author  wenzelm 
Thu, 29 Jun 2000 22:36:45 +0200  
changeset 9199  7a1a856f0571 
parent 9030  bb7622789bf2 
child 9233  8c8399b9ecaa 
permissions  rwrr 
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\chapter{Basic Isar Language Elements}\label{ch:puresyntax} 
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Subsequently, we introduce the main part of Pure Isar theory and proof 
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commands, together with fundamental proof methods and attributes. 
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Chapter~\ref{ch:gentools} describes further Isar elements provided by generic 
7 
tools and packages (such as the Simplifier) that are either part of Pure 

8 
Isabelle or preinstalled by most object logics. Chapter~\ref{ch:holtools} 

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refers to actual objectlogic specific elements of 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}. Some proof methods and attributes introduced later 
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are classified as improper as well. Improper Isar language elements, which 

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are subsequently marked by $^*$, are often helpful when developing proof 

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documents, while their use is discouraged for the final outcome. Typical 
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examples are diagnostic commands that print terms or theorems according to the 

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current context; other commands even emulate oldstyle tactical theorem 

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proving. 
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23 
\section{Theory commands} 

24 

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\subsection{Defining theories}\label{sec:beginthy} 
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\indexisarcmd{header}\indexisarcmd{theory}\indexisarcmd{end}\indexisarcmd{context} 
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\begin{matharray}{rcl} 
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\isarcmd{header} & : & \isarkeep{toplevel} \\ 
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\isarcmd{theory} & : & \isartrans{toplevel}{theory} \\ 
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\isarcmd{context}^* & : & \isartrans{toplevel}{theory} \\ 

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\isarcmd{end} & : & \isartrans{theory}{toplevel} \\ 

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\end{matharray} 
34 

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

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interactively. Both theorylevel specifications and proofs are handled 
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uniformly  occasionally definitional mechanisms even require some explicit 
38 
proof as well. In contrast, ``oldstyle'' Isabelle theories support batch 

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

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The first actual command of any theory has to be $\THEORY$, starting a new 
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theory based on the merge of existing ones. Just preceding $\THEORY$, there 

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may be an optional $\isarkeyword{header}$ declaration, which is relevant to 

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document preparation only; it acts very much like a special pretheory markup 

45 
command (cf.\ \S\ref{sec:markupthy} and \S\ref{sec:markupthy}). The theory 

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context may be also changed by $\CONTEXT$ without creating a new theory. In 

47 
both cases, $\END$ concludes the theory development; it has to be the very 

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last command of any theory file. 
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50 
\begin{rail} 

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'header' text 
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; 

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

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'context' name 

56 
; 

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'end' 

58 
;; 

59 

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filespecs: 'files' ((name  parname) +); 
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\end{rail} 
62 

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\begin{descr} 
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\item [$\isarkeyword{header}~text$] provides plain text markup just preceding 
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the formal beginning of a theory. In actual document preparation the 
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corresponding {\LaTeX} macro \verb,\isamarkupheader, may be redefined to 
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produce chapter or section headings. See also \S\ref{sec:markupthy} and 

68 
\S\ref{sec:markupprf} for further markup commands. 

69 

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\item [$\THEORY~A = B@1 + \cdots + B@n\colon$] commences a new theory $A$ 
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based on existing ones $B@1 + \cdots + B@n$. Isabelle's theory loader 

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system ensures that any of the base theories are properly loaded (and fully 

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uptodate when $\THEORY$ is executed interactively). The optional 

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$\isarkeyword{files}$ specification declares additional dependencies on ML 

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files. Unless put in parentheses, any file will be loaded immediately via 

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$\isarcmd{use}$ (see also \S\ref{sec:ML}). The optional ML file 

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\texttt{$A$.ML} that may be associated with any theory should \emph{not} be 

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included in $\isarkeyword{files}$, though. 

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\item [$\CONTEXT~B$] enters an existing theory context, basically in readonly 
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mode, so only a limited set of commands may be performed without destroying 
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the theory. Just as for $\THEORY$, the theory loader ensures that $B$ is 

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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, but the whole theory definition has 
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to be retracted. 

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\end{descr} 
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90 

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\subsection{Theory markup commands}\label{sec:markupthy} 
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\indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection} 
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\indexisarcmd{subsubsection}\indexisarcmd{text}\indexisarcmd{textraw} 

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\begin{matharray}{rcl} 
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\isarcmd{chapter} & : & \isartrans{theory}{theory} \\ 

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

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

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\isarcmd{text_raw} & : & \isartrans{theory}{theory} \\ 
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\end{matharray} 
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Apart from formal comments (see \S\ref{sec:comments}), markup commands provide 
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a structured way to insert text into the document generated from a theory (see 
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\cite{isabellesys} for more information on Isabelle's document preparation 
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tools). 

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\railalias{textraw}{text\_raw} 
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\railterm{textraw} 

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\begin{rail} 

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('chapter'  'section'  'subsection'  'subsubsection'  'text'  textraw) text 
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; 
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\end{rail} 

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\begin{descr} 
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\item [$\isarkeyword{chapter}$, $\isarkeyword{section}$, 
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$\isarkeyword{subsection}$, and $\isarkeyword{subsubsection}$] mark chapter 

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and section headings. 

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\item [$\TEXT$] specifies paragraphs of plain text, including references to 
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formal entities.\footnote{The latter feature is not yet supported. 

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Nevertheless, any source text of the form 

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``\texttt{\at\ttlbrace$\dots$\ttrbrace}'' should be considered as reserved 

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for future use.} 

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\item [$\isarkeyword{text_raw}$] inserts {\LaTeX} source into the output, 

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without additional markup. Thus the full range of document manipulations 

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becomes available. A typical application would be to emit 

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\verb,\begin{comment}, and \verb,\end{comment}, commands to exclude certain 

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parts from the final document.\footnote{This requires the \texttt{comment} 

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package to be included in {\LaTeX}, of course.} 
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\end{descr} 
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Any of these markup elements corresponds to a {\LaTeX} command with the name 
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prefixed by \verb,\isamarkup,. For the sectioning commands this is a plain 

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macro with a single argument, e.g.\ \verb,\isamarkupchapter{,\dots\verb,}, for 

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$\isarkeyword{chapter}$. The $\isarkeyword{text}$ markup results in a 

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{\LaTeX} environment \verb,\begin{isamarkuptext}, {\dots} 

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\verb,\end{isamarkuptext},, while $\isarkeyword{text_raw}$ causes the text 

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to be inserted directly into the {\LaTeX} source. 

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\medskip 
143 

144 
Additional markup commands are available for proofs (see 

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\S\ref{sec:markupprf}). Also note that the $\isarkeyword{header}$ 
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declaration (see \S\ref{sec:beginthy}) admits to insert section markup just 
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preceding the actual theory definition. 

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\subsection{Type classes and sorts}\label{sec:classes} 
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\indexisarcmd{classes}\indexisarcmd{classrel}\indexisarcmd{defaultsort} 

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\begin{matharray}{rcl} 

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

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

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

157 
\end{matharray} 

158 

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\begin{rail} 

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'classes' (classdecl comment? +) 
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; 
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'classrel' nameref '<' nameref comment? 

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; 

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'defaultsort' sort comment? 

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; 

166 
\end{rail} 

167 

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\begin{descr} 
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\item [$\isarkeyword{classes}~c<\vec c$] declares class $c$ to be a subclass 
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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 
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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 to 
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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 given without sort constraints. Usually, the default 
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sort would be only changed when defining new objectlogics. 
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\end{descr} 
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180 

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\subsection{Primitive types and type abbreviations}\label{sec:typespure} 
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183 
\indexisarcmd{typedecl}\indexisarcmd{types}\indexisarcmd{nonterminals}\indexisarcmd{arities} 

184 
\begin{matharray}{rcl} 

185 
\isarcmd{types} & : & \isartrans{theory}{theory} \\ 

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

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

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

189 
\end{matharray} 

190 

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\begin{rail} 

192 
'types' (typespec '=' type infix? comment? +) 

193 
; 

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'typedecl' typespec infix? comment? 

195 
; 

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'nonterminals' (name +) comment? 

197 
; 

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

199 
; 

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\end{rail} 

201 

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

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syntactic abbreviations without any logical significance. Internally, type 
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synonyms are fully expanded. 
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\item [$\isarkeyword{typedecl}~(\vec\alpha)t$] declares a new type constructor 
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$t$, intended as an actual logical type. Note that objectlogics such as 
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Isabelle/HOL override $\isarkeyword{typedecl}$ by their own version. 

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\item [$\isarkeyword{nonterminals}~\vec c$] declares $0$ary type constructors 
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$\vec c$ to act as purely syntactic types, i.e.\ nonterminal symbols of 

213 
Isabelle's inner syntax of terms or types. 

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\item [$\isarkeyword{arities}~t::(\vec s)s$] augments Isabelle's ordersorted 
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signature of types by new type constructor arities. This is done 

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

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\S\ref{sec:axclass}) provides a way to introduce proven type arities. 
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\end{descr} 
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220 

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\subsection{Constants and simple definitions}\label{sec:consts} 
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\indexisarcmd{consts}\indexisarcmd{defs}\indexisarcmd{constdefs}\indexoutertoken{constdecl} 
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\begin{matharray}{rcl} 
225 
\isarcmd{consts} & : & \isartrans{theory}{theory} \\ 

226 
\isarcmd{defs} & : & \isartrans{theory}{theory} \\ 

227 
\isarcmd{constdefs} & : & \isartrans{theory}{theory} \\ 

228 
\end{matharray} 

229 

230 
\begin{rail} 

231 
'consts' (constdecl +) 

232 
; 

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'defs' (axmdecl prop comment? +) 
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; 
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'constdefs' (constdecl prop comment? +) 

236 
; 

237 

238 
constdecl: name '::' type mixfix? comment? 

239 
; 

240 
\end{rail} 

241 

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\begin{descr} 
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\item [$\CONSTS~c::\sigma$] declares constant $c$ to have any instance of type 
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scheme $\sigma$. The optional mixfix annotations may attach concrete syntax 

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to the constants declared. 
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\item [$\DEFS~name: eqn$] introduces $eqn$ as a definitional axiom for some 
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existing constant. See \cite[\S6]{isabelleref} for more details on the 

248 
form of equations admitted as constant definitions. 

249 
\item [$\isarkeyword{constdefs}~c::\sigma~eqn$] combines declarations and 

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

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\end{descr} 
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254 

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\subsection{Syntax and translations}\label{sec:syntrans} 
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257 
\indexisarcmd{syntax}\indexisarcmd{translations} 

258 
\begin{matharray}{rcl} 

259 
\isarcmd{syntax} & : & \isartrans{theory}{theory} \\ 

260 
\isarcmd{translations} & : & \isartrans{theory}{theory} \\ 

261 
\end{matharray} 

262 

263 
\begin{rail} 

264 
'syntax' ('(' name 'output'? ')')? (constdecl +) 

265 
; 

266 
'translations' (transpat ('=='  '=>'  '<=') transpat comment? +) 

267 
; 

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

269 
; 

270 
\end{rail} 

271 

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\begin{descr} 
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\item [$\isarkeyword{syntax}~(mode)~decls$] is similar to $\CONSTS~decls$, 
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except that the actual logical signature extension is omitted. Thus the 

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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 
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the print mode that the grammar rules belong; unless the \texttt{output} 
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flag is given, all productions are added both to the input and output 

279 
grammar. 

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\item [$\isarkeyword{translations}~rules$] specifies syntactic translation 
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rules (i.e.\ \emph{macros}): parse~/ print rules (\texttt{==}), parse rules 
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(\texttt{=>}), or print rules (\texttt{<=}). Translation patterns may be 
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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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287 

288 
\subsection{Axioms and theorems} 

289 

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

291 
\begin{matharray}{rcl} 

292 
\isarcmd{axioms} & : & \isartrans{theory}{theory} \\ 

293 
\isarcmd{theorems} & : & \isartrans{theory}{theory} \\ 

294 
\isarcmd{lemmas} & : & \isartrans{theory}{theory} \\ 

295 
\end{matharray} 

296 

297 
\begin{rail} 

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'axioms' (axmdecl prop comment? +) 
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; 
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('theorems'  'lemmas') (thmdef? thmrefs comment? + 'and') 
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; 
302 
\end{rail} 

303 

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\begin{descr} 
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\item [$\isarkeyword{axioms}~a: \phi$] introduces arbitrary statements as 
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axioms of the metalogic. In fact, axioms are ``axiomatic theorems'', and 
307 
may be referred later just as any other theorem. 

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309 
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 proven theorems. 
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\item [$\isarkeyword{theorems}~a = \vec b$] stores lists of existing theorems. 
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Typical applications would also involve attributes, to declare Simplifier 
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rules, for example. 

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\item [$\isarkeyword{lemmas}$] is similar to $\isarkeyword{theorems}$, but 
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tags the results as ``lemma''. 

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\end{descr} 
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319 

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\subsection{Name spaces} 
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\indexisarcmd{global}\indexisarcmd{local}\indexisarcmd{hide} 
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\begin{matharray}{rcl} 
324 
\isarcmd{global} & : & \isartrans{theory}{theory} \\ 

325 
\isarcmd{local} & : & \isartrans{theory}{theory} \\ 

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\isarcmd{hide} & : & \isartrans{theory}{theory} \\ 
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\end{matharray} 
328 

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\begin{rail} 
330 
'global' comment? 

331 
; 

332 
'local' comment? 

333 
; 

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'hide' name (nameref + ) comment? 

335 
; 

336 
\end{rail} 

337 

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Isabelle organizes any kind of name declarations (of types, constants, 
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theorems etc.) by separate hierarchically structured name spaces. Normally 
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the user does not have to control the behavior of name spaces by hand, yet the 
341 
following commands provide some way to do so. 

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\begin{descr} 
344 
\item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current 

345 
name declaration mode. Initially, theories start in $\isarkeyword{local}$ 

346 
mode, causing all names to be automatically qualified by the theory name. 

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Changing this to $\isarkeyword{global}$ causes all names to be declared 
348 
without the theory prefix, until $\isarkeyword{local}$ is declared again. 

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350 
Note that global names are prone to get hidden accidently later, when 

351 
qualified names of the same base name are introduced. 

352 

353 
\item [$\isarkeyword{hide}~space~names$] removes declarations from a given 

354 
name space (which may be $class$, $type$, or $const$). Hidden objects 

355 
remain valid within the logic, but are inaccessible from user input. In 

356 
output, the special qualifier ``$\mathord?\mathord?$'' is prefixed to the 

357 
full internal name. 

358 

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Unqualified (global) names may not be hidden deliberately. 

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\end{descr} 
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362 

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\subsection{Incorporating ML code}\label{sec:ML} 
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\indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{MLcommand} 
366 
\indexisarcmd{MLsetup}\indexisarcmd{setup} 

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\indexisarcmd{methodsetup} 
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\begin{matharray}{rcl} 
369 
\isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\ 

370 
\isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\ 

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\isarcmd{ML_command} & : & \isartrans{\cdot}{\cdot} \\ 
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\isarcmd{ML_setup} & : & \isartrans{theory}{theory} \\ 
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\isarcmd{setup} & : & \isartrans{theory}{theory} \\ 
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\isarcmd{method_setup} & : & \isartrans{theory}{theory} \\ 
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\end{matharray} 
376 

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\railalias{MLsetup}{ML\_setup} 
378 
\railterm{MLsetup} 

379 

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\railalias{methodsetup}{method\_setup} 
381 
\railterm{methodsetup} 

382 

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\railalias{MLcommand}{ML\_command} 
384 
\railterm{MLcommand} 

385 

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\begin{rail} 
387 
'use' name 

388 
; 

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('ML'  MLcommand  MLsetup  'setup') text 
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; 
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methodsetup name '=' text text comment? 
392 
; 

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\end{rail} 
394 

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\begin{descr} 
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\item [$\isarkeyword{use}~file$] reads and executes ML commands from $file$. 
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The current theory context (if present) is passed down to the ML session, 
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but may not be modified. Furthermore, the file name is checked with the 
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$\isarkeyword{files}$ dependency declaration given in the theory header (see 
400 
also \S\ref{sec:beginthy}). 

401 

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\item [$\isarkeyword{ML}~text$ and $\isarkeyword{ML_command}~text$] execute ML 
403 
commands from $text$. The theory context is passed in the same way as for 

404 
$\isarkeyword{use}$, but may not be changed. Note that 

405 
$\isarkeyword{ML_command}$ is less verbose than plain $\isarkeyword{ML}$. 

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407 
\item [$\isarkeyword{ML_setup}~text$] executes ML commands from $text$. The 

408 
theory context is passed down to the ML session, and fetched back 

409 
afterwards. Thus $text$ may actually change the theory as a side effect. 

410 

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\item [$\isarkeyword{setup}~text$] changes the current theory context by 
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applying $text$, which refers to an ML expression of type 
413 
\texttt{(theory~>~theory)~list}. The $\isarkeyword{setup}$ command is the 

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canonical way to initialize any objectlogic specific tools and packages 
415 
written in ML. 

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417 
\item [$\isarkeyword{method_setup}~name = text~description$] defines a proof 

418 
method in the current theory. The given $text$ has to be an ML expression 

419 
of type \texttt{Args.src > Proof.context > Proof.method}. Parsing 

420 
concrete method syntax from \texttt{Args.src} input can be quite tedious in 

421 
general. The following simple examples are for methods without any explicit 

422 
arguments, or a list of theorems, respectively. 

423 

424 
{\footnotesize 

425 
\begin{verbatim} 

426 
Method.no_args (Method.METHOD (fn facts => foobar_tac)) 

427 
Method.thms_args (fn thms => (Method.METHOD (fn facts => foobar_tac))) 

428 
\end{verbatim} 

429 
} 

430 

431 
Note that mere tactic emulations may ignore the \texttt{facts} parameter 

432 
above. Proper proof methods would do something ``appropriate'' with the list 

433 
of current facts, though. Singlerule methods usually do strict 

434 
forwardchaining (e.g.\ by using \texttt{Method.multi_resolves}), while 

435 
automatic ones just insert the facts using \texttt{Method.insert_tac} before 

436 
applying the main tactic. 

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\end{descr} 
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439 

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\subsection{Syntax translation functions} 
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\indexisarcmd{parseasttranslation}\indexisarcmd{parsetranslation} 
443 
\indexisarcmd{printtranslation}\indexisarcmd{typedprinttranslation} 

444 
\indexisarcmd{printasttranslation}\indexisarcmd{tokentranslation} 

445 
\begin{matharray}{rcl} 

446 
\isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\ 

447 
\isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\ 

448 
\isarcmd{print_translation} & : & \isartrans{theory}{theory} \\ 

449 
\isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\ 

450 
\isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\ 

451 
\isarcmd{token_translation} & : & \isartrans{theory}{theory} \\ 

452 
\end{matharray} 

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Syntax translation functions written in ML admit almost arbitrary 
455 
manipulations of Isabelle's inner syntax. Any of the above commands have a 

456 
single \railqtoken{text} argument that refers to an ML expression of 

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appropriate type. 
458 

459 
\begin{ttbox} 

460 
val parse_ast_translation : (string * (ast list > ast)) list 

461 
val parse_translation : (string * (term list > term)) list 

462 
val print_translation : (string * (term list > term)) list 

463 
val typed_print_translation : 

464 
(string * (bool > typ > term list > term)) list 

465 
val print_ast_translation : (string * (ast list > ast)) list 

466 
val token_translation : 

467 
(string * string * (string > string * real)) list 

468 
\end{ttbox} 

469 
See \cite[\S8]{isabelleref} for more information on syntax transformations. 

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471 

472 
\subsection{Oracles} 

473 

474 
\indexisarcmd{oracle} 

475 
\begin{matharray}{rcl} 

476 
\isarcmd{oracle} & : & \isartrans{theory}{theory} \\ 

477 
\end{matharray} 

478 

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Oracles provide an interface to external reasoning systems, without giving up 
480 
control completely  each theorem carries a derivation object recording any 

481 
oracle invocation. See \cite[\S6]{isabelleref} for more information. 

482 

7134  483 
\begin{rail} 
484 
'oracle' name '=' text comment? 

485 
; 

486 
\end{rail} 

487 

7167  488 
\begin{descr} 
7175  489 
\item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML 
8379  490 
function $text$, which has to be of type 
491 
\texttt{Sign.sg~*~Object.T~>~term}. 

7167  492 
\end{descr} 
7134  493 

494 

495 
\section{Proof commands} 

496 

7987  497 
Proof commands perform transitions of Isar/VM machine configurations, which 
7315  498 
are blockstructured, consisting of a stack of nodes with three main 
7335  499 
components: logical proof context, current facts, and open goals. Isar/VM 
8547  500 
transitions are \emph{typed} according to the following three different modes 
501 
of operation: 

7167  502 
\begin{descr} 
503 
\item [$proof(prove)$] means that a new goal has just been stated that is now 

8547  504 
to be \emph{proven}; the next command may refine it by some proof method, 
505 
and enter a subproof to establish the actual result. 

7167  506 
\item [$proof(state)$] is like an internal theory mode: the context may be 
7987  507 
augmented by \emph{stating} additional assumptions, intermediate results 
508 
etc. 

7895  509 
\item [$proof(chain)$] is intermediate between $proof(state)$ and 
7987  510 
$proof(prove)$: existing facts (i.e.\ the contents of the special ``$this$'' 
511 
register) have been just picked up in order to be used when refining the 

512 
goal claimed next. 

7167  513 
\end{descr} 
7134  514 

7167  515 

7895  516 
\subsection{Proof markup commands}\label{sec:markupprf} 
7167  517 

7987  518 
\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsubsect} 
7895  519 
\indexisarcmd{txt}\indexisarcmd{txtraw} 
7134  520 
\begin{matharray}{rcl} 
8101  521 
\isarcmd{sect} & : & \isartrans{proof}{proof} \\ 
522 
\isarcmd{subsect} & : & \isartrans{proof}{proof} \\ 

523 
\isarcmd{subsubsect} & : & \isartrans{proof}{proof} \\ 

524 
\isarcmd{txt} & : & \isartrans{proof}{proof} \\ 

525 
\isarcmd{txt_raw} & : & \isartrans{proof}{proof} \\ 

7134  526 
\end{matharray} 
527 

7895  528 
These markup commands for proof mode closely correspond to the ones of theory 
8684  529 
mode (see \S\ref{sec:markupthy}). 
7895  530 

531 
\railalias{txtraw}{txt\_raw} 

532 
\railterm{txtraw} 

7175  533 

7134  534 
\begin{rail} 
7895  535 
('sect'  'subsect'  'subsubsect'  'txt'  txtraw) text 
7134  536 
; 
537 
\end{rail} 

538 

539 

7315  540 
\subsection{Proof context}\label{sec:proofcontext} 
7134  541 

7315  542 
\indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def} 
7134  543 
\begin{matharray}{rcl} 
544 
\isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\ 

545 
\isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\ 

546 
\isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\ 

547 
\isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\ 

548 
\end{matharray} 

549 

7315  550 
The logical proof context consists of fixed variables and assumptions. The 
551 
former closely correspond to Skolem constants, or metalevel universal 

552 
quantification as provided by the Isabelle/Pure logical framework. 

553 
Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in 

7987  554 
a local value that may be used in the subsequent proof as any other variable 
7895  555 
or constant. Furthermore, any result $\edrv \phi[x]$ exported from the 
7987  556 
context will be universally closed wrt.\ $x$ at the outermost level: $\edrv 
557 
\All x \phi$ (this is expressed using Isabelle's metavariables). 

7315  558 

559 
Similarly, introducing some assumption $\chi$ has two effects. On the one 

560 
hand, a local theorem is created that may be used as a fact in subsequent 

7895  561 
proof steps. On the other hand, any result $\chi \drv \phi$ exported from the 
562 
context becomes conditional wrt.\ the assumption: $\edrv \chi \Imp \phi$. 

563 
Thus, solving an enclosing goal using such a result would basically introduce 

564 
a new subgoal stemming from the assumption. How this situation is handled 

565 
depends on the actual version of assumption command used: while $\ASSUMENAME$ 

566 
insists on solving the subgoal by unification with some premise of the goal, 

567 
$\PRESUMENAME$ leaves the subgoal unchanged in order to be proved later by the 

568 
user. 

7315  569 

7319  570 
Local definitions, introduced by $\DEF{}{x \equiv t}$, are achieved by 
7987  571 
combining $\FIX x$ with another version of assumption that causes any 
572 
hypothetical equation $x \equiv t$ to be eliminated by the reflexivity rule. 

573 
Thus, exporting some result $x \equiv t \drv \phi[x]$ yields $\edrv \phi[t]$. 

7175  574 

7134  575 
\begin{rail} 
7431  576 
'fix' (vars + 'and') comment? 
7134  577 
; 
7315  578 
('assume'  'presume') (assm comment? + 'and') 
7134  579 
; 
7175  580 
'def' thmdecl? \\ var '==' term termpat? comment? 
7134  581 
; 
582 

583 
var: name ('::' type)? 

584 
; 

7458  585 
vars: (name+) ('::' type)? 
7431  586 
; 
7315  587 
assm: thmdecl? (prop proppat? +) 
588 
; 

7134  589 
\end{rail} 
590 

7167  591 
\begin{descr} 
8547  592 
\item [$\FIX{\vec x}$] introduces local \emph{arbitrary, but fixed} variables 
593 
$\vec x$. 

8515  594 
\item [$\ASSUME{a}{\vec\phi}$ and $\PRESUME{a}{\vec\phi}$] introduce local 
595 
theorems $\vec\phi$ by assumption. Subsequent results applied to an 

596 
enclosing goal (e.g.\ by $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ 

597 
expects to be able to unify with existing premises in the goal, while 

598 
$\PRESUMENAME$ leaves $\vec\phi$ as new subgoals. 

7335  599 

600 
Several lists of assumptions may be given (separated by 

7895  601 
$\isarkeyword{and}$); the resulting list of current facts consists of all of 
602 
these concatenated. 

7315  603 
\item [$\DEF{a}{x \equiv t}$] introduces a local (nonpolymorphic) definition. 
604 
In results exported from the context, $x$ is replaced by $t$. Basically, 

7987  605 
$\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\ASSUME{}{x \equiv t}$, with the 
7335  606 
resulting hypothetical equation solved by reflexivity. 
7431  607 

608 
The default name for the definitional equation is $x_def$. 

7167  609 
\end{descr} 
610 

7895  611 
The special name $prems$\indexisarthm{prems} refers to all assumptions of the 
612 
current context as a list of theorems. 

7315  613 

7167  614 

615 
\subsection{Facts and forward chaining} 

616 

617 
\indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with} 

618 
\begin{matharray}{rcl} 

619 
\isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\ 

620 
\isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\ 

621 
\isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\ 

622 
\isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\ 

623 
\end{matharray} 

624 

7319  625 
New facts are established either by assumption or proof of local statements. 
7335  626 
Any fact will usually be involved in further proofs, either as explicit 
8547  627 
arguments of proof methods, or when forward chaining towards the next goal via 
7335  628 
$\THEN$ (and variants). Note that the special theorem name 
7987  629 
$this$\indexisarthm{this} refers to the most recently established facts. 
7167  630 
\begin{rail} 
9199  631 
'note' (thmdef? thmrefs comment? + 'and') 
7167  632 
; 
633 
'then' comment? 

634 
; 

9199  635 
('from'  'with') (thmrefs comment? + 'and') 
7167  636 
; 
637 
\end{rail} 

638 

639 
\begin{descr} 

7175  640 
\item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result 
641 
as $a$. Note that attributes may be involved as well, both on the left and 

642 
right hand sides. 

7167  643 
\item [$\THEN$] indicates forward chaining by the current facts in order to 
7895  644 
establish the goal to be claimed next. The initial proof method invoked to 
645 
refine that will be offered the facts to do ``anything appropriate'' (cf.\ 

646 
also \S\ref{sec:proofsteps}). For example, method $rule$ (see 

8515  647 
\S\ref{sec:puremethatt}) would typically do an elimination rather than an 
7895  648 
introduction. Automatic methods usually insert the facts into the goal 
8547  649 
state before operation. This provides a simple scheme to control relevance 
650 
of facts in automated proof search. 

7335  651 
\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; thus $\THEN$ is 
7458  652 
equivalent to $\FROM{this}$. 
7175  653 
\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward 
654 
chaining is from earlier facts together with the current ones. 

7167  655 
\end{descr} 
656 

8515  657 
Basic proof methods (such as $rule$, see \S\ref{sec:puremethatt}) expect 
7895  658 
multiple facts to be given in their proper order, corresponding to a prefix of 
659 
the premises of the rule involved. Note that positions may be easily skipped 

8547  660 
using something like $\FROM{\text{\texttt{_}}~a~b}$, for example. This 
661 
involves the trivial rule $\PROP\psi \Imp \PROP\psi$, which happens to be 

662 
bound in Isabelle/Pure as ``\texttt{_}'' 

663 
(underscore).\indexisarthm{_@\texttt{_}} 

7389  664 

7167  665 

666 
\subsection{Goal statements} 

667 

668 
\indexisarcmd{theorem}\indexisarcmd{lemma} 

669 
\indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus} 

670 
\begin{matharray}{rcl} 

671 
\isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\ 

672 
\isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\ 

7987  673 
\isarcmd{have} & : & \isartrans{proof(state) ~~ proof(chain)}{proof(prove)} \\ 
674 
\isarcmd{show} & : & \isartrans{proof(state) ~~ proof(chain)}{proof(prove)} \\ 

7167  675 
\isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\ 
676 
\isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\ 

677 
\end{matharray} 

678 

7175  679 
Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$ 
7895  680 
and $\LEMMANAME$. New local goals may be claimed within proof mode as well. 
681 
Four variants are available, indicating whether the result is meant to solve 

8547  682 
some pending goal or whether forward chaining is indicated. 
7175  683 

7167  684 
\begin{rail} 
685 
('theorem'  'lemma') goal 

686 
; 

687 
('have'  'show'  'hence'  'thus') goal 

688 
; 

689 

8632  690 
goal: thmdecl? prop proppat? comment? 
7167  691 
; 
692 
\end{rail} 

693 

694 
\begin{descr} 

7335  695 
\item [$\THEOREM{a}{\phi}$] enters proof mode with $\phi$ as main goal, 
8547  696 
eventually resulting in some theorem $\turn \phi$ to be put back into the 
697 
theory. 

7987  698 
\item [$\LEMMA{a}{\phi}$] is similar to $\THEOREMNAME$, but tags the result as 
7167  699 
``lemma''. 
7335  700 
\item [$\HAVE{a}{\phi}$] claims a local goal, eventually resulting in a 
7167  701 
theorem with the current assumption context as hypotheses. 
7335  702 
\item [$\SHOW{a}{\phi}$] is similar to $\HAVE{a}{\phi}$, but solves some 
7895  703 
pending goal with the result \emph{exported} into the corresponding context 
704 
(cf.\ \S\ref{sec:proofcontext}). 

705 
\item [$\HENCENAME$] abbreviates $\THEN~\HAVENAME$, i.e.\ claims a local goal 

706 
to be proven by forward chaining the current facts. Note that $\HENCENAME$ 

707 
is also equivalent to $\FROM{this}~\HAVENAME$. 

708 
\item [$\THUSNAME$] abbreviates $\THEN~\SHOWNAME$. Note that $\THUSNAME$ is 

709 
also equivalent to $\FROM{this}~\SHOWNAME$. 

7167  710 
\end{descr} 
711 

8991  712 
Note that any goal statement causes some term abbreviations (such as 
713 
$\Var{thesis}$, $\dots$) to be bound automatically, see also 

714 
\S\ref{sec:termabbrev}. Furthermore, the local context of a (nonatomic) 

715 
goal is provided via the case name $antecedent$\indexisarcase{antecedent}, see 

716 
also \S\ref{sec:cases}. 

717 

7167  718 

719 
\subsection{Initial and terminal proof steps}\label{sec:proofsteps} 

720 

7175  721 
\indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by} 
722 
\indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry} 

723 
\begin{matharray}{rcl} 

724 
\isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\ 

725 
\isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~~ theory} \\ 

726 
\isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\ 

727 
\isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\ 

728 
\isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\ 

729 
\isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~~ theory} \\ 

730 
\end{matharray} 

731 

8547  732 
Arbitrary goal refinement via tactics is considered harmful. Properly, the 
7335  733 
Isar framework admits proof methods to be invoked in two places only. 
7167  734 
\begin{enumerate} 
7175  735 
\item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated 
7335  736 
goal to a number of subgoals that are to be solved later. Facts are passed 
7895  737 
to $m@1$ for forward chaining, if so indicated by $proof(chain)$ mode. 
7167  738 

7987  739 
\item A \emph{terminal} conclusion step $\QED{m@2}$ is intended to solve 
740 
remaining goals. No facts are passed to $m@2$. 

7167  741 
\end{enumerate} 
742 

8547  743 
The only other proper way to affect pending goals is by $\SHOWNAME$, which 
744 
involves an explicit statement of what is to be solved. 

7167  745 

7175  746 
\medskip 
747 

7167  748 
Also note that initial proof methods should either solve the goal completely, 
7895  749 
or constitute some wellunderstood reduction to new subgoals. Arbitrary 
750 
automatic proof tools that are prone leave a large number of badly structured 

751 
subgoals are no help in continuing the proof document in any intelligible 

7987  752 
way. 
7167  753 

7175  754 
\medskip 
755 

8547  756 
Unless given explicitly by the user, the default initial method is ``$rule$'', 
757 
which applies a single standard elimination or introduction rule according to 

758 
the topmost symbol involved. There is no separate default terminal method. 

759 
Any remaining goals are always solved by assumption in the very last step. 

7167  760 

761 
\begin{rail} 

762 
'proof' interest? meth? comment? 

763 
; 

764 
'qed' meth? comment? 

765 
; 

766 
'by' meth meth? comment? 

767 
; 

768 
('.'  '..'  'sorry') comment? 

769 
; 

770 

771 
meth: method interest? 

772 
; 

773 
\end{rail} 

774 

775 
\begin{descr} 

7335  776 
\item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for 
777 
forward chaining are passed if so indicated by $proof(chain)$ mode. 

778 
\item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and 

7895  779 
concludes the subproof by assumption. If the goal had been $\SHOWNAME$ (or 
780 
$\THUSNAME$), some pending subgoal is solved as well by the rule resulting 

781 
from the result \emph{exported} into the enclosing goal context. Thus 

782 
$\QEDNAME$ may fail for two reasons: either $m@2$ fails, or the resulting 

783 
rule does not fit to any pending goal\footnote{This includes any additional 

784 
``strong'' assumptions as introduced by $\ASSUMENAME$.} of the enclosing 

785 
context. Debugging such a situation might involve temporarily changing 

786 
$\SHOWNAME$ into $\HAVENAME$, or weakening the local context by replacing 

787 
some occurrences of $\ASSUMENAME$ by $\PRESUMENAME$. 

788 
\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}\index{proof!terminal}; it 

7987  789 
abbreviates $\PROOF{m@1}~\QED{m@2}$, with backtracking across both methods, 
790 
though. Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done 

7895  791 
by expanding its definition; in many cases $\PROOF{m@1}$ is already 
7175  792 
sufficient to see what is going wrong. 
7895  793 
\item [``$\DDOT$''] is a \emph{default proof}\index{proof!default}; it 
8515  794 
abbreviates $\BY{rule}$. 
7895  795 
\item [``$\DOT$''] is a \emph{trivial proof}\index{proof!trivial}; it 
8195  796 
abbreviates $\BY{this}$. 
8379  797 
\item [$\SORRY$] is a \emph{fake proof}\index{proof!fake}; provided that the 
798 
\texttt{quick_and_dirty} flag is enabled, $\SORRY$ pretends to solve the 

8515  799 
goal without further ado. Of course, the result would be a fake theorem 
800 
only, involving some oracle in its internal derivation object (this is 

801 
indicated as ``$[!]$'' in the printed result). The main application of 

802 
$\SORRY$ is to support experimentation and topdown proof development. 

803 
\end{descr} 

804 

805 

806 
\subsection{Fundamental methods and attributes}\label{sec:puremethatt} 

807 

8547  808 
The following proof methods and attributes refer to basic logical operations 
809 
of Isar. Further methods and attributes are provided by several generic and 

810 
objectlogic specific tools and packages (see chapters \ref{ch:gentools} and 

811 
\ref{ch:holtools}). 

8515  812 

813 
\indexisarmeth{assumption}\indexisarmeth{this}\indexisarmeth{rule}\indexisarmeth{$$} 

814 
\indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest} 

815 
\indexisaratt{OF}\indexisaratt{of} 

816 
\begin{matharray}{rcl} 

817 
assumption & : & \isarmeth \\ 

818 
this & : & \isarmeth \\ 

819 
rule & : & \isarmeth \\ 

820 
 & : & \isarmeth \\ 

821 
OF & : & \isaratt \\ 

822 
of & : & \isaratt \\ 

823 
intro & : & \isaratt \\ 

824 
elim & : & \isaratt \\ 

825 
dest & : & \isaratt \\ 

826 
delrule & : & \isaratt \\ 

827 
\end{matharray} 

828 

829 
\begin{rail} 

8547  830 
'rule' thmrefs? 
8515  831 
; 
832 
'OF' thmrefs 

833 
; 

8693  834 
'of' insts ('concl' ':' insts)? 
8515  835 
; 
836 
\end{rail} 

837 

838 
\begin{descr} 

839 
\item [$assumption$] solves some goal by a single assumption step. Any facts 

840 
given (${} \le 1$) are guaranteed to participate in the refinement. Recall 

841 
that $\QEDNAME$ (see \S\ref{sec:proofsteps}) already concludes any 

842 
remaining subgoals by assumption. 

843 
\item [$this$] applies all of the current facts directly as rules. Recall 

844 
that ``$\DOT$'' (dot) abbreviates $\BY{this}$. 

8547  845 
\item [$rule~\vec a$] applies some rule given as argument in backward manner; 
8515  846 
facts are used to reduce the rule before applying it to the goal. Thus 
847 
$rule$ without facts is plain \emph{introduction}, while with facts it 

848 
becomes \emph{elimination}. 

849 

8547  850 
When no arguments are given, the $rule$ method tries to pick appropriate 
851 
rules automatically, as declared in the current context using the $intro$, 

852 
$elim$, $dest$ attributes (see below). This is the default behavior of 

853 
$\PROOFNAME$ and ``$\DDOT$'' (doubledot) steps (see 

8515  854 
\S\ref{sec:proofsteps}). 
855 
\item [``$$''] does nothing but insert the forward chaining facts as premises 

856 
into the goal. Note that command $\PROOFNAME$ without any method actually 

857 
performs a single reduction step using the $rule$ method; thus a plain 

858 
\emph{donothing} proof step would be $\PROOF{}$ rather than $\PROOFNAME$ 

859 
alone. 

8547  860 
\item [$OF~\vec a$] applies some theorem to given rules $\vec a$ (in 
861 
parallel). This corresponds to the \texttt{MRS} operator in ML 

862 
\cite[\S5]{isabelleref}, but note the reversed order. Positions may be 

863 
skipped by including ``$\_$'' (underscore) as argument. 

864 
\item [$of~\vec t$] performs positional instantiation. The terms $\vec t$ are 

8515  865 
substituted for any schematic variables occurring in a theorem from left to 
866 
right; ``\texttt{_}'' (underscore) indicates to skip a position. Arguments 

867 
following a ``$concl\colon$'' specification refer to positions of the 

868 
conclusion of a rule. 

869 
\item [$intro$, $elim$, and $dest$] declare introduction, elimination, and 

870 
destruct rules, respectively. Note that the classical reasoner (see 

871 
\S\ref{sec:classicalbasic}) introduces different versions of these 

872 
attributes, and the $rule$ method, too. In objectlogics with classical 

873 
reasoning enabled, the latter version should be used all the time to avoid 

874 
confusion! 

875 
\item [$delrule$] undeclares introduction or elimination rules. 

7315  876 
\end{descr} 
877 

878 

879 
\subsection{Term abbreviations}\label{sec:termabbrev} 

880 

881 
\indexisarcmd{let} 

882 
\begin{matharray}{rcl} 

883 
\isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\ 

884 
\isarkeyword{is} & : & syntax \\ 

885 
\end{matharray} 

886 

887 
Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements, 

7987  888 
or by annotating assumptions or goal statements with a list of patterns 
889 
$\ISS{p@1\;\dots}{p@n}$. In both cases, higherorder matching is invoked to 

890 
bind extralogical term variables, which may be either named schematic 

891 
variables of the form $\Var{x}$, or nameless dummies ``\texttt{_}'' 

892 
(underscore).\indexisarvar{_@\texttt{_}} Note that in the $\LETNAME$ form the 

893 
patterns occur on the lefthand side, while the $\ISNAME$ patterns are in 

894 
postfix position. 

7315  895 

8620
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

896 
Polymorphism of term bindings is handled in HindleyMilner style, as in ML. 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

897 
Type variables referring to local assumptions or open goal statements are 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

898 
\emph{fixed}, while those of finished results or bound by $\LETNAME$ may occur 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

899 
in \emph{arbitrary} instances later. Even though actual polymorphism should 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

900 
be rarely used in practice, this mechanism is essential to achieve proper 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

901 
incremental typeinference, as the user proceeds to build up the Isar proof 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

902 
text. 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

903 

3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

904 
\medskip 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

905 

7319  906 
Term abbreviations are quite different from actual local definitions as 
907 
introduced via $\DEFNAME$ (see \S\ref{sec:proofcontext}). The latter are 

7315  908 
visible within the logic as actual equations, while abbreviations disappear 
8620
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

909 
during the input process just after type checking. Also note that $\DEFNAME$ 
3786d47f5570
support HindleyMilner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset

910 
does not support polymorphism. 
7315  911 

912 
\begin{rail} 

8664  913 
'let' ((term + 'and') '=' term comment? + 'and') 
7315  914 
; 
915 
\end{rail} 

916 

917 
The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or 

918 
\railnonterm{proppat} (see \S\ref{sec:termpats}). 

919 

920 
\begin{descr} 

921 
\item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$ 

922 
by simultaneous higherorder matching against terms $\vec t$. 

923 
\item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the 

924 
preceding statement. Also note that $\ISNAME$ is not a separate command, 

925 
but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.). 

926 
\end{descr} 

927 

7988  928 
A few \emph{automatic} term abbreviations\index{term abbreviations} for goals 
929 
and facts are available as well. For any open goal, 

7466  930 
$\Var{thesis_prop}$\indexisarvar{thesisprop} refers to the full proposition 
931 
(which may be a rule), $\Var{thesis_concl}$\indexisarvar{thesisconcl} to its 

932 
(atomic) conclusion, and $\Var{thesis}$\indexisarvar{thesis} to its 

8547  933 
objectlevel statement. The latter two abstract over any metalevel 
7987  934 
parameters. 
7315  935 

7466  936 
Fact statements resulting from assumptions or finished goals are bound as 
937 
$\Var{this_prop}$\indexisarvar{thisprop}, 

938 
$\Var{this_concl}$\indexisarvar{thisconcl}, and 

939 
$\Var{this}$\indexisarvar{this}, similar to $\Var{thesis}$ above. In case 

940 
$\Var{this}$ refers to an objectlogic statement that is an application 

7895  941 
$f(t)$, then $t$ is bound to the special text variable 
7466  942 
``$\dots$''\indexisarvar{\dots} (three dots). The canonical application of 
7987  943 
the latter are calculational proofs (see \S\ref{sec:calculation}). 
7315  944 

945 

7134  946 
\subsection{Block structure} 
947 

8896  948 
\indexisarcmd{next}\indexisarcmd{\{}\indexisarcmd{\}} 
7397  949 
\begin{matharray}{rcl} 
8448  950 
\NEXT & : & \isartrans{proof(state)}{proof(state)} \\ 
7974  951 
\BG & : & \isartrans{proof(state)}{proof(state)} \\ 
952 
\EN & : & \isartrans{proof(state)}{proof(state)} \\ 

7397  953 
\end{matharray} 
954 

9030  955 
\railalias{lbrace}{\ttlbrace} 
956 
\railterm{lbrace} 

957 

958 
\railalias{rbrace}{\ttrbrace} 

959 
\railterm{rbrace} 

960 

961 
\begin{rail} 

962 
'next' comment? 

963 
; 

964 
lbrace comment? 

965 
; 

966 
rbrace comment? 

967 
; 

968 
\end{rail} 

969 

7167  970 
While Isar is inherently blockstructured, opening and closing blocks is 
971 
mostly handled rather casually, with little explicit userintervention. Any 

972 
local goal statement automatically opens \emph{two} blocks, which are closed 

973 
again when concluding the subproof (by $\QEDNAME$ etc.). Sections of 

8448  974 
different context within a subproof may be switched via $\NEXT$, which is 
975 
just a single blockclose followed by blockopen again. Thus the effect of 

976 
$\NEXT$ to reset the local proof context. There is no goal focus involved 

977 
here! 

7167  978 

7175  979 
For slightly more advanced applications, there are explicit block parentheses 
7895  980 
as well. These typically achieve a stronger forward style of reasoning. 
7167  981 

982 
\begin{descr} 

8448  983 
\item [$\NEXT$] switches to a fresh block within a subproof, resetting the 
984 
local context to the initial one. 

8896  985 
\item [$\BG$ and $\EN$] explicitly open and close blocks. Any current facts 
986 
pass through ``$\BG$'' unchanged, while ``$\EN$'' causes any result to be 

7895  987 
\emph{exported} into the enclosing context. Thus fixed variables are 
988 
generalized, assumptions discharged, and local definitions unfolded (cf.\ 

989 
\S\ref{sec:proofcontext}). There is no difference of $\ASSUMENAME$ and 

990 
$\PRESUMENAME$ in this mode of forward reasoning  in contrast to plain 

991 
backward reasoning with the result exported at $\SHOWNAME$ time. 

7167  992 
\end{descr} 
7134  993 

994 

8533  995 
\subsection{Emulating tactic scripts}\label{sec:tacticalproof} 
8515  996 

997 
The following elements emulate unstructured tactic scripts to some extent. 

998 
While these are anathema for writing proper Isar proof documents, they might 

999 
come in handy for interactive exploration and debugging, or even actual 

1000 
tactical proof within newstyle theories (to benefit from document 

1001 
preparation, for example). 

1002 

8946  1003 
\indexisarcmd{apply}\indexisarcmd{done}\indexisarcmd{applyend} 
8515  1004 
\indexisarcmd{defer}\indexisarcmd{prefer}\indexisarcmd{back} 
1005 
\begin{matharray}{rcl} 

8533  1006 
\isarcmd{apply}^* & : & \isartrans{proof(prove)}{proof(prove)} \\ 
8946  1007 
\isarcmd{done}^* & : & \isartrans{proof(prove)}{proof(state)} \\ 
8533  1008 
\isarcmd{apply_end}^* & : & \isartrans{proof(state)}{proof(state)} \\ 
1009 
\isarcmd{defer}^* & : & \isartrans{proof}{proof} \\ 

1010 
\isarcmd{prefer}^* & : & \isartrans{proof}{proof} \\ 

1011 
\isarcmd{back}^* & : & \isartrans{proof}{proof} \\ 

8515  1012 
\end{matharray} 
1013 

1014 
\railalias{applyend}{apply\_end} 

1015 
\railterm{applyend} 

1016 

1017 
\begin{rail} 

8682  1018 
'apply' method comment? 
8515  1019 
; 
8946  1020 
'done' comment? 
1021 
; 

8682  1022 
applyend method comment? 
8515  1023 
; 
8682  1024 
'defer' nat? comment? 
8515  1025 
; 
8682  1026 
'prefer' nat comment? 
8515  1027 
; 
1028 
\end{rail} 

1029 

1030 
\begin{descr} 

8547  1031 
\item [$\isarkeyword{apply}~(m)$] applies proof method $m$ in initial 
1032 
position, but unlike $\PROOFNAME$ it retains ``$proof(prove)$'' mode. Thus 

8946  1033 
consecutive method applications may be given just as in tactic scripts. 
8515  1034 

8881  1035 
Facts are passed to $m$ as indicated by the goal's forwardchain mode, and 
1036 
are \emph{consumed} afterwards. Thus any further $\isarkeyword{apply}$ 

1037 
command would always work in a purely backward manner. 

8946  1038 

1039 
\item [$\isarkeyword{done}$] completes a proof script, provided that the 

8947  1040 
current goal state is already solved completely. Note that actual 
1041 
structured proof commands (e.g.\ ``$\DOT$'' or $\SORRY$) may be used to 

1042 
conclude proof scripts as well. 

8881  1043 

8515  1044 
\item [$\isarkeyword{apply_end}~(m)$] applies proof method $m$ as if in 
1045 
terminal position. Basically, this simulates a multistep tactic script for 

1046 
$\QEDNAME$, but may be given anywhere within the proof body. 

1047 

1048 
No facts are passed to $m$. Furthermore, the static context is that of the 

1049 
enclosing goal (as for actual $\QEDNAME$). Thus the proof method may not 

1050 
refer to any assumptions introduced in the current body, for example. 

1051 
\item [$\isarkeyword{defer}~n$ and $\isarkeyword{prefer}~n$] shuffle the list 

1052 
of pending goals: $defer$ puts off goal $n$ to the end of the list ($n = 1$ 

1053 
by default), while $prefer$ brings goal $n$ to the top. 

1054 
\item [$\isarkeyword{back}$] does backtracking over the result sequence of 

1055 
the latest proof command.\footnote{Unlike the ML function \texttt{back} 

1056 
\cite{isabelleref}, the Isar command does not search upwards for further 

1057 
branch points.} Basically, any proof command may return multiple results. 

9006  1058 
\end{descr} 
1059 

1060 
Any proper Isar proof method may be used with tactic script commands such as 

1061 
$\isarkeyword{apply}$. A few additional emulations of actual tactics are 

1062 
provided as well; these would be never used in actual structured proofs, of 

1063 
course. 

1064 

1065 
\indexisarmeth{tactic}\indexisarmeth{insert} 

1066 
\indexisarmeth{resinsttac}\indexisarmeth{eresinsttac} 

1067 
\indexisarmeth{dresinsttac}\indexisarmeth{forwinsttac} 

1068 
\indexisarmeth{subgoaltac} 

1069 
\begin{matharray}{rcl} 

1070 
tactic^* & : & \isarmeth \\ 

1071 
insert^* & : & \isarmeth \\ 

1072 
res_inst_tac^* & : & \isarmeth \\ 

1073 
eres_inst_tac^* & : & \isarmeth \\ 

1074 
dres_inst_tac^* & : & \isarmeth \\ 

1075 
forw_inst_tac^* & : & \isarmeth \\ 

1076 
subgoal_tac^* & : & \isarmeth \\ 

1077 
\end{matharray} 

1078 

1079 
\railalias{resinsttac}{res\_inst\_tac} 

1080 
\railterm{resinsttac} 

1081 

1082 
\railalias{eresinsttac}{eres\_inst\_tac} 

1083 
\railterm{eresinsttac} 

1084 

1085 
\railalias{dresinsttac}{dres\_inst\_tac} 

1086 
\railterm{dresinsttac} 

1087 

1088 
\railalias{forwinsttac}{forw\_inst\_tac} 

1089 
\railterm{forwinsttac} 

1090 

1091 
\railalias{subgoaltac}{subgoal\_tac} 

1092 
\railterm{subgoaltac} 

1093 

1094 
\begin{rail} 

1095 
'tactic' text 

1096 
; 

1097 
'insert' thmrefs 

1098 
; 

1099 
( resinsttac  eresinsttac  dresinsttac  forwinsttac ) goalspec? \\ 

1100 
((name '=' term) + 'and') 'in' thmref 

1101 
; 

1102 
subgoaltac goalspec? prop 

1103 
; 

1104 
\end{rail} 

1105 

1106 
\begin{descr} 

8515  1107 
\item [$tactic~text$] produces a proof method from any ML text of type 
8547  1108 
\texttt{tactic}. Apart from the usual ML environment and the current 
8515  1109 
implicit theory context, the ML code may refer to the following locally 
1110 
bound values: 

9006  1111 

8515  1112 
%%FIXME ttbox produces too much trailing space (why?) 
1113 
{\footnotesize\begin{verbatim} 

1114 
val ctxt : Proof.context 

1115 
val facts : thm list 

1116 
val thm : string > thm 

1117 
val thms : string > thm list 

1118 
\end{verbatim}} 

1119 
Here \texttt{ctxt} refers to the current proof context, \texttt{facts} 

1120 
indicates any current facts for forwardchaining, and 

1121 
\texttt{thm}~/~\texttt{thms} retrieve named facts (including global 

1122 
theorems) from the context. 

8696  1123 
\item [$insert~\vec a$] inserts theorems as facts into all goals of the proof 
1124 
state; the current facts indicated for forward chaining are ignored! 

8533  1125 
\item [$res_inst_tac$ etc.] do resolution of rules with explicit 
1126 
instantiation. This works the same way as the corresponding ML tactics, see 

1127 
\cite[\S3]{isabelleref}. 

1128 

1129 
It is very important to note that the instantiations are read and 

1130 
typechecked according to the dynamic goal state, rather than the static 

1131 
proof context! In particular, locally fixed variables and term 

1132 
abbreviations may not be included in the term specifications. 

8547  1133 
\item [$subgoal_tac~\phi$] emulates the ML tactic of the same name, see 
8533  1134 
\cite[\S3]{isabelleref}. Syntactically, the given proposition is handled 
1135 
as the instantiations in $res_inst_tac$ etc. 

1136 

1137 
Note that the proper Isar command $\PRESUMENAME$ achieves a similar effect 

1138 
as $subgoal_tac$. 

8515  1139 
\end{descr} 
1140 

1141 

1142 
\subsection{Metalinguistic features} 

1143 

1144 
\indexisarcmd{oops} 

1145 
\begin{matharray}{rcl} 

1146 
\isarcmd{oops} & : & \isartrans{proof}{theory} \\ 

1147 
\end{matharray} 

1148 

1149 
The $\OOPS$ command discontinues the current proof attempt, while considering 

1150 
the partial proof text as properly processed. This is conceptually quite 

1151 
different from ``faking'' actual proofs via $\SORRY$ (see 

1152 
\S\ref{sec:proofsteps}): $\OOPS$ does not observe the proof structure at all, 

1153 
but goes back right to the theory level. Furthermore, $\OOPS$ does not 

1154 
produce any result theorem  there is no claim to be able to complete the 

1155 
proof anyhow. 

1156 

1157 
A typical application of $\OOPS$ is to explain Isar proofs \emph{within} the 

1158 
system itself, in conjunction with the document preparation tools of Isabelle 

1159 
described in \cite{isabellesys}. Thus partial or even wrong proof attempts 

1160 
can be discussed in a logically sound manner. Note that the Isabelle {\LaTeX} 

1161 
macros can be easily adapted to print something like ``$\dots$'' instead of an 

1162 
``$\OOPS$'' keyword. 

1163 

8547  1164 
\medskip The $\OOPS$ command is undoable, unlike $\isarkeyword{kill}$ (see 
1165 
\S\ref{sec:history}). The effect is to get back to the theory \emph{before} 

1166 
the opening of the proof. 

8515  1167 

1168 

7134  1169 
\section{Other commands} 
1170 

8448  1171 
\subsection{Diagnostics}\label{sec:diag} 
7134  1172 

8485  1173 
\indexisarcmd{pr}\indexisarcmd{thm}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{typ} 
1174 
\indexisarcmd{printfacts}\indexisarcmd{printbinds} 

7134  1175 
\begin{matharray}{rcl} 
8515  1176 
\isarcmd{help}^* & : & \isarkeep{\cdot} \\ 
1177 
\isarcmd{pr}^* & : & \isarkeep{\cdot} \\ 

1178 
\isarcmd{thm}^* & : & \isarkeep{theory~~proof} \\ 

1179 
\isarcmd{term}^* & : & \isarkeep{theory~~proof} \\ 

1180 
\isarcmd{prop}^* & : & \isarkeep{theory~~proof} \\ 

1181 
\isarcmd{typ}^* & : & \isarkeep{theory~~proof} \\ 

1182 
\isarcmd{print_facts}^* & : & \isarkeep{proof} \\ 

1183 
\isarcmd{print_binds}^* & : & \isarkeep{proof} \\ 

7134  1184 
\end{matharray} 
1185 

7335  1186 
These commands are not part of the actual Isabelle/Isar syntax, but assist 
1187 
interactive development. Also note that $undo$ does not apply here, since the 

1188 
theory or proof configuration is not changed. 

1189 

7134  1190 
\begin{rail} 
8485  1191 
'pr' modes? nat? 
7134  1192 
; 
8485  1193 
'thm' modes? thmrefs 
1194 
; 

1195 
'term' modes? term 

7134  1196 
; 
8485  1197 
'prop' modes? prop 
7134  1198 
; 
8485  1199 
'typ' modes? type 
1200 
; 

1201 

1202 
modes: '(' (name + ) ')' 

7134  1203 
; 
1204 
\end{rail} 

1205 

7167  1206 
\begin{descr} 
8515  1207 
\item [$\isarkeyword{help}$] prints a list of available language elements. 
1208 
Note that methods and attributes depend on the current theory context. 

8883  1209 
\item [$\isarkeyword{pr}~n$] prints the current proof state (if present), 
1210 
including the proof context, current facts and goals. The optional argument 

1211 
$n$ affects the implicit limit of goals to be displayed, which is initially 

1212 
10. Omitting the limit leaves the current value unchanged. 

8547  1213 
\item [$\isarkeyword{thm}~\vec a$] retrieves theorems from the current theory 
1214 
or proof context. Note that any attributes included in the theorem 

7974  1215 
specifications are applied to a temporary context derived from the current 
8547  1216 
theory or proof; the result is discarded, i.e.\ attributes involved in $\vec 
1217 
a$ do not have any permanent effect. 

7987  1218 
\item [$\isarkeyword{term}~t$, $\isarkeyword{prop}~\phi$] read, typecheck and 
1219 
print terms or propositions according to the current theory or proof 

7895  1220 
context; the inferred type of $t$ is output as well. Note that these 
1221 
commands are also useful in inspecting the current environment of term 

1222 
abbreviations. 

7974  1223 
\item [$\isarkeyword{typ}~\tau$] reads and prints types of the metalogic 
1224 
according to the current theory or proof context. 

8379  1225 
\item [$\isarkeyword{print_facts}$] prints any named facts of the current 
1226 
context, including assumptions and local results. 

1227 
\item [$\isarkeyword{print_binds}$] prints all term abbreviations present in 

1228 
the context. 

8485  1229 
\end{descr} 
1230 

1231 
The basic diagnostic commands above admit a list of $modes$ to be specified, 

1232 
which is appended to the current print mode (see also \cite{isabelleref}). 

1233 
Thus the output behavior may be modified according particular print mode 

1234 
features. 

1235 

1236 
For example, $\isarkeyword{pr}~(latex~xsymbols~symbols)$ would print the 

1237 
current proof state with mathematical symbols and special characters 

1238 
represented in {\LaTeX} source, according to the Isabelle style 

8547  1239 
\cite{isabellesys}. The resulting text can be directly pasted into a 
1240 
\verb,\begin{isabelle},\dots\verb,\end{isabelle}, environment. Note that 

1241 
$\isarkeyword{pr}~(latex)$ is sufficient to achieve the same output, if the 

1242 
current Isabelle session has the other modes already activated, say due to 

1243 
some particular user interface configuration such as Proof~General 

8510  1244 
\cite{proofgeneral,Aspinall:TACAS:2000} with XSymbol mode \cite{xsymbol}. 
8485  1245 

1246 

1247 
\subsection{History commands}\label{sec:history} 

1248 

1249 
\indexisarcmd{undo}\indexisarcmd{redo}\indexisarcmd{kill} 

1250 
\begin{matharray}{rcl} 

1251 
\isarcmd{undo}^{{*}{*}} & : & \isarkeep{\cdot} \\ 

1252 
\isarcmd{redo}^{{*}{*}} & : & \isarkeep{\cdot} \\ 

1253 
\isarcmd{kill}^{{*}{*}} & : & \isarkeep{\cdot} \\ 

1254 
\end{matharray} 

1255 

1256 
The Isabelle/Isar toplevel maintains a twostage history, for theory and 

1257 
proof state transformation. Basically, any command can be undone using 

1258 
$\isarkeyword{undo}$, excluding mere diagnostic elements. Its effect may be 

1259 
revoked via $\isarkeyword{redo}$, unless the corresponding the 

1260 
$\isarkeyword{undo}$ step has crossed the beginning of a proof or theory. The 

1261 
$\isarkeyword{kill}$ command aborts the current history node altogether, 

1262 
discontinuing a proof or even the whole theory. This operation is \emph{not} 

1263 
undoable. 

1264 

1265 
\begin{warn} 

8547  1266 
History commands should never be used with user interfaces such as 
1267 
Proof~General \cite{proofgeneral,Aspinall:TACAS:2000}, which takes care of 

1268 
stepping forth and back itself. Interfering by manual $\isarkeyword{undo}$, 

8510  1269 
$\isarkeyword{redo}$, or even $\isarkeyword{kill}$ commands would quickly 
1270 
result in utter confusion. 

8485  1271 
\end{warn} 
1272 

8379  1273 

7134  1274 
\subsection{System operations} 
1275 

7167  1276 
\indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{usethy}\indexisarcmd{usethyonly} 
1277 
\indexisarcmd{updatethy}\indexisarcmd{updatethyonly} 

7134  1278 
\begin{matharray}{rcl} 
8515  1279 
\isarcmd{cd}^* & : & \isarkeep{\cdot} \\ 
1280 
\isarcmd{pwd}^* & : & \isarkeep{\cdot} \\ 

1281 
\isarcmd{use_thy}^* & : & \isarkeep{\cdot} \\ 

1282 
\isarcmd{use_thy_only}^* & : & \isarkeep{\cdot} \\ 

1283 
\isarcmd{update_thy}^* & : & \isarkeep{\cdot} \\ 

1284 
\isarcmd{update_thy_only}^* & : & \isarkeep{\cdot} \\ 

7134  1285 
\end{matharray} 
1286 

7167  1287 
\begin{descr} 
7134  1288 
\item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle 
1289 
process. 

1290 
\item [$\isarkeyword{pwd}~$] prints the current working directory. 

7175  1291 
\item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$, 
7987  1292 
$\isarkeyword{update_thy}$, $\isarkeyword{update_thy_only}$] load some 
7895  1293 
theory given as $name$ argument. These commands are basically the same as 
7987  1294 
the corresponding ML functions\footnote{The ML versions also change the 
1295 
implicit theory context to that of the theory loaded.} (see also 

1296 
\cite[\S1,\S6]{isabelleref}). Note that both the ML and Isar versions may 

1297 
load new and oldstyle theories alike. 

7167  1298 
\end{descr} 
7134  1299 

7987  1300 
These system commands are scarcely used when working with the Proof~General 
1301 
interface, since loading of theories is done fully transparently. 

7134  1302 

8379  1303 

7046  1304 
%%% Local Variables: 
1305 
%%% mode: latex 

1306 
%%% TeXmaster: "isarref" 

1307 
%%% End: 