author | wenzelm |
Wed, 12 Apr 2000 23:52:21 +0200 | |
changeset 8696 | 37cbb053791c |
parent 8693 | feb1f9af3836 |
child 8726 | 7b15f4bdd72f |
permissions | -rw-r--r-- |
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\chapter{Basic Isar Language Elements}\label{ch:pure-syntax} |
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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:gen-tools} describes further Isar elements provided by generic |
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tools and packages (such as the Simplifier) that are either part of Pure |
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Isabelle or pre-installed by most object logics. Chapter~\ref{ch:hol-tools} |
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refers to actual object-logic specific elements of Isabelle/HOL. |
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\medskip |
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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 old-style tactical theorem |
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proving. |
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\section{Theory commands} |
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\subsection{Defining theories}\label{sec:begin-thy} |
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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} |
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Isabelle/Isar ``new-style'' theories are either defined via theory files or |
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interactively. Both theory-level specifications and proofs are handled |
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uniformly --- occasionally definitional mechanisms even require some explicit |
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proof as well. In contrast, ``old-style'' 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 pre-theory markup |
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command (cf.\ \S\ref{sec:markup-thy} and \S\ref{sec:markup-thy}). The theory |
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context may be also changed by $\CONTEXT$ without creating a new theory. In |
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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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\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 |
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; |
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'end' |
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;; |
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||
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filespecs: 'files' ((name | parname) +); |
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\end{rail} |
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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:markup-thy} and |
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\S\ref{sec:markup-prf} for further markup commands. |
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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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up-to-date 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 read-only |
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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 up-to-date. |
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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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\subsection{Theory markup commands}\label{sec:markup-thy} |
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\indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection} |
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\indexisarcmd{subsubsection}\indexisarcmd{text}\indexisarcmd{text-raw} |
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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{isabelle-sys} 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 |
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Additional markup commands are available for proofs (see |
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\S\ref{sec:markup-prf}). Also note that the $\isarkeyword{header}$ |
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declaration (see \S\ref{sec:begin-thy}) 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} \\ |
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\end{matharray} |
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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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; |
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\end{rail} |
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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 object-logics. |
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\end{descr} |
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\subsection{Primitive types and type abbreviations}\label{sec:types-pure} |
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\indexisarcmd{typedecl}\indexisarcmd{types}\indexisarcmd{nonterminals}\indexisarcmd{arities} |
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\begin{matharray}{rcl} |
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\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} \\ |
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\end{matharray} |
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\begin{rail} |
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'types' (typespec '=' type infix? comment? +) |
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; |
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'typedecl' typespec infix? comment? |
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; |
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'nonterminals' (name +) comment? |
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; |
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'arities' (nameref '::' arity comment? +) |
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; |
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\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, |
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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 object-logics 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 |
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Isabelle's inner syntax of terms or types. |
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\item [$\isarkeyword{arities}~t::(\vec s)s$] augments Isabelle's order-sorted |
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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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\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} |
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\isarcmd{consts} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{defs} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{constdefs} & : & \isartrans{theory}{theory} \\ |
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\end{matharray} |
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\begin{rail} |
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'consts' (constdecl +) |
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; |
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'defs' (axmdecl prop comment? +) |
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; |
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'constdefs' (constdecl prop comment? +) |
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; |
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constdecl: name '::' type mixfix? comment? |
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; |
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\end{rail} |
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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]{isabelle-ref} for more details on the |
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form of equations admitted as constant definitions. |
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\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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\subsection{Syntax and translations}\label{sec:syn-trans} |
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\indexisarcmd{syntax}\indexisarcmd{translations} |
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\begin{matharray}{rcl} |
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\isarcmd{syntax} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{translations} & : & \isartrans{theory}{theory} \\ |
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\end{matharray} |
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\begin{rail} |
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'syntax' ('(' name 'output'? ')')? (constdecl +) |
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; |
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'translations' (transpat ('==' | '=>' | '<=') transpat comment? +) |
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; |
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transpat: ('(' nameref ')')? string |
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; |
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\end{rail} |
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||
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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 |
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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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\subsection{Axioms and theorems} |
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\indexisarcmd{axioms}\indexisarcmd{theorems}\indexisarcmd{lemmas} |
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\begin{matharray}{rcl} |
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\isarcmd{axioms} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{theorems} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{lemmas} & : & \isartrans{theory}{theory} \\ |
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\end{matharray} |
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\begin{rail} |
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'axioms' (axmdecl prop comment? +) |
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; |
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('theorems' | 'lemmas') thmdef? thmrefs |
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; |
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\end{rail} |
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||
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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 meta-logic. In fact, axioms are ``axiomatic theorems'', and |
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may be referred later just as any other theorem. |
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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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\subsection{Name spaces} |
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\indexisarcmd{global}\indexisarcmd{local} |
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\begin{matharray}{rcl} |
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\isarcmd{global} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{local} & : & \isartrans{theory}{theory} \\ |
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\end{matharray} |
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||
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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 never has to control the behavior of name space entry by hand, yet |
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the following commands provide some way to do so. |
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\begin{descr} |
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\item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current |
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name declaration mode. Initially, theories start in $\isarkeyword{local}$ |
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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 |
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without the theory prefix, until $\isarkeyword{local}$ is declared again. |
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\end{descr} |
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\subsection{Incorporating ML code}\label{sec:ML} |
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\indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{ML-command} |
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\indexisarcmd{ML-setup}\indexisarcmd{setup} |
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\begin{matharray}{rcl} |
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\isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\ |
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\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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\end{matharray} |
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\railalias{MLsetup}{ML\_setup} |
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\railterm{MLsetup} |
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||
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\railalias{MLcommand}{ML\_command} |
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\railterm{MLcommand} |
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||
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\begin{rail} |
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'use' name |
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; |
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('ML' | MLcommand | MLsetup | 'setup') text |
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; |
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\end{rail} |
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||
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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 |
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also \S\ref{sec:begin-thy}). |
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||
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\item [$\isarkeyword{ML}~text$ and $\isarkeyword{ML_command}~text$] execute ML |
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commands from $text$. The theory context is passed in the same way as for |
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$\isarkeyword{use}$, but may not be changed. Note that |
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$\isarkeyword{ML_command}$ is less verbose than plain $\isarkeyword{ML}$. |
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\item [$\isarkeyword{ML_setup}~text$] executes ML commands from $text$. The |
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theory context is passed down to the ML session, and fetched back |
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afterwards. Thus $text$ may actually change the theory as a side effect. |
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||
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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 |
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\texttt{(theory~->~theory)~list}. The $\isarkeyword{setup}$ command is the |
|
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canonical way to initialize any object-logic specific tools and packages |
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written in ML. |
|
7167 | 388 |
\end{descr} |
7134 | 389 |
|
390 |
||
8250 | 391 |
\subsection{Syntax translation functions} |
7134 | 392 |
|
8250 | 393 |
\indexisarcmd{parse-ast-translation}\indexisarcmd{parse-translation} |
394 |
\indexisarcmd{print-translation}\indexisarcmd{typed-print-translation} |
|
395 |
\indexisarcmd{print-ast-translation}\indexisarcmd{token-translation} |
|
396 |
\begin{matharray}{rcl} |
|
397 |
\isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\ |
|
398 |
\isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\ |
|
399 |
\isarcmd{print_translation} & : & \isartrans{theory}{theory} \\ |
|
400 |
\isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\ |
|
401 |
\isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\ |
|
402 |
\isarcmd{token_translation} & : & \isartrans{theory}{theory} \\ |
|
403 |
\end{matharray} |
|
7134 | 404 |
|
8250 | 405 |
Syntax translation functions written in ML admit almost arbitrary |
406 |
manipulations of Isabelle's inner syntax. Any of the above commands have a |
|
407 |
single \railqtoken{text} argument that refers to an ML expression of |
|
8379 | 408 |
appropriate type. |
409 |
||
410 |
\begin{ttbox} |
|
411 |
val parse_ast_translation : (string * (ast list -> ast)) list |
|
412 |
val parse_translation : (string * (term list -> term)) list |
|
413 |
val print_translation : (string * (term list -> term)) list |
|
414 |
val typed_print_translation : |
|
415 |
(string * (bool -> typ -> term list -> term)) list |
|
416 |
val print_ast_translation : (string * (ast list -> ast)) list |
|
417 |
val token_translation : |
|
418 |
(string * string * (string -> string * real)) list |
|
419 |
\end{ttbox} |
|
420 |
See \cite[\S8]{isabelle-ref} for more information on syntax transformations. |
|
7134 | 421 |
|
422 |
||
423 |
\subsection{Oracles} |
|
424 |
||
425 |
\indexisarcmd{oracle} |
|
426 |
\begin{matharray}{rcl} |
|
427 |
\isarcmd{oracle} & : & \isartrans{theory}{theory} \\ |
|
428 |
\end{matharray} |
|
429 |
||
7175 | 430 |
Oracles provide an interface to external reasoning systems, without giving up |
431 |
control completely --- each theorem carries a derivation object recording any |
|
432 |
oracle invocation. See \cite[\S6]{isabelle-ref} for more information. |
|
433 |
||
7134 | 434 |
\begin{rail} |
435 |
'oracle' name '=' text comment? |
|
436 |
; |
|
437 |
\end{rail} |
|
438 |
||
7167 | 439 |
\begin{descr} |
7175 | 440 |
\item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML |
8379 | 441 |
function $text$, which has to be of type |
442 |
\texttt{Sign.sg~*~Object.T~->~term}. |
|
7167 | 443 |
\end{descr} |
7134 | 444 |
|
445 |
||
446 |
\section{Proof commands} |
|
447 |
||
7987 | 448 |
Proof commands perform transitions of Isar/VM machine configurations, which |
7315 | 449 |
are block-structured, consisting of a stack of nodes with three main |
7335 | 450 |
components: logical proof context, current facts, and open goals. Isar/VM |
8547 | 451 |
transitions are \emph{typed} according to the following three different modes |
452 |
of operation: |
|
7167 | 453 |
\begin{descr} |
454 |
\item [$proof(prove)$] means that a new goal has just been stated that is now |
|
8547 | 455 |
to be \emph{proven}; the next command may refine it by some proof method, |
456 |
and enter a sub-proof to establish the actual result. |
|
7167 | 457 |
\item [$proof(state)$] is like an internal theory mode: the context may be |
7987 | 458 |
augmented by \emph{stating} additional assumptions, intermediate results |
459 |
etc. |
|
7895 | 460 |
\item [$proof(chain)$] is intermediate between $proof(state)$ and |
7987 | 461 |
$proof(prove)$: existing facts (i.e.\ the contents of the special ``$this$'' |
462 |
register) have been just picked up in order to be used when refining the |
|
463 |
goal claimed next. |
|
7167 | 464 |
\end{descr} |
7134 | 465 |
|
7167 | 466 |
|
7895 | 467 |
\subsection{Proof markup commands}\label{sec:markup-prf} |
7167 | 468 |
|
7987 | 469 |
\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsubsect} |
7895 | 470 |
\indexisarcmd{txt}\indexisarcmd{txt-raw} |
7134 | 471 |
\begin{matharray}{rcl} |
8101 | 472 |
\isarcmd{sect} & : & \isartrans{proof}{proof} \\ |
473 |
\isarcmd{subsect} & : & \isartrans{proof}{proof} \\ |
|
474 |
\isarcmd{subsubsect} & : & \isartrans{proof}{proof} \\ |
|
475 |
\isarcmd{txt} & : & \isartrans{proof}{proof} \\ |
|
476 |
\isarcmd{txt_raw} & : & \isartrans{proof}{proof} \\ |
|
7134 | 477 |
\end{matharray} |
478 |
||
7895 | 479 |
These markup commands for proof mode closely correspond to the ones of theory |
8684 | 480 |
mode (see \S\ref{sec:markup-thy}). |
7895 | 481 |
|
482 |
\railalias{txtraw}{txt\_raw} |
|
483 |
\railterm{txtraw} |
|
7175 | 484 |
|
7134 | 485 |
\begin{rail} |
7895 | 486 |
('sect' | 'subsect' | 'subsubsect' | 'txt' | txtraw) text |
7134 | 487 |
; |
488 |
\end{rail} |
|
489 |
||
490 |
||
7315 | 491 |
\subsection{Proof context}\label{sec:proof-context} |
7134 | 492 |
|
7315 | 493 |
\indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def} |
7134 | 494 |
\begin{matharray}{rcl} |
495 |
\isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\ |
|
496 |
\isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\ |
|
497 |
\isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\ |
|
498 |
\isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\ |
|
499 |
\end{matharray} |
|
500 |
||
7315 | 501 |
The logical proof context consists of fixed variables and assumptions. The |
502 |
former closely correspond to Skolem constants, or meta-level universal |
|
503 |
quantification as provided by the Isabelle/Pure logical framework. |
|
504 |
Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in |
|
7987 | 505 |
a local value that may be used in the subsequent proof as any other variable |
7895 | 506 |
or constant. Furthermore, any result $\edrv \phi[x]$ exported from the |
7987 | 507 |
context will be universally closed wrt.\ $x$ at the outermost level: $\edrv |
508 |
\All x \phi$ (this is expressed using Isabelle's meta-variables). |
|
7315 | 509 |
|
510 |
Similarly, introducing some assumption $\chi$ has two effects. On the one |
|
511 |
hand, a local theorem is created that may be used as a fact in subsequent |
|
7895 | 512 |
proof steps. On the other hand, any result $\chi \drv \phi$ exported from the |
513 |
context becomes conditional wrt.\ the assumption: $\edrv \chi \Imp \phi$. |
|
514 |
Thus, solving an enclosing goal using such a result would basically introduce |
|
515 |
a new subgoal stemming from the assumption. How this situation is handled |
|
516 |
depends on the actual version of assumption command used: while $\ASSUMENAME$ |
|
517 |
insists on solving the subgoal by unification with some premise of the goal, |
|
518 |
$\PRESUMENAME$ leaves the subgoal unchanged in order to be proved later by the |
|
519 |
user. |
|
7315 | 520 |
|
7319 | 521 |
Local definitions, introduced by $\DEF{}{x \equiv t}$, are achieved by |
7987 | 522 |
combining $\FIX x$ with another version of assumption that causes any |
523 |
hypothetical equation $x \equiv t$ to be eliminated by the reflexivity rule. |
|
524 |
Thus, exporting some result $x \equiv t \drv \phi[x]$ yields $\edrv \phi[t]$. |
|
7175 | 525 |
|
7134 | 526 |
\begin{rail} |
7431 | 527 |
'fix' (vars + 'and') comment? |
7134 | 528 |
; |
7315 | 529 |
('assume' | 'presume') (assm comment? + 'and') |
7134 | 530 |
; |
7175 | 531 |
'def' thmdecl? \\ var '==' term termpat? comment? |
7134 | 532 |
; |
533 |
||
534 |
var: name ('::' type)? |
|
535 |
; |
|
7458 | 536 |
vars: (name+) ('::' type)? |
7431 | 537 |
; |
7315 | 538 |
assm: thmdecl? (prop proppat? +) |
539 |
; |
|
7134 | 540 |
\end{rail} |
541 |
||
7167 | 542 |
\begin{descr} |
8547 | 543 |
\item [$\FIX{\vec x}$] introduces local \emph{arbitrary, but fixed} variables |
544 |
$\vec x$. |
|
8515 | 545 |
\item [$\ASSUME{a}{\vec\phi}$ and $\PRESUME{a}{\vec\phi}$] introduce local |
546 |
theorems $\vec\phi$ by assumption. Subsequent results applied to an |
|
547 |
enclosing goal (e.g.\ by $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ |
|
548 |
expects to be able to unify with existing premises in the goal, while |
|
549 |
$\PRESUMENAME$ leaves $\vec\phi$ as new subgoals. |
|
7335 | 550 |
|
551 |
Several lists of assumptions may be given (separated by |
|
7895 | 552 |
$\isarkeyword{and}$); the resulting list of current facts consists of all of |
553 |
these concatenated. |
|
7315 | 554 |
\item [$\DEF{a}{x \equiv t}$] introduces a local (non-polymorphic) definition. |
555 |
In results exported from the context, $x$ is replaced by $t$. Basically, |
|
7987 | 556 |
$\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\ASSUME{}{x \equiv t}$, with the |
7335 | 557 |
resulting hypothetical equation solved by reflexivity. |
7431 | 558 |
|
559 |
The default name for the definitional equation is $x_def$. |
|
7167 | 560 |
\end{descr} |
561 |
||
7895 | 562 |
The special name $prems$\indexisarthm{prems} refers to all assumptions of the |
563 |
current context as a list of theorems. |
|
7315 | 564 |
|
7167 | 565 |
|
566 |
\subsection{Facts and forward chaining} |
|
567 |
||
568 |
\indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with} |
|
569 |
\begin{matharray}{rcl} |
|
570 |
\isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\ |
|
571 |
\isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\ |
|
572 |
\isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\ |
|
573 |
\isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\ |
|
574 |
\end{matharray} |
|
575 |
||
7319 | 576 |
New facts are established either by assumption or proof of local statements. |
7335 | 577 |
Any fact will usually be involved in further proofs, either as explicit |
8547 | 578 |
arguments of proof methods, or when forward chaining towards the next goal via |
7335 | 579 |
$\THEN$ (and variants). Note that the special theorem name |
7987 | 580 |
$this$\indexisarthm{this} refers to the most recently established facts. |
7167 | 581 |
\begin{rail} |
582 |
'note' thmdef? thmrefs comment? |
|
583 |
; |
|
584 |
'then' comment? |
|
585 |
; |
|
586 |
('from' | 'with') thmrefs comment? |
|
587 |
; |
|
588 |
\end{rail} |
|
589 |
||
590 |
\begin{descr} |
|
7175 | 591 |
\item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result |
592 |
as $a$. Note that attributes may be involved as well, both on the left and |
|
593 |
right hand sides. |
|
7167 | 594 |
\item [$\THEN$] indicates forward chaining by the current facts in order to |
7895 | 595 |
establish the goal to be claimed next. The initial proof method invoked to |
596 |
refine that will be offered the facts to do ``anything appropriate'' (cf.\ |
|
597 |
also \S\ref{sec:proof-steps}). For example, method $rule$ (see |
|
8515 | 598 |
\S\ref{sec:pure-meth-att}) would typically do an elimination rather than an |
7895 | 599 |
introduction. Automatic methods usually insert the facts into the goal |
8547 | 600 |
state before operation. This provides a simple scheme to control relevance |
601 |
of facts in automated proof search. |
|
7335 | 602 |
\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; thus $\THEN$ is |
7458 | 603 |
equivalent to $\FROM{this}$. |
7175 | 604 |
\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward |
605 |
chaining is from earlier facts together with the current ones. |
|
7167 | 606 |
\end{descr} |
607 |
||
8515 | 608 |
Basic proof methods (such as $rule$, see \S\ref{sec:pure-meth-att}) expect |
7895 | 609 |
multiple facts to be given in their proper order, corresponding to a prefix of |
610 |
the premises of the rule involved. Note that positions may be easily skipped |
|
8547 | 611 |
using something like $\FROM{\text{\texttt{_}}~a~b}$, for example. This |
612 |
involves the trivial rule $\PROP\psi \Imp \PROP\psi$, which happens to be |
|
613 |
bound in Isabelle/Pure as ``\texttt{_}'' |
|
614 |
(underscore).\indexisarthm{_@\texttt{_}} |
|
7389 | 615 |
|
7167 | 616 |
|
617 |
\subsection{Goal statements} |
|
618 |
||
619 |
\indexisarcmd{theorem}\indexisarcmd{lemma} |
|
620 |
\indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus} |
|
621 |
\begin{matharray}{rcl} |
|
622 |
\isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\ |
|
623 |
\isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\ |
|
7987 | 624 |
\isarcmd{have} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\ |
625 |
\isarcmd{show} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\ |
|
7167 | 626 |
\isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\ |
627 |
\isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\ |
|
628 |
\end{matharray} |
|
629 |
||
7175 | 630 |
Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$ |
7895 | 631 |
and $\LEMMANAME$. New local goals may be claimed within proof mode as well. |
632 |
Four variants are available, indicating whether the result is meant to solve |
|
8547 | 633 |
some pending goal or whether forward chaining is indicated. |
7175 | 634 |
|
7167 | 635 |
\begin{rail} |
636 |
('theorem' | 'lemma') goal |
|
637 |
; |
|
638 |
('have' | 'show' | 'hence' | 'thus') goal |
|
639 |
; |
|
640 |
||
8632 | 641 |
goal: thmdecl? prop proppat? comment? |
7167 | 642 |
; |
643 |
\end{rail} |
|
644 |
||
645 |
\begin{descr} |
|
7335 | 646 |
\item [$\THEOREM{a}{\phi}$] enters proof mode with $\phi$ as main goal, |
8547 | 647 |
eventually resulting in some theorem $\turn \phi$ to be put back into the |
648 |
theory. |
|
7987 | 649 |
\item [$\LEMMA{a}{\phi}$] is similar to $\THEOREMNAME$, but tags the result as |
7167 | 650 |
``lemma''. |
7335 | 651 |
\item [$\HAVE{a}{\phi}$] claims a local goal, eventually resulting in a |
7167 | 652 |
theorem with the current assumption context as hypotheses. |
7335 | 653 |
\item [$\SHOW{a}{\phi}$] is similar to $\HAVE{a}{\phi}$, but solves some |
7895 | 654 |
pending goal with the result \emph{exported} into the corresponding context |
655 |
(cf.\ \S\ref{sec:proof-context}). |
|
656 |
\item [$\HENCENAME$] abbreviates $\THEN~\HAVENAME$, i.e.\ claims a local goal |
|
657 |
to be proven by forward chaining the current facts. Note that $\HENCENAME$ |
|
658 |
is also equivalent to $\FROM{this}~\HAVENAME$. |
|
659 |
\item [$\THUSNAME$] abbreviates $\THEN~\SHOWNAME$. Note that $\THUSNAME$ is |
|
660 |
also equivalent to $\FROM{this}~\SHOWNAME$. |
|
7167 | 661 |
\end{descr} |
662 |
||
663 |
||
664 |
\subsection{Initial and terminal proof steps}\label{sec:proof-steps} |
|
665 |
||
7175 | 666 |
\indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by} |
667 |
\indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry} |
|
668 |
\begin{matharray}{rcl} |
|
669 |
\isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\ |
|
670 |
\isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\ |
|
671 |
\isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ |
|
672 |
\isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ |
|
673 |
\isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ |
|
674 |
\isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\ |
|
675 |
\end{matharray} |
|
676 |
||
8547 | 677 |
Arbitrary goal refinement via tactics is considered harmful. Properly, the |
7335 | 678 |
Isar framework admits proof methods to be invoked in two places only. |
7167 | 679 |
\begin{enumerate} |
7175 | 680 |
\item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated |
7335 | 681 |
goal to a number of sub-goals that are to be solved later. Facts are passed |
7895 | 682 |
to $m@1$ for forward chaining, if so indicated by $proof(chain)$ mode. |
7167 | 683 |
|
7987 | 684 |
\item A \emph{terminal} conclusion step $\QED{m@2}$ is intended to solve |
685 |
remaining goals. No facts are passed to $m@2$. |
|
7167 | 686 |
\end{enumerate} |
687 |
||
8547 | 688 |
The only other proper way to affect pending goals is by $\SHOWNAME$, which |
689 |
involves an explicit statement of what is to be solved. |
|
7167 | 690 |
|
7175 | 691 |
\medskip |
692 |
||
7167 | 693 |
Also note that initial proof methods should either solve the goal completely, |
7895 | 694 |
or constitute some well-understood reduction to new sub-goals. Arbitrary |
695 |
automatic proof tools that are prone leave a large number of badly structured |
|
696 |
sub-goals are no help in continuing the proof document in any intelligible |
|
7987 | 697 |
way. |
7167 | 698 |
|
7175 | 699 |
\medskip |
700 |
||
8547 | 701 |
Unless given explicitly by the user, the default initial method is ``$rule$'', |
702 |
which applies a single standard elimination or introduction rule according to |
|
703 |
the topmost symbol involved. There is no separate default terminal method. |
|
704 |
Any remaining goals are always solved by assumption in the very last step. |
|
7167 | 705 |
|
706 |
\begin{rail} |
|
707 |
'proof' interest? meth? comment? |
|
708 |
; |
|
709 |
'qed' meth? comment? |
|
710 |
; |
|
711 |
'by' meth meth? comment? |
|
712 |
; |
|
713 |
('.' | '..' | 'sorry') comment? |
|
714 |
; |
|
715 |
||
716 |
meth: method interest? |
|
717 |
; |
|
718 |
\end{rail} |
|
719 |
||
720 |
\begin{descr} |
|
7335 | 721 |
\item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for |
722 |
forward chaining are passed if so indicated by $proof(chain)$ mode. |
|
723 |
\item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and |
|
7895 | 724 |
concludes the sub-proof by assumption. If the goal had been $\SHOWNAME$ (or |
725 |
$\THUSNAME$), some pending sub-goal is solved as well by the rule resulting |
|
726 |
from the result \emph{exported} into the enclosing goal context. Thus |
|
727 |
$\QEDNAME$ may fail for two reasons: either $m@2$ fails, or the resulting |
|
728 |
rule does not fit to any pending goal\footnote{This includes any additional |
|
729 |
``strong'' assumptions as introduced by $\ASSUMENAME$.} of the enclosing |
|
730 |
context. Debugging such a situation might involve temporarily changing |
|
731 |
$\SHOWNAME$ into $\HAVENAME$, or weakening the local context by replacing |
|
732 |
some occurrences of $\ASSUMENAME$ by $\PRESUMENAME$. |
|
733 |
\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}\index{proof!terminal}; it |
|
7987 | 734 |
abbreviates $\PROOF{m@1}~\QED{m@2}$, with backtracking across both methods, |
735 |
though. Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done |
|
7895 | 736 |
by expanding its definition; in many cases $\PROOF{m@1}$ is already |
7175 | 737 |
sufficient to see what is going wrong. |
7895 | 738 |
\item [``$\DDOT$''] is a \emph{default proof}\index{proof!default}; it |
8515 | 739 |
abbreviates $\BY{rule}$. |
7895 | 740 |
\item [``$\DOT$''] is a \emph{trivial proof}\index{proof!trivial}; it |
8195 | 741 |
abbreviates $\BY{this}$. |
8379 | 742 |
\item [$\SORRY$] is a \emph{fake proof}\index{proof!fake}; provided that the |
743 |
\texttt{quick_and_dirty} flag is enabled, $\SORRY$ pretends to solve the |
|
8515 | 744 |
goal without further ado. Of course, the result would be a fake theorem |
745 |
only, involving some oracle in its internal derivation object (this is |
|
746 |
indicated as ``$[!]$'' in the printed result). The main application of |
|
747 |
$\SORRY$ is to support experimentation and top-down proof development. |
|
748 |
\end{descr} |
|
749 |
||
750 |
||
751 |
\subsection{Fundamental methods and attributes}\label{sec:pure-meth-att} |
|
752 |
||
8547 | 753 |
The following proof methods and attributes refer to basic logical operations |
754 |
of Isar. Further methods and attributes are provided by several generic and |
|
755 |
object-logic specific tools and packages (see chapters \ref{ch:gen-tools} and |
|
756 |
\ref{ch:hol-tools}). |
|
8515 | 757 |
|
758 |
\indexisarmeth{assumption}\indexisarmeth{this}\indexisarmeth{rule}\indexisarmeth{$-$} |
|
759 |
\indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest} |
|
760 |
\indexisaratt{OF}\indexisaratt{of} |
|
761 |
\begin{matharray}{rcl} |
|
762 |
assumption & : & \isarmeth \\ |
|
763 |
this & : & \isarmeth \\ |
|
764 |
rule & : & \isarmeth \\ |
|
765 |
- & : & \isarmeth \\ |
|
766 |
OF & : & \isaratt \\ |
|
767 |
of & : & \isaratt \\ |
|
768 |
intro & : & \isaratt \\ |
|
769 |
elim & : & \isaratt \\ |
|
770 |
dest & : & \isaratt \\ |
|
771 |
delrule & : & \isaratt \\ |
|
772 |
\end{matharray} |
|
773 |
||
774 |
\begin{rail} |
|
8547 | 775 |
'rule' thmrefs? |
8515 | 776 |
; |
777 |
'OF' thmrefs |
|
778 |
; |
|
8693 | 779 |
'of' insts ('concl' ':' insts)? |
8515 | 780 |
; |
781 |
\end{rail} |
|
782 |
||
783 |
\begin{descr} |
|
784 |
\item [$assumption$] solves some goal by a single assumption step. Any facts |
|
785 |
given (${} \le 1$) are guaranteed to participate in the refinement. Recall |
|
786 |
that $\QEDNAME$ (see \S\ref{sec:proof-steps}) already concludes any |
|
787 |
remaining sub-goals by assumption. |
|
788 |
\item [$this$] applies all of the current facts directly as rules. Recall |
|
789 |
that ``$\DOT$'' (dot) abbreviates $\BY{this}$. |
|
8547 | 790 |
\item [$rule~\vec a$] applies some rule given as argument in backward manner; |
8515 | 791 |
facts are used to reduce the rule before applying it to the goal. Thus |
792 |
$rule$ without facts is plain \emph{introduction}, while with facts it |
|
793 |
becomes \emph{elimination}. |
|
794 |
||
8547 | 795 |
When no arguments are given, the $rule$ method tries to pick appropriate |
796 |
rules automatically, as declared in the current context using the $intro$, |
|
797 |
$elim$, $dest$ attributes (see below). This is the default behavior of |
|
798 |
$\PROOFNAME$ and ``$\DDOT$'' (double-dot) steps (see |
|
8515 | 799 |
\S\ref{sec:proof-steps}). |
800 |
\item [``$-$''] does nothing but insert the forward chaining facts as premises |
|
801 |
into the goal. Note that command $\PROOFNAME$ without any method actually |
|
802 |
performs a single reduction step using the $rule$ method; thus a plain |
|
803 |
\emph{do-nothing} proof step would be $\PROOF{-}$ rather than $\PROOFNAME$ |
|
804 |
alone. |
|
8547 | 805 |
\item [$OF~\vec a$] applies some theorem to given rules $\vec a$ (in |
806 |
parallel). This corresponds to the \texttt{MRS} operator in ML |
|
807 |
\cite[\S5]{isabelle-ref}, but note the reversed order. Positions may be |
|
808 |
skipped by including ``$\_$'' (underscore) as argument. |
|
809 |
\item [$of~\vec t$] performs positional instantiation. The terms $\vec t$ are |
|
8515 | 810 |
substituted for any schematic variables occurring in a theorem from left to |
811 |
right; ``\texttt{_}'' (underscore) indicates to skip a position. Arguments |
|
812 |
following a ``$concl\colon$'' specification refer to positions of the |
|
813 |
conclusion of a rule. |
|
814 |
\item [$intro$, $elim$, and $dest$] declare introduction, elimination, and |
|
815 |
destruct rules, respectively. Note that the classical reasoner (see |
|
816 |
\S\ref{sec:classical-basic}) introduces different versions of these |
|
817 |
attributes, and the $rule$ method, too. In object-logics with classical |
|
818 |
reasoning enabled, the latter version should be used all the time to avoid |
|
819 |
confusion! |
|
820 |
\item [$delrule$] undeclares introduction or elimination rules. |
|
7315 | 821 |
\end{descr} |
822 |
||
823 |
||
824 |
\subsection{Term abbreviations}\label{sec:term-abbrev} |
|
825 |
||
826 |
\indexisarcmd{let} |
|
827 |
\begin{matharray}{rcl} |
|
828 |
\isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\ |
|
829 |
\isarkeyword{is} & : & syntax \\ |
|
830 |
\end{matharray} |
|
831 |
||
832 |
Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements, |
|
7987 | 833 |
or by annotating assumptions or goal statements with a list of patterns |
834 |
$\ISS{p@1\;\dots}{p@n}$. In both cases, higher-order matching is invoked to |
|
835 |
bind extra-logical term variables, which may be either named schematic |
|
836 |
variables of the form $\Var{x}$, or nameless dummies ``\texttt{_}'' |
|
837 |
(underscore).\indexisarvar{_@\texttt{_}} Note that in the $\LETNAME$ form the |
|
838 |
patterns occur on the left-hand side, while the $\ISNAME$ patterns are in |
|
839 |
postfix position. |
|
7315 | 840 |
|
8620
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
841 |
Polymorphism of term bindings is handled in Hindley-Milner style, as in ML. |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
842 |
Type variables referring to local assumptions or open goal statements are |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
843 |
\emph{fixed}, while those of finished results or bound by $\LETNAME$ may occur |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
844 |
in \emph{arbitrary} instances later. Even though actual polymorphism should |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
845 |
be rarely used in practice, this mechanism is essential to achieve proper |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
846 |
incremental type-inference, as the user proceeds to build up the Isar proof |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
847 |
text. |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
848 |
|
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
849 |
\medskip |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
850 |
|
7319 | 851 |
Term abbreviations are quite different from actual local definitions as |
852 |
introduced via $\DEFNAME$ (see \S\ref{sec:proof-context}). The latter are |
|
7315 | 853 |
visible within the logic as actual equations, while abbreviations disappear |
8620
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
854 |
during the input process just after type checking. Also note that $\DEFNAME$ |
3786d47f5570
support Hindley-Milner polymorphisms in results and bindings;
wenzelm
parents:
8547
diff
changeset
|
855 |
does not support polymorphism. |
7315 | 856 |
|
857 |
\begin{rail} |
|
8664 | 858 |
'let' ((term + 'and') '=' term comment? + 'and') |
7315 | 859 |
; |
860 |
\end{rail} |
|
861 |
||
862 |
The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or |
|
863 |
\railnonterm{proppat} (see \S\ref{sec:term-pats}). |
|
864 |
||
865 |
\begin{descr} |
|
866 |
\item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$ |
|
867 |
by simultaneous higher-order matching against terms $\vec t$. |
|
868 |
\item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the |
|
869 |
preceding statement. Also note that $\ISNAME$ is not a separate command, |
|
870 |
but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.). |
|
871 |
\end{descr} |
|
872 |
||
7988 | 873 |
A few \emph{automatic} term abbreviations\index{term abbreviations} for goals |
874 |
and facts are available as well. For any open goal, |
|
7466 | 875 |
$\Var{thesis_prop}$\indexisarvar{thesis-prop} refers to the full proposition |
876 |
(which may be a rule), $\Var{thesis_concl}$\indexisarvar{thesis-concl} to its |
|
877 |
(atomic) conclusion, and $\Var{thesis}$\indexisarvar{thesis} to its |
|
8547 | 878 |
object-level statement. The latter two abstract over any meta-level |
7987 | 879 |
parameters. |
7315 | 880 |
|
7466 | 881 |
Fact statements resulting from assumptions or finished goals are bound as |
882 |
$\Var{this_prop}$\indexisarvar{this-prop}, |
|
883 |
$\Var{this_concl}$\indexisarvar{this-concl}, and |
|
884 |
$\Var{this}$\indexisarvar{this}, similar to $\Var{thesis}$ above. In case |
|
885 |
$\Var{this}$ refers to an object-logic statement that is an application |
|
7895 | 886 |
$f(t)$, then $t$ is bound to the special text variable |
7466 | 887 |
``$\dots$''\indexisarvar{\dots} (three dots). The canonical application of |
7987 | 888 |
the latter are calculational proofs (see \S\ref{sec:calculation}). |
7315 | 889 |
|
890 |
||
7134 | 891 |
\subsection{Block structure} |
892 |
||
7397 | 893 |
\indexisarcmd{next}\indexisarcmd{\{\{}\indexisarcmd{\}\}} |
894 |
\begin{matharray}{rcl} |
|
8448 | 895 |
\NEXT & : & \isartrans{proof(state)}{proof(state)} \\ |
7974 | 896 |
\BG & : & \isartrans{proof(state)}{proof(state)} \\ |
897 |
\EN & : & \isartrans{proof(state)}{proof(state)} \\ |
|
7397 | 898 |
\end{matharray} |
899 |
||
7167 | 900 |
While Isar is inherently block-structured, opening and closing blocks is |
901 |
mostly handled rather casually, with little explicit user-intervention. Any |
|
902 |
local goal statement automatically opens \emph{two} blocks, which are closed |
|
903 |
again when concluding the sub-proof (by $\QEDNAME$ etc.). Sections of |
|
8448 | 904 |
different context within a sub-proof may be switched via $\NEXT$, which is |
905 |
just a single block-close followed by block-open again. Thus the effect of |
|
906 |
$\NEXT$ to reset the local proof context. There is no goal focus involved |
|
907 |
here! |
|
7167 | 908 |
|
7175 | 909 |
For slightly more advanced applications, there are explicit block parentheses |
7895 | 910 |
as well. These typically achieve a stronger forward style of reasoning. |
7167 | 911 |
|
912 |
\begin{descr} |
|
8448 | 913 |
\item [$\NEXT$] switches to a fresh block within a sub-proof, resetting the |
914 |
local context to the initial one. |
|
7167 | 915 |
\item [$\isarkeyword{\{\{}$ and $\isarkeyword{\}\}}$] explicitly open and |
7895 | 916 |
close blocks. Any current facts pass through ``$\isarkeyword{\{\{}$'' |
917 |
unchanged, while ``$\isarkeyword{\}\}}$'' causes any result to be |
|
918 |
\emph{exported} into the enclosing context. Thus fixed variables are |
|
919 |
generalized, assumptions discharged, and local definitions unfolded (cf.\ |
|
920 |
\S\ref{sec:proof-context}). There is no difference of $\ASSUMENAME$ and |
|
921 |
$\PRESUMENAME$ in this mode of forward reasoning --- in contrast to plain |
|
922 |
backward reasoning with the result exported at $\SHOWNAME$ time. |
|
7167 | 923 |
\end{descr} |
7134 | 924 |
|
925 |
||
8533 | 926 |
\subsection{Emulating tactic scripts}\label{sec:tactical-proof} |
8515 | 927 |
|
928 |
The following elements emulate unstructured tactic scripts to some extent. |
|
929 |
While these are anathema for writing proper Isar proof documents, they might |
|
930 |
come in handy for interactive exploration and debugging, or even actual |
|
931 |
tactical proof within new-style theories (to benefit from document |
|
932 |
preparation, for example). |
|
933 |
||
934 |
\indexisarcmd{apply}\indexisarcmd{apply-end} |
|
935 |
\indexisarcmd{defer}\indexisarcmd{prefer}\indexisarcmd{back} |
|
8693 | 936 |
\indexisarmeth{tactic}\indexisarmeth{insert} |
8533 | 937 |
\indexisarmeth{res-inst-tac}\indexisarmeth{eres-inst-tac} |
938 |
\indexisarmeth{dres-inst-tac}\indexisarmeth{forw-inst-tac} |
|
939 |
\indexisarmeth{subgoal-tac} |
|
8515 | 940 |
\begin{matharray}{rcl} |
8533 | 941 |
\isarcmd{apply}^* & : & \isartrans{proof(prove)}{proof(prove)} \\ |
942 |
\isarcmd{apply_end}^* & : & \isartrans{proof(state)}{proof(state)} \\ |
|
943 |
\isarcmd{defer}^* & : & \isartrans{proof}{proof} \\ |
|
944 |
\isarcmd{prefer}^* & : & \isartrans{proof}{proof} \\ |
|
945 |
\isarcmd{back}^* & : & \isartrans{proof}{proof} \\ |
|
946 |
tactic^* & : & \isarmeth \\ |
|
8693 | 947 |
insert^* & : & \isarmeth \\ |
8533 | 948 |
res_inst_tac^* & : & \isarmeth \\ |
949 |
eres_inst_tac^* & : & \isarmeth \\ |
|
950 |
dres_inst_tac^* & : & \isarmeth \\ |
|
951 |
forw_inst_tac^* & : & \isarmeth \\ |
|
952 |
subgoal_tac^* & : & \isarmeth \\ |
|
8515 | 953 |
\end{matharray} |
954 |
||
955 |
\railalias{applyend}{apply\_end} |
|
956 |
\railterm{applyend} |
|
957 |
||
8533 | 958 |
\railalias{resinsttac}{res\_inst\_tac} |
959 |
\railterm{resinsttac} |
|
960 |
||
961 |
\railalias{eresinsttac}{eres\_inst\_tac} |
|
962 |
\railterm{eresinsttac} |
|
963 |
||
964 |
\railalias{dresinsttac}{dres\_inst\_tac} |
|
965 |
\railterm{dresinsttac} |
|
966 |
||
967 |
\railalias{forwinsttac}{forw\_inst\_tac} |
|
968 |
\railterm{forwinsttac} |
|
969 |
||
970 |
\railalias{subgoaltac}{subgoal\_tac} |
|
971 |
\railterm{subgoaltac} |
|
972 |
||
8515 | 973 |
\begin{rail} |
8682 | 974 |
'apply' method comment? |
8515 | 975 |
; |
8682 | 976 |
applyend method comment? |
8515 | 977 |
; |
8682 | 978 |
'defer' nat? comment? |
8515 | 979 |
; |
8682 | 980 |
'prefer' nat comment? |
8515 | 981 |
; |
982 |
'tactic' text |
|
983 |
; |
|
8693 | 984 |
'insert' thmrefs |
985 |
; |
|
8533 | 986 |
( resinsttac | eresinsttac | dresinsttac | forwinsttac ) goalspec? ((name '=' term) + 'and') |
987 |
; |
|
988 |
subgoaltac goalspec? prop |
|
989 |
; |
|
8515 | 990 |
\end{rail} |
991 |
||
992 |
\begin{descr} |
|
8547 | 993 |
\item [$\isarkeyword{apply}~(m)$] applies proof method $m$ in initial |
994 |
position, but unlike $\PROOFNAME$ it retains ``$proof(prove)$'' mode. Thus |
|
8515 | 995 |
consecutive method applications may be given just as in tactic scripts. In |
996 |
order to complete the proof properly, any of the actual structured proof |
|
997 |
commands (e.g.\ ``$\DOT$'') has to be given eventually. |
|
998 |
||
999 |
Facts are passed to $m$ as indicated by the goal's forward-chain mode. |
|
1000 |
Common use of $\isarkeyword{apply}$ would be in a purely backward manner, |
|
1001 |
though. |
|
1002 |
\item [$\isarkeyword{apply_end}~(m)$] applies proof method $m$ as if in |
|
1003 |
terminal position. Basically, this simulates a multi-step tactic script for |
|
1004 |
$\QEDNAME$, but may be given anywhere within the proof body. |
|
1005 |
||
1006 |
No facts are passed to $m$. Furthermore, the static context is that of the |
|
1007 |
enclosing goal (as for actual $\QEDNAME$). Thus the proof method may not |
|
1008 |
refer to any assumptions introduced in the current body, for example. |
|
1009 |
\item [$\isarkeyword{defer}~n$ and $\isarkeyword{prefer}~n$] shuffle the list |
|
1010 |
of pending goals: $defer$ puts off goal $n$ to the end of the list ($n = 1$ |
|
1011 |
by default), while $prefer$ brings goal $n$ to the top. |
|
1012 |
\item [$\isarkeyword{back}$] does back-tracking over the result sequence of |
|
1013 |
the latest proof command.\footnote{Unlike the ML function \texttt{back} |
|
1014 |
\cite{isabelle-ref}, the Isar command does not search upwards for further |
|
1015 |
branch points.} Basically, any proof command may return multiple results. |
|
1016 |
\item [$tactic~text$] produces a proof method from any ML text of type |
|
8547 | 1017 |
\texttt{tactic}. Apart from the usual ML environment and the current |
8515 | 1018 |
implicit theory context, the ML code may refer to the following locally |
1019 |
bound values: |
|
1020 |
%%FIXME ttbox produces too much trailing space (why?) |
|
1021 |
{\footnotesize\begin{verbatim} |
|
1022 |
val ctxt : Proof.context |
|
1023 |
val facts : thm list |
|
1024 |
val thm : string -> thm |
|
1025 |
val thms : string -> thm list |
|
1026 |
\end{verbatim}} |
|
1027 |
Here \texttt{ctxt} refers to the current proof context, \texttt{facts} |
|
1028 |
indicates any current facts for forward-chaining, and |
|
1029 |
\texttt{thm}~/~\texttt{thms} retrieve named facts (including global |
|
1030 |
theorems) from the context. |
|
8696 | 1031 |
\item [$insert~\vec a$] inserts theorems as facts into all goals of the proof |
1032 |
state; the current facts indicated for forward chaining are ignored! |
|
8533 | 1033 |
\item [$res_inst_tac$ etc.] do resolution of rules with explicit |
1034 |
instantiation. This works the same way as the corresponding ML tactics, see |
|
1035 |
\cite[\S3]{isabelle-ref}. |
|
1036 |
||
1037 |
It is very important to note that the instantiations are read and |
|
1038 |
type-checked according to the dynamic goal state, rather than the static |
|
1039 |
proof context! In particular, locally fixed variables and term |
|
1040 |
abbreviations may not be included in the term specifications. |
|
8547 | 1041 |
\item [$subgoal_tac~\phi$] emulates the ML tactic of the same name, see |
8533 | 1042 |
\cite[\S3]{isabelle-ref}. Syntactically, the given proposition is handled |
1043 |
as the instantiations in $res_inst_tac$ etc. |
|
1044 |
||
1045 |
Note that the proper Isar command $\PRESUMENAME$ achieves a similar effect |
|
1046 |
as $subgoal_tac$. |
|
8515 | 1047 |
\end{descr} |
1048 |
||
1049 |
||
1050 |
\subsection{Meta-linguistic features} |
|
1051 |
||
1052 |
\indexisarcmd{oops} |
|
1053 |
\begin{matharray}{rcl} |
|
1054 |
\isarcmd{oops} & : & \isartrans{proof}{theory} \\ |
|
1055 |
\end{matharray} |
|
1056 |
||
1057 |
The $\OOPS$ command discontinues the current proof attempt, while considering |
|
1058 |
the partial proof text as properly processed. This is conceptually quite |
|
1059 |
different from ``faking'' actual proofs via $\SORRY$ (see |
|
1060 |
\S\ref{sec:proof-steps}): $\OOPS$ does not observe the proof structure at all, |
|
1061 |
but goes back right to the theory level. Furthermore, $\OOPS$ does not |
|
1062 |
produce any result theorem --- there is no claim to be able to complete the |
|
1063 |
proof anyhow. |
|
1064 |
||
1065 |
A typical application of $\OOPS$ is to explain Isar proofs \emph{within} the |
|
1066 |
system itself, in conjunction with the document preparation tools of Isabelle |
|
1067 |
described in \cite{isabelle-sys}. Thus partial or even wrong proof attempts |
|
1068 |
can be discussed in a logically sound manner. Note that the Isabelle {\LaTeX} |
|
1069 |
macros can be easily adapted to print something like ``$\dots$'' instead of an |
|
1070 |
``$\OOPS$'' keyword. |
|
1071 |
||
8547 | 1072 |
\medskip The $\OOPS$ command is undoable, unlike $\isarkeyword{kill}$ (see |
1073 |
\S\ref{sec:history}). The effect is to get back to the theory \emph{before} |
|
1074 |
the opening of the proof. |
|
8515 | 1075 |
|
1076 |
||
7134 | 1077 |
\section{Other commands} |
1078 |
||
8448 | 1079 |
\subsection{Diagnostics}\label{sec:diag} |
7134 | 1080 |
|
8485 | 1081 |
\indexisarcmd{pr}\indexisarcmd{thm}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{typ} |
1082 |
\indexisarcmd{print-facts}\indexisarcmd{print-binds} |
|
7134 | 1083 |
\begin{matharray}{rcl} |
8515 | 1084 |
\isarcmd{help}^* & : & \isarkeep{\cdot} \\ |
1085 |
\isarcmd{pr}^* & : & \isarkeep{\cdot} \\ |
|
1086 |
\isarcmd{thm}^* & : & \isarkeep{theory~|~proof} \\ |
|
1087 |
\isarcmd{term}^* & : & \isarkeep{theory~|~proof} \\ |
|
1088 |
\isarcmd{prop}^* & : & \isarkeep{theory~|~proof} \\ |
|
1089 |
\isarcmd{typ}^* & : & \isarkeep{theory~|~proof} \\ |
|
1090 |
\isarcmd{print_facts}^* & : & \isarkeep{proof} \\ |
|
1091 |
\isarcmd{print_binds}^* & : & \isarkeep{proof} \\ |
|
7134 | 1092 |
\end{matharray} |
1093 |
||
7335 | 1094 |
These commands are not part of the actual Isabelle/Isar syntax, but assist |
1095 |
interactive development. Also note that $undo$ does not apply here, since the |
|
1096 |
theory or proof configuration is not changed. |
|
1097 |
||
7134 | 1098 |
\begin{rail} |
8485 | 1099 |
'pr' modes? nat? |
7134 | 1100 |
; |
8485 | 1101 |
'thm' modes? thmrefs |
1102 |
; |
|
1103 |
'term' modes? term |
|
7134 | 1104 |
; |
8485 | 1105 |
'prop' modes? prop |
7134 | 1106 |
; |
8485 | 1107 |
'typ' modes? type |
1108 |
; |
|
1109 |
||
1110 |
modes: '(' (name + ) ')' |
|
7134 | 1111 |
; |
1112 |
\end{rail} |
|
1113 |
||
7167 | 1114 |
\begin{descr} |
8515 | 1115 |
\item [$\isarkeyword{help}$] prints a list of available language elements. |
1116 |
Note that methods and attributes depend on the current theory context. |
|
8485 | 1117 |
\item [$\isarkeyword{pr}~n$] prints the current top-level state, i.e.\ the |
1118 |
theory identifier or proof state. The latter includes the proof context, |
|
1119 |
current facts and goals. The optional argument $n$ affects the implicit |
|
1120 |
limit of goals to be displayed, which is initially 10. Omitting the limit |
|
1121 |
leaves the value unchanged. |
|
8547 | 1122 |
\item [$\isarkeyword{thm}~\vec a$] retrieves theorems from the current theory |
1123 |
or proof context. Note that any attributes included in the theorem |
|
7974 | 1124 |
specifications are applied to a temporary context derived from the current |
8547 | 1125 |
theory or proof; the result is discarded, i.e.\ attributes involved in $\vec |
1126 |
a$ do not have any permanent effect. |
|
7987 | 1127 |
\item [$\isarkeyword{term}~t$, $\isarkeyword{prop}~\phi$] read, type-check and |
1128 |
print terms or propositions according to the current theory or proof |
|
7895 | 1129 |
context; the inferred type of $t$ is output as well. Note that these |
1130 |
commands are also useful in inspecting the current environment of term |
|
1131 |
abbreviations. |
|
7974 | 1132 |
\item [$\isarkeyword{typ}~\tau$] reads and prints types of the meta-logic |
1133 |
according to the current theory or proof context. |
|
8379 | 1134 |
\item [$\isarkeyword{print_facts}$] prints any named facts of the current |
1135 |
context, including assumptions and local results. |
|
1136 |
\item [$\isarkeyword{print_binds}$] prints all term abbreviations present in |
|
1137 |
the context. |
|
8485 | 1138 |
\end{descr} |
1139 |
||
1140 |
The basic diagnostic commands above admit a list of $modes$ to be specified, |
|
1141 |
which is appended to the current print mode (see also \cite{isabelle-ref}). |
|
1142 |
Thus the output behavior may be modified according particular print mode |
|
1143 |
features. |
|
1144 |
||
1145 |
For example, $\isarkeyword{pr}~(latex~xsymbols~symbols)$ would print the |
|
1146 |
current proof state with mathematical symbols and special characters |
|
1147 |
represented in {\LaTeX} source, according to the Isabelle style |
|
8547 | 1148 |
\cite{isabelle-sys}. The resulting text can be directly pasted into a |
1149 |
\verb,\begin{isabelle},\dots\verb,\end{isabelle}, environment. Note that |
|
1150 |
$\isarkeyword{pr}~(latex)$ is sufficient to achieve the same output, if the |
|
1151 |
current Isabelle session has the other modes already activated, say due to |
|
1152 |
some particular user interface configuration such as Proof~General |
|
8510 | 1153 |
\cite{proofgeneral,Aspinall:TACAS:2000} with X-Symbol mode \cite{x-symbol}. |
8485 | 1154 |
|
1155 |
||
1156 |
\subsection{History commands}\label{sec:history} |
|
1157 |
||
1158 |
\indexisarcmd{undo}\indexisarcmd{redo}\indexisarcmd{kill} |
|
1159 |
\begin{matharray}{rcl} |
|
1160 |
\isarcmd{undo}^{{*}{*}} & : & \isarkeep{\cdot} \\ |
|
1161 |
\isarcmd{redo}^{{*}{*}} & : & \isarkeep{\cdot} \\ |
|
1162 |
\isarcmd{kill}^{{*}{*}} & : & \isarkeep{\cdot} \\ |
|
1163 |
\end{matharray} |
|
1164 |
||
1165 |
The Isabelle/Isar top-level maintains a two-stage history, for theory and |
|
1166 |
proof state transformation. Basically, any command can be undone using |
|
1167 |
$\isarkeyword{undo}$, excluding mere diagnostic elements. Its effect may be |
|
1168 |
revoked via $\isarkeyword{redo}$, unless the corresponding the |
|
1169 |
$\isarkeyword{undo}$ step has crossed the beginning of a proof or theory. The |
|
1170 |
$\isarkeyword{kill}$ command aborts the current history node altogether, |
|
1171 |
discontinuing a proof or even the whole theory. This operation is \emph{not} |
|
1172 |
undoable. |
|
1173 |
||
1174 |
\begin{warn} |
|
8547 | 1175 |
History commands should never be used with user interfaces such as |
1176 |
Proof~General \cite{proofgeneral,Aspinall:TACAS:2000}, which takes care of |
|
1177 |
stepping forth and back itself. Interfering by manual $\isarkeyword{undo}$, |
|
8510 | 1178 |
$\isarkeyword{redo}$, or even $\isarkeyword{kill}$ commands would quickly |
1179 |
result in utter confusion. |
|
8485 | 1180 |
\end{warn} |
1181 |
||
8515 | 1182 |
%FIXME remove |
1183 |
% \begin{descr} |
|
1184 |
% \item [$\isarkeyword{undo}$] revokes the latest state-transforming command. |
|
1185 |
% \item [$\isarkeyword{redo}$] undos the latest $\isarkeyword{undo}$. |
|
1186 |
% \item [$\isarkeyword{kill}$] aborts the current history level. |
|
1187 |
% \end{descr} |
|
8485 | 1188 |
|
8379 | 1189 |
|
7134 | 1190 |
\subsection{System operations} |
1191 |
||
7167 | 1192 |
\indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{use-thy}\indexisarcmd{use-thy-only} |
1193 |
\indexisarcmd{update-thy}\indexisarcmd{update-thy-only} |
|
7134 | 1194 |
\begin{matharray}{rcl} |
8515 | 1195 |
\isarcmd{cd}^* & : & \isarkeep{\cdot} \\ |
1196 |
\isarcmd{pwd}^* & : & \isarkeep{\cdot} \\ |
|
1197 |
\isarcmd{use_thy}^* & : & \isarkeep{\cdot} \\ |
|
1198 |
\isarcmd{use_thy_only}^* & : & \isarkeep{\cdot} \\ |
|
1199 |
\isarcmd{update_thy}^* & : & \isarkeep{\cdot} \\ |
|
1200 |
\isarcmd{update_thy_only}^* & : & \isarkeep{\cdot} \\ |
|
7134 | 1201 |
\end{matharray} |
1202 |
||
7167 | 1203 |
\begin{descr} |
7134 | 1204 |
\item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle |
1205 |
process. |
|
1206 |
\item [$\isarkeyword{pwd}~$] prints the current working directory. |
|
7175 | 1207 |
\item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$, |
7987 | 1208 |
$\isarkeyword{update_thy}$, $\isarkeyword{update_thy_only}$] load some |
7895 | 1209 |
theory given as $name$ argument. These commands are basically the same as |
7987 | 1210 |
the corresponding ML functions\footnote{The ML versions also change the |
1211 |
implicit theory context to that of the theory loaded.} (see also |
|
1212 |
\cite[\S1,\S6]{isabelle-ref}). Note that both the ML and Isar versions may |
|
1213 |
load new- and old-style theories alike. |
|
7167 | 1214 |
\end{descr} |
7134 | 1215 |
|
7987 | 1216 |
These system commands are scarcely used when working with the Proof~General |
1217 |
interface, since loading of theories is done fully transparently. |
|
7134 | 1218 |
|
8379 | 1219 |
|
7046 | 1220 |
%%% Local Variables: |
1221 |
%%% mode: latex |
|
1222 |
%%% TeX-master: "isar-ref" |
|
1223 |
%%% End: |