\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}'' on goal \\

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

Try this: \textbf{by} (\textit{metis last\_ConsL}). \\

To minimize: \textbf{sledgehammer} \textit{min} [\textit{e}] (\textit{last\_ConsL}). \\[3\smallskipamount]

%

Sledgehammer: ``\textit{vampire}'' on goal \\

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

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

To minimize: \textbf{sledgehammer} \textit{min} [\textit{vampire}] (\textit{hd.simps}). \\[3\smallskipamount]

%

Sledgehammer: ``\textit{spass}'' on goal \\

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

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

To minimize: \textbf{sledgehammer} \textit{min} [\textit{spass}]~(\textit{list.inject}). \\[3\smallskipamount]

%

Sledgehammer: ``\textit{remote\_waldmeister}'' on goal \\

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

Try this: \textbf{by} (\textit{metis hd.simps insert\_Nil}). \\

To minimize: \textbf{sledgehammer} \textit{min} [\textit{remote\_waldmeister}] \\

\phantom{To minimize: \textbf{sledgehammer}~}(\textit{hd.simps insert\_Nil}). \\[3\smallskipamount]

%

Sledgehammer: ``\textit{remote\_sine\_e}'' on goal \\

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

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

To minimize: \textbf{sledgehammer} \textit{min} [\textit{remote\_sine\_e}]~(\textit{hd.simps}). \\[3\smallskipamount]

%

Sledgehammer: ``\textit{remote\_z3}'' on goal \\

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

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

To minimize: \textbf{sledgehammer} \textit{min} [\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}''). For convenience, you can omit ``\textit{provers}~=''

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

spass remote\_vampire}'').

\item[$\bullet$] \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}). The command also prints the list of all available

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

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

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

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

\section{Frequently Asked Questions}

\label{frequently-asked-questions}

This sections answers frequently (and infrequently) asked questions about

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

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

send them to the author at \authoremail.

\point{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

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

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

\end{enum}

In some rare cases, 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 generated 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 re-find 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{Which facts are passed to the automatic provers?}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(\S\ref{relevance-filter}) to, say, 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 generated Isar proofs so ugly/detailed/broken?}

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 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 on 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 would be unlikely to re-find 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{A strange error occurred---what should I do?}

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

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

\prew

\slshape

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

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

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

want to report this to the Isabelle developers.

\postw

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

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

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

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

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

\point{Why are there so many options?}

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

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

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

you are so inclined.

\section{Command Syntax}

\label{command-syntax}

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{min}:} 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\defl{\{}

\def\defr{\}}

\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}$\bigr]$\enskip \defl\textit{true}\defr\hfill (neg.: \textit{#2})}\nopagebreak\\[\parskip]}

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

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

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

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

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

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

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

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

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

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

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

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

(\S\ref{output-format}), 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{smart\_bool\/}: \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{smart\_int\/}: 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 braces. 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}

\opnodefaultbrk{provers}{string}

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

``\textit{e}~\textit{spass}~\textit{remote\_vampire}''). 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}{smart\_int}

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}).}