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doc-src/Sledgehammer/sledgehammer.tex

author | blanchet |

Fri, 27 May 2011 10:30:07 +0200 | |

changeset 43007 | b48aa3492f0b |

parent 43005 | c96f06bffd90 |

child 43008 | bb212c2ad238 |

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

minor doc adjustments

\documentclass[a4paper,12pt]{article} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amssymb} \usepackage[english,french]{babel} \usepackage{color} \usepackage{footmisc} \usepackage{graphicx} %\usepackage{mathpazo} \usepackage{multicol} \usepackage{stmaryrd} %\usepackage[scaled=.85]{beramono} \usepackage{../../lib/texinputs/isabelle,../iman,../pdfsetup} %\oddsidemargin=4.6mm %\evensidemargin=4.6mm %\textwidth=150mm %\topmargin=4.6mm %\headheight=0mm %\headsep=0mm %\textheight=234mm \def\Colon{\mathord{:\mkern-1.5mu:}} %\def\lbrakk{\mathopen{\lbrack\mkern-3.25mu\lbrack}} %\def\rbrakk{\mathclose{\rbrack\mkern-3.255mu\rbrack}} \def\lparr{\mathopen{(\mkern-4mu\mid}} \def\rparr{\mathclose{\mid\mkern-4mu)}} \def\unk{{?}} \def\undef{(\lambda x.\; \unk)} %\def\unr{\textit{others}} \def\unr{\ldots} \def\Abs#1{\hbox{\rm{\flqq}}{\,#1\,}\hbox{\rm{\frqq}}} \def\Q{{\smash{\lower.2ex\hbox{$\scriptstyle?$}}}} \urlstyle{tt} \begin{document} \selectlanguage{english} \title{\includegraphics[scale=0.5]{isabelle_sledgehammer} \\[4ex] Hammering Away \\[\smallskipamount] \Large A User's Guide to Sledgehammer for Isabelle/HOL} \author{\hbox{} \\ Jasmin Christian Blanchette \\ {\normalsize Institut f\"ur Informatik, Technische Universit\"at M\"unchen} \\[4\smallskipamount] {\normalsize with contributions from} \\[4\smallskipamount] Lawrence C. Paulson \\ {\normalsize Computer Laboratory, University of Cambridge} \\ \hbox{}} \maketitle \tableofcontents \setlength{\parskip}{.7em plus .2em minus .1em} \setlength{\parindent}{0pt} \setlength{\abovedisplayskip}{\parskip} \setlength{\abovedisplayshortskip}{.9\parskip} \setlength{\belowdisplayskip}{\parskip} \setlength{\belowdisplayshortskip}{.9\parskip} % General-purpose enum environment with correct spacing \newenvironment{enum}% {\begin{list}{}{% \setlength{\topsep}{.1\parskip}% \setlength{\partopsep}{.1\parskip}% \setlength{\itemsep}{\parskip}% \advance\itemsep by-\parsep}} {\end{list}} \def\pre{\begingroup\vskip0pt plus1ex\advance\leftskip by\leftmargin \advance\rightskip by\leftmargin} \def\post{\vskip0pt plus1ex\endgroup} \def\prew{\pre\advance\rightskip by-\leftmargin} \def\postw{\post} \section{Introduction} \label{introduction} Sledgehammer is a tool that applies automatic theorem provers (ATPs) and satisfiability-modulo-theories (SMT) solvers on the current goal. The supported ATPs are E \cite{schulz-2002}, LEO-II \cite{leo2}, Satallax \cite{satallax}, SInE-E \cite{sine}, SNARK \cite{snark}, SPASS \cite{weidenbach-et-al-2009}, ToFoF-E \cite{tofof}, Vampire \cite{riazanov-voronkov-2002}, and Waldmeister \cite{waldmeister}. The ATPs are run either locally or remotely via the System\-On\-TPTP web service \cite{sutcliffe-2000}. In addition to the ATPs, the SMT solvers Z3 \cite{z3} is used by default, and you can tell Sledgehammer to try CVC3 \cite{cvc3} and Yices \cite{yices} as well; these are run either locally or on a server at the TU M\"unchen. The problem passed to the automatic provers consists of your current goal together with a heuristic selection of hundreds of facts (theorems) from the current theory context, filtered by relevance. Because jobs are run in the background, you can continue to work on your proof by other means. Provers can be run in parallel. Any reply (which may arrive half a minute later) will appear in the Proof General response buffer. The result of a successful proof search is some source text that usually (but not always) reconstructs the proof within Isabelle. For ATPs, the reconstructed proof relies on the general-purpose Metis prover \cite{metis}, which is fully integrated into Isabelle/HOL, with explicit inferences going through the kernel. Thus its results are correct by construction. In this manual, we will explicitly invoke the \textbf{sledgehammer} command. Sledgehammer also provides an automatic mode that can be enabled via the ``Auto Sledgehammer'' option from the ``Isabelle'' menu in Proof General. In this mode, Sledgehammer is run on every newly entered theorem. The time limit for Auto Sledgehammer and other automatic tools can be set using the ``Auto Tools Time Limit'' option. \newbox\boxA \setbox\boxA=\hbox{\texttt{nospam}} \newcommand\authoremail{\texttt{blan{\color{white}nospam}\kern-\wd\boxA{}chette@\allowbreak in.\allowbreak tum.\allowbreak de}} To run Sledgehammer, you must make sure that the theory \textit{Sledgehammer} is imported---this is rarely a problem in practice since it is part of \textit{Main}. Examples of Sledgehammer use can be found in Isabelle's \texttt{src/HOL/Metis\_Examples} directory. Comments and bug reports concerning Sledgehammer or this manual should be directed to the author at \authoremail. \vskip2.5\smallskipamount %\textbf{Acknowledgment.} The author would like to thank Mark Summerfield for %suggesting several textual improvements. \section{Installation} \label{installation} Sledgehammer is part of Isabelle, so you don't need to install it. However, it relies on third-party automatic theorem provers (ATPs) and SMT solvers. \subsection{Installing ATPs} Currently, E, SPASS, and Vampire can be run locally; in addition, E, Vampire, LEO-II, Satallax, SInE-E, SNARK, ToFoF-E, and Waldmeister are available remotely via System\-On\-TPTP \cite{sutcliffe-2000}. If you want better performance, you should at least install E and SPASS locally. There are three main ways to install ATPs on your machine: \begin{enum} \item[$\bullet$] If you installed an official Isabelle package with everything inside, it should already include properly setup executables for E and SPASS, ready to use.% \footnote{Vampire's license prevents us from doing the same for this otherwise wonderful tool.} \item[$\bullet$] Alternatively, you can download the Isabelle-aware E and SPASS binary packages from Isabelle's download page. Extract the archives, then add a line to your \texttt{\$ISABELLE\_HOME\_USER/etc/components}% \footnote{The variable \texttt{\$ISABELLE\_HOME\_USER} is set by Isabelle at startup. Its value can be retrieved by invoking \texttt{isabelle} \texttt{getenv} \texttt{ISABELLE\_HOME\_USER} on the command line.} file with the absolute path to E or SPASS. For example, if the \texttt{components} does not exist yet and you extracted SPASS to \texttt{/usr/local/spass-3.7}, create the \texttt{components} file with the single line \prew \texttt{/usr/local/spass-3.7} \postw in it. \item[$\bullet$] If you prefer to build E or SPASS yourself, or obtained a Vampire executable from somewhere (e.g., \url{http://www.vprover.org/}), set the environment variable \texttt{E\_HOME}, \texttt{SPASS\_HOME}, or \texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{eproof}, \texttt{SPASS}, or \texttt{vampire} executable. Sledgehammer has been tested with E 1.0 and 1.2, SPASS 3.5 and 3.7, and Vampire 0.6 and 1.0% \footnote{Following the rewrite of Vampire, the counter for version numbers was reset to 0; hence the (new) Vampire versions 0.6 and 1.0 are more recent than, say, Vampire 11.5.}% . Since the ATPs' output formats are neither documented nor stable, other versions of the ATPs might or might not work well with Sledgehammer. Ideally, also set \texttt{E\_VERSION}, \texttt{SPASS\_VERSION}, or \texttt{VAMPIRE\_VERSION} to the ATP's version number (e.g., ``1.2''). \end{enum} To check whether E and SPASS are successfully installed, follow the example in \S\ref{first-steps}. If the remote versions of E and SPASS are used (identified by the prefix ``\emph{remote\_}''), or if the local versions fail to solve the easy goal presented there, this is a sign that something is wrong with your installation. Remote ATP invocation via the SystemOnTPTP web service requires Perl with the World Wide Web Library (\texttt{libwww-perl}) installed. If you must use a proxy server to access the Internet, set the \texttt{http\_proxy} environment variable to the proxy, either in the environment in which Isabelle is launched or in your \texttt{\char`\~/\$ISABELLE\_HOME\_USER/etc/settings} file. Here are a few examples: \prew \texttt{http\_proxy=http://proxy.example.org} \\ \texttt{http\_proxy=http://proxy.example.org:8080} \\ \texttt{http\_proxy=http://joeblow:pAsSwRd@proxy.example.org} \postw \subsection{Installing SMT Solvers} CVC3, Yices, and Z3 can be run locally or (for CVC3 and Z3) remotely on a TU M\"unchen server. If you want better performance and get the ability to replay proofs that rely on the \emph{smt} proof method, you should at least install Z3 locally. There are two main ways of installing SMT solvers locally. \begin{enum} \item[$\bullet$] If you installed an official Isabelle package with everything inside, it should already include properly setup executables for CVC3 and Z3, ready to use.% \footnote{Yices's license prevents us from doing the same for this otherwise wonderful tool.} For Z3, you additionally need to set the environment variable \texttt{Z3\_NON\_COMMERCIAL} to ``yes'' to confirm that you are a noncommercial user. \item[$\bullet$] Otherwise, follow the instructions documented in the \emph{SMT} theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}). \end{enum} \section{First Steps} \label{first-steps} To illustrate Sledgehammer in context, let us start a theory file and attempt to prove a simple lemma: \prew \textbf{theory}~\textit{Scratch} \\ \textbf{imports}~\textit{Main} \\ \textbf{begin} \\[2\smallskipamount] % \textbf{lemma} ``$[a] = [b] \,\Longrightarrow\, a = b$'' \\ \textbf{sledgehammer} \postw Instead of issuing the \textbf{sledgehammer} command, you can also find Sledgehammer in the ``Commands'' submenu of the ``Isabelle'' menu in Proof General or press the Emacs key sequence C-c C-a C-s. Either way, Sledgehammer produces the following output after a few seconds: \prew \slshape Sledgehammer: ``\textit{e}'' for subgoal 1: \\ $[a] = [b] \,\Longrightarrow\, a = b$ \\ Try this command: \textbf{by} (\textit{metis last\_ConsL}). \\ To minimize: \textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{e}] (\textit{last\_ConsL}). \\[3\smallskipamount] % Sledgehammer: ``\textit{vampire}'' for subgoal 1: \\ $[a] = [b] \,\Longrightarrow\, a = b$ \\ Try this command: \textbf{by} (\textit{metis hd.simps}). \\ To minimize: \textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{vampire}] (\textit{hd.simps}). \\[3\smallskipamount] % Sledgehammer: ``\textit{spass}'' for subgoal 1: \\ $[a] = [b] \,\Longrightarrow\, a = b$ \\ Try this command: \textbf{by} (\textit{metis list.inject}). \\ To minimize: \textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{spass}]~(\textit{list.inject}). \\[3\smallskipamount] % %Sledgehammer: ``\textit{remote\_waldmeister}'' for subgoal 1: \\ %$[a] = [b] \,\Longrightarrow\, a = b$ \\ %Try this command: \textbf{by} (\textit{metis hd.simps insert\_Nil}). \\ %To minimize: \textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{remote\_waldmeister}] \\ %\phantom{\textbf{sledgehammer}~}(\textit{hd.simps insert\_Nil}). \\[3\smallskipamount] % Sledgehammer: ``\textit{remote\_sine\_e}'' for subgoal 1: \\ $[a] = [b] \,\Longrightarrow\, a = b$ \\ Try this command: \textbf{by} (\textit{metis hd.simps}). \\ To minimize: \textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{remote\_sine\_e}] \\ \phantom{To minimize: }(\textit{hd.simps}). \\[3\smallskipamount] % Sledgehammer: ``\textit{remote\_z3}'' for subgoal 1: \\ $[a] = [b] \,\Longrightarrow\, a = b$ \\ Try this command: \textbf{by} (\textit{metis hd.simps}). \\ To minimize: \textbf{sledgehammer} \textit{minimize} [\textit{prover} = \textit{remote\_z3}]~(\textit{hd.simps}). \postw Sledgehammer ran E, SInE-E, SPASS, Vampire, %Waldmeister, and Z3 in parallel. Depending on which provers are installed and how many processor cores are available, some of the provers might be missing or present with a \textit{remote\_} prefix. %Waldmeister is run only for unit equational problems, %where the goal's conclusion is a (universally quantified) equation. For each successful prover, Sledgehammer gives a one-liner proof that uses the \textit{metis} or \textit{smt} method. You can click the proof to insert it into the theory text. You can click the ``\textbf{sledgehammer} \textit{minimize}'' command if you want to look for a shorter (and probably faster) proof. But here the proof found by E looks perfect, so click it to finish the proof. You can ask Sledgehammer for an Isar text proof by passing the \textit{isar\_proof} option (\S\ref{output-format}): \prew \textbf{sledgehammer} [\textit{isar\_proof}] \postw When Isar proof construction is successful, it can yield proofs that are more readable and also faster than the \textit{metis} one-liners. This feature is experimental and is only available for ATPs. \section{Hints} \label{hints} This section presents a few hints that should help you get the most out of Sledgehammer and Metis. Frequently (and infrequently) asked questions are answered in \S\ref{frequently-asked-questions}. \newcommand\point[1]{\medskip\par{\sl\bfseries#1}\par\nopagebreak} \point{Presimplify the goal} For best results, first simplify your problem by calling \textit{auto} or at least \textit{safe} followed by \textit{simp\_all}. The SMT solvers provide arithmetic decision procedures, but the ATPs typically do not (or if they do, Sledgehammer does not use it yet). Apart from Waldmeister, they are not especially good at heavy rewriting, but because they regard equations as undirected, they often prove theorems that require the reverse orientation of a \textit{simp} rule. Higher-order problems can be tackled, but the success rate is better for first-order problems. Hence, you may get better results if you first simplify the problem to remove higher-order features. \point{Make sure at least E, SPASS, Vampire, and Z3 are installed} Locally installed provers are faster and more reliable than those running on servers. See \S\ref{installation} for details on how to install them. \point{Familiarize yourself with the most important options} Sledgehammer's options are fully documented in \S\ref{command-syntax}. Many of the options are very specialized, but serious users of the tool should at least familiarize themselves with the following options: \begin{enum} \item[$\bullet$] \textbf{\textit{provers}} (\S\ref{mode-of-operation}) specifies the automatic provers (ATPs and SMT solvers) that should be run whenever Sledgehammer is invoked (e.g., ``\textit{provers}~= \textit{e spass remote\_vampire}''). \item[$\bullet$] \textbf{\textit{timeout}} (\S\ref{mode-of-operation}) controls the provers' time limit. It is set to 30 seconds, but since Sledgehammer runs asynchronously you should not hesitate to raise this limit to 60 or 120 seconds if you are the kind of user who can think clearly while ATPs are active. \item[$\bullet$] \textbf{\textit{full\_types}} (\S\ref{problem-encoding}) specifies whether type-sound encodings should be used. By default, Sledgehammer employs a mixture of type-sound and type-unsound encodings, occasionally yielding unsound ATP proofs. In contrast, SMT solver proofs should always be sound. \item[$\bullet$] \textbf{\textit{max\_relevant}} (\S\ref{relevance-filter}) specifies the maximum number of facts that should be passed to the provers. By default, the value is prover-dependent but varies between about 150 and 1000. If the provers time out, you can try lowering this value to, say, 100 or 50 and see if that helps. \item[$\bullet$] \textbf{\textit{isar\_proof}} (\S\ref{output-format}) specifies that Isar proofs should be generated, instead of one-liner Metis proofs. The length of the Isar proofs can be controlled by setting \textit{isar\_shrink\_factor} (\S\ref{output-format}). \end{enum} Options can be set globally using \textbf{sledgehammer\_params} (\S\ref{command-syntax}). Fact selection can be influenced by specifying ``$(\textit{add}{:}~\textit{my\_facts})$'' after the \textbf{sledgehammer} call to ensure that certain facts are included, or simply ``$(\textit{my\_facts})$'' to force Sledgehammer to run only with $\textit{my\_facts}$. \section{Frequently Asked Questions} \label{frequently-asked-questions} This sections answers frequently (and infrequently) asked questions about Sledgehammer. It is a good idea to skim over it now even if you don't have any questions at this stage. And if you have any further questions not listed here, send them to the author at \authoremail. \point{Why does Metis fail to reconstruct the proof?} There are many reasons. If Metis runs seemingly forever, that is a sign that the proof is too difficult for it. Metis is complete, so it should eventually find it, but that's little consolation. There are several possible solutions: \begin{enum} \item[$\bullet$] Try the \textit{isar\_proof} option (\S\ref{output-format}) to obtain a step-by-step Isar proof where each step is justified by Metis. Since the steps are fairly small, Metis is more likely to be able to replay them. \item[$\bullet$] Try the \textit{smt} proof method instead of \textit{metis}. It is usually stronger, but you need to have Z3 available to replay the proofs, trust the SMT solver, or use certificates. See the documentation in the \emph{SMT} theory (\texttt{\$ISABELLE\_HOME/src/HOL/SMT.thy}) for details. \item[$\bullet$] Try the \textit{blast} or \textit{auto} proof methods, passing facts via \textbf{unfolding}, \textbf{using}, \textit{intro}{:}, \textit{elim}{:}, \textit{dest}{:}, or \textit{simp}{:}, as appropriate. \end{enum} In some rare cases, Metis fails fairly quickly. This usually indicates that Sledgehammer found a type-incorrect proof. Sledgehammer erases some type information to speed up the search. Try Sledgehammer again with full type information: \textit{full\_types} (\S\ref{problem-encoding}), or choose a specific type encoding with \textit{type\_sys} (\S\ref{problem-encoding}). Older versions of Sledgehammer were frequent victims of this problem. Now this should very seldom be an issue, but if you notice many unsound proofs, contact the author at \authoremail. \point{How can I tell whether a Sledgehammer proof is sound?} First, if \emph{metis} (or \emph{metisFT}) can reconstruct it, the proof is sound (modulo soundness of Isabelle's inference kernel). If it fails or runs seemingly forever, you can try \prew \textbf{apply}~\textbf{--} \\ \textbf{sledgehammer} [\textit{type\_sys} = \textit{poly\_tags}] (\textit{metis\_facts}) \postw where \textit{metis\_facts} is the list of facts appearing in the suggested Metis call. The automatic provers should be able to refind the proof very quickly if it is sound, and the \textit{type\_sys} $=$ \textit{poly\_tags} option (\S\ref{problem-encoding}) ensures that no unsound proofs are found. The \textit{full\_types} option (\S\ref{problem-encoding}) can also be used here, but it is unsound in extremely rare degenerate cases such as the following: \prew \textbf{lemma} ``$\forall x\> y\Colon{'}\!a.\ x = y \,\Longrightarrow \exists f\> g\Colon\mathit{nat} \Rightarrow {'}\!a.\ f \not= g$'' \\ \textbf{sledgehammer} [\textit{full\_types}] (\textit{nat.distinct\/}(1)) \postw \point{How does Sledgehammer select the facts that should be passed to the automatic provers?} Briefly, the relevance filter assigns a score to every available fact (lemma, theorem, definition, or axiom)\ based upon how many constants that fact shares with the conjecture. This process iterates to include facts relevant to those just accepted, but with a decay factor to ensure termination. The constants are weighted to give unusual ones greater significance. The relevance filter copes best when the conjecture contains some unusual constants; if all the constants are common, it is unable to discriminate among the hundreds of facts that are picked up. The relevance filter is also memoryless: It has no information about how many times a particular fact has been used in a proof, and it cannot learn. The number of facts included in a problem varies from prover to prover, since some provers get overwhelmed quicker than others. You can show the number of facts given using the \textit{verbose} option (\S\ref{output-format}) and the actual facts using \textit{debug} (\S\ref{output-format}). Sledgehammer is good at finding short proofs combining a handful of existing lemmas. If you are looking for longer proofs, you must typically restrict the number of facts, by setting the \textit{max\_relevant} option (\S\ref{relevance-filter}) to, say, 50 or 100. You can also influence which facts are actually selected in a number of ways. If you simply want to ensure that a fact is included, you can specify it using the ``$(\textit{add}{:}~\textit{my\_facts})$'' syntax. For example: % \prew \textbf{sledgehammer} (\textit{add}: \textit{hd.simps} \textit{tl.simps}) \postw % The specified facts then replace the least relevant facts that would otherwise be included; the other selected facts remain the same. If you want to direct the selection in a particular direction, you can specify the facts via \textbf{using}: % \prew \textbf{using} \textit{hd.simps} \textit{tl.simps} \\ \textbf{sledgehammer} \postw % The facts are then more likely to be selected than otherwise, and if they are selected at iteration $j$ they also influence which facts are selected at iterations $j + 1$, $j + 2$, etc. To give them even more weight, try % \prew \textbf{using} \textit{hd.simps} \textit{tl.simps} \\ \textbf{apply}~\textbf{--} \\ \textbf{sledgehammer} \postw \point{Why are the Isar proofs generated by Sledgehammer so ugly?} The current implementation is experimental and explodes exponentially in the worst case. Work on a new implementation has begun. There is a large body of research into transforming resolution proofs into natural deduction proofs (such as Isar proofs), which we hope to leverage. In the meantime, a workaround is to set the \textit{isar\_shrink\_factor} option (\S\ref{output-format}) to a larger value or to try several provers and keep the nicest-looking proof. \point{Should I let Sledgehammer minimize the number of lemmas?} In general, minimization is a good idea, because proofs involving fewer lemmas tend to be shorter as well, and hence easier to re-find by Metis. But the opposite is sometimes the case. \point{Why does the minimizer sometimes starts of its own?} There are two scenarios in which this can happen. First, some provers (notably CVC3, Satallax, and Yices) do not provide proofs or sometimes provide incomplete proofs. The minimizer is then invoked to find out which facts are actually needed from the (large) set of facts that was initinally given to the prover. Second, if a prover returns a proof with lots of facts, the minimizer is invoked automatically since Metis is unlikely to refind the proof. \point{What is metisFT?} The \textit{metisFT} proof method is the fully-typed version of Metis. It is much slower than \textit{metis}, but the proof search is fully typed, and it also includes more powerful rules such as the axiom ``$x = \mathit{True} \mathrel{\lor} x = \mathit{False}$'' for reasoning in higher-order places (e.g., in set comprehensions). The method kicks in automatically as a fallback when \textit{metis} fails, and it is sometimes generated by Sledgehammer instead of \textit{metis} if the proof obviously requires type information. If you see the warning \prew \slshape Metis: Falling back on ``\textit{metisFT\/}''. \postw in a successful Metis proof, you can advantageously replace the \textit{metis} call with \textit{metisFT}. \point{I got a strange error from Sledgehammer---what should I do?} Sledgehammer tries to give informative error messages. Please report any strange error to the author at \authoremail. This applies double if you get the message \prew \slshape The prover found a type-unsound proof involving ``\textit{foo}'', ``\textit{bar}'', and ``\textit{baz}'' even though a supposedly type-sound encoding was used (or, less likely, your axioms are inconsistent). You might want to report this to the Isabelle developers. \postw \point{Auto can solve it---why not Sledgehammer?} Problems can be easy for \textit{auto} and difficult for automatic provers, but the reverse is also true, so don't be discouraged if your first attempts fail. Because the system refers to all theorems known to Isabelle, it is particularly suitable when your goal has a short proof from lemmas that you don't know about. \point{Why are there so many options?} Sledgehammer's philosophy should work out of the box, without user guidance. Many of the options are meant to be used mostly by the Sledgehammer developers for experimentation purposes. Of course, feel free to experiment with them if you are so inclined. \section{Command Syntax} \label{command-syntax} Sledgehammer can be invoked at any point when there is an open goal by entering the \textbf{sledgehammer} command in the theory file. Its general syntax is as follows: \prew \textbf{sledgehammer} \textit{subcommand\/$^?$ options\/$^?$ facts\_override\/$^?$ num\/$^?$} \postw For convenience, Sledgehammer is also available in the ``Commands'' submenu of the ``Isabelle'' menu in Proof General or by pressing the Emacs key sequence C-c C-a C-s. This is equivalent to entering the \textbf{sledgehammer} command with no arguments in the theory text. In the general syntax, the \textit{subcommand} may be any of the following: \begin{enum} \item[$\bullet$] \textbf{\textit{run} (the default):} Runs Sledgehammer on subgoal number \textit{num} (1 by default), with the given options and facts. \item[$\bullet$] \textbf{\textit{minimize}:} Attempts to minimize the provided facts (specified in the \textit{facts\_override} argument) to obtain a simpler proof involving fewer facts. The options and goal number are as for \textit{run}. \item[$\bullet$] \textbf{\textit{messages}:} Redisplays recent messages issued by Sledgehammer. This allows you to examine results that might have been lost due to Sledgehammer's asynchronous nature. The \textit{num} argument specifies a limit on the number of messages to display (5 by default). \item[$\bullet$] \textbf{\textit{supported\_provers}:} Prints the list of automatic provers supported by Sledgehammer. See \S\ref{installation} and \S\ref{mode-of-operation} for more information on how to install automatic provers. \item[$\bullet$] \textbf{\textit{running\_provers}:} Prints information about currently running automatic provers, including elapsed runtime and remaining time until timeout. \item[$\bullet$] \textbf{\textit{kill\_provers}:} Terminates all running automatic provers. \item[$\bullet$] \textbf{\textit{refresh\_tptp}:} Refreshes the list of remote ATPs available at System\-On\-TPTP \cite{sutcliffe-2000}. \end{enum} Sledgehammer's behavior can be influenced by various \textit{options}, which can be specified in brackets after the \textbf{sledgehammer} command. The \textit{options} are a list of key--value pairs of the form ``[$k_1 = v_1, \ldots, k_n = v_n$]''. For Boolean options, ``= \textit{true}'' is optional. For example: \prew \textbf{sledgehammer} [\textit{isar\_proof}, \,\textit{timeout} = 120$\,s$] \postw Default values can be set using \textbf{sledgehammer\_\allowbreak params}: \prew \textbf{sledgehammer\_params} \textit{options} \postw The supported options are described in \S\ref{option-reference}. The \textit{facts\_override} argument lets you alter the set of facts that go through the relevance filter. It may be of the form ``(\textit{facts})'', where \textit{facts} is a space-separated list of Isabelle facts (theorems, local assumptions, etc.), in which case the relevance filter is bypassed and the given facts are used. It may also be of the form ``(\textit{add}:\ \textit{facts}$_1$)'', ``(\textit{del}:\ \textit{facts}$_2$)'', or ``(\textit{add}:\ \textit{facts}$_1$\ \textit{del}:\ \textit{facts}$_2$)'', where the relevance filter is instructed to proceed as usual except that it should consider \textit{facts}$_1$ highly-relevant and \textit{facts}$_2$ fully irrelevant. You can instruct Sledgehammer to run automatically on newly entered theorems by enabling the ``Auto Sledgehammer'' option from the ``Isabelle'' menu in Proof General. For automatic runs, only the first prover set using \textit{provers} (\S\ref{mode-of-operation}) is considered, fewer facts are passed to the prover, \textit{slicing} (\S\ref{mode-of-operation}) is disabled, \textit{timeout} (\S\ref{mode-of-operation}) is superseded by the ``Auto Tools Time Limit'' in Proof General's ``Isabelle'' menu, \textit{full\_types} (\S\ref{problem-encoding}) is enabled, and \textit{verbose} (\S\ref{output-format}) and \textit{debug} (\S\ref{output-format}) are disabled. Sledgehammer's output is also more concise. \section{Option Reference} \label{option-reference} \def\flushitem#1{\item[]\noindent\kern-\leftmargin \textbf{#1}} \def\qty#1{$\left<\textit{#1}\right>$} \def\qtybf#1{$\mathbf{\left<\textbf{\textit{#1}}\right>}$} \def\optrue#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{true}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} \def\opfalse#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool}$\bigr]$\quad [\textit{false}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} \def\opsmart#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} \def\opsmartx#1#2{\flushitem{\textit{#1} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\quad [\textit{smart}]\hfill\\\hbox{}\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]} \def\opnodefault#1#2{\flushitem{\textit{#1} = \qtybf{#2}} \nopagebreak\\[\parskip]} \def\opdefault#1#2#3{\flushitem{\textit{#1} = \qtybf{#2}\quad [\textit{#3}]} \nopagebreak\\[\parskip]} \def\oparg#1#2#3{\flushitem{\textit{#1} \qtybf{#2} = \qtybf{#3}} \nopagebreak\\[\parskip]} \def\opargbool#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]} \def\opargboolorsmart#1#2#3{\flushitem{\textit{#1} \qtybf{#2} $\bigl[$= \qtybf{bool\_or\_smart}$\bigr]$\hfill (neg.: \textit{#3})}\nopagebreak\\[\parskip]} Sledgehammer's options are categorized as follows:\ mode of operation (\S\ref{mode-of-operation}), problem encoding (\S\ref{problem-encoding}), relevance filter (\S\ref{relevance-filter}), output format (\S\ref{output-format}), and authentication (\S\ref{authentication}). The descriptions below refer to the following syntactic quantities: \begin{enum} \item[$\bullet$] \qtybf{string}: A string. \item[$\bullet$] \qtybf{bool\/}: \textit{true} or \textit{false}. \item[$\bullet$] \qtybf{bool\_or\_smart\/}: \textit{true}, \textit{false}, or \textit{smart}. \item[$\bullet$] \qtybf{int\/}: An integer. %\item[$\bullet$] \qtybf{float\/}: A floating-point number (e.g., 2.5). \item[$\bullet$] \qtybf{float\_pair\/}: A pair of floating-point numbers (e.g., 0.6 0.95). \item[$\bullet$] \qtybf{int\_or\_smart\/}: An integer or \textit{smart}. \item[$\bullet$] \qtybf{float\_or\_none\/}: An integer (e.g., 60) or floating-point number (e.g., 0.5) expressing a number of seconds, or the keyword \textit{none} ($\infty$ seconds). \end{enum} Default values are indicated in square brackets. Boolean options have a negated counterpart (e.g., \textit{blocking} vs.\ \textit{non\_blocking}). When setting Boolean options, ``= \textit{true}'' may be omitted. \subsection{Mode of Operation} \label{mode-of-operation} \begin{enum} \opnodefault{provers}{string} Specifies the automatic provers to use as a space-separated list (e.g., ``\textit{e}~\textit{spass}''). The following local provers are supported: \begin{enum} \item[$\bullet$] \textbf{\textit{cvc3}:} CVC3 is an SMT solver developed by Clark Barrett, Cesare Tinelli, and their colleagues \cite{cvc3}. To use CVC3, set the environment variable \texttt{CVC3\_SOLVER} to the complete path of the executable, including the file name. Sledgehammer has been tested with version 2.2. \item[$\bullet$] \textbf{\textit{e}:} E is a first-order resolution prover developed by Stephan Schulz \cite{schulz-2002}. To use E, set the environment variable \texttt{E\_HOME} to the directory that contains the \texttt{eproof} executable, or install the prebuilt E package from Isabelle's download page. See \S\ref{installation} for details. \item[$\bullet$] \textbf{\textit{spass}:} SPASS is a first-order resolution prover developed by Christoph Weidenbach et al.\ \cite{weidenbach-et-al-2009}. To use SPASS, set the environment variable \texttt{SPASS\_HOME} to the directory that contains the \texttt{SPASS} executable, or install the prebuilt SPASS package from Isabelle's download page. Sledgehammer requires version 3.5 or above. See \S\ref{installation} for details. \item[$\bullet$] \textbf{\textit{yices}:} Yices is an SMT solver developed at SRI \cite{yices}. To use Yices, set the environment variable \texttt{YICES\_SOLVER} to the complete path of the executable, including the file name. Sledgehammer has been tested with version 1.0. \item[$\bullet$] \textbf{\textit{vampire}:} Vampire is a first-order resolution prover developed by Andrei Voronkov and his colleagues \cite{riazanov-voronkov-2002}. To use Vampire, set the environment variable \texttt{VAMPIRE\_HOME} to the directory that contains the \texttt{vampire} executable. Sledgehammer has been tested with versions 11, 0.6, and 1.0. \item[$\bullet$] \textbf{\textit{z3}:} Z3 is an SMT solver developed at Microsoft Research \cite{z3}. To use Z3, set the environment variable \texttt{Z3\_SOLVER} to the complete path of the executable, including the file name, and set \texttt{Z3\_NON\_COMMERCIAL=yes} to confirm that you are a noncommercial user. Sledgehammer has been tested with versions 2.7 to 2.18. \item[$\bullet$] \textbf{\textit{z3\_atp}:} This version of Z3 pretends to be an ATP, exploiting Z3's undocumented support for the TPTP format. It is included for experimental purposes. It requires version 2.18 or above. \end{enum} In addition, the following remote provers are supported: \begin{enum} \item[$\bullet$] \textbf{\textit{remote\_cvc3}:} The remote version of CVC3 runs on servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to point). \item[$\bullet$] \textbf{\textit{remote\_e}:} The remote version of E runs on Geoff Sutcliffe's Miami servers \cite{sutcliffe-2000}. \item[$\bullet$] \textbf{\textit{remote\_leo2}:} LEO-II is an automatic higher-order prover developed by Christoph Benzm\"uller et al. \cite{leo2}. The remote version of LEO-II runs on Geoff Sutcliffe's Miami servers. In the current setup, the problems given to LEO-II are only mildly higher-order. \item[$\bullet$] \textbf{\textit{remote\_satallax}:} Satallax is an automatic higher-order prover developed by Chad Brown et al. \cite{satallax}. The remote version of Satallax runs on Geoff Sutcliffe's Miami servers. In the current setup, the problems given to Satallax are only mildly higher-order. \item[$\bullet$] \textbf{\textit{remote\_sine\_e}:} SInE-E is a metaprover developed by Kry\v stof Hoder \cite{sine} based on E. The remote version of SInE runs on Geoff Sutcliffe's Miami servers. \item[$\bullet$] \textbf{\textit{remote\_snark}:} SNARK is a first-order resolution prover developed by Stickel et al.\ \cite{snark}. The remote version of SNARK runs on Geoff Sutcliffe's Miami servers. \item[$\bullet$] \textbf{\textit{remote\_tofof\_e}:} ToFoF-E is a metaprover developed by Geoff Sutcliffe \cite{tofof} based on E running on his Miami servers. This ATP supports a fragment of the TPTP many-typed first-order format (TFF). It is supported primarily for experimenting with the \textit{type\_sys} $=$ \textit{simple} option (\S\ref{problem-encoding}). \item[$\bullet$] \textbf{\textit{remote\_vampire}:} The remote version of Vampire runs on Geoff Sutcliffe's Miami servers. Version 9 is used. \item[$\bullet$] \textbf{\textit{remote\_waldmeister}:} Waldmeister is a unit equality prover developed by Hillenbrand et al.\ \cite{waldmeister}. It can be used to prove universally quantified equations using unconditional equations. The remote version of Waldmeister runs on Geoff Sutcliffe's Miami servers. \item[$\bullet$] \textbf{\textit{remote\_z3}:} The remote version of Z3 runs on servers at the TU M\"unchen (or wherever \texttt{REMOTE\_SMT\_URL} is set to point). \item[$\bullet$] \textbf{\textit{remote\_z3\_atp}:} The remote version of ``Z3 as an ATP'' runs on Geoff Sutcliffe's Miami servers. \end{enum} By default, Sledgehammer will run E, SPASS, Vampire, SInE-E, and Z3 (or whatever the SMT module's \textit{smt\_solver} configuration option is set to) in parallel---either locally or remotely, depending on the number of processor cores available. For historical reasons, the default value of this option can be overridden using the option ``Sledgehammer: Provers'' from the ``Isabelle'' menu in Proof General. It is a good idea to run several provers in parallel, although it could slow down your machine. Running E, SPASS, and Vampire for 5~seconds yields a similar success rate to running the most effective of these for 120~seconds \cite{boehme-nipkow-2010}. \opnodefault{prover}{string} Alias for \textit{provers}. %\opnodefault{atps}{string} %Legacy alias for \textit{provers}. %\opnodefault{atp}{string} %Legacy alias for \textit{provers}. \opdefault{timeout}{float\_or\_none}{\upshape 30} Specifies the maximum number of seconds that the automatic provers should spend searching for a proof. This excludes problem preparation and is a soft limit. For historical reasons, the default value of this option can be overridden using the option ``Sledgehammer: Time Limit'' from the ``Isabelle'' menu in Proof General. \opfalse{blocking}{non\_blocking} Specifies whether the \textbf{sledgehammer} command should operate synchronously. The asynchronous (non-blocking) mode lets the user start proving the putative theorem manually while Sledgehammer looks for a proof, but it can also be more confusing. Irrespective of the value of this option, Sledgehammer is always run synchronously for the new jEdit-based user interface or if \textit{debug} (\S\ref{output-format}) is enabled. \optrue{slicing}{no\_slicing} Specifies whether the time allocated to a prover should be sliced into several segments, each of which has its own set of possibly prover-dependent options. For SPASS and Vampire, the first slice tries the fast but incomplete set-of-support (SOS) strategy, whereas the second slice runs without it. For E, up to three slices are tried, with different weighted search strategies and number of facts. For SMT solvers, several slices are tried with the same options each time but fewer and fewer facts. According to benchmarks with a timeout of 30 seconds, slicing is a valuable optimization, and you should probably leave it enabled unless you are conducting experiments. This option is implicitly disabled for (short) automatic runs. \nopagebreak {\small See also \textit{verbose} (\S\ref{output-format}).} \opfalse{overlord}{no\_overlord} Specifies whether Sledgehammer should put its temporary files in \texttt{\$ISA\-BELLE\_\allowbreak HOME\_\allowbreak USER}, which is useful for debugging Sledgehammer but also unsafe if several instances of the tool are run simultaneously. The files are identified by the prefix \texttt{prob\_}; you may safely remove them after Sledgehammer has run. \nopagebreak {\small See also \textit{debug} (\S\ref{output-format}).} \end{enum} \subsection{Problem Encoding} \label{problem-encoding} \begin{enum} \opfalse{explicit\_apply}{implicit\_apply} Specifies whether function application should be encoded as an explicit ``apply'' operator in ATP problems. If the option is set to \textit{false}, each function will be directly applied to as many arguments as possible. Enabling this option can sometimes help discover higher-order proofs that otherwise would not be found. \opfalse{full\_types}{partial\_types} Specifies whether full type information is encoded in ATP problems. Enabling this option prevents the discovery of type-incorrect proofs, but it can slow down the ATP slightly. This option is implicitly enabled for automatic runs. For historical reasons, the default value of this option can be overridden using the option ``Sledgehammer: Full Types'' from the ``Isabelle'' menu in Proof General. \opdefault{type\_sys}{string}{smart} Specifies the type system to use in ATP problems. Some of the type systems are unsound, meaning that they can give rise to spurious proofs (unreconstructible using Metis). The supported type systems are listed below, with an indication of their soundness in parentheses: \begin{enum} \item[$\bullet$] \textbf{\textit{erased} (very unsound):} No type information is supplied to the ATP. Types are simply erased. \item[$\bullet$] \textbf{\textit{poly\_preds} (sound):} Types are encoded using a predicate \textit{has\_\allowbreak type\/}$(\tau, t)$ that restricts the range of bound variables. Constants are annotated with their types, supplied as extra arguments, to resolve overloading. \item[$\bullet$] \textbf{\textit{poly\_tags} (sound):} Each term and subterm is tagged with its type using a function $\mathit{type\_info\/}(\tau, t)$. This coincides with the encoding used by the \textit{metisFT} command. \item[$\bullet$] \textbf{\textit{poly\_args} (unsound):} Like for \textit{poly\_preds} constants are annotated with their types to resolve overloading, but otherwise no type information is encoded. This coincides with the encoding used by the \textit{metis} command (before it falls back on \textit{metisFT}). \item[$\bullet$] \textbf{% \textit{mono\_preds}, \textit{mono\_tags} (sound); \textit{mono\_args} (unsound):} \\ Similar to \textit{poly\_preds}, \textit{poly\_tags}, and \textit{poly\_args}, respectively, but the problem is additionally monomorphized, meaning that type variables are instantiated with heuristically chosen ground types. Monomorphization can simplify reasoning but also leads to larger fact bases, which can slow down the ATPs. \item[$\bullet$] \textbf{% \textit{mangled\_preds}, \textit{mangled\_tags} (sound); \\ \textit{mangled\_args} (unsound):} \\ Similar to \textit{mono\_preds}, \textit{mono\_tags}, and \textit{mono\_args}, respectively but types are mangled in constant names instead of being supplied as ground term arguments. The binary predicate $\mathit{has\_type\/}(\tau, t)$ becomes a unary predicate $\mathit{has\_type\_}\tau(t)$, and the binary function $\mathit{type\_info\/}(\tau, t)$ becomes a unary function $\mathit{type\_info\_}\tau(t)$. \item[$\bullet$] \textbf{\textit{simple} (sound):} Use the prover's support for simple types if available; otherwise, fall back on \textit{mangled\_preds}. The problem is monomorphized. \item[$\bullet$] \textbf{% \textit{poly\_preds}?, \textit{poly\_tags}?, \textit{mono\_preds}?, \textit{mono\_tags}?, \\ \textit{mangled\_preds}?, \textit{mangled\_tags}?, \textit{simple}? (quasi-sound):} \\ The type systems \textit{poly\_preds}, \textit{poly\_tags}, \textit{mono\_preds}, \textit{mono\_tags}, \textit{mangled\_preds}, \textit{mangled\_tags}, and \textit{simple} are fully typed and sound. For each of these, Sledgehammer also provides a lighter, virtually sound variant identified by a question mark (`{?}')\ that detects and erases monotonic types, notably infinite types. (For \textit{simple}, the types are not actually erased but rather replaced by a shared uniform type of individuals.) \item[$\bullet$] \textbf{% \textit{poly\_preds}!, \textit{poly\_tags}!, \textit{mono\_preds}!, \textit{mono\_tags}!, \\ \textit{mangled\_preds}!, \textit{mangled\_tags}!, \textit{simple}! \\ (mildly unsound):} \\ The type systems \textit{poly\_preds}, \textit{poly\_tags}, \textit{mono\_preds}, \textit{mono\_tags}, \textit{mangled\_preds}, \textit{mangled\_tags}, and \textit{simple} also admit a mildly unsound (but very efficient) variant identified by an exclamation mark (`{!}') that detects and erases erases all types except those that are clearly finite (e.g., \textit{bool}). (For \textit{simple}, the types are not actually erased but rather replaced by a shared uniform type of individuals.) \item[$\bullet$] \textbf{\textit{smart}:} If \textit{full\_types} is enabled, uses a sound or virtually sound encoding; otherwise, uses any encoding. The actual encoding used depends on the ATP and should be the most efficient for that ATP. \end{enum} In addition, all the \textit{preds} and \textit{tags} type systems are available in two variants, a lightweight and a heavyweight variant. The lightweight variants are generally more efficient and are the default; the heavyweight variants are identified by a \textit{\_heavy} suffix (e.g., \textit{mangled\_preds\_heavy}{?}). For SMT solvers and ToFoF-E, the type system is always \textit{simple}, irrespective of the value of this option. \nopagebreak {\small See also \textit{max\_new\_mono\_instances} (\S\ref{relevance-filter}) and \textit{max\_mono\_iters} (\S\ref{relevance-filter}).} \end{enum} \subsection{Relevance Filter} \label{relevance-filter} \begin{enum} \opdefault{relevance\_thresholds}{float\_pair}{\upshape 0.45~0.85} Specifies the thresholds above which facts are considered relevant by the relevance filter. The first threshold is used for the first iteration of the relevance filter and the second threshold is used for the last iteration (if it is reached). The effective threshold is quadratically interpolated for the other iterations. Each threshold ranges from 0 to 1, where 0 means that all theorems are relevant and 1 only theorems that refer to previously seen constants. \opsmart{max\_relevant}{int\_or\_smart} Specifies the maximum number of facts that may be returned by the relevance filter. If the option is set to \textit{smart}, it is set to a value that was empirically found to be appropriate for the prover. A typical value would be 300. \opdefault{max\_new\_mono\_instances}{int}{\upshape 400} Specifies the maximum number of monomorphic instances to generate beyond \textit{max\_relevant}. The higher this limit is, the more monomorphic instances are potentially generated. Whether monomorphization takes place depends on the type system used. \nopagebreak {\small See also \textit{type\_sys} (\S\ref{problem-encoding}).} \opdefault{max\_mono\_iters}{int}{\upshape 3} Specifies the maximum number of iterations for the monomorphization fixpoint construction. The higher this limit is, the more monomorphic instances are potentially generated. Whether monomorphization takes place depends on the type system used. \nopagebreak {\small See also \textit{type\_sys} (\S\ref{problem-encoding}).} \end{enum} \subsection{Output Format} \label{output-format} \begin{enum} \opfalse{verbose}{quiet} Specifies whether the \textbf{sledgehammer} command should explain what it does. This option is implicitly disabled for automatic runs. \opfalse{debug}{no\_debug} Specifies whether Sledgehammer should display additional debugging information beyond what \textit{verbose} already displays. Enabling \textit{debug} also enables \textit{verbose} and \textit{blocking} (\S\ref{mode-of-operation}) behind the scenes. The \textit{debug} option is implicitly disabled for automatic runs. \nopagebreak {\small See also \textit{overlord} (\S\ref{mode-of-operation}).} \opfalse{isar\_proof}{no\_isar\_proof} Specifies whether Isar proofs should be output in addition to one-liner \textit{metis} proofs. Isar proof construction is still experimental and often fails; however, they are usually faster and sometimes more robust than \textit{metis} proofs. \opdefault{isar\_shrink\_factor}{int}{\upshape 1} Specifies the granularity of the Isar proof. A value of $n$ indicates that each Isar proof step should correspond to a group of up to $n$ consecutive proof steps in the ATP proof. \end{enum} \subsection{Authentication} \label{authentication} \begin{enum} \opnodefault{expect}{string} Specifies the expected outcome, which must be one of the following: \begin{enum} \item[$\bullet$] \textbf{\textit{some}:} Sledgehammer found a (potentially unsound) proof. \item[$\bullet$] \textbf{\textit{none}:} Sledgehammer found no proof. \item[$\bullet$] \textbf{\textit{unknown}:} Sledgehammer encountered some problem. \end{enum} Sledgehammer emits an error (if \textit{blocking} is enabled) or a warning (otherwise) if the actual outcome differs from the expected outcome. This option is useful for regression testing. \nopagebreak {\small See also \textit{blocking} (\S\ref{mode-of-operation}).} \end{enum} \let\em=\sl \bibliography{../manual}{} \bibliographystyle{abbrv} \end{document}