   248 Either way, Sledgehammer produces the following output after a few seconds:
   249 
   250 \prew
   251 \slshape
   252 Sledgehammer: ``\textit{e}'' on goal \\
   253 $[a] = [b] \,\Longrightarrow\, a = b$ \\
   254 Try this: \textbf{by} (\textit{metis last\_ConsL}) (64 ms). \\[3\smallskipamount]
   255 %
   256 Sledgehammer: ``\textit{vampire}'' on goal \\
   257 $[a] = [b] \,\Longrightarrow\, a = b$ \\
   258 Try this: \textbf{by} (\textit{metis hd.simps}) (14 ms). \\[3\smallskipamount]
   259 %
   260 Sledgehammer: ``\textit{spass}'' on goal \\
   261 $[a] = [b] \,\Longrightarrow\, a = b$ \\
   262 Try this: \textbf{by} (\textit{metis list.inject}) (17 ms). \\[3\smallskipamount]
   263 %
   264 Sledgehammer: ``\textit{remote\_waldmeister}'' on goal \\
   265 $[a] = [b] \,\Longrightarrow\, a = b$ \\
   266 Try this: \textbf{by} (\textit{metis hd.simps}) (15 ms). \\[3\smallskipamount]
   267 %
   268 Sledgehammer: ``\textit{remote\_e\_sine}'' on goal \\
   269 $[a] = [b] \,\Longrightarrow\, a = b$ \\
   270 Try this: \textbf{by} (\textit{metis hd.simps}) (18 ms). \\[3\smallskipamount]
   271 %
   272 Sledgehammer: ``\textit{remote\_z3}'' on goal \\
   273 $[a] = [b] \,\Longrightarrow\, a = b$ \\
   274 Try this: \textbf{by} (\textit{metis list.inject}) (20 ms).
   275 \postw
   276 
   277 Sledgehammer ran E, E-SInE, SPASS, Vampire, Waldmeister, and Z3 in parallel.
   278 Depending on which provers are installed and how many processor cores are
   279 available, some of the provers might be missing or present with a
   280 \textit{remote\_} prefix. Waldmeister is run only for unit equational problems,
   281 where the goal's conclusion is a (universally quantified) equation.
   282 
   283 For each successful prover, Sledgehammer gives a one-liner proof that uses Metis
   284 or the \textit{smt} proof method. For Metis, approximate timings are shown in
   285 parentheses, indicating how fast the call is. You can click the proof to insert
   286 it into the theory text.
   287 
   288 In addition, you can ask Sledgehammer for an Isar text proof by passing the
   289 \textit{isar\_proof} option (\S\ref{output-format}):
   290 
   291 \prew
   292 \textbf{sledgehammer} [\textit{isar\_proof}]
   293 \postw
   294 
   295 When Isar proof construction is successful, it can yield proofs that are more
   296 readable and also faster than the Metis one-liners. This feature is experimental
   297 and is only available for ATPs.
   298 
   299 \section{Hints}
   300 \label{hints}
   301 
   302 This section presents a few hints that should help you get the most out of
   303 Sledgehammer and Metis. Frequently (and infrequently) asked questions are
   304 answered in \S\ref{frequently-asked-questions}.
   305 
   306 \newcommand\point[1]{\medskip\par{\sl\bfseries#1}\par\nopagebreak}
   307 
   308 \point{Presimplify the goal}
   309 
   310 For best results, first simplify your problem by calling \textit{auto} or at
   311 least \textit{safe} followed by \textit{simp\_all}. The SMT solvers provide
   312 arithmetic decision procedures, but the ATPs typically do not (or if they do,
   313 Sledgehammer does not use it yet). Apart from Waldmeister, they are not
   314 especially good at heavy rewriting, but because they regard equations as
   315 undirected, they often prove theorems that require the reverse orientation of a
   316 \textit{simp} rule. Higher-order problems can be tackled, but the success rate
   317 is better for first-order problems. Hence, you may get better results if you
   318 first simplify the problem to remove higher-order features.
   319 
   320 \point{Make sure at least E, SPASS, Vampire, and Z3 are installed}
   321 
   322 Locally installed provers are faster and more reliable than those running on
   323 servers. See \S\ref{installation} for details on how to install them.
   324 
   325 \point{Familiarize yourself with the most important options}
   326 
   327 Sledgehammer's options are fully documented in \S\ref{command-syntax}. Many of
   328 the options are very specialized, but serious users of the tool should at least
   329 familiarize themselves with the following options:
   330 
   331 \begin{enum}
   332 \item[$\bullet$] \textbf{\textit{provers}} (\S\ref{mode-of-operation}) specifies
   333 the automatic provers (ATPs and SMT solvers) that should be run whenever
   334 Sledgehammer is invoked (e.g., ``\textit{provers}~= \textit{e spass
   335 remote\_vampire}''). For convenience, you can omit ``\textit{provers}~=''
   336 and simply write the prover names as a space-separated list (e.g., ``\textit{e
   337 spass remote\_vampire}'').
   338 
   339 \item[$\bullet$] \textbf{\textit{max\_relevant}} (\S\ref{relevance-filter})
   340 specifies the maximum number of facts that should be passed to the provers. By
   341 default, the value is prover-dependent but varies between about 150 and 1000. If
   342 the provers time out, you can try lowering this value to, say, 100 or 50 and see
   343 if that helps.
   344 
   345 \item[$\bullet$] \textbf{\textit{isar\_proof}} (\S\ref{output-format}) specifies
   346 that Isar proofs should be generated, instead of one-liner Metis proofs. The
   347 length of the Isar proofs can be controlled by setting
   348 \textit{isar\_shrink\_factor} (\S\ref{output-format}).
   349 
   350 \item[$\bullet$] \textbf{\textit{timeout}} (\S\ref{timeouts}) controls the
   351 provers' time limit. It is set to 30 seconds, but since Sledgehammer runs
   352 asynchronously you should not hesitate to raise this limit to 60 or 120 seconds
   353 if you are the kind of user who can think clearly while ATPs are active.
   354 \end{enum}
   355 
   356 Options can be set globally using \textbf{sledgehammer\_params}
   357 (\S\ref{command-syntax}). The command also prints the list of all available
   358 options with their current value. Fact selection can be influenced by specifying
   359 ``$(\textit{add}{:}~\textit{my\_facts})$'' after the \textbf{sledgehammer} call
   360 to ensure that certain facts are included, or simply ``$(\textit{my\_facts})$''
   361 to force Sledgehammer to run only with $\textit{my\_facts}$.
   362 
   363 \section{Frequently Asked Questions}
   364 \label{frequently-asked-questions}
   365 
   366 This sections answers frequently (and infrequently) asked questions about
   367 Sledgehammer. It is a good idea to skim over it now even if you don't have any
   368 questions at this stage. And if you have any further questions not listed here,
   369 send them to the author at \authoremail.
   370 
   371 \point{Why does Metis fail to reconstruct the proof?}
   372 
   373 There are many reasons. If Metis runs seemingly forever, that is a sign that the
   374 proof is too difficult for it. Metis's search is complete, so it should
   375 eventually find it, but that's little consolation. There are several possible
   376 solutions:
   377 
   378 \begin{enum}
   379 \item[$\bullet$] Try the \textit{isar\_proof} option (\S\ref{output-format}) to
   380 obtain a step-by-step Isar proof where each step is justified by Metis. Since
   381 the steps are fairly small, Metis is more likely to be able to replay them.
   382 
   383 \item[$\bullet$] Try the \textit{smt} proof method instead of Metis. It is
   384 usually stronger, but you need to have Z3 available to replay the proofs, trust
   385 the SMT solver, or use certificates. See the documentation in the \emph{SMT}
   386 theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}) for details.
   387 
   388 \item[$\bullet$] Try the \textit{blast} or \textit{auto} proof methods, passing
   389 the necessary facts via \textbf{unfolding}, \textbf{using}, \textit{intro}{:},
   390 \textit{elim}{:}, \textit{dest}{:}, or \textit{simp}{:}, as appropriate.
   391 \end{enum}
   392 
   393 In some rare cases, Metis fails fairly quickly, and you get the error message
   394 
   395 \prew
   396 \slshape
   397 Proof reconstruction failed.
   398 \postw
   399 
   400 This message usually indicates that Sledgehammer found a type-incorrect proof.
   401 This was a frequent issue with older versions of Sledgehammer, which did not
   402 supply enough typing information to the ATPs by default. If you notice many
   403 unsound proofs and are not using \textit{type\_enc} (\S\ref{problem-encoding}),
   404 contact the author at \authoremail.
   405 
   406 \point{How can I tell whether a generated proof is sound?}
   407 
   408 First, if Metis can reconstruct it, the proof is sound (assuming Isabelle's
   409 inference kernel is sound). If it fails or runs seemingly forever, you can try
   410 
   411 \prew
   412 \textbf{apply}~\textbf{--} \\
   413 \textbf{sledgehammer} [\textit{sound}] (\textit{metis\_facts})
   414 \postw
   415 
   416 where \textit{metis\_facts} is the list of facts appearing in the suggested
   417 Metis call. The automatic provers should be able to re-find the proof quickly if
   418 it is sound, and the \textit{sound} option (\S\ref{problem-encoding}) ensures
   419 that no unsound proofs are found.
   420 
   421 \point{Which facts are passed to the automatic provers?}
   422 
   423 The relevance filter assigns a score to every available fact (lemma, theorem,
   424 definition, or axiom)\ based upon how many constants that fact shares with the
   425 conjecture. This process iterates to include facts relevant to those just
   426 accepted, but with a decay factor to ensure termination. The constants are
   427 weighted to give unusual ones greater significance. The relevance filter copes
   428 best when the conjecture contains some unusual constants; if all the constants
   429 are common, it is unable to discriminate among the hundreds of facts that are
   430 picked up. The relevance filter is also memoryless: It has no information about
   431 how many times a particular fact has been used in a proof, and it cannot learn.
   432 
   433 The number of facts included in a problem varies from prover to prover, since
   434 some provers get overwhelmed more easily than others. You can show the number of
   435 facts given using the \textit{verbose} option (\S\ref{output-format}) and the
   436 actual facts using \textit{debug} (\S\ref{output-format}).
   437 
   438 Sledgehammer is good at finding short proofs combining a handful of existing
   439 lemmas. If you are looking for longer proofs, you must typically restrict the
   440 number of facts, by setting the \textit{max\_relevant} option
   441 (\S\ref{relevance-filter}) to, say, 25 or 50.
   442 
   443 You can also influence which facts are actually selected in a number of ways. If
   444 you simply want to ensure that a fact is included, you can specify it using the
   445 ``$(\textit{add}{:}~\textit{my\_facts})$'' syntax. For example:
   446 %
   447 \prew
   448 \textbf{sledgehammer} (\textit{add}: \textit{hd.simps} \textit{tl.simps})
   449 \postw
   450 %
   451 The specified facts then replace the least relevant facts that would otherwise be
   452 included; the other selected facts remain the same.
   453 If you want to direct the selection in a particular direction, you can specify
   454 the facts via \textbf{using}:
   455 %
   456 \prew
   457 \textbf{using} \textit{hd.simps} \textit{tl.simps} \\
   458 \textbf{sledgehammer}
   459 \postw
   460 %
   461 The facts are then more likely to be selected than otherwise, and if they are
   462 selected at iteration $j$ they also influence which facts are selected at
   463 iterations $j + 1$, $j + 2$, etc. To give them even more weight, try
   464 %
   465 \prew
   466 \textbf{using} \textit{hd.simps} \textit{tl.simps} \\
   467 \textbf{apply}~\textbf{--} \\
   468 \textbf{sledgehammer}
   469 \postw
   470 
   471 \point{Why are the generated Isar proofs so ugly/detailed/broken?}
   472 
   473 The current implementation is experimental and explodes exponentially in the
   474 worst case. Work on a new implementation has begun. There is a large body of
   475 research into transforming resolution proofs into natural deduction proofs (such
   476 as Isar proofs), which we hope to leverage. In the meantime, a workaround is to
   477 set the \textit{isar\_shrink\_factor} option (\S\ref{output-format}) to a larger
   478 value or to try several provers and keep the nicest-looking proof.
   479 
   480 \point{What are the \textit{full\_types} and \textit{no\_types} arguments to
   481 Metis?}
   482 
   483 The \textit{metis}~(\textit{full\_types}) proof method is the fully-typed
   484 version of Metis. It is somewhat slower than \textit{metis}, but the proof
   485 search is fully typed, and it also includes more powerful rules such as the
   486 axiom ``$x = \mathit{True} \mathrel{\lor} x = \mathit{False}$'' for reasoning in
   487 higher-order places (e.g., in set comprehensions). The method kicks in
   488 automatically as a fallback when \textit{metis} fails, and it is sometimes
   489 generated by Sledgehammer instead of \textit{metis} if the proof obviously
   490 requires type information or if \textit{metis} failed when Sledgehammer
   491 preplayed the proof. (By default, Sledgehammer tries to run \textit{metis} with
   492 various options for up to 4 seconds to ensure that the generated one-line proofs
   493 actually work and to display timing information. This can be configured using
   494 the \textit{preplay\_timeout} option (\S\ref{timeouts}).)
   495 
   496 At the other end of the soundness spectrum, \textit{metis} (\textit{no\_types})
   497 uses no type information at all during the proof search, which is more efficient
   498 but often fails. Calls to \textit{metis} (\textit{no\_types}) are occasionally
   499 generated by Sledgehammer.
   500 
   501 Incidentally, if you see the warning
   502 
   503 \prew
   504 \slshape
   505 Metis: Falling back on ``\textit{metis} (\textit{full\_types})''.
   506 \postw
   507 
   508 in a successful Metis proof, you can advantageously pass the
   509 \textit{full\_types} option to \textit{metis} directly.
   510 
   511 \point{Are generated proofs minimal?}
   512 
   513 Automatic provers frequently use many more facts than are necessary.
   514 Sledgehammer inclues a minimization tool that takes a set of facts returned by a
   515 given prover and repeatedly calls the same prover or Metis with subsets of those
   516 axioms in order to find a minimal set. Reducing the number of axioms typically
   517 improves Metis's speed and success rate, while also removing superfluous clutter
   518 from the proof scripts.
   519 
   520 In earlier versions of Sledgehammer, generated proofs were systematically
   521 accompanied by a suggestion to invoke the minimization tool. This step is now
   522 performed implicitly if it can be done in a reasonable amount of time (something
   523 that can be guessed from the number of facts in the original proof and the time
   524 it took to find it or replay it).
   525 
   526 In addition, some provers (notably CVC3, Satallax, and Yices) do not provide
   527 proofs or sometimes produce incomplete proofs. The minimizer is then invoked to
   528 find out which facts are actually needed from the (large) set of facts that was
   529 initinally given to the prover. Finally, if a prover returns a proof with lots
   530 of facts, the minimizer is invoked automatically since Metis would be unlikely
   531 to re-find the proof.
   532 
   533 \point{A strange error occurred---what should I do?}
   534 
   535 Sledgehammer tries to give informative error messages. Please report any strange
   536 error to the author at \authoremail. This applies double if you get the message
   537 
   538 \prew
   539 \slshape
   540 The prover found a type-unsound proof involving ``\textit{foo}'',
   541 ``\textit{bar}'', and ``\textit{baz}'' even though a supposedly type-sound
   542 encoding was used (or, less likely, your axioms are inconsistent). You might
   543 want to report this to the Isabelle developers.
   544 \postw
   545 
   546 \point{Auto can solve it---why not Sledgehammer?}
   547 
   548 Problems can be easy for \textit{auto} and difficult for automatic provers, but
   549 the reverse is also true, so don't be discouraged if your first attempts fail.
   550 Because the system refers to all theorems known to Isabelle, it is particularly
   551 suitable when your goal has a short proof from lemmas that you don't know about.
   552 
   553 \point{Why are there so many options?}
   554 
   555 Sledgehammer's philosophy should work out of the box, without user guidance.
   556 Many of the options are meant to be used mostly by the Sledgehammer developers
   557 for experimentation purposes. Of course, feel free to experiment with them if
   558 you are so inclined.
   559 
   560 \section{Command Syntax}
   561 \label{command-syntax}
   562 
   563 Sledgehammer can be invoked at any point when there is an open goal by entering
   564 the \textbf{sledgehammer} command in the theory file. Its general syntax is as
   565 follows:
   566 
   567 \prew
   568 \textbf{sledgehammer} \qty{subcommand}$^?$ \qty{options}$^?$ \qty{facts\_override}$^?$ \qty{num}$^?$
   569 \postw
   570 
   571 For convenience, Sledgehammer is also available in the ``Commands'' submenu of
   572 the ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c
   573 C-a C-s. This is equivalent to entering the \textbf{sledgehammer} command with
   574 no arguments in the theory text.
   575 
   576 In the general syntax, the \qty{subcommand} may be any of the following:
   577 
   578 \begin{enum}
   579 \item[$\bullet$] \textbf{\textit{run} (the default):} Runs Sledgehammer on
   580 subgoal number \qty{num} (1 by default), with the given options and facts.
   581 
   582 \item[$\bullet$] \textbf{\textit{min}:} Attempts to minimize the facts
   583 specified in the \qty{facts\_override} argument to obtain a simpler proof
   584 involving fewer facts. The options and goal number are as for \textit{run}.
   585 
   586 \item[$\bullet$] \textbf{\textit{messages}:} Redisplays recent messages issued
   587 by Sledgehammer. This allows you to examine results that might have been lost
   588 due to Sledgehammer's asynchronous nature. The \qty{num} argument specifies a
   589 limit on the number of messages to display (5 by default).
   590 
   591 \item[$\bullet$] \textbf{\textit{supported\_provers}:} Prints the list of
   592 automatic provers supported by Sledgehammer. See \S\ref{installation} and
   593 \S\ref{mode-of-operation} for more information on how to install automatic
   594 provers.
   595 
   596 \item[$\bullet$] \textbf{\textit{running\_provers}:} Prints information about
   597 currently running automatic provers, including elapsed runtime and remaining
   598 time until timeout.
   599 
   600 \item[$\bullet$] \textbf{\textit{kill\_provers}:} Terminates all running
   601 automatic provers.
   602 
   603 \item[$\bullet$] \textbf{\textit{refresh\_tptp}:} Refreshes the list of remote
   604 ATPs available at System\-On\-TPTP \cite{sutcliffe-2000}.
   605 \end{enum}
   606 
   607 Sledgehammer's behavior can be influenced by various \qty{options}, which can be
   608 specified in brackets after the \textbf{sledgehammer} command. The
   609 \qty{options} are a list of key--value pairs of the form ``[$k_1 = v_1,
   610 \ldots, k_n = v_n$]''. For Boolean options, ``= \textit{true}'' is optional. For
   611 example:
   612 
   613 \prew
   614 \textbf{sledgehammer} [\textit{isar\_proof}, \,\textit{timeout} = 120]
   615 \postw
   616 
   617 Default values can be set using \textbf{sledgehammer\_\allowbreak params}:
   618 
   619 \prew
   620 \textbf{sledgehammer\_params} \qty{options}
   621 \postw
   622 
   623 The supported options are described in \S\ref{option-reference}.
   624 
   625 The \qty{facts\_override} argument lets you alter the set of facts that go
   626 through the relevance filter. It may be of the form ``(\qty{facts})'', where
   627 \qty{facts} is a space-separated list of Isabelle facts (theorems, local
   628 assumptions, etc.), in which case the relevance filter is bypassed and the given
   629 facts are used. It may also be of the form ``(\textit{add}:\ \qty{facts\/_{\mathrm{1}}})'',
   630 ``(\textit{del}:\ \qty{facts\/_{\mathrm{2}}})'', or ``(\textit{add}:\ \qty{facts\/_{\mathrm{1}}}\
   631 \textit{del}:\ \qty{facts\/_{\mathrm{2}}})'', where the relevance filter is instructed to
   632 proceed as usual except that it should consider \qty{facts\/_{\mathrm{1}}}
   633 highly-relevant and \qty{facts\/_{\mathrm{2}}} fully irrelevant.
   634 
   635 You can instruct Sledgehammer to run automatically on newly entered theorems by
   636 enabling the ``Auto Sledgehammer'' option in Proof General's ``Isabelle'' menu.
   637 For automatic runs, only the first prover set using \textit{provers}
   638 (\S\ref{mode-of-operation}) is considered, fewer facts are passed to the prover,
   639 \textit{slicing} (\S\ref{mode-of-operation}) is disabled, \textit{sound}
   640 (\S\ref{problem-encoding}) is enabled, \textit{verbose} (\S\ref{output-format})
   641 and \textit{debug} (\S\ref{output-format}) are disabled, and \textit{timeout}
   642 (\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in Proof
   643 General's ``Isabelle'' menu. Sledgehammer's output is also more concise.
   644 
   645 The \textit{metis} proof method has the syntax
   646 
   647 \prew
   648 \textbf{\textit{metis}}~(\qty{type\_enc})${}^?$~\qty{facts}${}^?$
   649 \postw
   650 
   651 where \qty{type\_enc} is a type encoding specification with the same semantics
   652 as Sledgehammer's \textit{type\_enc} option (\S\ref{problem-encoding}) and
   653 \qty{facts} is a list of arbitrary facts. In addition to the values listed in
   654 \S\ref{problem-encoding}, \qty{type\_enc} may also be \textit{full\_types}, in
   655 which case an appropriate type-sound encoding is chosen, \textit{partial\_types}
   656 (the default type-unsound encoding), or \textit{no\_types}, a synonym for
   657 \textit{erased}.
   658 
   659 \section{Option Reference}
   660 \label{option-reference}
   661 
   662 \def\defl{\{}
   663 \def\defr{\}}
   664 
   665 \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}}
   666 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{true}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
   667 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{false}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
   668 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\enskip \defl\textit{smart}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}
   669 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}
   670 \def\opnodefaultbrk#1#2{\flushitem{$\bigl[$\textit{#1} =$\bigr]$ \qtybf{#2}} \nopagebreak\\[\parskip]}
   671 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\enskip \defl\textit{#3}\defr} \nopagebreak\\[\parskip]}
   672 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}
   673 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
   674 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}
   675 
   676 Sledgehammer's options are categorized as follows:\ mode of operation
   677 (\S\ref{mode-of-operation}), problem encoding (\S\ref{problem-encoding}),
   678 relevance filter (\S\ref{relevance-filter}), output format
   679 (\S\ref{output-format}), authentication (\S\ref{authentication}), and timeouts
   680 (\S\ref{timeouts}).
   681 
   682 The descriptions below refer to the following syntactic quantities:
   683 
   684 \begin{enum}
   685 \item[$\bullet$] \qtybf{string}: A string.
   686 \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.
   687 \item[$\bullet$] \qtybf{smart\_bool\/}: \textit{true}, \textit{false}, or
   688 \textit{smart}.
   689 \item[$\bullet$] \qtybf{int\/}: An integer.
   690 %\item[$\bullet$] \qtybf{float\/}: A floating-point number (e.g., 2.5).
   691 \item[$\bullet$] \qtybf{float\_pair\/}: A pair of floating-point numbers
   692 (e.g., 0.6 0.95).
   693 \item[$\bullet$] \qtybf{smart\_int\/}: An integer or \textit{smart}.
   694 \item[$\bullet$] \qtybf{float\_or\_none\/}: A floating-point number (e.g., 60 or
   695 0.5) expressing a number of seconds, or the keyword \textit{none} ($\infty$
   696 seconds).
   697 \end{enum}
   698 
   699 Default values are indicated in curly brackets (\textrm{\{\}}). Boolean options
   700 have a negated counterpart (e.g., \textit{blocking} vs.\
   701 \textit{non\_blocking}). When setting them, ``= \textit{true}'' may be omitted.
   702 
   703 \subsection{Mode of Operation}
   704 \label{mode-of-operation}
   705 
   706 \begin{enum}
   707 \opnodefaultbrk{provers}{string}
   708 Specifies the automatic provers to use as a space-separated list (e.g.,
   709 ``\textit{e}~\textit{spass}~\textit{remote\_vampire}''). The following local
   710 provers are supported:
   711 
   712 \begin{enum}
   713 \item[$\bullet$] \textbf{\textit{cvc3}:} CVC3 is an SMT solver developed by
   714 Clark Barrett, Cesare Tinelli, and their colleagues \cite{cvc3}. To use CVC3,
   715 set the environment variable \texttt{CVC3\_SOLVER} to the complete path of the
   716 executable, including the file name. Sledgehammer has been tested with version
   717 2.2.
   718 
   719 \item[$\bullet$] \textbf{\textit{e}:} E is a first-order resolution prover
   720 developed by Stephan Schulz \cite{schulz-2002}. To use E, set the environment
   721 variable \texttt{E\_HOME} to the directory that contains the \texttt{eproof}
   722 executable, or install the prebuilt E package from Isabelle's download page. See
   723 \S\ref{installation} for details.
   724 
   725 \item[$\bullet$] \textbf{\textit{leo2}:} LEO-II is an automatic
   726 higher-order prover developed by Christoph Benzm\"uller et al.\ \cite{leo2},
   727 with support for the TPTP many-typed higher-order syntax (THF0).
   728 
   729 \item[$\bullet$] \textbf{\textit{metis}:} Although it is much less powerful than
   730 the external provers, Metis itself can be used for proof search.
   731 
   732 \item[$\bullet$] \textbf{\textit{metis\_full\_types}:} Fully typed version of
   733 Metis, corresponding to \textit{metis} (\textit{full\_types}).
   734 
   735 \item[$\bullet$] \textbf{\textit{metis\_no\_types}:} Untyped version of Metis,
   736 corresponding to \textit{metis} (\textit{no\_types}).
   737 
   738 \item[$\bullet$] \textbf{\textit{satallax}:} Satallax is an automatic
   739 higher-order prover developed by Chad Brown et al.\ \cite{satallax}, with
   740 support for the TPTP many-typed higher-order syntax (THF0).
   741 
   742 \item[$\bullet$] \textbf{\textit{spass}:} SPASS is a first-order resolution
   743 prover developed by Christoph Weidenbach et al.\ \cite{weidenbach-et-al-2009}.
   744 To use SPASS, set the environment variable \texttt{SPASS\_HOME} to the directory
   745 that contains the \texttt{SPASS} executable, or install the prebuilt SPASS
   746 package from Isabelle's download page. Sledgehammer requires version 3.5 or
   747 above. See \S\ref{installation} for details.
   748 
   749 \item[$\bullet$] \textbf{\textit{vampire}:} Vampire is a first-order resolution
   750 prover developed by Andrei Voronkov and his colleagues
   751 \cite{riazanov-voronkov-2002}. To use Vampire, set the environment variable
   752 \texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{vampire}
   753 executable and \texttt{VAMPIRE\_VERSION} to the version number (e.g., ``1.8'').
   754 Sledgehammer has been tested with versions 0.6, 1.0, and 1.8. Vampire 1.8
   755 supports the TPTP many-typed first-order format (TFF0).
   756 
   757 \item[$\bullet$] \textbf{\textit{yices}:} Yices is an SMT solver developed at
   758 SRI \cite{yices}. To use Yices, set the environment variable
   759 \texttt{YICES\_SOLVER} to the complete path of the executable, including the
   760 file name. Sledgehammer has been tested with version 1.0.
   761 
   762 \item[$\bullet$] \textbf{\textit{z3}:} Z3 is an SMT solver developed at
   763 Microsoft Research \cite{z3}. To use Z3, set the environment variable
   764 \texttt{Z3\_SOLVER} to the complete path of the executable, including the file
   765 name, and set \texttt{Z3\_NON\_COMMERCIAL} to ``yes'' to confirm that you are a
   766 noncommercial user. Sledgehammer has been tested with versions 2.7 to 2.18.
   767 
   768 \item[$\bullet$] \textbf{\textit{z3\_tptp}:} This version of Z3 pretends to be
   769 an ATP, exploiting Z3's support for the TPTP untyped and many-typed first-order
   770 formats (FOF and TFF0). It is included for experimental purposes. It requires
   771 version 3.0 or above.
   772 \end{enum}
   773 
   774 In addition, the following remote provers are supported:
   775 
   776 \begin{enum}
   777 \item[$\bullet$] \textbf{\textit{remote\_cvc3}:} The remote version of CVC3 runs
   778 on servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to
   779 point).
   780 
   781 \item[$\bullet$] \textbf{\textit{remote\_e}:} The remote version of E runs
   782 on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.
   783 
   784 \item[$\bullet$] \textbf{\textit{remote\_e\_sine}:} E-SInE is a metaprover
   785 developed by Kry\v stof Hoder \cite{sine} based on E. The remote version of
   786 SInE runs on Geoff Sutcliffe's Miami servers.
   787 
   788 \item[$\bullet$] \textbf{\textit{remote\_e\_tofof}:} E-ToFoF is a metaprover
   789 developed by Geoff Sutcliffe \cite{tofof} based on E running on his Miami
   790 servers. This ATP supports the TPTP many-typed first-order format (TFF0). The
   791 remote version of E-ToFoF runs on Geoff Sutcliffe's Miami servers.
   792 
   793 \item[$\bullet$] \textbf{\textit{remote\_leo2}:} The remote version of LEO-II
   794 runs on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.
   795 
   796 \item[$\bullet$] \textbf{\textit{remote\_satallax}:} The remote version of
   797 Satallax runs on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}.
   798 
   799 \item[$\bullet$] \textbf{\textit{remote\_snark}:} SNARK is a first-order
   800 resolution prover developed by Stickel et al.\ \cite{snark}. It supports the
   801 TPTP many-typed first-order format (TFF0). The remote version of SNARK runs on
   802 Geoff Sutcliffe's Miami servers.
   803 
   804 \item[$\bullet$] \textbf{\textit{remote\_vampire}:} The remote version of
   805 Vampire runs on Geoff Sutcliffe's Miami servers. Version 1.8 is used.
   806 
   807 \item[$\bullet$] \textbf{\textit{remote\_waldmeister}:} Waldmeister is a unit
   808 equality prover developed by Hillenbrand et al.\ \cite{waldmeister}. It can be
   809 used to prove universally quantified equations using unconditional equations,
   810 corresponding to the TPTP CNF UEQ division. The remote version of Waldmeister
   811 runs on Geoff Sutcliffe's Miami servers.
   812 
   813 \item[$\bullet$] \textbf{\textit{remote\_z3}:} The remote version of Z3 runs on
   814 servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to
   815 point).
   816 
   817 \item[$\bullet$] \textbf{\textit{remote\_z3\_tptp}:} The remote version of ``Z3
   818 with TPTP syntax'' runs on Geoff Sutcliffe's Miami servers.
   819 \end{enum}
   820 
   821 By default, Sledgehammer runs E, E-SInE, SPASS, Vampire, Z3 (or whatever
   822 the SMT module's \textit{smt\_solver} configuration option is set to), and (if
   823 appropriate) Waldmeister in parallel---either locally or remotely, depending on
   824 the number of processor cores available. For historical reasons, the default
   825 value of this option can be overridden using the option ``Sledgehammer:
   826 Provers'' in Proof General's ``Isabelle'' menu.
   827 
   828 It is generally a good idea to run several provers in parallel. Running E,
   829 SPASS, and Vampire for 5~seconds yields a similar success rate to running the
   830 most effective of these for 120~seconds \cite{boehme-nipkow-2010}.
   831 
   832 For the \textit{min} subcommand, the default prover is \textit{metis}. If
   833 several provers are set, the first one is used.
   834 
   835 \opnodefault{prover}{string}
   836 Alias for \textit{provers}.
   837 
   838 %\opnodefault{atps}{string}
   839 %Legacy alias for \textit{provers}.
   840 
   841 %\opnodefault{atp}{string}
   842 %Legacy alias for \textit{provers}.
   843 
   844 \opfalse{blocking}{non\_blocking}
   845 Specifies whether the \textbf{sledgehammer} command should operate
   846 synchronously. The asynchronous (non-blocking) mode lets the user start proving
   847 the putative theorem manually while Sledgehammer looks for a proof, but it can
   848 also be more confusing. Irrespective of the value of this option, Sledgehammer
   849 is always run synchronously for the new jEdit-based user interface or if
   850 \textit{debug} (\S\ref{output-format}) is enabled.
   851 
   852 \optrue{slicing}{no\_slicing}
   853 Specifies whether the time allocated to a prover should be sliced into several
   854 segments, each of which has its own set of possibly prover-dependent options.
   855 For SPASS and Vampire, the first slice tries the fast but incomplete
   856 set-of-support (SOS) strategy, whereas the second slice runs without it. For E,
   857 up to three slices are tried, with different weighted search strategies and
   858 number of facts. For SMT solvers, several slices are tried with the same options
   859 each time but fewer and fewer facts. According to benchmarks with a timeout of
   860 30 seconds, slicing is a valuable optimization, and you should probably leave it
   861 enabled unless you are conducting experiments. This option is implicitly
   862 disabled for (short) automatic runs.
   863 
   864 \nopagebreak
   865 {\small See also \textit{verbose} (\S\ref{output-format}).}
   866 
   867 \opfalse{overlord}{no\_overlord}
   868 Specifies whether Sledgehammer should put its temporary files in
   869 \texttt{\$ISA\-BELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for
   870 debugging Sledgehammer but also unsafe if several instances of the tool are run
   871 simultaneously. The files are identified by the prefix \texttt{prob\_}; you may
   872 safely remove them after Sledgehammer has run.
   873 
   874 \nopagebreak
   875 {\small See also \textit{debug} (\S\ref{output-format}).}
   876 \end{enum}
   877 
   878 \subsection{Problem Encoding}
   879 \label{problem-encoding}
   880 
   881 \begin{enum}
   882 \opdefault{type\_enc}{string}{smart}
   883 Specifies the type encoding to use in ATP problems. Some of the type encodings
   884 are unsound, meaning that they can give rise to spurious proofs
   885 (unreconstructible using Metis). The supported type encodings are listed below,
   886 with an indication of their soundness in parentheses:
   887 
   888 \begin{enum}
   889 \item[$\bullet$] \textbf{\textit{erased} (very unsound):} No type information is
   890 supplied to the ATP. Types are simply erased.
   891 
   892 \item[$\bullet$] \textbf{\textit{poly\_guards} (sound):} Types are encoded using
   893 a predicate \textit{has\_\allowbreak type\/}$(\tau, t)$ that guards bound
   894 variables. Constants are annotated with their types, supplied as additional
   895 arguments, to resolve overloading.
   896 
   897 \item[$\bullet$] \textbf{\textit{poly\_tags} (sound):} Each term and subterm is
   898 tagged with its type using a function $\mathit{type\/}(\tau, t)$.
   899 
   900 \item[$\bullet$] \textbf{\textit{poly\_args} (unsound):}
   901 Like for \textit{poly\_guards} constants are annotated with their types to
   902 resolve overloading, but otherwise no type information is encoded. This
   903 coincides with the default encoding used by the \textit{metis} command.
   904 
   905 \item[$\bullet$]
   906 \textbf{%
   907 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags} (sound); \\
   908 \textit{raw\_mono\_args} (unsound):} \\
   909 Similar to \textit{poly\_guards}, \textit{poly\_tags}, and \textit{poly\_args},
   910 respectively, but the problem is additionally monomorphized, meaning that type
   911 variables are instantiated with heuristically chosen ground types.
   912 Monomorphization can simplify reasoning but also leads to larger fact bases,
   913 which can slow down the ATPs.
   914 
   915 \item[$\bullet$]
   916 \textbf{%
   917 \textit{mono\_guards}, \textit{mono\_tags} (sound);
   918 \textit{mono\_args} (unsound):} \\
   919 Similar to
   920 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, and
   921 \textit{raw\_mono\_args}, respectively but types are mangled in constant names
   922 instead of being supplied as ground term arguments. The binary predicate
   923 $\mathit{has\_type\/}(\tau, t)$ becomes a unary predicate
   924 $\mathit{has\_type\_}\tau(t)$, and the binary function
   925 $\mathit{type\/}(\tau, t)$ becomes a unary function
   926 $\mathit{type\_}\tau(t)$.
   927 
   928 \item[$\bullet$] \textbf{\textit{mono\_simple} (sound):} Exploits simple
   929 first-order types if the prover supports the TFF0 or THF0 syntax; otherwise,
   930 falls back on \textit{mono\_guards}. The problem is monomorphized.
   931 
   932 \item[$\bullet$] \textbf{\textit{mono\_simple\_higher} (sound):} Exploits simple
   933 higher-order types if the prover supports the THF0 syntax; otherwise, falls back
   934 on \textit{mono\_simple} or \textit{mono\_guards}. The problem is monomorphized.
   935 
   936 \item[$\bullet$]
   937 \textbf{%
   938 \textit{poly\_guards}?, \textit{poly\_tags}?, \textit{raw\_mono\_guards}?, \\
   939 \textit{raw\_mono\_tags}?, \textit{mono\_guards}?, \textit{mono\_tags}?, \\
   940 \textit{mono\_simple}? (quasi-sound):} \\
   941 The type encodings \textit{poly\_guards}, \textit{poly\_tags},
   942 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, \textit{mono\_guards},
   943 \textit{mono\_tags}, and \textit{mono\_simple} are fully
   944 typed and sound. For each of these, Sledgehammer also provides a lighter,
   945 virtually sound variant identified by a question mark (`\hbox{?}')\ that detects
   946 and erases monotonic types, notably infinite types. (For \textit{mono\_simple},
   947 the types are not actually erased but rather replaced by a shared uniform type
   948 of individuals.) As argument to the \textit{metis} proof method, the question
   949 mark is replaced by a \hbox{``\textit{\_query}''} suffix. If the \emph{sound}
   950 option is enabled, these encodings are fully sound.
   951 
   952 \item[$\bullet$]
   953 \textbf{%
   954 \textit{poly\_guards}??, \textit{poly\_tags}??, \textit{raw\_mono\_guards}??, \\
   955 \textit{raw\_mono\_tags}??, \textit{mono\_guards}??, \textit{mono\_tags}?? \\
   956 (quasi-sound):} \\
   957 Even lighter versions of the `\hbox{?}' encodings. As argument to the
   958 \textit{metis} proof method, the `\hbox{??}' suffix is replaced by
   959 \hbox{``\textit{\_query\_query}''}.
   960 
   961 \item[$\bullet$]
   962 \textbf{%
   963 \textit{poly\_guards}@?, \textit{poly\_tags}@?, \textit{raw\_mono\_guards}@?, \\
   964 \textit{raw\_mono\_tags}@? (quasi-sound):} \\
   965 Alternative versions of the `\hbox{??}' encodings. As argument to the
   966 \textit{metis} proof method, the `\hbox{@?}' suffix is replaced by
   967 \hbox{``\textit{\_at\_query}''}.
   968 
   969 \item[$\bullet$]
   970 \textbf{%
   971 \textit{poly\_guards}!, \textit{poly\_tags}!, \textit{raw\_mono\_guards}!, \\
   972 \textit{raw\_mono\_tags}!, \textit{mono\_guards}!, \textit{mono\_tags}!, \\
   973 \textit{mono\_simple}!, \textit{mono\_simple\_higher}! (mildly unsound):} \\
   974 The type encodings \textit{poly\_guards}, \textit{poly\_tags},
   975 \textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, \textit{mono\_guards},
   976 \textit{mono\_tags}, \textit{mono\_simple}, and \textit{mono\_simple\_higher}
   977 also admit a mildly unsound (but very efficient) variant identified by an
   978 exclamation mark (`\hbox{!}') that detects and erases erases all types except
   979 those that are clearly finite (e.g., \textit{bool}). (For \textit{mono\_simple}
   980 and \textit{mono\_simple\_higher}, the types are not actually erased but rather
   981 replaced by a shared uniform type of individuals.) As argument to the
   982 \textit{metis} proof method, the exclamation mark is replaced by the suffix
   983 \hbox{``\textit{\_bang}''}.
   984 
   985 \item[$\bullet$]
   986 \textbf{%
   987 \textit{poly\_guards}!!, \textit{poly\_tags}!!, \textit{raw\_mono\_guards}!!, \\
   988 \textit{raw\_mono\_tags}!!, \textit{mono\_guards}!!, \textit{mono\_tags}!! \\
   989 (mildly unsound):} \\
   990 Even lighter versions of the `\hbox{!}' encodings. As argument to the
   991 \textit{metis} proof method, the `\hbox{!!}' suffix is replaced by
   992 \hbox{``\textit{\_bang\_bang}''}.
   993 
   994 \item[$\bullet$]
   995 \textbf{%
   996 \textit{poly\_guards}@!, \textit{poly\_tags}@!, \textit{raw\_mono\_guards}@!, \\
   997 \textit{raw\_mono\_tags}@! (mildly unsound):} \\
   998 Alternative versions of the `\hbox{!!}' encodings. As argument to the
   999 \textit{metis} proof method, the `\hbox{@!}' suffix is replaced by
  1000 \hbox{``\textit{\_at\_bang}''}.
  1001 
  1002 \item[$\bullet$] \textbf{\textit{smart}:} The actual encoding used depends on
  1003 the ATP and should be the most efficient virtually sound encoding for that ATP.
  1004 \end{enum}
  1005 
  1006 For SMT solvers, the type encoding is always \textit{mono\_simple}, irrespective
  1007 of the value of this option.
  1008 
  1009 \nopagebreak
  1010 {\small See also \textit{max\_new\_mono\_instances} (\S\ref{relevance-filter})
  1011 and \textit{max\_mono\_iters} (\S\ref{relevance-filter}).}
  1012 
  1013 \opfalse{sound}{unsound}
  1014 Specifies whether Sledgehammer should run in its fully sound mode. In that mode,
  1015 quasi-sound type encodings (which are the default) are made fully sound, at the
  1016 cost of some clutter in the generated problems. This option is ignored if
  1017 \textit{type\_enc} is explicitly set to an unsound encoding.
  1018 \end{enum}
  1019 
  1020 \subsection{Relevance Filter}
  1021 \label{relevance-filter}
  1022 
  1023 \begin{enum}
  1024 \opdefault{relevance\_thresholds}{float\_pair}{\upshape 0.45~0.85}
  1025 Specifies the thresholds above which facts are considered relevant by the
  1026 relevance filter. The first threshold is used for the first iteration of the
  1027 relevance filter and the second threshold is used for the last iteration (if it
  1028 is reached). The effective threshold is quadratically interpolated for the other
  1029 iterations. Each threshold ranges from 0 to 1, where 0 means that all theorems
  1030 are relevant and 1 only theorems that refer to previously seen constants.
  1031 
  1032 \opdefault{max\_relevant}{smart\_int}{smart}
  1033 Specifies the maximum number of facts that may be returned by the relevance
  1034 filter. If the option is set to \textit{smart}, it is set to a value that was
  1035 empirically found to be appropriate for the prover. A typical value would be
  1036 250.
  1037 
  1038 \opdefault{max\_new\_mono\_instances}{int}{\upshape 200}
  1039 Specifies the maximum number of monomorphic instances to generate beyond
  1040 \textit{max\_relevant}. The higher this limit is, the more monomorphic instances
  1041 are potentially generated. Whether monomorphization takes place depends on the
  1042 type encoding used.
  1043 
  1044 \nopagebreak
  1045 {\small See also \textit{type\_enc} (\S\ref{problem-encoding}).}
  1046 
  1047 \opdefault{max\_mono\_iters}{int}{\upshape 3}
  1048 Specifies the maximum number of iterations for the monomorphization fixpoint
  1049 construction. The higher this limit is, the more monomorphic instances are
  1050 potentially generated. Whether monomorphization takes place depends on the
  1051 type encoding used.
  1052 
  1053 \nopagebreak
  1054 {\small See also \textit{type\_enc} (\S\ref{problem-encoding}).}
  1055 \end{enum}
  1056 
  1057 \subsection{Output Format}
  1058 \label{output-format}
  1059 
  1060 \begin{enum}
  1061 
  1062 \opfalse{verbose}{quiet}
  1063 Specifies whether the \textbf{sledgehammer} command should explain what it does.
  1064 This option is implicitly disabled for automatic runs.
  1065 
  1066 \opfalse{debug}{no\_debug}
  1067 Specifies whether Sledgehammer should display additional debugging information
  1068 beyond what \textit{verbose} already displays. Enabling \textit{debug} also
  1069 enables \textit{verbose} and \textit{blocking} (\S\ref{mode-of-operation})
  1070 behind the scenes. The \textit{debug} option is implicitly disabled for
  1071 automatic runs.
  1072 
  1073 \nopagebreak
  1074 {\small See also \textit{overlord} (\S\ref{mode-of-operation}).}
  1075 
  1076 \opfalse{isar\_proof}{no\_isar\_proof}
  1077 Specifies whether Isar proofs should be output in addition to one-liner
  1078 \textit{metis} proofs. Isar proof construction is still experimental and often
  1079 fails; however, they are usually faster and sometimes more robust than
  1080 \textit{metis} proofs.
  1081 
  1082 \opdefault{isar\_shrink\_factor}{int}{\upshape 1}
  1083 Specifies the granularity of the Isar proof. A value of $n$ indicates that each
  1084 Isar proof step should correspond to a group of up to $n$ consecutive proof
  1085 steps in the ATP proof.
  1086 \end{enum}
  1087 
  1088 \subsection{Authentication}
  1089 \label{authentication}
  1090 
  1091 \begin{enum}
  1092 \opnodefault{expect}{string}
  1093 Specifies the expected outcome, which must be one of the following:
  1094 
  1095 \begin{enum}
  1096 \item[$\bullet$] \textbf{\textit{some}:} Sledgehammer found a (potentially
  1097 unsound) proof.
  1098 \item[$\bullet$] \textbf{\textit{none}:} Sledgehammer found no proof.
  1099 \item[$\bullet$] \textbf{\textit{timeout}:} Sledgehammer timed out.
  1100 \item[$\bullet$] \textbf{\textit{unknown}:} Sledgehammer encountered some
  1101 problem.
  1102 \end{enum}
  1103 
  1104 Sledgehammer emits an error (if \textit{blocking} is enabled) or a warning
  1105 (otherwise) if the actual outcome differs from the expected outcome. This option
  1106 is useful for regression testing.
  1107 
  1108 \nopagebreak
  1109 {\small See also \textit{blocking} (\S\ref{mode-of-operation}) and
  1110 \textit{timeout} (\S\ref{timeouts}).}
  1111 \end{enum}
  1112 
  1113 \subsection{Timeouts}
  1114 \label{timeouts}
  1115 
  1116 \begin{enum}
  1117 \opdefault{timeout}{float\_or\_none}{\upshape 30}
  1118 Specifies the maximum number of seconds that the automatic provers should spend
  1119 searching for a proof. This excludes problem preparation and is a soft limit.
  1120 For historical reasons, the default value of this option can be overridden using
  1121 the option ``Sledgehammer: Time Limit'' in Proof General's ``Isabelle'' menu.
  1122 
  1123 \opdefault{preplay\_timeout}{float\_or\_none}{\upshape 4}
  1124 Specifies the maximum number of seconds that Metis should be spent trying to
  1125 ``preplay'' the found proof. If this option is set to 0, no preplaying takes
  1126 place, and no timing information is displayed next to the suggested Metis calls.
  1127 \end{enum}
  1128 
  1129 \let\em=\sl
  1130 \bibliography{../manual}{}
  1131 \bibliographystyle{abbrv}
  1132 
  1133 \end{document}