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\chapter{Basic Isar Elements}\label{ch:puresyntax}

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3 

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4 
Subsequently, we introduce the main part of the basic Isar theory and proof


5 
commands as provided by Isabelle/Pure. Chapter~\ref{ch:gentools} describes

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6 
further Isar elements as provided by generic tools and packages (such as the


7 
Simplifier) that are either part of Pure Isabelle, or preloaded by most


8 
object logics. See chapter~\ref{ch:holtools} for actual objectlogic


9 
specific elements (for Isabelle/HOL).

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\medskip


12 


13 
Isar commands may be either \emph{proper} document constructors, or

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\emph{improper commands} (indicated by $^*$). Some proof methods and

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attributes introduced later are classified as improper as well. Improper Isar


16 
language elements might be helpful when developing proof documents, while

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their use is strongly discouraged for the final version. Typical examples are


18 
diagnostic commands that print terms or theorems according to the current


19 
context; other commands even emulate oldstyle tactical theorem proving, which


20 
facilitates porting of legacy proof scripts.

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22 


23 
\section{Theory commands}


24 

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\subsection{Defining theories}\label{sec:beginthy}

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26 


27 
\indexisarcmd{theory}\indexisarcmd{end}\indexisarcmd{context}


28 
\begin{matharray}{rcl}


29 
\isarcmd{theory} & : & \isartrans{\cdot}{theory} \\


30 
\isarcmd{context}^* & : & \isartrans{\cdot}{theory} \\


31 
\isarcmd{end} & : & \isartrans{theory}{\cdot} \\


32 
\end{matharray}


33 


34 
Isabelle/Isar ``newstyle'' theories are either defined via theory files or

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interactively. Both actual theory specifications and proofs are handled

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uniformly  occasionally definitional mechanisms even require some explicit


37 
proof as well. In contrast, ``oldstyle'' Isabelle theories support batch


38 
processing only, with the proof scripts collected in separate ML files.

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40 
The first command of any theory has to be $\THEORY$, starting a new theory

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based on the merge of existing ones. The theory context may be also changed


42 
by $\CONTEXT$ without creating a new theory. In both cases, $\END$ concludes


43 
the theory development; it has to be the very last command of any proper


44 
theory file.

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45 


46 
\begin{rail}


47 
'theory' name '=' (name + '+') filespecs? ':'


48 
;


49 
'context' name


50 
;


51 
'end'


52 
;;


53 

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filespecs: 'files' ((name  parname) +);

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\end{rail}


56 

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\begin{descr}

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\item [$\THEORY~A = B@1 + \cdots + B@n$] commences a new theory $A$ based on

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existing ones $B@1 + \cdots + B@n$. Isabelle's theory loader system ensures


60 
that any of the base theories are properly loaded (and fully uptodate when


61 
$\THEORY$ is executed interactively). The optional $\isarkeyword{files}$


62 
specification declares additional dependencies on ML files. Unless put in


63 
parentheses, any file will be loaded immediately via $\isarcmd{use}$ (see

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also \S\ref{sec:ML}). The optional ML file \texttt{$A$.ML} that may be


65 
associated with any theory should \emph{not} be included in


66 
$\isarkeyword{files}$.

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\item [$\CONTEXT~B$] enters an existing theory context $B$, basically in

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readonly mode, so only a limited set of commands may be performed. Just as


70 
for $\THEORY$, the theory loader ensures that $B$ is loaded and uptodate.

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\item [$\END$] concludes the current theory definition or context switch.

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Note that this command cannot be undone, instead the theory definition


74 
itself has to be retracted.

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\end{descr}

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\subsection{Formal comments}\label{sec:formalcmtthy}

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\indexisarcmd{title}\indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection}


81 
\indexisarcmd{subsubsection}\indexisarcmd{text}

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\begin{matharray}{rcl}


83 
\isarcmd{title} & : & \isartrans{theory}{theory} \\


84 
\isarcmd{chapter} & : & \isartrans{theory}{theory} \\

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\isarcmd{section} & : & \isartrans{theory}{theory} \\

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\isarcmd{subsection} & : & \isartrans{theory}{theory} \\


87 
\isarcmd{subsubsection} & : & \isartrans{theory}{theory} \\


88 
\isarcmd{text} & : & \isartrans{theory}{theory} \\


89 
\end{matharray}


90 

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There are several commands to include \emph{formal comments} in theory


92 
specification (a few more are available for proofs, see


93 
\S\ref{sec:formalcmtprf}). In contrast to sourcelevel comments

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\verb(*\dots\verb*), which are stripped at the lexical level, any text


95 
given as formal comment is meant to be part of the actual document.


96 
Consequently, it would be included in the final printed version.


97 


98 
Apart from plain prose, formal comments may also refer to logical entities of

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the theory context (types, terms, theorems etc.). Proper processing of the


100 
text would then include some further consistency checks with the items


101 
declared in the current theory, e.g.\ typechecking of included


102 
terms.\footnote{The current version of Isabelle/Isar does not process formal

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comments in any such way. This will be available as part of the automatic

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theory and proof document preparation system (using (PDF){\LaTeX}) that is

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planned for the near future.}


106 


107 
\begin{rail}


108 
'title' text text? text?


109 
;

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('chapter'  'section'  'subsection'  'subsubsection'  'text') text

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;


112 
\end{rail}


113 

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\begin{descr}

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\item [$\isarkeyword{title}~title~author~date$] specifies the document title

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just as in typical {\LaTeX} documents.

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\item [$\isarkeyword{chapter}$, $\isarkeyword{section}$,


118 
$\isarkeyword{subsection}$, and $\isarkeyword{subsubsection}$] mark chapter


119 
and section headings.


120 
\item [$\TEXT$] specifies an actual body of prose text, including references


121 
to formal entities.\footnote{The latter feature is not yet exploited.


122 
Nevertheless, any text of the form \texttt{\at\ttlbrace\dots\ttrbrace}


123 
should be considered as reserved for future use.}

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\end{descr}

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125 


126 

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\subsection{Type classes and sorts}\label{sec:classes}

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129 
\indexisarcmd{classes}\indexisarcmd{classrel}\indexisarcmd{defaultsort}


130 
\begin{matharray}{rcl}


131 
\isarcmd{classes} & : & \isartrans{theory}{theory} \\


132 
\isarcmd{classrel} & : & \isartrans{theory}{theory} \\


133 
\isarcmd{defaultsort} & : & \isartrans{theory}{theory} \\


134 
\end{matharray}


135 


136 
\begin{rail}

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'classes' (classdecl comment? +)

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;


139 
'classrel' nameref '<' nameref comment?


140 
;


141 
'defaultsort' sort comment?


142 
;


143 
\end{rail}


144 

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\begin{descr}

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\item [$\isarkeyword{classes}~c<\vec c$] declares class $c$ to be a subclass


147 
of existing classes $\vec c$. Cyclic class structures are ruled out.

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\item [$\isarkeyword{classrel}~c@1<c@2$] states a subclass relation between


149 
existing classes $c@1$ and $c@2$. This is done axiomatically! The

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$\isarkeyword{instance}$ command (see \S\ref{sec:axclass}) provides a way


151 
introduce proven class relations.

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\item [$\isarkeyword{defaultsort}~s$] makes sort $s$ the new default sort for

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any type variables input without sort constraints. Usually, the default

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sort would be only changed when defining new logics.

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\end{descr}

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156 


157 

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\subsection{Primitive types and type abbreviations}\label{sec:typespure}

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160 
\indexisarcmd{typedecl}\indexisarcmd{types}\indexisarcmd{nonterminals}\indexisarcmd{arities}


161 
\begin{matharray}{rcl}


162 
\isarcmd{types} & : & \isartrans{theory}{theory} \\


163 
\isarcmd{typedecl} & : & \isartrans{theory}{theory} \\


164 
\isarcmd{nonterminals} & : & \isartrans{theory}{theory} \\


165 
\isarcmd{arities} & : & \isartrans{theory}{theory} \\


166 
\end{matharray}


167 


168 
\begin{rail}


169 
'types' (typespec '=' type infix? comment? +)


170 
;


171 
'typedecl' typespec infix? comment?


172 
;


173 
'nonterminals' (name +) comment?


174 
;


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'arities' (nameref '::' arity comment? +)


176 
;


177 
\end{rail}


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\begin{descr}

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\item [$\TYPES~(\vec\alpha)t = \tau$] introduces \emph{type synonym}

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$(\vec\alpha)t$ for existing type $\tau$. Unlike actual type definitions,


182 
as are available in Isabelle/HOL for example, type synonyms are just purely


183 
syntactic abbreviations, without any logical significance. Internally, type


184 
synonyms are fully expanded, as may be observed when printing terms or


185 
theorems.


186 
\item [$\isarkeyword{typedecl}~(\vec\alpha)t$] declares a new type constructor


187 
$t$, intended as an actual logical type. Note that some logics such as


188 
Isabelle/HOL provide their own version of $\isarkeyword{typedecl}$.

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\item [$\isarkeyword{nonterminals}~\vec c$] declares $0$ary type constructors


190 
$\vec c$ to act as purely syntactic types, i.e.\ nonterminal symbols of


191 
Isabelle's inner syntax of terms or types.

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\item [$\isarkeyword{arities}~t::(\vec s)s$] augments Isabelle's ordersorted


193 
signature of types by new type constructor arities. This is done


194 
axiomatically! The $\isarkeyword{instance}$ command (see

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\S\ref{sec:axclass}) provides a way introduce proven type arities.

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\end{descr}

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197 


198 


199 
\subsection{Constants and simple definitions}


200 

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\indexisarcmd{consts}\indexisarcmd{defs}\indexisarcmd{constdefs}\indexoutertoken{constdecl}

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\begin{matharray}{rcl}


203 
\isarcmd{consts} & : & \isartrans{theory}{theory} \\


204 
\isarcmd{defs} & : & \isartrans{theory}{theory} \\


205 
\isarcmd{constdefs} & : & \isartrans{theory}{theory} \\


206 
\end{matharray}


207 


208 
\begin{rail}


209 
'consts' (constdecl +)


210 
;


211 
'defs' (thmdecl? prop comment? +)


212 
;


213 
'constdefs' (constdecl prop comment? +)


214 
;


215 


216 
constdecl: name '::' type mixfix? comment?


217 
;


218 
\end{rail}


219 

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\begin{descr}

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\item [$\CONSTS~c::\sigma$] declares constant $c$ to have any instance of type


222 
scheme $\sigma$. The optional mixfix annotations may attach concrete syntax


223 
constants.


224 
\item [$\DEFS~name: eqn$] introduces $eqn$ as a definitional axiom for some


225 
existing constant. See \cite[\S6]{isabelleref} for more details on the


226 
form of equations admitted as constant definitions.


227 
\item [$\isarkeyword{constdefs}~c::\sigma~eqn$] combines declarations and


228 
definitions of constants, using canonical name $c_def$ for the definitional


229 
axiom.

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\end{descr}

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232 

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\subsection{Syntax and translations}

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234 


235 
\indexisarcmd{syntax}\indexisarcmd{translations}


236 
\begin{matharray}{rcl}


237 
\isarcmd{syntax} & : & \isartrans{theory}{theory} \\


238 
\isarcmd{translations} & : & \isartrans{theory}{theory} \\


239 
\end{matharray}


240 


241 
\begin{rail}


242 
'syntax' ('(' name 'output'? ')')? (constdecl +)


243 
;


244 
'translations' (transpat ('=='  '=>'  '<=') transpat comment? +)


245 
;


246 
transpat: ('(' nameref ')')? string


247 
;


248 
\end{rail}


249 

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\begin{descr}

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\item [$\isarkeyword{syntax}~(mode)~decls$] is similar to $\CONSTS~decls$,


252 
except that the actual logical signature extension is omitted. Thus the


253 
context free grammar of Isabelle's inner syntax may be augmented in

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arbitrary ways, independently of the logic. The $mode$ argument refers to


255 
the print mode that the grammar rules belong; unless there is the


256 
\texttt{output} flag given, all productions are added both to the input and


257 
output grammar.

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\item [$\isarkeyword{translations}~rules$] specifies syntactic translation


259 
rules (also known as \emph{macros}): parse/print rules (\texttt{==}), parse

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rules (\texttt{=>}), or print rules (\texttt{<=}). Translation patterns may


261 
be prefixed by the syntactic category to be used for parsing; the default is

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\texttt{logic}.

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\end{descr}

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265 


266 
\subsection{Axioms and theorems}


267 


268 
\indexisarcmd{axioms}\indexisarcmd{theorems}\indexisarcmd{lemmas}


269 
\begin{matharray}{rcl}


270 
\isarcmd{axioms} & : & \isartrans{theory}{theory} \\


271 
\isarcmd{theorems} & : & \isartrans{theory}{theory} \\


272 
\isarcmd{lemmas} & : & \isartrans{theory}{theory} \\


273 
\end{matharray}


274 


275 
\begin{rail}

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'axioms' (axmdecl prop comment? +)

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;


278 
('theorems'  'lemmas') thmdef? thmrefs


279 
;


280 
\end{rail}


281 

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\begin{descr}

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\item [$\isarkeyword{axioms}~a: \phi$] introduces arbitrary statements as


284 
logical axioms. In fact, axioms are ``axiomatic theorems'', and may be


285 
referred later just as any other theorem.

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287 
Axioms are usually only introduced when declaring new logical systems.

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Everyday work is typically done the hard way, with proper definitions and

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actual theorems.

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\item [$\isarkeyword{theorems}~a = \vec b$] stores lists of existing theorems.


291 
Typical applications would also involve attributes, to augment the default


292 
Simplifier context, for example.

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\item [$\isarkeyword{lemmas}$] is similar to $\isarkeyword{theorems}$, but


294 
tags the results as ``lemma''.

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\end{descr}

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297 

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\subsection{Name spaces}

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\indexisarcmd{global}\indexisarcmd{local}

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\begin{matharray}{rcl}


302 
\isarcmd{global} & : & \isartrans{theory}{theory} \\


303 
\isarcmd{local} & : & \isartrans{theory}{theory} \\


304 
\end{matharray}


305 

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Isabelle organizes any kind of names (of types, constants, theorems etc.) by

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hierarchically structured name spaces. Normally the user never has to control

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the behavior of name space entry by hand, yet the following commands provide

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some way to do so.


310 

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\begin{descr}


312 
\item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current


313 
name declaration mode. Initially, theories start in $\isarkeyword{local}$


314 
mode, causing all names to be automatically qualified by the theory name.


315 
Changing this to $\isarkeyword{global}$ causes all names to be declared as

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base names only, until $\isarkeyword{local}$ is declared again.

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\end{descr}

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319 

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\subsection{Incorporating ML code}\label{sec:ML}

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322 
\indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{setup}


323 
\begin{matharray}{rcl}


324 
\isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\


325 
\isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\

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\isarcmd{setup} & : & \isartrans{theory}{theory} \\

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\end{matharray}


328 


329 
\begin{rail}


330 
'use' name


331 
;


332 
'ML' text


333 
;


334 
'setup' text


335 
;


336 
\end{rail}


337 

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\begin{descr}

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\item [$\isarkeyword{use}~file$] reads and executes ML commands from $file$.


340 
The current theory context (if present) is passed down to the ML session.


341 
Furthermore, the file name is checked with the $\isarkeyword{files}$


342 
dependency declaration given in the theory header (see also


343 
\S\ref{sec:beginthy}).


344 
\item [$\isarkeyword{ML}~text$] reads and executes ML commands from $text$.


345 
The theory context is passed just as for $\isarkeyword{use}$.

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%FIXME picked up again!?

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\item [$\isarkeyword{setup}~text$] changes the current theory context by

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applying setup functions $text$, which has to be an ML expression of type


349 
$(theory \to theory)~list$. The $\isarkeyword{setup}$ command is the usual

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way to initialize objectlogic specific tools and packages written in ML.

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\end{descr}

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353 

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\subsection{Syntax translation functions}

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\indexisarcmd{parseasttranslation}\indexisarcmd{parsetranslation}


357 
\indexisarcmd{printtranslation}\indexisarcmd{typedprinttranslation}


358 
\indexisarcmd{printasttranslation}\indexisarcmd{tokentranslation}

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\begin{matharray}{rcl}


360 
\isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\


361 
\isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\


362 
\isarcmd{print_translation} & : & \isartrans{theory}{theory} \\


363 
\isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\


364 
\isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\


365 
\isarcmd{token_translation} & : & \isartrans{theory}{theory} \\


366 
\end{matharray}


367 


368 
Syntax translation functions written in ML admit almost arbitrary


369 
manipulations of Isabelle's inner syntax. Any of the above commands have a


370 
single \railqtoken{text} argument that refers to an ML expression of


371 
appropriate type. See \cite[\S8]{isabelleref} for more information on syntax


372 
transformations.


373 


374 


375 
\subsection{Oracles}


376 


377 
\indexisarcmd{oracle}


378 
\begin{matharray}{rcl}


379 
\isarcmd{oracle} & : & \isartrans{theory}{theory} \\


380 
\end{matharray}


381 

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Oracles provide an interface to external reasoning systems, without giving up


383 
control completely  each theorem carries a derivation object recording any


384 
oracle invocation. See \cite[\S6]{isabelleref} for more information.


385 

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\begin{rail}


387 
'oracle' name '=' text comment?


388 
;


389 
\end{rail}


390 

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\begin{descr}

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\item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML

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393 
function $text$, which has to be of type $Sign\mathord.sg \times

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394 
Object\mathord.T \to term$.

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395 
\end{descr}

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396 


397 


398 
\section{Proof commands}


399 

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400 
Proof commands provide transitions of Isar/VM machine configurations, which


401 
are blockstructured, consisting of a stack of nodes with three main

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components: logical proof context, current facts, and open goals. Isar/VM


403 
transitions are \emph{typed} according to the following three three different


404 
modes of operation:

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405 
\begin{descr}


406 
\item [$proof(prove)$] means that a new goal has just been stated that is now


407 
to be \emph{proven}; the next command may refine it by some proof method

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408 
($\approx$ tactic), and enter a subproof to establish the final result.

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409 
\item [$proof(state)$] is like an internal theory mode: the context may be

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410 
augmented by \emph{stating} additional assumptions, intermediate result etc.


411 
\item [$proof(chain)$] is an intermediate mode between $proof(state)$ and

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412 
$proof(prove)$: existing facts have been just picked up in order to use them

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413 
when refining the goal claimed next.

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414 
\end{descr}

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415 

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416 


417 
\subsection{Formal comments}\label{sec:formalcmtprf}


418 


419 
\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsect}\indexisarcmd{txt}

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420 
\begin{matharray}{rcl}

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421 
\isarcmd{sect} & : & \isartrans{proof(state)}{proof(state)} \\


422 
\isarcmd{subsect} & : & \isartrans{proof(state)}{proof(state)} \\


423 
\isarcmd{subsubsect} & : & \isartrans{proof(state)}{proof(state)} \\


424 
\isarcmd{txt} & : & \isartrans{proof(state)}{proof(state)} \\

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425 
\end{matharray}


426 

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427 
These formal comments in proof mode closely correspond to the ones of theory


428 
mode (see \S\ref{sec:formalcmtthy}).


429 

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430 
\begin{rail}

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431 
('sect'  'subsect'  'subsubsect'  'txt') text

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432 
;


433 
\end{rail}


434 


435 

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436 
\subsection{Proof context}\label{sec:proofcontext}

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437 

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438 
\indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def}

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439 
\begin{matharray}{rcl}


440 
\isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\


441 
\isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\


442 
\isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\


443 
\isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\


444 
\end{matharray}


445 

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446 
The logical proof context consists of fixed variables and assumptions. The


447 
former closely correspond to Skolem constants, or metalevel universal


448 
quantification as provided by the Isabelle/Pure logical framework.


449 
Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in

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450 
a local object that may be used in the subsequent proof as any other variable

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451 
or constant. Furthermore, any result $\phi[x]$ exported from the current


452 
context will be universally closed wrt.\ $x$ at the outermost level (this is


453 
expressed using Isabelle's metavariables).


454 


455 
Similarly, introducing some assumption $\chi$ has two effects. On the one


456 
hand, a local theorem is created that may be used as a fact in subsequent


457 
proof steps. On the other hand, any result $\phi$ exported from the context


458 
becomes conditional wrt.\ the assumption. Thus, solving an enclosing goal

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using such a result would basically introduce a new subgoal stemming from the

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460 
assumption. How this situation is handled depends on the actual version of


461 
assumption command used: while $\ASSUMENAME$ solves the subgoal by unifying


462 
with some premise of the goal, $\PRESUMENAME$ leaves the subgoal unchanged to


463 
be proved later by the user.


464 

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465 
Local definitions, introduced by $\DEF{}{x \equiv t}$, are achieved by


466 
combining $\FIX x$ with another kind of assumption that causes any

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467 
hypothetical equation $x = t$ to be eliminated by reflexivity. Thus,


468 
exporting some result $\phi[x]$ simply yields $\phi[t]$.

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469 

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470 
\begin{rail}


471 
'fix' (var +) comment?


472 
;

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473 
('assume'  'presume') (assm comment? + 'and')

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474 
;

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475 
'def' thmdecl? \\ var '==' term termpat? comment?

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476 
;


477 


478 
var: name ('::' type)?


479 
;

7315

480 
assm: thmdecl? (prop proppat? +)


481 
;

7134

482 
\end{rail}


483 

7167

484 
\begin{descr}

7315

485 
\item [$\FIX{x}$] introduces a local \emph{arbitrary, but fixed} variable $x$.


486 
\item [$\ASSUME{a}{\Phi}$ and $\PRESUME{a}{\Phi}$] introduce local theorems

7335

487 
$\Phi$ by assumption. Subsequent results applied to an enclosing goal


488 
(e.g.\ via $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ expects to be


489 
able to unify with existing premises in the goal, while $\PRESUMENAME$


490 
leaves $\Phi$ as new subgoals.


491 


492 
Several lists of assumptions may be given (separated by


493 
$\isarkeyword{and}$); the resulting list of facts consists of all of these


494 
concatenated.

7315

495 
\item [$\DEF{a}{x \equiv t}$] introduces a local (nonpolymorphic) definition.


496 
In results exported from the context, $x$ is replaced by $t$. Basically,

7335

497 
$\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\PRESUME{}{x \equiv t}$, with the


498 
resulting hypothetical equation solved by reflexivity.

7167

499 
\end{descr}


500 

7389

501 
The special theorem name $prems$\indexisarthm{prems} refers to all current

7335

502 
assumptions.

7315

503 

7167

504 


505 
\subsection{Facts and forward chaining}


506 


507 
\indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with}


508 
\begin{matharray}{rcl}


509 
\isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\


510 
\isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\


511 
\isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\


512 
\isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\


513 
\end{matharray}


514 

7319

515 
New facts are established either by assumption or proof of local statements.

7335

516 
Any fact will usually be involved in further proofs, either as explicit


517 
arguments of proof methods or when forward chaining towards the next goal via


518 
$\THEN$ (and variants). Note that the special theorem name

7389

519 
$facts$.\indexisarthm{facts} refers to the most recently established facts.

7167

520 
\begin{rail}


521 
'note' thmdef? thmrefs comment?


522 
;


523 
'then' comment?


524 
;


525 
('from'  'with') thmrefs comment?


526 
;


527 
\end{rail}


528 


529 
\begin{descr}

7175

530 
\item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result


531 
as $a$. Note that attributes may be involved as well, both on the left and


532 
right hand sides.

7167

533 
\item [$\THEN$] indicates forward chaining by the current facts in order to

7335

534 
establish the goal claimed next. The initial proof method invoked to refine


535 
that will be offered these facts to do ``anything appropriate'' (see also


536 
\S\ref{sec:proofsteps}). For example, method $rule$ (see

7167

537 
\S\ref{sec:puremeth}) would do an elimination rather than an introduction.

7335

538 
\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; thus $\THEN$ is


539 
equivalent to $\FROM{facts}$.

7175

540 
\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward


541 
chaining is from earlier facts together with the current ones.

7167

542 
\end{descr}


543 

7389

544 
Basic proof methods (such as $rule$, see \S\ref{sec:puremeth}) expect


545 
multiple facts to be given in proper order, corresponding to a prefix of the


546 
premises of the rule involved. Note that positions may be easily skipped


547 
using a form like $\FROM{_~a~b}$, for example. This involves the rule


548 
$PROP~\psi \Imp PROP~\psi$, which is bound in Isabelle/Pure as ``_''


549 
(underscore).\indexisarthm{_@\texttt{_}}


550 

7167

551 


552 
\subsection{Goal statements}


553 


554 
\indexisarcmd{theorem}\indexisarcmd{lemma}


555 
\indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus}


556 
\begin{matharray}{rcl}


557 
\isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\


558 
\isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\


559 
\isarcmd{have} & : & \isartrans{proof(state)}{proof(prove)} \\


560 
\isarcmd{show} & : & \isartrans{proof(state)}{proof(prove)} \\


561 
\isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\


562 
\isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\


563 
\end{matharray}


564 

7175

565 
Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$


566 
and $\LEMMANAME$. New local goals may be claimed within proof mode: four


567 
variants are available, indicating whether the result is meant to solve some


568 
pending goal and whether forward chaining is employed.


569 

7167

570 
\begin{rail}


571 
('theorem'  'lemma') goal


572 
;


573 
('have'  'show'  'hence'  'thus') goal


574 
;


575 


576 
goal: thmdecl? proppat comment?


577 
;


578 
\end{rail}


579 


580 
\begin{descr}

7335

581 
\item [$\THEOREM{a}{\phi}$] enters proof mode with $\phi$ as main goal,

7175

582 
eventually resulting in some theorem $\turn \phi$, which will be stored in


583 
the theory.

7167

584 
\item [$\LEMMANAME$] is similar to $\THEOREMNAME$, but tags the result as


585 
``lemma''.

7335

586 
\item [$\HAVE{a}{\phi}$] claims a local goal, eventually resulting in a

7167

587 
theorem with the current assumption context as hypotheses.

7335

588 
\item [$\SHOW{a}{\phi}$] is similar to $\HAVE{a}{\phi}$, but solves some

7175

589 
pending goal with the result \emph{exported} into the corresponding context.

7335

590 
\item [$\HENCE{a}{\phi}$] abbreviates $\THEN~\HAVE{a}{\phi}$, i.e.\ claims a


591 
local goal to be proven by forward chaining the current facts.


592 
\item [$\THUS{a}{\phi}$] abbreviates $\THEN~\SHOW{a}{\phi}$.

7167

593 
\end{descr}


594 


595 


596 
\subsection{Initial and terminal proof steps}\label{sec:proofsteps}


597 

7175

598 
\indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by}


599 
\indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry}


600 
\begin{matharray}{rcl}


601 
\isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\


602 
\isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~~ theory} \\


603 
\isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\


604 
\isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\


605 
\isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\


606 
\isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\


607 
\end{matharray}


608 

7335

609 
Arbitrary goal refinement via tactics is considered harmful. Consequently the


610 
Isar framework admits proof methods to be invoked in two places only.

7167

611 
\begin{enumerate}

7175

612 
\item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated

7335

613 
goal to a number of subgoals that are to be solved later. Facts are passed


614 
to $m@1$ for forward chaining if so indicated by $proof(chain)$ mode.

7167

615 

7335

616 
\item A \emph{terminal} conclusion step $\QED{m@2}$ solves any remaining goals


617 
completely. No facts are passed to $m@2$.

7167

618 
\end{enumerate}


619 

7335

620 
The only other proper way to affect pending goals is by $\SHOWNAME$ (or


621 
$\THUSNAME$), which involves an explicit statement of what is to be solved.

7167

622 

7175

623 
\medskip


624 

7167

625 
Also note that initial proof methods should either solve the goal completely,


626 
or constitute some wellunderstood deterministic reduction to new subgoals.


627 
Arbitrary automatic proof tools that are prone leave a large number of badly


628 
structured subgoals are no help in continuing the proof document in any

7175

629 
intelligible way. A much better technique would be to $\SHOWNAME$ some


630 
nontrivial reduction as an explicit rule, which is solved completely by some


631 
automated method, and then applied to some pending goal.

7167

632 

7175

633 
\medskip


634 


635 
Unless given explicitly by the user, the default initial method is


636 
``$default$'', which is usually set up to apply a single standard elimination


637 
or introduction rule according to the topmost symbol involved. The default


638 
terminal method is ``$finish$''; it solves all goals by assumption.

7167

639 


640 
\begin{rail}


641 
'proof' interest? meth? comment?


642 
;


643 
'qed' meth? comment?


644 
;


645 
'by' meth meth? comment?


646 
;


647 
('.'  '..'  'sorry') comment?


648 
;


649 


650 
meth: method interest?


651 
;


652 
\end{rail}


653 


654 
\begin{descr}

7335

655 
\item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for


656 
forward chaining are passed if so indicated by $proof(chain)$ mode.


657 
\item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and


658 
concludes the subproof. If the goal had been $\SHOWNAME$ (or $\THUSNAME$),


659 
some pending subgoal is solved as well by the rule resulting from the


660 
result exported to the enclosing goal context. Thus $\QEDNAME$ may fail for


661 
two reasons: either $m@2$ fails to solve all remaining goals completely, or


662 
the resulting rule does not resolve with any enclosing goal. Debugging such


663 
a situation might involve temporarily changing $\SHOWNAME$ into $\HAVENAME$,


664 
or softening the local context by replacing $\ASSUMENAME$ by $\PRESUMENAME$.

7175

665 
\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}; it abbreviates


666 
$\PROOF{m@1}~\QED{m@2}$, with automatic backtracking across both methods.


667 
Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done by simply


668 
expanding the abbreviation by hand; note that $\PROOF{m@1}$ is usually


669 
sufficient to see what is going wrong.

7321

670 
\item [``$\DDOT$''] is a \emph{default proof}; it abbreviates $\BY{default}$.

7335

671 
\item [``$\DOT$''] is a \emph{trivial proof}; it abbreviates $\BY{}$.

7167

672 
\item [$\isarkeyword{sorry}$] is a \emph{fake proof}; provided that


673 
\texttt{quick_and_dirty} is enabled, $\isarkeyword{sorry}$ pretends to solve

7335

674 
the goal without further ado. Of course, the result is a fake theorem only,

7175

675 
involving some oracle in its internal derivation object (this is indicated

7319

676 
as $[!]$ in the printed result). The main application of

7167

677 
$\isarkeyword{sorry}$ is to support topdown proof development.


678 
\end{descr}

7134

679 


680 

7315

681 
\subsection{Improper proof steps}


682 


683 
The following commands emulate unstructured tactic scripts to some extent.


684 
While these are anathema for writing proper Isar proof documents, they might


685 
come in handy for exploring and debugging.


686 


687 
\indexisarcmd{apply}\indexisarcmd{thenapply}\indexisarcmd{back}


688 
\begin{matharray}{rcl}


689 
\isarcmd{apply}^* & : & \isartrans{proof}{proof} \\


690 
\isarcmd{then_apply}^* & : & \isartrans{proof}{proof} \\


691 
\isarcmd{back}^* & : & \isartrans{proof}{proof} \\


692 
\end{matharray}


693 


694 
\railalias{thenapply}{then\_apply}


695 
\railterm{thenapply}


696 


697 
\begin{rail}


698 
'apply' method


699 
;


700 
thenapply method


701 
;


702 
'back'


703 
;


704 
\end{rail}


705 


706 
\begin{descr}

7335

707 
\item [$\isarkeyword{apply}~(m)$] applies proof method $m$ in the

7315

708 
plainoldtactic sense. Facts for forward chaining are ignored.

7335

709 
\item [$\isarkeyword{then_apply}~(m)$] is similar to $\isarkeyword{apply}$,


710 
but observes the goal's facts.

7315

711 
\item [$\isarkeyword{back}$] does backtracking over the result sequence of

7389

712 
the latest proof command.\footnote{Unlike the ML function \texttt{back}


713 
\cite{isabelleref}, the Isar command does not search upwards for further


714 
branch points.} Basically, any proof command may return multiple results.

7315

715 
\end{descr}


716 


717 


718 
\subsection{Term abbreviations}\label{sec:termabbrev}


719 


720 
\indexisarcmd{let}


721 
\begin{matharray}{rcl}


722 
\isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\


723 
\isarkeyword{is} & : & syntax \\


724 
\end{matharray}


725 


726 
Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements,


727 
or by annotating assumptions or goal statements ($\ASSUMENAME$, $\SHOWNAME$


728 
etc.) with a list of patterns $\IS{p@1 \dots p@n}$. In both cases,


729 
higherorder matching is applied to bind extralogical text


730 
variables\index{text variables}, which may be either of the form $\VVar{x}$


731 
(token class \railtoken{textvar}, see \S\ref{sec:lexsyntax}) or nameless


732 
dummies ``\verb,_,'' (underscore).\index{dummy variables} Note that in the


733 
$\LETNAME$ form the patterns occur on the lefthand side, while the $\ISNAME$


734 
patterns are in postfix position.


735 

7319

736 
Term abbreviations are quite different from actual local definitions as


737 
introduced via $\DEFNAME$ (see \S\ref{sec:proofcontext}). The latter are

7315

738 
visible within the logic as actual equations, while abbreviations disappear


739 
during the input process just after type checking.


740 


741 
\begin{rail}


742 
'let' ((term + 'as') '=' term comment? + 'and')


743 
;


744 
\end{rail}


745 


746 
The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or


747 
\railnonterm{proppat} (see \S\ref{sec:termpats}).


748 


749 
\begin{descr}


750 
\item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$


751 
by simultaneous higherorder matching against terms $\vec t$.


752 
\item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the


753 
preceding statement. Also note that $\ISNAME$ is not a separate command,


754 
but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.).


755 
\end{descr}


756 

7319

757 
A few \emph{automatic} term abbreviations\index{automatic abbreviation} for

7335

758 
goals and facts are available as well. For any open goal,


759 
$\VVar{thesis_prop}$ refers to the full proposition (which may be a rule),


760 
$\VVar{thesis_concl}$ to its (atomic) conclusion, and $\VVar{thesis}$ to its


761 
objectlogical statement. The latter two abstract over any metalevel


762 
parameters.

7315

763 


764 
Facts (i.e.\ assumptions and finished goals) that have an application $f(x)$


765 
as objectlogic statement get $x$ bound to the special text variable


766 
``$\dots$'' (three dots). The canonical application of this feature are

7335

767 
calculational proofs (see \S\ref{sec:calculation}).

7315

768 


769 

7134

770 
\subsection{Block structure}


771 

7397

772 
\indexisarcmd{next}\indexisarcmd{\{\{}\indexisarcmd{\}\}}


773 
\begin{matharray}{rcl}


774 
\isarcmd{next} & : & \isartrans{proof(state)}{proof(state)} \\


775 
\isarcmd{\{\{} & : & \isartrans{proof(state)}{proof(state)} \\


776 
\isarcmd{\}\}} & : & \isartrans{proof(state)}{proof(state)} \\


777 
\end{matharray}


778 

7167

779 
While Isar is inherently blockstructured, opening and closing blocks is


780 
mostly handled rather casually, with little explicit userintervention. Any


781 
local goal statement automatically opens \emph{two} blocks, which are closed


782 
again when concluding the subproof (by $\QEDNAME$ etc.). Sections of


783 
different context within a subproof are typically switched via


784 
$\isarkeyword{next}$, which is just a single blockclose followed by


785 
blockopen again. Thus the effect of $\isarkeyword{next}$ is to reset the


786 
proof context to that of the head of the subproof. Note that there is no

7175

787 
goal focus involved here!

7167

788 

7175

789 
For slightly more advanced applications, there are explicit block parentheses


790 
as well. These typically achieve a strong forward style of reasoning.

7167

791 


792 
\begin{descr}


793 
\item [$\isarkeyword{next}$] switches to a fresh block within a subproof,


794 
resetting the context to the initial one.


795 
\item [$\isarkeyword{\{\{}$ and $\isarkeyword{\}\}}$] explicitly open and


796 
close blocks. Any current facts pass through $\isarkeyword{\{\{}$


797 
unchanged, while $\isarkeyword{\}\}}$ causes them to be \emph{exported} into

7335

798 
the enclosing context. Thus fixed variables are generalized, assumptions


799 
discharged, and local definitions eliminated. There is no difference of


800 
$\ASSUMENAME$ and $\PRESUMENAME$ here.

7167

801 
\end{descr}

7134

802 


803 


804 
\section{Other commands}


805 


806 
\subsection{Diagnostics}


807 


808 
\indexisarcmd{typ}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{thm}


809 
\begin{matharray}{rcl}


810 
\isarcmd{typ} & : & \isarkeep{theory~~proof} \\


811 
\isarcmd{term} & : & \isarkeep{theory~~proof} \\


812 
\isarcmd{prop} & : & \isarkeep{theory~~proof} \\


813 
\isarcmd{thm} & : & \isarkeep{theory~~proof} \\


814 
\end{matharray}


815 

7335

816 
These commands are not part of the actual Isabelle/Isar syntax, but assist


817 
interactive development. Also note that $undo$ does not apply here, since the


818 
theory or proof configuration is not changed.


819 

7134

820 
\begin{rail}


821 
'typ' type


822 
;


823 
'term' term


824 
;


825 
'prop' prop


826 
;


827 
'thm' thmrefs


828 
;


829 
\end{rail}


830 

7167

831 
\begin{descr}

7134

832 
\item [$\isarkeyword{typ}~\tau$, $\isarkeyword{term}~t$,


833 
$\isarkeyword{prop}~\phi$] read and print types / terms / propositions


834 
according to the current theory or proof context.


835 
\item [$\isarkeyword{thm}~thms$] retrieves lists of theorems from the current


836 
theory or proof context. Note that any attributes included in the theorem

7175

837 
specifications are applied to a temporary context derived from the current

7335

838 
theory or proof; the result is discarded, i.e.\ attributes involved in


839 
$thms$ only have a temporary effect.

7167

840 
\end{descr}

7134

841 


842 


843 
\subsection{System operations}


844 

7167

845 
\indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{usethy}\indexisarcmd{usethyonly}


846 
\indexisarcmd{updatethy}\indexisarcmd{updatethyonly}

7134

847 
\begin{matharray}{rcl}


848 
\isarcmd{cd} & : & \isarkeep{\cdot} \\


849 
\isarcmd{pwd} & : & \isarkeep{\cdot} \\


850 
\isarcmd{use_thy} & : & \isarkeep{\cdot} \\


851 
\isarcmd{use_thy_only} & : & \isarkeep{\cdot} \\


852 
\isarcmd{update_thy} & : & \isarkeep{\cdot} \\


853 
\isarcmd{update_thy_only} & : & \isarkeep{\cdot} \\


854 
\end{matharray}


855 

7167

856 
\begin{descr}

7134

857 
\item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle


858 
process.


859 
\item [$\isarkeyword{pwd}~$] prints the current working directory.

7175

860 
\item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$,


861 
$\isarkeyword{update_thy}$, and $\isarkeyword{update_thy_only}$] load some


862 
theory given as $name$ argument. These commands are exactly the same as the

7335

863 
corresponding ML functions (see also \cite[\S1,\S6]{isabelleref}). Note

7397

864 
that both the ML and Isar versions may load new and oldstyle theories


865 
alike.

7167

866 
\end{descr}

7134

867 

7335

868 
Note that these system commands are scarcely used when working with


869 
Proof~General, since loading of theories is done fully automatic.


870 

7134

871 

7046

872 
%%% Local Variables:


873 
%%% mode: latex


874 
%%% TeXmaster: "isarref"


875 
%%% End:
