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doc-src/Sledgehammer/sledgehammer.tex

author | blanchet |

Thu Sep 22 19:42:06 2011 +0200 (2011-09-22) | |

changeset 45048 | 59ca831deef4 |

parent 44816 | efa1f532508b |

child 45063 | b3b50d8b535a |

permissions | -rw-r--r-- |

take out remote E-SInE -- it's broken and Geoff says it might take quite a while before he gets to it, plus it's fairly obsolete in the meantime

1 \documentclass[a4paper,12pt]{article}

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39 \urlstyle{tt}

41 \begin{document}

43 \selectlanguage{english}

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{}}

56 \maketitle

58 \tableofcontents

60 \setlength{\parskip}{.7em plus .2em minus .1em}

61 \setlength{\parindent}{0pt}

62 \setlength{\abovedisplayskip}{\parskip}

63 \setlength{\abovedisplayshortskip}{.9\parskip}

64 \setlength{\belowdisplayskip}{\parskip}

65 \setlength{\belowdisplayshortskip}{.9\parskip}

67 % General-purpose enum environment with correct spacing

68 \newenvironment{enum}%

69 {\begin{list}{}{%

70 \setlength{\topsep}{.1\parskip}%

71 \setlength{\partopsep}{.1\parskip}%

72 \setlength{\itemsep}{\parskip}%

73 \advance\itemsep by-\parsep}}

74 {\end{list}}

76 \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin

77 \advance\rightskip by\leftmargin}

78 \def\post{\vskip0pt plus1ex\endgroup}

80 \def\prew{\pre\advance\rightskip by-\leftmargin}

81 \def\postw{\post}

83 \section{Introduction}

84 \label{introduction}

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.

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.

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.

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.

117 \newbox\boxA

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

120 \newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak

121 in.\allowbreak tum.\allowbreak de}}

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.

130 \vskip2.5\smallskipamount

132 %\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for

133 %suggesting several textual improvements.

135 \section{Installation}

136 \label{installation}

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.

141 \subsection{Installing ATPs}

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.

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

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.}

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

168 \prew

169 \texttt{/usr/local/spass-3.7}

170 \postw

172 in it.

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 1.4, 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}

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.

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:

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

207 \subsection{Installing SMT Solvers}

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.

214 There are two main ways of installing SMT solvers locally.

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.

226 \item[$\bullet$] Otherwise, follow the instructions documented in the \emph{SMT}

227 theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}).

228 \end{enum}

230 \section{First Steps}

231 \label{first-steps}

233 To illustrate Sledgehammer in context, let us start a theory file and

234 attempt to prove a simple lemma:

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

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:

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\_z3}'' on goal \\

269 $[a] = [b] \,\Longrightarrow\, a = b$ \\

270 Try this: \textbf{by} (\textit{metis list.inject}) (20 ms).

271 \postw

273 Sledgehammer ran E, SPASS, Vampire, Waldmeister, and Z3 in parallel.

274 Depending on which provers are installed and how many processor cores are

275 available, some of the provers might be missing or present with a

276 \textit{remote\_} prefix. Waldmeister is run only for unit equational problems,

277 where the goal's conclusion is a (universally quantified) equation.

279 For each successful prover, Sledgehammer gives a one-liner proof that uses Metis

280 or the \textit{smt} proof method. For Metis, approximate timings are shown in

281 parentheses, indicating how fast the call is. You can click the proof to insert

282 it into the theory text.

284 In addition, you can ask Sledgehammer for an Isar text proof by passing the

285 \textit{isar\_proof} option (\S\ref{output-format}):

287 \prew

288 \textbf{sledgehammer} [\textit{isar\_proof}]

289 \postw

291 When Isar proof construction is successful, it can yield proofs that are more

292 readable and also faster than the Metis one-liners. This feature is experimental

293 and is only available for ATPs.

295 \section{Hints}

296 \label{hints}

298 This section presents a few hints that should help you get the most out of

299 Sledgehammer and Metis. Frequently (and infrequently) asked questions are

300 answered in \S\ref{frequently-asked-questions}.

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

304 \point{Presimplify the goal}

306 For best results, first simplify your problem by calling \textit{auto} or at

307 least \textit{safe} followed by \textit{simp\_all}. The SMT solvers provide

308 arithmetic decision procedures, but the ATPs typically do not (or if they do,

309 Sledgehammer does not use it yet). Apart from Waldmeister, they are not

310 especially good at heavy rewriting, but because they regard equations as

311 undirected, they often prove theorems that require the reverse orientation of a

312 \textit{simp} rule. Higher-order problems can be tackled, but the success rate

313 is better for first-order problems. Hence, you may get better results if you

314 first simplify the problem to remove higher-order features.

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

318 Locally installed provers are faster and more reliable than those running on

319 servers. See \S\ref{installation} for details on how to install them.

321 \point{Familiarize yourself with the most important options}

323 Sledgehammer's options are fully documented in \S\ref{command-syntax}. Many of

324 the options are very specialized, but serious users of the tool should at least

325 familiarize themselves with the following options:

327 \begin{enum}

328 \item[$\bullet$] \textbf{\textit{provers}} (\S\ref{mode-of-operation}) specifies

329 the automatic provers (ATPs and SMT solvers) that should be run whenever

330 Sledgehammer is invoked (e.g., ``\textit{provers}~= \textit{e spass

331 remote\_vampire}''). For convenience, you can omit ``\textit{provers}~=''

332 and simply write the prover names as a space-separated list (e.g., ``\textit{e

333 spass remote\_vampire}'').

335 \item[$\bullet$] \textbf{\textit{max\_relevant}} (\S\ref{relevance-filter})

336 specifies the maximum number of facts that should be passed to the provers. By

337 default, the value is prover-dependent but varies between about 150 and 1000. If

338 the provers time out, you can try lowering this value to, say, 100 or 50 and see

339 if that helps.

341 \item[$\bullet$] \textbf{\textit{isar\_proof}} (\S\ref{output-format}) specifies

342 that Isar proofs should be generated, instead of one-liner Metis proofs. The

343 length of the Isar proofs can be controlled by setting

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

346 \item[$\bullet$] \textbf{\textit{timeout}} (\S\ref{timeouts}) controls the

347 provers' time limit. It is set to 30 seconds, but since Sledgehammer runs

348 asynchronously you should not hesitate to raise this limit to 60 or 120 seconds

349 if you are the kind of user who can think clearly while ATPs are active.

350 \end{enum}

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

353 (\S\ref{command-syntax}). The command also prints the list of all available

354 options with their current value. Fact selection can be influenced by specifying

355 ``$(\textit{add}{:}~\textit{my\_facts})$'' after the \textbf{sledgehammer} call

356 to ensure that certain facts are included, or simply ``$(\textit{my\_facts})$''

357 to force Sledgehammer to run only with $\textit{my\_facts}$.

359 \section{Frequently Asked Questions}

360 \label{frequently-asked-questions}

362 This sections answers frequently (and infrequently) asked questions about

363 Sledgehammer. It is a good idea to skim over it now even if you don't have any

364 questions at this stage. And if you have any further questions not listed here,

365 send them to the author at \authoremail.

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

369 There are many reasons. If Metis runs seemingly forever, that is a sign that the

370 proof is too difficult for it. Metis's search is complete, so it should

371 eventually find it, but that's little consolation. There are several possible

372 solutions:

374 \begin{enum}

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

376 obtain a step-by-step Isar proof where each step is justified by Metis. Since

377 the steps are fairly small, Metis is more likely to be able to replay them.

379 \item[$\bullet$] Try the \textit{smt} proof method instead of Metis. It is

380 usually stronger, but you need to have Z3 available to replay the proofs, trust

381 the SMT solver, or use certificates. See the documentation in the \emph{SMT}

382 theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}) for details.

384 \item[$\bullet$] Try the \textit{blast} or \textit{auto} proof methods, passing

385 the necessary facts via \textbf{unfolding}, \textbf{using}, \textit{intro}{:},

386 \textit{elim}{:}, \textit{dest}{:}, or \textit{simp}{:}, as appropriate.

387 \end{enum}

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

391 \prew

392 \slshape

393 Proof reconstruction failed.

394 \postw

396 This message usually indicates that Sledgehammer found a type-incorrect proof.

397 This was a frequent issue with older versions of Sledgehammer, which did not

398 supply enough typing information to the ATPs by default. If you notice many

399 unsound proofs and are not using \textit{type\_enc} (\S\ref{problem-encoding}),

400 contact the author at \authoremail.

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

404 First, if Metis can reconstruct it, the proof is sound (assuming Isabelle's

405 inference kernel is sound). If it fails or runs seemingly forever, you can try

407 \prew

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

409 \textbf{sledgehammer} [\textit{sound}] (\textit{metis\_facts})

410 \postw

412 where \textit{metis\_facts} is the list of facts appearing in the suggested

413 Metis call. The automatic provers should be able to re-find the proof quickly if

414 it is sound, and the \textit{sound} option (\S\ref{problem-encoding}) ensures

415 that no unsound proofs are found.

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

419 The relevance filter assigns a score to every available fact (lemma, theorem,

420 definition, or axiom)\ based upon how many constants that fact shares with the

421 conjecture. This process iterates to include facts relevant to those just

422 accepted, but with a decay factor to ensure termination. The constants are

423 weighted to give unusual ones greater significance. The relevance filter copes

424 best when the conjecture contains some unusual constants; if all the constants

425 are common, it is unable to discriminate among the hundreds of facts that are

426 picked up. The relevance filter is also memoryless: It has no information about

427 how many times a particular fact has been used in a proof, and it cannot learn.

429 The number of facts included in a problem varies from prover to prover, since

430 some provers get overwhelmed more easily than others. You can show the number of

431 facts given using the \textit{verbose} option (\S\ref{output-format}) and the

432 actual facts using \textit{debug} (\S\ref{output-format}).

434 Sledgehammer is good at finding short proofs combining a handful of existing

435 lemmas. If you are looking for longer proofs, you must typically restrict the

436 number of facts, by setting the \textit{max\_relevant} option

437 (\S\ref{relevance-filter}) to, say, 25 or 50.

439 You can also influence which facts are actually selected in a number of ways. If

440 you simply want to ensure that a fact is included, you can specify it using the

441 ``$(\textit{add}{:}~\textit{my\_facts})$'' syntax. For example:

442 %

443 \prew

444 \textbf{sledgehammer} (\textit{add}: \textit{hd.simps} \textit{tl.simps})

445 \postw

446 %

447 The specified facts then replace the least relevant facts that would otherwise be

448 included; the other selected facts remain the same.

449 If you want to direct the selection in a particular direction, you can specify

450 the facts via \textbf{using}:

451 %

452 \prew

453 \textbf{using} \textit{hd.simps} \textit{tl.simps} \\

454 \textbf{sledgehammer}

455 \postw

456 %

457 The facts are then more likely to be selected than otherwise, and if they are

458 selected at iteration $j$ they also influence which facts are selected at

459 iterations $j + 1$, $j + 2$, etc. To give them even more weight, try

460 %

461 \prew

462 \textbf{using} \textit{hd.simps} \textit{tl.simps} \\

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

464 \textbf{sledgehammer}

465 \postw

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

469 The current implementation is experimental and explodes exponentially in the

470 worst case. Work on a new implementation has begun. There is a large body of

471 research into transforming resolution proofs into natural deduction proofs (such

472 as Isar proofs), which we hope to leverage. In the meantime, a workaround is to

473 set the \textit{isar\_shrink\_factor} option (\S\ref{output-format}) to a larger

474 value or to try several provers and keep the nicest-looking proof.

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

477 Metis?}

479 The \textit{metis}~(\textit{full\_types}) proof method is the fully-typed

480 version of Metis. It is somewhat slower than \textit{metis}, but the proof

481 search is fully typed, and it also includes more powerful rules such as the

482 axiom ``$x = \mathit{True} \mathrel{\lor} x = \mathit{False}$'' for reasoning in

483 higher-order places (e.g., in set comprehensions). The method kicks in

484 automatically as a fallback when \textit{metis} fails, and it is sometimes

485 generated by Sledgehammer instead of \textit{metis} if the proof obviously

486 requires type information or if \textit{metis} failed when Sledgehammer

487 preplayed the proof. (By default, Sledgehammer tries to run \textit{metis} with

488 various options for up to 4 seconds to ensure that the generated one-line proofs

489 actually work and to display timing information. This can be configured using

490 the \textit{preplay\_timeout} option (\S\ref{timeouts}).)

492 At the other end of the soundness spectrum, \textit{metis} (\textit{no\_types})

493 uses no type information at all during the proof search, which is more efficient

494 but often fails. Calls to \textit{metis} (\textit{no\_types}) are occasionally

495 generated by Sledgehammer.

497 Incidentally, if you see the warning

499 \prew

500 \slshape

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

502 \postw

504 for a successful Metis proof, you can advantageously pass the

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

507 \point{Are generated proofs minimal?}

509 Automatic provers frequently use many more facts than are necessary.

510 Sledgehammer inclues a minimization tool that takes a set of facts returned by a

511 given prover and repeatedly calls the same prover or Metis with subsets of those

512 axioms in order to find a minimal set. Reducing the number of axioms typically

513 improves Metis's speed and success rate, while also removing superfluous clutter

514 from the proof scripts.

516 In earlier versions of Sledgehammer, generated proofs were systematically

517 accompanied by a suggestion to invoke the minimization tool. This step is now

518 performed implicitly if it can be done in a reasonable amount of time (something

519 that can be guessed from the number of facts in the original proof and the time

520 it took to find it or replay it).

522 In addition, some provers (notably CVC3, Satallax, and Yices) do not provide

523 proofs or sometimes produce incomplete proofs. The minimizer is then invoked to

524 find out which facts are actually needed from the (large) set of facts that was

525 initinally given to the prover. Finally, if a prover returns a proof with lots

526 of facts, the minimizer is invoked automatically since Metis would be unlikely

527 to re-find the proof.

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

531 Sledgehammer tries to give informative error messages. Please report any strange

532 error to the author at \authoremail. This applies double if you get the message

534 \prew

535 \slshape

536 The prover found a type-unsound proof involving ``\textit{foo}'',

537 ``\textit{bar}'', and ``\textit{baz}'' even though a supposedly type-sound

538 encoding was used (or, less likely, your axioms are inconsistent). You might

539 want to report this to the Isabelle developers.

540 \postw

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

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

545 the reverse is also true, so don't be discouraged if your first attempts fail.

546 Because the system refers to all theorems known to Isabelle, it is particularly

547 suitable when your goal has a short proof from lemmas that you don't know about.

549 \point{Why are there so many options?}

551 Sledgehammer's philosophy should work out of the box, without user guidance.

552 Many of the options are meant to be used mostly by the Sledgehammer developers

553 for experimentation purposes. Of course, feel free to experiment with them if

554 you are so inclined.

556 \section{Command Syntax}

557 \label{command-syntax}

559 Sledgehammer can be invoked at any point when there is an open goal by entering

560 the \textbf{sledgehammer} command in the theory file. Its general syntax is as

561 follows:

563 \prew

564 \textbf{sledgehammer} \qty{subcommand}$^?$ \qty{options}$^?$ \qty{facts\_override}$^?$ \qty{num}$^?$

565 \postw

567 For convenience, Sledgehammer is also available in the ``Commands'' submenu of

568 the ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c

569 C-a C-s. This is equivalent to entering the \textbf{sledgehammer} command with

570 no arguments in the theory text.

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

574 \begin{enum}

575 \item[$\bullet$] \textbf{\textit{run} (the default):} Runs Sledgehammer on

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

578 \item[$\bullet$] \textbf{\textit{min}:} Attempts to minimize the facts

579 specified in the \qty{facts\_override} argument to obtain a simpler proof

580 involving fewer facts. The options and goal number are as for \textit{run}.

582 \item[$\bullet$] \textbf{\textit{messages}:} Redisplays recent messages issued

583 by Sledgehammer. This allows you to examine results that might have been lost

584 due to Sledgehammer's asynchronous nature. The \qty{num} argument specifies a

585 limit on the number of messages to display (5 by default).

587 \item[$\bullet$] \textbf{\textit{supported\_provers}:} Prints the list of

588 automatic provers supported by Sledgehammer. See \S\ref{installation} and

589 \S\ref{mode-of-operation} for more information on how to install automatic

590 provers.

592 \item[$\bullet$] \textbf{\textit{running\_provers}:} Prints information about

593 currently running automatic provers, including elapsed runtime and remaining

594 time until timeout.

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

597 automatic provers.

599 \item[$\bullet$] \textbf{\textit{refresh\_tptp}:} Refreshes the list of remote

600 ATPs available at System\-On\-TPTP \cite{sutcliffe-2000}.

601 \end{enum}

603 Sledgehammer's behavior can be influenced by various \qty{options}, which can be

604 specified in brackets after the \textbf{sledgehammer} command. The

605 \qty{options} are a list of key--value pairs of the form ``[$k_1 = v_1,

606 \ldots, k_n = v_n$]''. For Boolean options, ``= \textit{true}'' is optional. For

607 example:

609 \prew

610 \textbf{sledgehammer} [\textit{isar\_proof}, \,\textit{timeout} = 120]

611 \postw

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

615 \prew

616 \textbf{sledgehammer\_params} \qty{options}

617 \postw

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

621 The \qty{facts\_override} argument lets you alter the set of facts that go

622 through the relevance filter. It may be of the form ``(\qty{facts})'', where

623 \qty{facts} is a space-separated list of Isabelle facts (theorems, local

624 assumptions, etc.), in which case the relevance filter is bypassed and the given

625 facts are used. It may also be of the form ``(\textit{add}:\ \qty{facts\/_{\mathrm{1}}})'',

626 ``(\textit{del}:\ \qty{facts\/_{\mathrm{2}}})'', or ``(\textit{add}:\ \qty{facts\/_{\mathrm{1}}}\

627 \textit{del}:\ \qty{facts\/_{\mathrm{2}}})'', where the relevance filter is instructed to

628 proceed as usual except that it should consider \qty{facts\/_{\mathrm{1}}}

629 highly-relevant and \qty{facts\/_{\mathrm{2}}} fully irrelevant.

631 You can instruct Sledgehammer to run automatically on newly entered theorems by

632 enabling the ``Auto Sledgehammer'' option in Proof General's ``Isabelle'' menu.

633 For automatic runs, only the first prover set using \textit{provers}

634 (\S\ref{mode-of-operation}) is considered, fewer facts are passed to the prover,

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

636 (\S\ref{problem-encoding}) is enabled, \textit{verbose} (\S\ref{output-format})

637 and \textit{debug} (\S\ref{output-format}) are disabled, and \textit{timeout}

638 (\S\ref{timeouts}) is superseded by the ``Auto Tools Time Limit'' in Proof

639 General's ``Isabelle'' menu. Sledgehammer's output is also more concise.

641 The \textit{metis} proof method has the syntax

643 \prew

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

645 \postw

647 where \qty{type\_enc} is a type encoding specification with the same semantics

648 as Sledgehammer's \textit{type\_enc} option (\S\ref{problem-encoding}) and

649 \qty{facts} is a list of arbitrary facts. In addition to the values listed in

650 \S\ref{problem-encoding}, \qty{type\_enc} may also be \textit{full\_types}, in

651 which case an appropriate type-sound encoding is chosen, \textit{partial\_types}

652 (the default type-unsound encoding), or \textit{no\_types}, a synonym for

653 \textit{erased}.

655 \section{Option Reference}

656 \label{option-reference}

658 \def\defl{\{}

659 \def\defr{\}}

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

662 \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{true}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}

663 \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\enskip \defl\textit{false}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}

664 \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\enskip \defl\textit{smart}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}

665 \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]}

666 \def\opnodefaultbrk#1#2{\flushitem{$\bigl[$\textit{#1} =$\bigr]$ \qtybf{#2}} \nopagebreak\\[\parskip]}

667 \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\enskip \defl\textit{#3}\defr} \nopagebreak\\[\parskip]}

668 \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]}

669 \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}

670 \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{smart\_bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]}

672 Sledgehammer's options are categorized as follows:\ mode of operation

673 (\S\ref{mode-of-operation}), problem encoding (\S\ref{problem-encoding}),

674 relevance filter (\S\ref{relevance-filter}), output format

675 (\S\ref{output-format}), authentication (\S\ref{authentication}), and timeouts

676 (\S\ref{timeouts}).

678 The descriptions below refer to the following syntactic quantities:

680 \begin{enum}

681 \item[$\bullet$] \qtybf{string}: A string.

682 \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}.

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

684 \textit{smart}.

685 \item[$\bullet$] \qtybf{int\/}: An integer.

686 %\item[$\bullet$] \qtybf{float\/}: A floating-point number (e.g., 2.5).

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

688 (e.g., 0.6 0.95).

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

690 \item[$\bullet$] \qtybf{float\_or\_none\/}: A floating-point number (e.g., 60 or

691 0.5) expressing a number of seconds, or the keyword \textit{none} ($\infty$

692 seconds).

693 \end{enum}

695 Default values are indicated in curly brackets (\textrm{\{\}}). Boolean options

696 have a negated counterpart (e.g., \textit{blocking} vs.\

697 \textit{non\_blocking}). When setting them, ``= \textit{true}'' may be omitted.

699 \subsection{Mode of Operation}

700 \label{mode-of-operation}

702 \begin{enum}

703 \opnodefaultbrk{provers}{string}

704 Specifies the automatic provers to use as a space-separated list (e.g.,

705 ``\textit{e}~\textit{spass}~\textit{remote\_vampire}''). The following local

706 provers are supported:

708 \begin{enum}

709 \item[$\bullet$] \textbf{\textit{cvc3}:} CVC3 is an SMT solver developed by

710 Clark Barrett, Cesare Tinelli, and their colleagues \cite{cvc3}. To use CVC3,

711 set the environment variable \texttt{CVC3\_SOLVER} to the complete path of the

712 executable, including the file name. Sledgehammer has been tested with version

713 2.2.

715 \item[$\bullet$] \textbf{\textit{e}:} E is a first-order resolution prover

716 developed by Stephan Schulz \cite{schulz-2002}. To use E, set the environment

717 variable \texttt{E\_HOME} to the directory that contains the \texttt{eproof}

718 executable, or install the prebuilt E package from Isabelle's download page. See

719 \S\ref{installation} for details.

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

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

723 with support for the TPTP many-typed higher-order syntax (THF0).

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

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

728 \item[$\bullet$] \textbf{\textit{metis\_full\_types}:} Fully typed version of

729 Metis, corresponding to \textit{metis} (\textit{full\_types}).

731 \item[$\bullet$] \textbf{\textit{metis\_no\_types}:} Untyped version of Metis,

732 corresponding to \textit{metis} (\textit{no\_types}).

734 \item[$\bullet$] \textbf{\textit{satallax}:} Satallax is an automatic

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

736 support for the TPTP many-typed higher-order syntax (THF0).

738 \item[$\bullet$] \textbf{\textit{spass}:} SPASS is a first-order resolution

739 prover developed by Christoph Weidenbach et al.\ \cite{weidenbach-et-al-2009}.

740 To use SPASS, set the environment variable \texttt{SPASS\_HOME} to the directory

741 that contains the \texttt{SPASS} executable, or install the prebuilt SPASS

742 package from Isabelle's download page. Sledgehammer requires version 3.5 or

743 above. See \S\ref{installation} for details.

745 \item[$\bullet$] \textbf{\textit{vampire}:} Vampire is a first-order resolution

746 prover developed by Andrei Voronkov and his colleagues

747 \cite{riazanov-voronkov-2002}. To use Vampire, set the environment variable

748 \texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{vampire}

749 executable and \texttt{VAMPIRE\_VERSION} to the version number (e.g., ``1.8'').

750 Sledgehammer has been tested with versions 0.6, 1.0, and 1.8. Vampire 1.8

751 supports the TPTP many-typed first-order format (TFF0).

753 \item[$\bullet$] \textbf{\textit{yices}:} Yices is an SMT solver developed at

754 SRI \cite{yices}. To use Yices, set the environment variable

755 \texttt{YICES\_SOLVER} to the complete path of the executable, including the

756 file name. Sledgehammer has been tested with version 1.0.

758 \item[$\bullet$] \textbf{\textit{z3}:} Z3 is an SMT solver developed at

759 Microsoft Research \cite{z3}. To use Z3, set the environment variable

760 \texttt{Z3\_SOLVER} to the complete path of the executable, including the file

761 name, and set \texttt{Z3\_NON\_COMMERCIAL} to ``yes'' to confirm that you are a

762 noncommercial user. Sledgehammer has been tested with versions 2.7 to 2.18.

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

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

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

767 version 3.0 or above.

768 \end{enum}

770 In addition, the following remote provers are supported:

772 \begin{enum}

773 \item[$\bullet$] \textbf{\textit{remote\_cvc3}:} The remote version of CVC3 runs

774 on servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to

775 point).

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

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

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

781 developed by Kry\v stof Hoder \cite{sine} based on E. The remote version of

782 SInE runs on Geoff Sutcliffe's Miami servers.

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

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

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

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

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

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

792 \item[$\bullet$] \textbf{\textit{remote\_satallax}:} The remote version of

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

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

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

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

798 Geoff Sutcliffe's Miami servers.

800 \item[$\bullet$] \textbf{\textit{remote\_vampire}:} The remote version of

801 Vampire runs on Geoff Sutcliffe's Miami servers. Version 1.8 is used.

803 \item[$\bullet$] \textbf{\textit{remote\_waldmeister}:} Waldmeister is a unit

804 equality prover developed by Hillenbrand et al.\ \cite{waldmeister}. It can be

805 used to prove universally quantified equations using unconditional equations,

806 corresponding to the TPTP CNF UEQ division. The remote version of Waldmeister

807 runs on Geoff Sutcliffe's Miami servers.

809 \item[$\bullet$] \textbf{\textit{remote\_z3}:} The remote version of Z3 runs on

810 servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to

811 point).

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

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

815 \end{enum}

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

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

819 appropriate) Waldmeister in parallel---either locally or remotely, depending on

820 the number of processor cores available. For historical reasons, the default

821 value of this option can be overridden using the option ``Sledgehammer:

822 Provers'' in Proof General's ``Isabelle'' menu.

824 It is generally a good idea to run several provers in parallel. Running E,

825 SPASS, and Vampire for 5~seconds yields a similar success rate to running the

826 most effective of these for 120~seconds \cite{boehme-nipkow-2010}.

828 For the \textit{min} subcommand, the default prover is \textit{metis}. If

829 several provers are set, the first one is used.

831 \opnodefault{prover}{string}

832 Alias for \textit{provers}.

834 %\opnodefault{atps}{string}

835 %Legacy alias for \textit{provers}.

837 %\opnodefault{atp}{string}

838 %Legacy alias for \textit{provers}.

840 \opfalse{blocking}{non\_blocking}

841 Specifies whether the \textbf{sledgehammer} command should operate

842 synchronously. The asynchronous (non-blocking) mode lets the user start proving

843 the putative theorem manually while Sledgehammer looks for a proof, but it can

844 also be more confusing. Irrespective of the value of this option, Sledgehammer

845 is always run synchronously for the new jEdit-based user interface or if

846 \textit{debug} (\S\ref{output-format}) is enabled.

848 \optrue{slicing}{no\_slicing}

849 Specifies whether the time allocated to a prover should be sliced into several

850 segments, each of which has its own set of possibly prover-dependent options.

851 For SPASS and Vampire, the first slice tries the fast but incomplete

852 set-of-support (SOS) strategy, whereas the second slice runs without it. For E,

853 up to three slices are tried, with different weighted search strategies and

854 number of facts. For SMT solvers, several slices are tried with the same options

855 each time but fewer and fewer facts. According to benchmarks with a timeout of

856 30 seconds, slicing is a valuable optimization, and you should probably leave it

857 enabled unless you are conducting experiments. This option is implicitly

858 disabled for (short) automatic runs.

860 \nopagebreak

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

863 \opfalse{overlord}{no\_overlord}

864 Specifies whether Sledgehammer should put its temporary files in

865 \texttt{\$ISA\-BELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for

866 debugging Sledgehammer but also unsafe if several instances of the tool are run

867 simultaneously. The files are identified by the prefix \texttt{prob\_}; you may

868 safely remove them after Sledgehammer has run.

870 \nopagebreak

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

872 \end{enum}

874 \subsection{Problem Encoding}

875 \label{problem-encoding}

877 \begin{enum}

878 \opdefault{type\_enc}{string}{smart}

879 Specifies the type encoding to use in ATP problems. Some of the type encodings

880 are unsound, meaning that they can give rise to spurious proofs

881 (unreconstructible using Metis). The supported type encodings are listed below,

882 with an indication of their soundness in parentheses:

884 \begin{enum}

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

886 supplied to the ATP. Types are simply erased.

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

889 a predicate \textit{has\_\allowbreak type\/}$(\tau, t)$ that guards bound

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

891 arguments, to resolve overloading.

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

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

896 \item[$\bullet$] \textbf{\textit{poly\_args} (unsound):}

897 Like for \textit{poly\_guards} constants are annotated with their types to

898 resolve overloading, but otherwise no type information is encoded. This

899 coincides with the default encoding used by the \textit{metis} command.

901 \item[$\bullet$]

902 \textbf{%

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

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

905 Similar to \textit{poly\_guards}, \textit{poly\_tags}, and \textit{poly\_args},

906 respectively, but the problem is additionally monomorphized, meaning that type

907 variables are instantiated with heuristically chosen ground types.

908 Monomorphization can simplify reasoning but also leads to larger fact bases,

909 which can slow down the ATPs.

911 \item[$\bullet$]

912 \textbf{%

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

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

915 Similar to

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

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

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

919 $\mathit{has\_type\/}(\tau, t)$ becomes a unary predicate

920 $\mathit{has\_type\_}\tau(t)$, and the binary function

921 $\mathit{type\/}(\tau, t)$ becomes a unary function

922 $\mathit{type\_}\tau(t)$.

924 \item[$\bullet$] \textbf{\textit{mono\_simple} (sound):} Exploits simple

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

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

928 \item[$\bullet$] \textbf{\textit{mono\_simple\_higher} (sound):} Exploits simple

929 higher-order types if the prover supports the THF0 syntax; otherwise, falls back

930 on \textit{mono\_simple} or \textit{mono\_guards}. The problem is monomorphized.

932 \item[$\bullet$]

933 \textbf{%

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

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

936 \textit{mono\_simple}? (quasi-sound):} \\

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

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

939 \textit{mono\_tags}, and \textit{mono\_simple} are fully

940 typed and sound. For each of these, Sledgehammer also provides a lighter,

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

942 and erases monotonic types, notably infinite types. (For \textit{mono\_simple},

943 the types are not actually erased but rather replaced by a shared uniform type

944 of individuals.) As argument to the \textit{metis} proof method, the question

945 mark is replaced by a \hbox{``\textit{\_query}''} suffix. If the \emph{sound}

946 option is enabled, these encodings are fully sound.

948 \item[$\bullet$]

949 \textbf{%

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

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

952 (quasi-sound):} \\

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

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

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

957 \item[$\bullet$]

958 \textbf{%

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

960 \textit{raw\_mono\_tags}@? (quasi-sound):} \\

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

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

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

965 \item[$\bullet$]

966 \textbf{%

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

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

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

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

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

972 \textit{mono\_tags}, \textit{mono\_simple}, and \textit{mono\_simple\_higher}

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

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

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

976 and \textit{mono\_simple\_higher}, the types are not actually erased but rather

977 replaced by a shared uniform type of individuals.) As argument to the

978 \textit{metis} proof method, the exclamation mark is replaced by the suffix

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

981 \item[$\bullet$]

982 \textbf{%

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

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

985 (mildly unsound):} \\

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

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

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

990 \item[$\bullet$]

991 \textbf{%

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

993 \textit{raw\_mono\_tags}@! (mildly unsound):} \\

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

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

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

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

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

1000 \end{enum}

1002 For SMT solvers, the type encoding is always \textit{mono\_simple}, irrespective

1003 of the value of this option.

1005 \nopagebreak

1006 {\small See also \textit{max\_new\_mono\_instances} (\S\ref{relevance-filter})

1007 and \textit{max\_mono\_iters} (\S\ref{relevance-filter}).}

1009 \opfalse{sound}{unsound}

1010 Specifies whether Sledgehammer should run in its fully sound mode. In that mode,

1011 quasi-sound type encodings (which are the default) are made fully sound, at the

1012 cost of some clutter in the generated problems. This option is ignored if

1013 \textit{type\_enc} is explicitly set to an unsound encoding.

1014 \end{enum}

1016 \subsection{Relevance Filter}

1017 \label{relevance-filter}

1019 \begin{enum}

1020 \opdefault{relevance\_thresholds}{float\_pair}{\upshape 0.45~0.85}

1021 Specifies the thresholds above which facts are considered relevant by the

1022 relevance filter. The first threshold is used for the first iteration of the

1023 relevance filter and the second threshold is used for the last iteration (if it

1024 is reached). The effective threshold is quadratically interpolated for the other

1025 iterations. Each threshold ranges from 0 to 1, where 0 means that all theorems

1026 are relevant and 1 only theorems that refer to previously seen constants.

1028 \opdefault{max\_relevant}{smart\_int}{smart}

1029 Specifies the maximum number of facts that may be returned by the relevance

1030 filter. If the option is set to \textit{smart}, it is set to a value that was

1031 empirically found to be appropriate for the prover. A typical value would be

1032 250.

1034 \opdefault{max\_new\_mono\_instances}{int}{\upshape 200}

1035 Specifies the maximum number of monomorphic instances to generate beyond

1036 \textit{max\_relevant}. The higher this limit is, the more monomorphic instances

1037 are potentially generated. Whether monomorphization takes place depends on the

1038 type encoding used.

1040 \nopagebreak

1041 {\small See also \textit{type\_enc} (\S\ref{problem-encoding}).}

1043 \opdefault{max\_mono\_iters}{int}{\upshape 3}

1044 Specifies the maximum number of iterations for the monomorphization fixpoint

1045 construction. The higher this limit is, the more monomorphic instances are

1046 potentially generated. Whether monomorphization takes place depends on the

1047 type encoding used.

1049 \nopagebreak

1050 {\small See also \textit{type\_enc} (\S\ref{problem-encoding}).}

1051 \end{enum}

1053 \subsection{Output Format}

1054 \label{output-format}

1056 \begin{enum}

1058 \opfalse{verbose}{quiet}

1059 Specifies whether the \textbf{sledgehammer} command should explain what it does.

1060 This option is implicitly disabled for automatic runs.

1062 \opfalse{debug}{no\_debug}

1063 Specifies whether Sledgehammer should display additional debugging information

1064 beyond what \textit{verbose} already displays. Enabling \textit{debug} also

1065 enables \textit{verbose} and \textit{blocking} (\S\ref{mode-of-operation})

1066 behind the scenes. The \textit{debug} option is implicitly disabled for

1067 automatic runs.

1069 \nopagebreak

1070 {\small See also \textit{overlord} (\S\ref{mode-of-operation}).}

1072 \opfalse{isar\_proof}{no\_isar\_proof}

1073 Specifies whether Isar proofs should be output in addition to one-liner

1074 \textit{metis} proofs. Isar proof construction is still experimental and often

1075 fails; however, they are usually faster and sometimes more robust than

1076 \textit{metis} proofs.

1078 \opdefault{isar\_shrink\_factor}{int}{\upshape 1}

1079 Specifies the granularity of the Isar proof. A value of $n$ indicates that each

1080 Isar proof step should correspond to a group of up to $n$ consecutive proof

1081 steps in the ATP proof.

1082 \end{enum}

1084 \subsection{Authentication}

1085 \label{authentication}

1087 \begin{enum}

1088 \opnodefault{expect}{string}

1089 Specifies the expected outcome, which must be one of the following:

1091 \begin{enum}

1092 \item[$\bullet$] \textbf{\textit{some}:} Sledgehammer found a (potentially

1093 unsound) proof.

1094 \item[$\bullet$] \textbf{\textit{none}:} Sledgehammer found no proof.

1095 \item[$\bullet$] \textbf{\textit{timeout}:} Sledgehammer timed out.

1096 \item[$\bullet$] \textbf{\textit{unknown}:} Sledgehammer encountered some

1097 problem.

1098 \end{enum}

1100 Sledgehammer emits an error (if \textit{blocking} is enabled) or a warning

1101 (otherwise) if the actual outcome differs from the expected outcome. This option

1102 is useful for regression testing.

1104 \nopagebreak

1105 {\small See also \textit{blocking} (\S\ref{mode-of-operation}) and

1106 \textit{timeout} (\S\ref{timeouts}).}

1107 \end{enum}

1109 \subsection{Timeouts}

1110 \label{timeouts}

1112 \begin{enum}

1113 \opdefault{timeout}{float\_or\_none}{\upshape 30}

1114 Specifies the maximum number of seconds that the automatic provers should spend

1115 searching for a proof. This excludes problem preparation and is a soft limit.

1116 For historical reasons, the default value of this option can be overridden using

1117 the option ``Sledgehammer: Time Limit'' in Proof General's ``Isabelle'' menu.

1119 \opdefault{preplay\_timeout}{float\_or\_none}{\upshape 4}

1120 Specifies the maximum number of seconds that Metis should be spent trying to

1121 ``preplay'' the found proof. If this option is set to 0, no preplaying takes

1122 place, and no timing information is displayed next to the suggested Metis calls.

1123 \end{enum}

1125 \let\em=\sl

1126 \bibliography{../manual}{}

1127 \bibliographystyle{abbrv}

1129 \end{document}