doc-src/IsarRef/pure.tex
author wenzelm
Wed Mar 08 23:43:11 2000 +0100 (2000-03-08)
changeset 8379 4c7659e98eb9
parent 8250 f4029c34adef
child 8448 e7df316491d4
permissions -rw-r--r--
tuned ML types;
improved translation functions;
'case' command;
'oops' command;
"Emulating tactic scripts";
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\chapter{Basic Isar Language Elements}\label{ch:pure-syntax}
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Subsequently, we introduce the main part of the basic Isar theory and proof
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commands as provided by Isabelle/Pure.  Chapter~\ref{ch:gen-tools} describes
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further Isar elements provided by generic tools and packages (such as the
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Simplifier) that are either part of Pure Isabelle or pre-loaded by most object
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logics.  Chapter~\ref{ch:hol-tools} refers to actual object-logic specific
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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, which facilitates porting of legacy proof scripts.
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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{\cdot}{theory} \\
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  \isarcmd{context}^* & : & \isartrans{\cdot}{theory} \\
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  \isarcmd{end} & : & \isartrans{theory}{\cdot} \\
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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 in a 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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  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 begin 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}.}
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\end{descr}
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Any markup command (except $\isarkeyword{text_raw}$) corresponds to a {\LaTeX}
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macro with the name prefixed by \verb,\isamarkup, (e.g.\ 
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\verb,\isamarkupchapter, for $\isarkeyword{chapter}$). The \railqtoken{text}
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argument is passed to that macro unchanged, i.e.\ further {\LaTeX} commands
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may be included here as well.
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\medskip 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 document markup
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elements just 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 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 canonical name $c_def$ for the definitional
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  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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\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 there is the
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  \texttt{output} flag given, all productions are added both to the input and
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  output 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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\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 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 augment the
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  Simplifier context, 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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   319
\begin{matharray}{rcl}
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   320
  \isarcmd{global} & : & \isartrans{theory}{theory} \\
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   321
  \isarcmd{local} & : & \isartrans{theory}{theory} \\
wenzelm@7134
   322
\end{matharray}
wenzelm@7134
   323
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   324
Isabelle organizes any kind of name declarations (of types, constants,
wenzelm@7895
   325
theorems etc.)  by hierarchically structured name spaces.  Normally the user
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   326
never has to control the behavior of name space entry by hand, yet the
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   327
following commands provide some way to do so.
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   328
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   329
\begin{descr}
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   330
\item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current
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   331
  name declaration mode.  Initially, theories start in $\isarkeyword{local}$
wenzelm@7167
   332
  mode, causing all names to be automatically qualified by the theory name.
wenzelm@7895
   333
  Changing this to $\isarkeyword{global}$ causes all names to be declared
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   334
  without the theory prefix, until $\isarkeyword{local}$ is declared again.
wenzelm@7167
   335
\end{descr}
wenzelm@7134
   336
wenzelm@7134
   337
wenzelm@7167
   338
\subsection{Incorporating ML code}\label{sec:ML}
wenzelm@7134
   339
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   340
\indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{ML-setup}\indexisarcmd{setup}
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   341
\begin{matharray}{rcl}
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   342
  \isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\
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   343
  \isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\
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   344
  \isarcmd{ML_setup} & : & \isartrans{theory}{theory} \\
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   345
  \isarcmd{setup} & : & \isartrans{theory}{theory} \\
wenzelm@7134
   346
\end{matharray}
wenzelm@7134
   347
wenzelm@7895
   348
\railalias{MLsetup}{ML\_setup}
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   349
\railterm{MLsetup}
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   350
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   351
\begin{rail}
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   352
  'use' name
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   353
  ;
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   354
  ('ML' | MLsetup | 'setup') text
wenzelm@7134
   355
  ;
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   356
\end{rail}
wenzelm@7134
   357
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   358
\begin{descr}
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   359
\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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   361
  but may not be modified.  Furthermore, the file name is checked with the
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   362
  $\isarkeyword{files}$ dependency declaration given in the theory header (see
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   363
  also \S\ref{sec:begin-thy}).
wenzelm@7466
   364
  
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   365
\item [$\isarkeyword{ML}~text$] executes ML commands from $text$.  The theory
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   366
  context is passed in the same way as for $\isarkeyword{use}$.
wenzelm@7895
   367
  
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   368
\item [$\isarkeyword{ML_setup}~text$] executes ML commands from $text$.  The
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   369
  theory context is passed down to the ML session, and fetched back
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   370
  afterwards.  Thus $text$ may actually change the theory as a side effect.
wenzelm@7895
   371
  
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\item [$\isarkeyword{setup}~text$] changes the current theory context by
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   373
  applying $text$, which refers to an ML expression of type
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   374
  \texttt{(theory~->~theory)~list}.  The $\isarkeyword{setup}$ command is the
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   375
  canonical way to initialize object-logic specific tools and packages written
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   376
  in ML.
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   377
\end{descr}
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   378
wenzelm@7134
   379
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   380
\subsection{Syntax translation functions}
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   381
wenzelm@8250
   382
\indexisarcmd{parse-ast-translation}\indexisarcmd{parse-translation}
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   383
\indexisarcmd{print-translation}\indexisarcmd{typed-print-translation}
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   384
\indexisarcmd{print-ast-translation}\indexisarcmd{token-translation}
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   385
\begin{matharray}{rcl}
wenzelm@8250
   386
  \isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\
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   387
  \isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\
wenzelm@8250
   388
  \isarcmd{print_translation} & : & \isartrans{theory}{theory} \\
wenzelm@8250
   389
  \isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\
wenzelm@8250
   390
  \isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\
wenzelm@8250
   391
  \isarcmd{token_translation} & : & \isartrans{theory}{theory} \\
wenzelm@8250
   392
\end{matharray}
wenzelm@7134
   393
wenzelm@8250
   394
Syntax translation functions written in ML admit almost arbitrary
wenzelm@8250
   395
manipulations of Isabelle's inner syntax.  Any of the above commands have a
wenzelm@8250
   396
single \railqtoken{text} argument that refers to an ML expression of
wenzelm@8379
   397
appropriate type.
wenzelm@8379
   398
wenzelm@8379
   399
\begin{ttbox}
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   400
val parse_ast_translation   : (string * (ast list -> ast)) list
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   401
val parse_translation       : (string * (term list -> term)) list
wenzelm@8379
   402
val print_translation       : (string * (term list -> term)) list
wenzelm@8379
   403
val typed_print_translation :
wenzelm@8379
   404
  (string * (bool -> typ -> term list -> term)) list
wenzelm@8379
   405
val print_ast_translation   : (string * (ast list -> ast)) list
wenzelm@8379
   406
val token_translation       :
wenzelm@8379
   407
  (string * string * (string -> string * real)) list
wenzelm@8379
   408
\end{ttbox}
wenzelm@8379
   409
See \cite[\S8]{isabelle-ref} for more information on syntax transformations.
wenzelm@7134
   410
wenzelm@7134
   411
wenzelm@7134
   412
\subsection{Oracles}
wenzelm@7134
   413
wenzelm@7134
   414
\indexisarcmd{oracle}
wenzelm@7134
   415
\begin{matharray}{rcl}
wenzelm@7134
   416
  \isarcmd{oracle} & : & \isartrans{theory}{theory} \\
wenzelm@7134
   417
\end{matharray}
wenzelm@7134
   418
wenzelm@7175
   419
Oracles provide an interface to external reasoning systems, without giving up
wenzelm@7175
   420
control completely --- each theorem carries a derivation object recording any
wenzelm@7175
   421
oracle invocation.  See \cite[\S6]{isabelle-ref} for more information.
wenzelm@7175
   422
wenzelm@7134
   423
\begin{rail}
wenzelm@7134
   424
  'oracle' name '=' text comment?
wenzelm@7134
   425
  ;
wenzelm@7134
   426
\end{rail}
wenzelm@7134
   427
wenzelm@7167
   428
\begin{descr}
wenzelm@7175
   429
\item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML
wenzelm@8379
   430
  function $text$, which has to be of type
wenzelm@8379
   431
  \texttt{Sign.sg~*~Object.T~->~term}.
wenzelm@7167
   432
\end{descr}
wenzelm@7134
   433
wenzelm@7134
   434
wenzelm@7134
   435
\section{Proof commands}
wenzelm@7134
   436
wenzelm@7987
   437
Proof commands perform transitions of Isar/VM machine configurations, which
wenzelm@7315
   438
are block-structured, consisting of a stack of nodes with three main
wenzelm@7335
   439
components: logical proof context, current facts, and open goals.  Isar/VM
wenzelm@7335
   440
transitions are \emph{typed} according to the following three three different
wenzelm@7335
   441
modes of operation:
wenzelm@7167
   442
\begin{descr}
wenzelm@7167
   443
\item [$proof(prove)$] means that a new goal has just been stated that is now
wenzelm@7167
   444
  to be \emph{proven}; the next command may refine it by some proof method
wenzelm@7895
   445
  (read: tactic), and enter a sub-proof to establish the actual result.
wenzelm@7167
   446
\item [$proof(state)$] is like an internal theory mode: the context may be
wenzelm@7987
   447
  augmented by \emph{stating} additional assumptions, intermediate results
wenzelm@7987
   448
  etc.
wenzelm@7895
   449
\item [$proof(chain)$] is intermediate between $proof(state)$ and
wenzelm@7987
   450
  $proof(prove)$: existing facts (i.e.\ the contents of the special ``$this$''
wenzelm@7987
   451
  register) have been just picked up in order to be used when refining the
wenzelm@7987
   452
  goal claimed next.
wenzelm@7167
   453
\end{descr}
wenzelm@7134
   454
wenzelm@7167
   455
wenzelm@7895
   456
\subsection{Proof markup commands}\label{sec:markup-prf}
wenzelm@7167
   457
wenzelm@7987
   458
\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsubsect}
wenzelm@7895
   459
\indexisarcmd{txt}\indexisarcmd{txt-raw}
wenzelm@7134
   460
\begin{matharray}{rcl}
wenzelm@8101
   461
  \isarcmd{sect} & : & \isartrans{proof}{proof} \\
wenzelm@8101
   462
  \isarcmd{subsect} & : & \isartrans{proof}{proof} \\
wenzelm@8101
   463
  \isarcmd{subsubsect} & : & \isartrans{proof}{proof} \\
wenzelm@8101
   464
  \isarcmd{txt} & : & \isartrans{proof}{proof} \\
wenzelm@8101
   465
  \isarcmd{txt_raw} & : & \isartrans{proof}{proof} \\
wenzelm@7134
   466
\end{matharray}
wenzelm@7134
   467
wenzelm@7895
   468
These markup commands for proof mode closely correspond to the ones of theory
wenzelm@7895
   469
mode (see \S\ref{sec:markup-thy}).  Note that $\isarkeyword{txt_raw}$ is
wenzelm@7895
   470
special in the same way as $\isarkeyword{text_raw}$.
wenzelm@7895
   471
wenzelm@7895
   472
\railalias{txtraw}{txt\_raw}
wenzelm@7895
   473
\railterm{txtraw}
wenzelm@7175
   474
wenzelm@7134
   475
\begin{rail}
wenzelm@7895
   476
  ('sect' | 'subsect' | 'subsubsect' | 'txt' | txtraw) text
wenzelm@7134
   477
  ;
wenzelm@7134
   478
\end{rail}
wenzelm@7134
   479
wenzelm@7134
   480
wenzelm@7315
   481
\subsection{Proof context}\label{sec:proof-context}
wenzelm@7134
   482
wenzelm@7315
   483
\indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def}
wenzelm@8379
   484
\indexisarcmd{case}
wenzelm@7134
   485
\begin{matharray}{rcl}
wenzelm@7134
   486
  \isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7134
   487
  \isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7134
   488
  \isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7134
   489
  \isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@8379
   490
  \isarcmd{case} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7134
   491
\end{matharray}
wenzelm@7134
   492
wenzelm@7315
   493
The logical proof context consists of fixed variables and assumptions.  The
wenzelm@7315
   494
former closely correspond to Skolem constants, or meta-level universal
wenzelm@7315
   495
quantification as provided by the Isabelle/Pure logical framework.
wenzelm@7315
   496
Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in
wenzelm@7987
   497
a local value that may be used in the subsequent proof as any other variable
wenzelm@7895
   498
or constant.  Furthermore, any result $\edrv \phi[x]$ exported from the
wenzelm@7987
   499
context will be universally closed wrt.\ $x$ at the outermost level: $\edrv
wenzelm@7987
   500
\All x \phi$ (this is expressed using Isabelle's meta-variables).
wenzelm@7315
   501
wenzelm@7315
   502
Similarly, introducing some assumption $\chi$ has two effects.  On the one
wenzelm@7315
   503
hand, a local theorem is created that may be used as a fact in subsequent
wenzelm@7895
   504
proof steps.  On the other hand, any result $\chi \drv \phi$ exported from the
wenzelm@7895
   505
context becomes conditional wrt.\ the assumption: $\edrv \chi \Imp \phi$.
wenzelm@7895
   506
Thus, solving an enclosing goal using such a result would basically introduce
wenzelm@7895
   507
a new subgoal stemming from the assumption.  How this situation is handled
wenzelm@7895
   508
depends on the actual version of assumption command used: while $\ASSUMENAME$
wenzelm@7895
   509
insists on solving the subgoal by unification with some premise of the goal,
wenzelm@7895
   510
$\PRESUMENAME$ leaves the subgoal unchanged in order to be proved later by the
wenzelm@7895
   511
user.
wenzelm@7315
   512
wenzelm@7319
   513
Local definitions, introduced by $\DEF{}{x \equiv t}$, are achieved by
wenzelm@7987
   514
combining $\FIX x$ with another version of assumption that causes any
wenzelm@7987
   515
hypothetical equation $x \equiv t$ to be eliminated by the reflexivity rule.
wenzelm@7987
   516
Thus, exporting some result $x \equiv t \drv \phi[x]$ yields $\edrv \phi[t]$.
wenzelm@7175
   517
wenzelm@8379
   518
\medskip Basically, Isar proof contexts have to be built up explicitly using
wenzelm@8379
   519
any of the above commands.  In typical verification tasks this can become hard
wenzelm@8379
   520
to manage, though, with a large number of local contexts emerging from case
wenzelm@8379
   521
analysis or induction, for example.  The $\CASENAME$ command provides a
wenzelm@8379
   522
shorthand to refer to certain parts of logical context symbolically.  Proof
wenzelm@8379
   523
methods may provide an environment of named ``cases'' of the form $c\colon
wenzelm@8379
   524
\vec x, \vec \chi$.  Then the effect of $\CASE{c}$ is exactly the same as
wenzelm@8379
   525
$\FIX{\vec x}~\ASSUME{c}{\vec\chi}$.
wenzelm@8379
   526
wenzelm@8379
   527
It is important to note that $\CASENAME$ does \emph{not} provide means to peek
wenzelm@8379
   528
at the current goal state, which is considered strictly non-observable in
wenzelm@8379
   529
Isar.  Instead, the cases considered here typically emerge in a canonical way
wenzelm@8379
   530
from certain pieces of specification that appears in the theory somewhere,
wenzelm@8379
   531
such as an inductive definition, or recursive function.  See \S\ref{sec:FIXME}
wenzelm@8379
   532
for more details of how this works in HOL.
wenzelm@8379
   533
wenzelm@7134
   534
\begin{rail}
wenzelm@7431
   535
  'fix' (vars + 'and') comment?
wenzelm@7134
   536
  ;
wenzelm@7315
   537
  ('assume' | 'presume') (assm comment? + 'and')
wenzelm@7134
   538
  ;
wenzelm@7175
   539
  'def' thmdecl? \\ var '==' term termpat? comment?
wenzelm@7134
   540
  ;
wenzelm@8379
   541
  'case' name
wenzelm@8379
   542
  ;
wenzelm@7134
   543
wenzelm@7134
   544
  var: name ('::' type)?
wenzelm@7134
   545
  ;
wenzelm@7458
   546
  vars: (name+) ('::' type)?
wenzelm@7431
   547
  ;
wenzelm@7315
   548
  assm: thmdecl? (prop proppat? +)
wenzelm@7315
   549
  ;
wenzelm@7134
   550
\end{rail}
wenzelm@7134
   551
wenzelm@7167
   552
\begin{descr}
wenzelm@7315
   553
\item [$\FIX{x}$] introduces a local \emph{arbitrary, but fixed} variable $x$.
wenzelm@7315
   554
\item [$\ASSUME{a}{\Phi}$ and $\PRESUME{a}{\Phi}$] introduce local theorems
wenzelm@7335
   555
  $\Phi$ by assumption.  Subsequent results applied to an enclosing goal
wenzelm@7895
   556
  (e.g.\ by $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ expects to be
wenzelm@7335
   557
  able to unify with existing premises in the goal, while $\PRESUMENAME$
wenzelm@7335
   558
  leaves $\Phi$ as new subgoals.
wenzelm@7335
   559
  
wenzelm@7335
   560
  Several lists of assumptions may be given (separated by
wenzelm@7895
   561
  $\isarkeyword{and}$); the resulting list of current facts consists of all of
wenzelm@7895
   562
  these concatenated.
wenzelm@7315
   563
\item [$\DEF{a}{x \equiv t}$] introduces a local (non-polymorphic) definition.
wenzelm@7315
   564
  In results exported from the context, $x$ is replaced by $t$.  Basically,
wenzelm@7987
   565
  $\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\ASSUME{}{x \equiv t}$, with the
wenzelm@7335
   566
  resulting hypothetical equation solved by reflexivity.
wenzelm@7431
   567
  
wenzelm@7431
   568
  The default name for the definitional equation is $x_def$.
wenzelm@8379
   569
\item [$\CASE{c}$] invokes local context $c\colon \vec x, \vec \chi$, as
wenzelm@8379
   570
  provided by an appropriate proof method.  This abbreviates $\FIX{\vec
wenzelm@8379
   571
    x}~\ASSUME{c}{\vec\chi}$.
wenzelm@7167
   572
\end{descr}
wenzelm@7167
   573
wenzelm@7895
   574
The special name $prems$\indexisarthm{prems} refers to all assumptions of the
wenzelm@7895
   575
current context as a list of theorems.
wenzelm@7315
   576
wenzelm@7167
   577
wenzelm@7167
   578
\subsection{Facts and forward chaining}
wenzelm@7167
   579
wenzelm@7167
   580
\indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with}
wenzelm@7167
   581
\begin{matharray}{rcl}
wenzelm@7167
   582
  \isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7167
   583
  \isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\
wenzelm@7167
   584
  \isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\
wenzelm@7167
   585
  \isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\
wenzelm@7167
   586
\end{matharray}
wenzelm@7167
   587
wenzelm@7319
   588
New facts are established either by assumption or proof of local statements.
wenzelm@7335
   589
Any fact will usually be involved in further proofs, either as explicit
wenzelm@7335
   590
arguments of proof methods or when forward chaining towards the next goal via
wenzelm@7335
   591
$\THEN$ (and variants).  Note that the special theorem name
wenzelm@7987
   592
$this$\indexisarthm{this} refers to the most recently established facts.
wenzelm@7167
   593
\begin{rail}
wenzelm@7167
   594
  'note' thmdef? thmrefs comment?
wenzelm@7167
   595
  ;
wenzelm@7167
   596
  'then' comment?
wenzelm@7167
   597
  ;
wenzelm@7167
   598
  ('from' | 'with') thmrefs comment?
wenzelm@7167
   599
  ;
wenzelm@7167
   600
\end{rail}
wenzelm@7167
   601
wenzelm@7167
   602
\begin{descr}
wenzelm@7175
   603
\item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result
wenzelm@7175
   604
  as $a$.  Note that attributes may be involved as well, both on the left and
wenzelm@7175
   605
  right hand sides.
wenzelm@7167
   606
\item [$\THEN$] indicates forward chaining by the current facts in order to
wenzelm@7895
   607
  establish the goal to be claimed next.  The initial proof method invoked to
wenzelm@7895
   608
  refine that will be offered the facts to do ``anything appropriate'' (cf.\ 
wenzelm@7895
   609
  also \S\ref{sec:proof-steps}).  For example, method $rule$ (see
wenzelm@7895
   610
  \S\ref{sec:pure-meth}) would typically do an elimination rather than an
wenzelm@7895
   611
  introduction.  Automatic methods usually insert the facts into the goal
wenzelm@7895
   612
  state before operation.
wenzelm@7335
   613
\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; thus $\THEN$ is
wenzelm@7458
   614
  equivalent to $\FROM{this}$.
wenzelm@7175
   615
\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward
wenzelm@7175
   616
  chaining is from earlier facts together with the current ones.
wenzelm@7167
   617
\end{descr}
wenzelm@7167
   618
wenzelm@7389
   619
Basic proof methods (such as $rule$, see \S\ref{sec:pure-meth}) expect
wenzelm@7895
   620
multiple facts to be given in their proper order, corresponding to a prefix of
wenzelm@7895
   621
the premises of the rule involved.  Note that positions may be easily skipped
wenzelm@7458
   622
using a form like $\FROM{\text{\texttt{_}}~a~b}$, for example.  This involves
wenzelm@7895
   623
the trivial rule $\PROP\psi \Imp \PROP\psi$, which is bound in Isabelle/Pure
wenzelm@7895
   624
as ``\texttt{_}'' (underscore).\indexisarthm{_@\texttt{_}}
wenzelm@7389
   625
wenzelm@7167
   626
wenzelm@7167
   627
\subsection{Goal statements}
wenzelm@7167
   628
wenzelm@7167
   629
\indexisarcmd{theorem}\indexisarcmd{lemma}
wenzelm@7167
   630
\indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus}
wenzelm@7167
   631
\begin{matharray}{rcl}
wenzelm@7167
   632
  \isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@7167
   633
  \isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@7987
   634
  \isarcmd{have} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
wenzelm@7987
   635
  \isarcmd{show} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
wenzelm@7167
   636
  \isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\
wenzelm@7167
   637
  \isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\
wenzelm@7167
   638
\end{matharray}
wenzelm@7167
   639
wenzelm@7175
   640
Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$
wenzelm@7895
   641
and $\LEMMANAME$.  New local goals may be claimed within proof mode as well.
wenzelm@7895
   642
Four variants are available, indicating whether the result is meant to solve
wenzelm@7987
   643
some pending goal or whether forward chaining is employed.
wenzelm@7175
   644
wenzelm@7167
   645
\begin{rail}
wenzelm@7167
   646
  ('theorem' | 'lemma') goal
wenzelm@7167
   647
  ;
wenzelm@7167
   648
  ('have' | 'show' | 'hence' | 'thus') goal
wenzelm@7167
   649
  ;
wenzelm@7167
   650
wenzelm@7167
   651
  goal: thmdecl? proppat comment?
wenzelm@7167
   652
  ;
wenzelm@7167
   653
\end{rail}
wenzelm@7167
   654
wenzelm@7167
   655
\begin{descr}
wenzelm@7335
   656
\item [$\THEOREM{a}{\phi}$] enters proof mode with $\phi$ as main goal,
wenzelm@7895
   657
  eventually resulting in some theorem $\turn \phi$ put back into the theory.
wenzelm@7987
   658
\item [$\LEMMA{a}{\phi}$] is similar to $\THEOREMNAME$, but tags the result as
wenzelm@7167
   659
  ``lemma''.
wenzelm@7335
   660
\item [$\HAVE{a}{\phi}$] claims a local goal, eventually resulting in a
wenzelm@7167
   661
  theorem with the current assumption context as hypotheses.
wenzelm@7335
   662
\item [$\SHOW{a}{\phi}$] is similar to $\HAVE{a}{\phi}$, but solves some
wenzelm@7895
   663
  pending goal with the result \emph{exported} into the corresponding context
wenzelm@7895
   664
  (cf.\ \S\ref{sec:proof-context}).
wenzelm@7895
   665
\item [$\HENCENAME$] abbreviates $\THEN~\HAVENAME$, i.e.\ claims a local goal
wenzelm@7895
   666
  to be proven by forward chaining the current facts.  Note that $\HENCENAME$
wenzelm@7895
   667
  is also equivalent to $\FROM{this}~\HAVENAME$.
wenzelm@7895
   668
\item [$\THUSNAME$] abbreviates $\THEN~\SHOWNAME$.  Note that $\THUSNAME$ is
wenzelm@7895
   669
  also equivalent to $\FROM{this}~\SHOWNAME$.
wenzelm@7167
   670
\end{descr}
wenzelm@7167
   671
wenzelm@7167
   672
wenzelm@7167
   673
\subsection{Initial and terminal proof steps}\label{sec:proof-steps}
wenzelm@7167
   674
wenzelm@7175
   675
\indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by}
wenzelm@7175
   676
\indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry}
wenzelm@7175
   677
\begin{matharray}{rcl}
wenzelm@7175
   678
  \isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\
wenzelm@7175
   679
  \isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\
wenzelm@7175
   680
  \isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm@7175
   681
  \isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm@7175
   682
  \isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm@7175
   683
  \isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm@7175
   684
\end{matharray}
wenzelm@7175
   685
wenzelm@7335
   686
Arbitrary goal refinement via tactics is considered harmful.  Consequently the
wenzelm@7335
   687
Isar framework admits proof methods to be invoked in two places only.
wenzelm@7167
   688
\begin{enumerate}
wenzelm@7175
   689
\item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated
wenzelm@7335
   690
  goal to a number of sub-goals that are to be solved later.  Facts are passed
wenzelm@7895
   691
  to $m@1$ for forward chaining, if so indicated by $proof(chain)$ mode.
wenzelm@7167
   692
  
wenzelm@7987
   693
\item A \emph{terminal} conclusion step $\QED{m@2}$ is intended to solve
wenzelm@7987
   694
  remaining goals.  No facts are passed to $m@2$.
wenzelm@7167
   695
\end{enumerate}
wenzelm@7167
   696
wenzelm@7335
   697
The only other proper way to affect pending goals is by $\SHOWNAME$ (or
wenzelm@7335
   698
$\THUSNAME$), which involves an explicit statement of what is to be solved.
wenzelm@7167
   699
wenzelm@7175
   700
\medskip
wenzelm@7175
   701
wenzelm@7167
   702
Also note that initial proof methods should either solve the goal completely,
wenzelm@7895
   703
or constitute some well-understood reduction to new sub-goals.  Arbitrary
wenzelm@7895
   704
automatic proof tools that are prone leave a large number of badly structured
wenzelm@7895
   705
sub-goals are no help in continuing the proof document in any intelligible
wenzelm@7987
   706
way.
wenzelm@7987
   707
%FIXME
wenzelm@7987
   708
%A more appropriate technique would be to $\SHOWNAME$ some non-trivial
wenzelm@7987
   709
%reduction as an explicit rule, which is solved completely by some automated
wenzelm@7987
   710
%method, and then applied to some pending goal.
wenzelm@7167
   711
wenzelm@7175
   712
\medskip
wenzelm@7175
   713
wenzelm@7175
   714
Unless given explicitly by the user, the default initial method is
wenzelm@7175
   715
``$default$'', which is usually set up to apply a single standard elimination
wenzelm@7458
   716
or introduction rule according to the topmost symbol involved.  There is no
wenzelm@7987
   717
separate default terminal method.  In any case, any goals left after that are
wenzelm@7987
   718
solved by assumption as the very last step.
wenzelm@7167
   719
wenzelm@7167
   720
\begin{rail}
wenzelm@7167
   721
  'proof' interest? meth? comment?
wenzelm@7167
   722
  ;
wenzelm@7167
   723
  'qed' meth? comment?
wenzelm@7167
   724
  ;
wenzelm@7167
   725
  'by' meth meth? comment?
wenzelm@7167
   726
  ;
wenzelm@7167
   727
  ('.' | '..' | 'sorry') comment?
wenzelm@7167
   728
  ;
wenzelm@7167
   729
wenzelm@7167
   730
  meth: method interest?
wenzelm@7167
   731
  ;
wenzelm@7167
   732
\end{rail}
wenzelm@7167
   733
wenzelm@7167
   734
\begin{descr}
wenzelm@7335
   735
\item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for
wenzelm@7335
   736
  forward chaining are passed if so indicated by $proof(chain)$ mode.
wenzelm@7335
   737
\item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and
wenzelm@7895
   738
  concludes the sub-proof by assumption.  If the goal had been $\SHOWNAME$ (or
wenzelm@7895
   739
  $\THUSNAME$), some pending sub-goal is solved as well by the rule resulting
wenzelm@7895
   740
  from the result \emph{exported} into the enclosing goal context.  Thus
wenzelm@7895
   741
  $\QEDNAME$ may fail for two reasons: either $m@2$ fails, or the resulting
wenzelm@7895
   742
  rule does not fit to any pending goal\footnote{This includes any additional
wenzelm@7895
   743
    ``strong'' assumptions as introduced by $\ASSUMENAME$.} of the enclosing
wenzelm@7895
   744
  context.  Debugging such a situation might involve temporarily changing
wenzelm@7895
   745
  $\SHOWNAME$ into $\HAVENAME$, or weakening the local context by replacing
wenzelm@7895
   746
  some occurrences of $\ASSUMENAME$ by $\PRESUMENAME$.
wenzelm@7895
   747
\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}\index{proof!terminal}; it
wenzelm@7987
   748
  abbreviates $\PROOF{m@1}~\QED{m@2}$, with backtracking across both methods,
wenzelm@7987
   749
  though.  Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done
wenzelm@7895
   750
  by expanding its definition; in many cases $\PROOF{m@1}$ is already
wenzelm@7175
   751
  sufficient to see what is going wrong.
wenzelm@7895
   752
\item [``$\DDOT$''] is a \emph{default proof}\index{proof!default}; it
wenzelm@7895
   753
  abbreviates $\BY{default}$.
wenzelm@7895
   754
\item [``$\DOT$''] is a \emph{trivial proof}\index{proof!trivial}; it
wenzelm@8195
   755
  abbreviates $\BY{this}$.
wenzelm@8379
   756
\item [$\SORRY$] is a \emph{fake proof}\index{proof!fake}; provided that the
wenzelm@8379
   757
  \texttt{quick_and_dirty} flag is enabled, $\SORRY$ pretends to solve the
wenzelm@8379
   758
  goal without further ado.  Of course, the result is a fake theorem only,
wenzelm@8379
   759
  involving some oracle in its internal derivation object (this is indicated
wenzelm@8379
   760
  as ``$[!]$'' in the printed result).  The main application of $\SORRY$ is to
wenzelm@8379
   761
  support experimentation and top-down proof development.
wenzelm@7315
   762
\end{descr}
wenzelm@7315
   763
wenzelm@7315
   764
wenzelm@7315
   765
\subsection{Term abbreviations}\label{sec:term-abbrev}
wenzelm@7315
   766
wenzelm@7315
   767
\indexisarcmd{let}
wenzelm@7315
   768
\begin{matharray}{rcl}
wenzelm@7315
   769
  \isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7315
   770
  \isarkeyword{is} & : & syntax \\
wenzelm@7315
   771
\end{matharray}
wenzelm@7315
   772
wenzelm@7315
   773
Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements,
wenzelm@7987
   774
or by annotating assumptions or goal statements with a list of patterns
wenzelm@7987
   775
$\ISS{p@1\;\dots}{p@n}$.  In both cases, higher-order matching is invoked to
wenzelm@7987
   776
bind extra-logical term variables, which may be either named schematic
wenzelm@7987
   777
variables of the form $\Var{x}$, or nameless dummies ``\texttt{_}''
wenzelm@7987
   778
(underscore).\indexisarvar{_@\texttt{_}} Note that in the $\LETNAME$ form the
wenzelm@7987
   779
patterns occur on the left-hand side, while the $\ISNAME$ patterns are in
wenzelm@7987
   780
postfix position.
wenzelm@7315
   781
wenzelm@7319
   782
Term abbreviations are quite different from actual local definitions as
wenzelm@7319
   783
introduced via $\DEFNAME$ (see \S\ref{sec:proof-context}).  The latter are
wenzelm@7315
   784
visible within the logic as actual equations, while abbreviations disappear
wenzelm@7315
   785
during the input process just after type checking.
wenzelm@7315
   786
wenzelm@7315
   787
\begin{rail}
wenzelm@7315
   788
  'let' ((term + 'as') '=' term comment? + 'and')
wenzelm@7315
   789
  ;  
wenzelm@7315
   790
\end{rail}
wenzelm@7315
   791
wenzelm@7315
   792
The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or
wenzelm@7315
   793
\railnonterm{proppat} (see \S\ref{sec:term-pats}).
wenzelm@7315
   794
wenzelm@7315
   795
\begin{descr}
wenzelm@7315
   796
\item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$
wenzelm@7315
   797
  by simultaneous higher-order matching against terms $\vec t$.
wenzelm@7315
   798
\item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the
wenzelm@7315
   799
  preceding statement.  Also note that $\ISNAME$ is not a separate command,
wenzelm@7315
   800
  but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.).
wenzelm@7315
   801
\end{descr}
wenzelm@7315
   802
wenzelm@7988
   803
A few \emph{automatic} term abbreviations\index{term abbreviations} for goals
wenzelm@7988
   804
and facts are available as well.  For any open goal,
wenzelm@7466
   805
$\Var{thesis_prop}$\indexisarvar{thesis-prop} refers to the full proposition
wenzelm@7466
   806
(which may be a rule), $\Var{thesis_concl}$\indexisarvar{thesis-concl} to its
wenzelm@7466
   807
(atomic) conclusion, and $\Var{thesis}$\indexisarvar{thesis} to its
wenzelm@7335
   808
object-logical statement.  The latter two abstract over any meta-level
wenzelm@7987
   809
parameters.
wenzelm@7315
   810
wenzelm@7466
   811
Fact statements resulting from assumptions or finished goals are bound as
wenzelm@7466
   812
$\Var{this_prop}$\indexisarvar{this-prop},
wenzelm@7466
   813
$\Var{this_concl}$\indexisarvar{this-concl}, and
wenzelm@7466
   814
$\Var{this}$\indexisarvar{this}, similar to $\Var{thesis}$ above.  In case
wenzelm@7466
   815
$\Var{this}$ refers to an object-logic statement that is an application
wenzelm@7895
   816
$f(t)$, then $t$ is bound to the special text variable
wenzelm@7466
   817
``$\dots$''\indexisarvar{\dots} (three dots).  The canonical application of
wenzelm@7987
   818
the latter are calculational proofs (see \S\ref{sec:calculation}).
wenzelm@7315
   819
wenzelm@7315
   820
wenzelm@7134
   821
\subsection{Block structure}
wenzelm@7134
   822
wenzelm@7397
   823
\indexisarcmd{next}\indexisarcmd{\{\{}\indexisarcmd{\}\}}
wenzelm@7397
   824
\begin{matharray}{rcl}
wenzelm@7397
   825
  \isarcmd{next} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7974
   826
  \BG & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7974
   827
  \EN & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@7397
   828
\end{matharray}
wenzelm@7397
   829
wenzelm@7167
   830
While Isar is inherently block-structured, opening and closing blocks is
wenzelm@7167
   831
mostly handled rather casually, with little explicit user-intervention.  Any
wenzelm@7167
   832
local goal statement automatically opens \emph{two} blocks, which are closed
wenzelm@7167
   833
again when concluding the sub-proof (by $\QEDNAME$ etc.).  Sections of
wenzelm@7895
   834
different context within a sub-proof may be switched via $\isarkeyword{next}$,
wenzelm@7895
   835
which is just a single block-close followed by block-open again.  Thus the
wenzelm@7987
   836
effect of $\isarkeyword{next}$ to reset the local proof context. There is no
wenzelm@7987
   837
goal focus involved here!
wenzelm@7167
   838
wenzelm@7175
   839
For slightly more advanced applications, there are explicit block parentheses
wenzelm@7895
   840
as well.  These typically achieve a stronger forward style of reasoning.
wenzelm@7167
   841
wenzelm@7167
   842
\begin{descr}
wenzelm@7167
   843
\item [$\isarkeyword{next}$] switches to a fresh block within a sub-proof,
wenzelm@7895
   844
  resetting the local context to the initial one.
wenzelm@7167
   845
\item [$\isarkeyword{\{\{}$ and $\isarkeyword{\}\}}$] explicitly open and
wenzelm@7895
   846
  close blocks.  Any current facts pass through ``$\isarkeyword{\{\{}$''
wenzelm@7895
   847
  unchanged, while ``$\isarkeyword{\}\}}$'' causes any result to be
wenzelm@7895
   848
  \emph{exported} into the enclosing context.  Thus fixed variables are
wenzelm@7895
   849
  generalized, assumptions discharged, and local definitions unfolded (cf.\ 
wenzelm@7895
   850
  \S\ref{sec:proof-context}).  There is no difference of $\ASSUMENAME$ and
wenzelm@7895
   851
  $\PRESUMENAME$ in this mode of forward reasoning --- in contrast to plain
wenzelm@7895
   852
  backward reasoning with the result exported at $\SHOWNAME$ time.
wenzelm@7167
   853
\end{descr}
wenzelm@7134
   854
wenzelm@7134
   855
wenzelm@7134
   856
\section{Other commands}
wenzelm@7134
   857
wenzelm@7134
   858
\subsection{Diagnostics}
wenzelm@7134
   859
wenzelm@7974
   860
\indexisarcmd{thm}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{typ}
wenzelm@8379
   861
\indexisarcmd{print-facts}\indexisarcmd{print-binds}\indexisarcmd{print-cases}
wenzelm@7134
   862
\begin{matharray}{rcl}
wenzelm@7974
   863
  \isarcmd{thm} & : & \isarkeep{theory~|~proof} \\
wenzelm@7134
   864
  \isarcmd{term} & : & \isarkeep{theory~|~proof} \\
wenzelm@7134
   865
  \isarcmd{prop} & : & \isarkeep{theory~|~proof} \\
wenzelm@7974
   866
  \isarcmd{typ} & : & \isarkeep{theory~|~proof} \\
wenzelm@8379
   867
  \isarcmd{print_facts} & : & \isarkeep{proof} \\
wenzelm@8379
   868
  \isarcmd{print_binds} & : & \isarkeep{proof} \\
wenzelm@8379
   869
  \isarcmd{print_cases} & : & \isarkeep{proof} \\
wenzelm@7134
   870
\end{matharray}
wenzelm@7134
   871
wenzelm@7335
   872
These commands are not part of the actual Isabelle/Isar syntax, but assist
wenzelm@7335
   873
interactive development.  Also note that $undo$ does not apply here, since the
wenzelm@7335
   874
theory or proof configuration is not changed.
wenzelm@7335
   875
wenzelm@7134
   876
\begin{rail}
wenzelm@7974
   877
  'thm' thmrefs
wenzelm@7134
   878
  ;
wenzelm@7134
   879
  'term' term
wenzelm@7134
   880
  ;
wenzelm@7134
   881
  'prop' prop
wenzelm@7134
   882
  ;
wenzelm@7974
   883
  'typ' type
wenzelm@7134
   884
  ;
wenzelm@7134
   885
\end{rail}
wenzelm@7134
   886
wenzelm@7167
   887
\begin{descr}
wenzelm@7974
   888
\item [$\isarkeyword{thm}~thms$] retrieves lists of theorems from the current
wenzelm@7974
   889
  theory or proof context.  Note that any attributes included in the theorem
wenzelm@7974
   890
  specifications are applied to a temporary context derived from the current
wenzelm@7974
   891
  theory or proof; the result is discarded, i.e.\ attributes involved in
wenzelm@7974
   892
  $thms$ do not have any permanent effect.
wenzelm@7987
   893
\item [$\isarkeyword{term}~t$, $\isarkeyword{prop}~\phi$] read, type-check and
wenzelm@7987
   894
  print terms or propositions according to the current theory or proof
wenzelm@7895
   895
  context; the inferred type of $t$ is output as well.  Note that these
wenzelm@7895
   896
  commands are also useful in inspecting the current environment of term
wenzelm@7895
   897
  abbreviations.
wenzelm@7974
   898
\item [$\isarkeyword{typ}~\tau$] reads and prints types of the meta-logic
wenzelm@7974
   899
  according to the current theory or proof context.
wenzelm@8379
   900
\item [$\isarkeyword{print_facts}$] prints any named facts of the current
wenzelm@8379
   901
  context, including assumptions and local results.
wenzelm@8379
   902
\item [$\isarkeyword{print_binds}$] prints all term abbreviations present in
wenzelm@8379
   903
  the context.
wenzelm@8379
   904
\item [$\isarkeyword{print_cases}$] prints all local contexts (also known as
wenzelm@8379
   905
  ``cases'') of the current goal context.
wenzelm@7167
   906
\end{descr}
wenzelm@7134
   907
wenzelm@7134
   908
wenzelm@8379
   909
\subsection{Meta-linguistic features}
wenzelm@8379
   910
wenzelm@8379
   911
\indexisarcmd{oops}
wenzelm@8379
   912
\begin{matharray}{rcl}
wenzelm@8379
   913
  \isarcmd{oops}^* & : & \isartrans{proof}{theory} \\
wenzelm@8379
   914
\end{matharray}
wenzelm@8379
   915
wenzelm@8379
   916
The $\OOPS$ command discontinues the current proof attempt, while considering
wenzelm@8379
   917
the partial proof text as properly processed.  This is conceptually quite
wenzelm@8379
   918
different from ``faking'' actual proofs via $\SORRY$ (see
wenzelm@8379
   919
\S\ref{sec:proof-steps}): $\OOPS$ does not observe the proof structure at all,
wenzelm@8379
   920
but goes back right to the theory level.  Furthermore, $\OOPS$ does not
wenzelm@8379
   921
produce any result theorem --- there is no claim to be able to complete the
wenzelm@8379
   922
proof anyhow.
wenzelm@8379
   923
wenzelm@8379
   924
A typical application of $\OOPS$ is to explain Isar proofs \emph{within} the
wenzelm@8379
   925
system itself, in conjunction with the document preparation tools of Isabelle
wenzelm@8379
   926
described in \cite{isabelle-sys}.  Thus partial or even wrong proof attempts
wenzelm@8379
   927
can be discussed in a logically sound manner.  Note that the Isabelle {\LaTeX}
wenzelm@8379
   928
macros can be easily adapted to print something like ``$\dots$'' instead of an
wenzelm@8379
   929
``$\OOPS$'' keyword.
wenzelm@8379
   930
wenzelm@8379
   931
wenzelm@7134
   932
\subsection{System operations}
wenzelm@7134
   933
wenzelm@7167
   934
\indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{use-thy}\indexisarcmd{use-thy-only}
wenzelm@7167
   935
\indexisarcmd{update-thy}\indexisarcmd{update-thy-only}
wenzelm@7134
   936
\begin{matharray}{rcl}
wenzelm@7134
   937
  \isarcmd{cd} & : & \isarkeep{\cdot} \\
wenzelm@7134
   938
  \isarcmd{pwd} & : & \isarkeep{\cdot} \\
wenzelm@7134
   939
  \isarcmd{use_thy} & : & \isarkeep{\cdot} \\
wenzelm@7134
   940
  \isarcmd{use_thy_only} & : & \isarkeep{\cdot} \\
wenzelm@7134
   941
  \isarcmd{update_thy} & : & \isarkeep{\cdot} \\
wenzelm@7134
   942
  \isarcmd{update_thy_only} & : & \isarkeep{\cdot} \\
wenzelm@7134
   943
\end{matharray}
wenzelm@7134
   944
wenzelm@7167
   945
\begin{descr}
wenzelm@7134
   946
\item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle
wenzelm@7134
   947
  process.
wenzelm@7134
   948
\item [$\isarkeyword{pwd}~$] prints the current working directory.
wenzelm@7175
   949
\item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$,
wenzelm@7987
   950
  $\isarkeyword{update_thy}$, $\isarkeyword{update_thy_only}$] load some
wenzelm@7895
   951
  theory given as $name$ argument.  These commands are basically the same as
wenzelm@7987
   952
  the corresponding ML functions\footnote{The ML versions also change the
wenzelm@7987
   953
    implicit theory context to that of the theory loaded.}  (see also
wenzelm@7987
   954
  \cite[\S1,\S6]{isabelle-ref}).  Note that both the ML and Isar versions may
wenzelm@7987
   955
  load new- and old-style theories alike.
wenzelm@7167
   956
\end{descr}
wenzelm@7134
   957
wenzelm@7987
   958
These system commands are scarcely used when working with the Proof~General
wenzelm@7987
   959
interface, since loading of theories is done fully transparently.
wenzelm@7134
   960
wenzelm@8379
   961
wenzelm@8379
   962
\subsection{Emulating tactic scripts}
wenzelm@8379
   963
wenzelm@8379
   964
The following elements emulate unstructured tactic scripts to some extent.
wenzelm@8379
   965
While these are anathema for writing proper Isar proof documents, they might
wenzelm@8379
   966
come in handy for interactive exploration and debugging.
wenzelm@8379
   967
wenzelm@8379
   968
\indexisarcmd{apply}\indexisarcmd{apply-end}\indexisarcmd{back}\indexisarmeth{tactic}
wenzelm@8379
   969
\begin{matharray}{rcl}
wenzelm@8379
   970
  \isarcmd{apply} & : & \isartrans{proof(prove)}{proof(prove)} \\
wenzelm@8379
   971
  \isarcmd{apply_end}^* & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm@8379
   972
  tactic & : & \isarmeth \\
wenzelm@8379
   973
  \isarcmd{back}^* & : & \isartrans{proof}{proof} \\
wenzelm@8379
   974
\end{matharray}
wenzelm@8379
   975
wenzelm@8379
   976
\railalias{applyend}{apply\_end}
wenzelm@8379
   977
\railterm{applyend}
wenzelm@8379
   978
wenzelm@8379
   979
\begin{rail}
wenzelm@8379
   980
  'apply' method
wenzelm@8379
   981
  ;
wenzelm@8379
   982
  applyend method
wenzelm@8379
   983
  ;
wenzelm@8379
   984
  'tactic' text
wenzelm@8379
   985
  ;
wenzelm@8379
   986
  'back'
wenzelm@8379
   987
  ;
wenzelm@8379
   988
\end{rail}
wenzelm@8379
   989
wenzelm@8379
   990
\begin{descr}
wenzelm@8379
   991
\item [$\isarkeyword{apply}~(m)$] applies proof method $m$ in an initial
wenzelm@8379
   992
  position, but retains ``$prove$'' mode (unlike $\PROOFNAME$).  Thus
wenzelm@8379
   993
  consecutive method applications may be given just as in tactic scripts.  In
wenzelm@8379
   994
  order to complete the proof properly, any of the actual structured proof
wenzelm@8379
   995
  commands (e.g.\ ``$\DOT$'') has to be given eventually.
wenzelm@8379
   996
  
wenzelm@8379
   997
  Facts are passed to $m$ as indicated by the goal's forward-chain mode.
wenzelm@8379
   998
  Common use of $\isarkeyword{apply}$ would be in a purely backward manner,
wenzelm@8379
   999
  though.
wenzelm@8379
  1000
\item [$\isarkeyword{apply_end}~(m)$] applies proof method $m$ as if in
wenzelm@8379
  1001
  terminal position.  Basically, this simulates a multi-step tactic script for
wenzelm@8379
  1002
  $\QEDNAME$, but may be given anywhere within the proof body.
wenzelm@8379
  1003
  
wenzelm@8379
  1004
  No facts are passed to $m$.  Furthermore, the static context is that of the
wenzelm@8379
  1005
  enclosing goal (as for actual $\QEDNAME$).  Thus the proof method may not
wenzelm@8379
  1006
  refer to any assumptions introduced in the current body, for example.
wenzelm@8379
  1007
\item [$tactic~text$] produces a proof method from any ML text of type
wenzelm@8379
  1008
  \texttt{tactic}.  Apart from the usual ML environment, and the current
wenzelm@8379
  1009
  implicit theory context, the ML code may refer to the following locally
wenzelm@8379
  1010
  bound values:
wenzelm@8379
  1011
  \begin{ttbox}
wenzelm@8379
  1012
val ctxt  : Proof.context
wenzelm@8379
  1013
val facts : thm list
wenzelm@8379
  1014
val thm   : string -> thm
wenzelm@8379
  1015
val thms  : string -> thm list
wenzelm@8379
  1016
  \end{ttbox}
wenzelm@8379
  1017
  Here \texttt{ctxt} refers to the current proof context, \texttt{facts}
wenzelm@8379
  1018
  indicates any current facts for forward-chaining, and
wenzelm@8379
  1019
  \texttt{thm}~/~\texttt{thms} retrieve named facts (including global
wenzelm@8379
  1020
  theorems) from the context.
wenzelm@8379
  1021
\item [$\isarkeyword{back}$] does back-tracking over the result sequence of
wenzelm@8379
  1022
  the latest proof command.\footnote{Unlike the ML function \texttt{back}
wenzelm@8379
  1023
    \cite{isabelle-ref}, the Isar command does not search upwards for further
wenzelm@8379
  1024
    branch points.} Basically, any proof command may return multiple results.
wenzelm@8379
  1025
\end{descr}
wenzelm@8379
  1026
wenzelm@8379
  1027
wenzelm@7046
  1028
%%% Local Variables: 
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  1029
%%% mode: latex
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  1030
%%% TeX-master: "isar-ref"
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  1031
%%% End: