doc-src/IsarRef/pure.tex
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\chapter{Basic Isar 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 as provided by generic tools and packages that are
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either part of Pure Isabelle, or pre-loaded by most object logics (such as the
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Simplifier).  See chapter~\ref{ch:hol-tools} for actual object-logic specific
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elements (for 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} (indicated by $^*$).  Some proof methods and
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attributes introduced later may be classified as improper as well.  Improper
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Isar language elements might be helpful when developing proof documents, while
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their use is strongly discouraged for the final version.  Typical examples are
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diagnostic commands that print terms or theorems according to the current
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context; other commands even emulate old-style tactical theorem proving, which
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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{theory}\indexisarcmd{end}\indexisarcmd{context}
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\begin{matharray}{rcl}
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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 actual theory specifications and proofs are handled
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uniformly --- occasionally definitional mechanisms even require some manual
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proof.  In contrast, ``old-style'' Isabelle theories support batch processing
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only, with the proof scripts collected in separate ML files.
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The first command of any theory has to be $\THEORY$, starting a new theory
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based on the merge of existing ones.  The theory context may be also changed
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by $\CONTEXT$ without creating a new theory.  In both cases, $\END$ concludes
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the theory development; it has to be the very last command of any proper
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theory file.
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\begin{rail}
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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 [$\THEORY~A = B@1 + \cdots + B@n$] commences a new theory $A$ based on
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  existing ones $B@1 + \cdots + B@n$.  Isabelle's theory loader system ensures
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  that any of the base theories are properly loaded (and fully up-to-date when
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  $\THEORY$ is executed interactively).  The optional $\isarkeyword{files}$
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  specification declares additional dependencies on ML files.  Unless put in
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  parentheses, any file will be loaded immediately via $\isarcmd{use}$ (see
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  also \S\ref{sec:ML}).
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\item [$\CONTEXT~B$] enters an existing theory context $B$, basically in
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  read-only mode, so only a limited set of commands may be performed.  Just as
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  for $\THEORY$, the theory loader ensures that $B$ is 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, instead the theory definition
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  itself has to be retracted.
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\end{descr}
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\subsection{Formal comments}\label{sec:formal-cmt-thy}
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\indexisarcmd{title}\indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection}
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\indexisarcmd{subsubsection}\indexisarcmd{text}
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\begin{matharray}{rcl}
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  \isarcmd{title} & : & \isartrans{theory}{theory} \\
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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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\end{matharray}
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There are several commands to include \emph{formal comments} in theory
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specification (a few more are available for proofs, see
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\S\ref{sec:formal-cmt-prf}).  In contrast to source-level comments
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\verb|(*|\dots\verb|*)|, which are stripped at the lexical level, any text
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given as formal comment is meant to be part of the actual document.
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Consequently, it would be included in the final printed version.
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Apart from plain prose, formal comments may also refer to logical entities of
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the theory context (types, terms, theorems etc.).  Proper processing of the
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text would then include some further consistency checks with the items
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declared in the current theory, e.g.\ type-checking of included
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terms.\footnote{The current version of Isabelle/Isar does not process formal
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  comments in any such way.  This will be available as part of the automatic
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  theory and proof document preparation system (using (PDF){\LaTeX}) that is
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  planned for the near future.}
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\begin{rail}
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  'title' text text? text?
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  ;
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  ('chapter' | 'section' | 'subsection' | 'subsubsection' | 'text') text
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  ;
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\end{rail}
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\begin{descr}
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\item [$\isarkeyword{title}~title~author~date$] specifies the document title
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  just as in typical {\LaTeX} documents.
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\item [$\isarkeyword{chapter}~text$, $\isarkeyword{section}~text$,
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  $\isarkeyword{subsection}~text$, and $\isarkeyword{subsubsection}~text$]
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  mark chapter and section headings.
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\item [$\TEXT~text$] specifies an actual body of prose text, including
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  references to formal entities.\footnote{The latter feature is not yet
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    exploited.  Nevertheless, any text of the form \texttt{\at\{\dots\}}
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    should be considered as reserved for future use.}
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\end{descr}
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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 [$\isarkeyword{classes}~c<\vec c ~\dots$] declares class $c$ to be a
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  subclass of existing classes $\vec c$.  Cyclic class structures are ruled
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  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
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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 input without sort constraints.  Usually, the default
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  sort would be only changed when defining new logics.
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\end{descr}
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\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 [$\TYPES~(\vec\alpha)t = \tau~\dots$] 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, as may be observed when printing terms or
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  theorems.
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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 some logics such as
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  Isabelle/HOL provide their own version of $\isarkeyword{typedecl}$.
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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 [$\isarkeyword{arities}~t::(\vec s)s~\dots$] augments Isabelle's
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  order-sorted signature of types by new type constructor arities.  This is
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  done axiomatically!  The $\isarkeyword{instance}$ command (see
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  \S\ref{sec:axclass}) provides a way introduce proven type arities.
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\end{descr}
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\subsection{Constants and simple definitions}
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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' (thmdecl? 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 [$\CONSTS~c::\sigma~\dots$] declares constant $c$ to have any instance
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  of type scheme $\sigma$.  The optional mixfix annotations may attach
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  concrete syntax constants.
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\item [$\DEFS~name: eqn~\dots$] introduces $eqn$ as a definitional axiom for
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  some existing constant.  See \cite[\S6]{isabelle-ref} for more details on
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  the form of equations admitted as constant definitions.
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\item [$\isarkeyword{constdefs}~c::\sigma~eqn~\dots$] combines constant
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  declarations and definitions, using 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}
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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.  The $mode$ argument refers to the print mode that the
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  grammar rules belong; unless there is the \texttt{output} flag given, all
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  productions are added both to the input and output grammar.
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\item [$\isarkeyword{translations}~rules$] specifies syntactic translation
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  rules (also known as \emph{macros}): parse/print rules (\texttt{==}), parse
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  rules (\texttt{=>}), 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 [$\isarkeyword{axioms}~name: \phi~\dots$] introduces arbitrary
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  statements as logical axioms.  In fact, axioms are ``axiomatic theorems'',
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  and may be referred just as any other theorem later.
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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 [$\isarkeyword{theorems}~name = thms$] stores lists of existing theorems
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  as $name$.  Typical applications would also involve attributes (to augment
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  the default simpset, 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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Isabelle organises any kind of names (of types, constants, theorems etc.)  by
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hierarchically structured name spaces.  Normally the user never has to control
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the behaviour of name space entry by hand, yet the following commands provide
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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 as
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  base names only, 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{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{setup} & : & \isartrans{theory}{theory} \\
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\end{matharray}
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\begin{rail}
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  'use' name
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  ;
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  'ML' text
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  ;
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  'setup' text
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  ;
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\end{rail}
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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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  Furthermore, the file name is checked with the $\isarkeyword{files}$
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  dependency declaration given in the theory header (see also
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  \S\ref{sec:begin-thy}).
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\item [$\isarkeyword{ML}~text$] reads and executes ML commands from $text$.
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  The theory context is passed just as for $\isarkeyword{use}$.
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\item [$\isarkeyword{setup}~text$] changes the current theory context by
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  applying setup functions $text$, which has to be an ML expression of type
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  $(theory \to theory)~list$.  The $\isarkeyword{setup}$ command is the usual
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  way to initialise object-logic specific tools and packages written in ML.
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\end{descr}
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\subsection{Syntax translation functions}
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\indexisarcmd{parse-ast-translation}\indexisarcmd{parse-translation}
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\indexisarcmd{print-translation}\indexisarcmd{typed-print-translation}
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\indexisarcmd{print-ast-translation}\indexisarcmd{token-translation}
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\begin{matharray}{rcl}
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  \isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\
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  \isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\
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  \isarcmd{print_translation} & : & \isartrans{theory}{theory} \\
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  \isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\
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  \isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\
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  \isarcmd{token_translation} & : & \isartrans{theory}{theory} \\
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\end{matharray}
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Syntax translation functions written in ML admit almost arbitrary
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manipulations of Isabelle's inner syntax.  Any of the above commands have a
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single \railqtoken{text} argument that refers to an ML expression of
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appropriate type.  See \cite[\S8]{isabelle-ref} for more information on syntax
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transformations.
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\subsection{Oracles}
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\indexisarcmd{oracle}
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\begin{matharray}{rcl}
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  \isarcmd{oracle} & : & \isartrans{theory}{theory} \\
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\end{matharray}
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Oracles provide an interface to external reasoning systems, without giving up
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control completely --- each theorem carries a derivation object recording any
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oracle invocation.  See \cite[\S6]{isabelle-ref} for more information.
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\begin{rail}
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  'oracle' name '=' text comment?
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  ;
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\end{rail}
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\begin{descr}
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\item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   390
  function $text$, which has to be of type $Sign\mathord.sg \times
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   391
  Object\mathord.T \to term)$.
7167
wenzelm
parents: 7141
diff changeset
   392
\end{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   393
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   394
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   395
\section{Proof commands}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   396
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   397
Proof commands provide transitions of Isar/VM machine configurations, which
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   398
are block-structured, consisting of a stack of nodes with three main
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   399
components: logical \emph{proof context}, local \emph{facts}, and open
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   400
\emph{goals}.  Isar/VM transitions are \emph{typed} according to the following
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   401
three three different modes of operation:
7167
wenzelm
parents: 7141
diff changeset
   402
\begin{descr}
wenzelm
parents: 7141
diff changeset
   403
\item [$proof(prove)$] means that a new goal has just been stated that is now
wenzelm
parents: 7141
diff changeset
   404
  to be \emph{proven}; the next command may refine it by some proof method
7175
wenzelm
parents: 7167
diff changeset
   405
  ($\approx$ tactic), and enter a sub-proof to establish the final result.
7167
wenzelm
parents: 7141
diff changeset
   406
\item [$proof(state)$] is like an internal theory mode: the context may be
7175
wenzelm
parents: 7167
diff changeset
   407
  augmented by \emph{stating} additional assumptions, intermediate result etc.
wenzelm
parents: 7167
diff changeset
   408
\item [$proof(chain)$] is an intermediate mode between $proof(state)$ and
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   409
  $proof(prove)$: existing facts have been just picked up in order to use them
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   410
  when refining the goal to be claimed next.
7167
wenzelm
parents: 7141
diff changeset
   411
\end{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   412
7167
wenzelm
parents: 7141
diff changeset
   413
wenzelm
parents: 7141
diff changeset
   414
\subsection{Formal comments}\label{sec:formal-cmt-prf}
wenzelm
parents: 7141
diff changeset
   415
wenzelm
parents: 7141
diff changeset
   416
\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsect}\indexisarcmd{txt}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   417
\begin{matharray}{rcl}
7167
wenzelm
parents: 7141
diff changeset
   418
  \isarcmd{sect} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm
parents: 7141
diff changeset
   419
  \isarcmd{subsect} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm
parents: 7141
diff changeset
   420
  \isarcmd{subsubsect} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm
parents: 7141
diff changeset
   421
  \isarcmd{txt} & : & \isartrans{proof(state)}{proof(state)} \\
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   422
\end{matharray}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   423
7175
wenzelm
parents: 7167
diff changeset
   424
These formal comments in proof mode closely correspond to the ones of theory
wenzelm
parents: 7167
diff changeset
   425
mode (see \S\ref{sec:formal-cmt-thy}).
wenzelm
parents: 7167
diff changeset
   426
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   427
\begin{rail}
7167
wenzelm
parents: 7141
diff changeset
   428
  ('sect' | 'subsect' | 'subsubsect' | 'txt') text
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   429
  ;
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   430
\end{rail}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   431
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   432
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   433
\subsection{Proof context}\label{sec:proof-context}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   434
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   435
\indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   436
\begin{matharray}{rcl}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   437
  \isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   438
  \isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   439
  \isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   440
  \isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   441
\end{matharray}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   442
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   443
The logical proof context consists of fixed variables and assumptions.  The
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   444
former closely correspond to Skolem constants, or meta-level universal
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   445
quantification as provided by the Isabelle/Pure logical framework.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   446
Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   447
a local entity that may be used in the subsequent proof as any other variable
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   448
or constant.  Furthermore, any result $\phi[x]$ exported from the current
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   449
context will be universally closed wrt.\ $x$ at the outermost level (this is
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   450
expressed using Isabelle's meta-variables).
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   451
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   452
Similarly, introducing some assumption $\chi$ has two effects.  On the one
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   453
hand, a local theorem is created that may be used as a fact in subsequent
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   454
proof steps.  On the other hand, any result $\phi$ exported from the context
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   455
becomes conditional wrt.\ the assumption.  Thus, solving an enclosing goal
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   456
using this result would basically introduce a new subgoal stemming from the
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   457
assumption.  How this situation is handled depends on the actual version of
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   458
assumption command used: while $\ASSUMENAME$ solves the subgoal by unifying
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   459
with some premise of the goal, $\PRESUMENAME$ leaves the subgoal unchanged to
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   460
be proved later by the user.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   461
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   462
Local definitions, introduced by $\DEF{a}{x \equiv t}$, are achieved by
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   463
combining $\FIX x$ with another version of assumption that causes any
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   464
hypothetical equation $x = t$ to be eliminated by reflexivity.  Thus,
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   465
exporting some result $\phi[x]$ simply yields $\phi[t]$.
7175
wenzelm
parents: 7167
diff changeset
   466
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   467
\begin{rail}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   468
  'fix' (var +) comment?
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   469
  ;
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   470
  ('assume' | 'presume') (assm comment? + 'and')
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   471
  ;
7175
wenzelm
parents: 7167
diff changeset
   472
  'def' thmdecl? \\ var '==' term termpat? comment?
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   473
  ;
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   474
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   475
  var: name ('::' type)?
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   476
  ;
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   477
  assm: thmdecl? (prop proppat? +)
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   478
  ;
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   479
\end{rail}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   480
7167
wenzelm
parents: 7141
diff changeset
   481
\begin{descr}
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   482
\item [$\FIX{x}$] introduces a local \emph{arbitrary, but fixed} variable $x$.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   483
\item [$\ASSUME{a}{\Phi}$ and $\PRESUME{a}{\Phi}$] introduce local theorems
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   484
  $\Phi$.  Subsequent results used to solve some enclosing goal (e.g.\ via
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   485
  $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ expects to be able to
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   486
  unify with existing premises in the goal, while $\PRESUMENAME$ leaves $\Phi$
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   487
  as new subgoals.  Note that several lists of assumptions may be given
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   488
  (separated by \railterm{and}); the resulting list of current facts consists
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   489
  of all of these.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   490
\item [$\DEF{a}{x \equiv t}$] introduces a local (non-polymorphic) definition.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   491
  In results exported from the context, $x$ is replaced by $t$.  Basically,
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   492
  $\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\PRESUME{}{x \equiv t}$ (the
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   493
  resulting hypothetical equation is solved by reflexivity, though).
7167
wenzelm
parents: 7141
diff changeset
   494
\end{descr}
wenzelm
parents: 7141
diff changeset
   495
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   496
The internal register $prems$\indexisarreg{prems} refers to all current
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   497
assumptions as a list of theorems.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   498
7167
wenzelm
parents: 7141
diff changeset
   499
wenzelm
parents: 7141
diff changeset
   500
\subsection{Facts and forward chaining}
wenzelm
parents: 7141
diff changeset
   501
wenzelm
parents: 7141
diff changeset
   502
\indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with}
wenzelm
parents: 7141
diff changeset
   503
\begin{matharray}{rcl}
wenzelm
parents: 7141
diff changeset
   504
  \isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\
wenzelm
parents: 7141
diff changeset
   505
  \isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\
wenzelm
parents: 7141
diff changeset
   506
  \isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\
wenzelm
parents: 7141
diff changeset
   507
  \isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\
wenzelm
parents: 7141
diff changeset
   508
\end{matharray}
wenzelm
parents: 7141
diff changeset
   509
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   510
New facts are established either by assumption or proof of local statements
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   511
(via $\HAVENAME$ or $\SHOWNAME$).  Any facts will usually be involved in
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   512
proofs of further results: either as explicit arguments of proof methods or
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   513
when forward chaining towards the next goal via $\THEN$ (and variants).
7175
wenzelm
parents: 7167
diff changeset
   514
7167
wenzelm
parents: 7141
diff changeset
   515
\begin{rail}
wenzelm
parents: 7141
diff changeset
   516
  'note' thmdef? thmrefs comment?
wenzelm
parents: 7141
diff changeset
   517
  ;
wenzelm
parents: 7141
diff changeset
   518
  'then' comment?
wenzelm
parents: 7141
diff changeset
   519
  ;
wenzelm
parents: 7141
diff changeset
   520
  ('from' | 'with') thmrefs comment?
wenzelm
parents: 7141
diff changeset
   521
  ;
wenzelm
parents: 7141
diff changeset
   522
\end{rail}
wenzelm
parents: 7141
diff changeset
   523
wenzelm
parents: 7141
diff changeset
   524
\begin{descr}
7175
wenzelm
parents: 7167
diff changeset
   525
\item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result
wenzelm
parents: 7167
diff changeset
   526
  as $a$.  Note that attributes may be involved as well, both on the left and
wenzelm
parents: 7167
diff changeset
   527
  right hand sides.
7167
wenzelm
parents: 7141
diff changeset
   528
\item [$\THEN$] indicates forward chaining by the current facts in order to
7175
wenzelm
parents: 7167
diff changeset
   529
  establish the goal to be claimed next.  The initial proof method invoked to
wenzelm
parents: 7167
diff changeset
   530
  solve that will be offered these facts to do ``anything appropriate'' (see
wenzelm
parents: 7167
diff changeset
   531
  also \S\ref{sec:proof-steps}).  For example, method $rule$ (see
7167
wenzelm
parents: 7141
diff changeset
   532
  \S\ref{sec:pure-meth}) would do an elimination rather than an introduction.
7175
wenzelm
parents: 7167
diff changeset
   533
\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; also note that
7167
wenzelm
parents: 7141
diff changeset
   534
  $\THEN$ is equivalent to $\FROM{facts}$.
7175
wenzelm
parents: 7167
diff changeset
   535
\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward
wenzelm
parents: 7167
diff changeset
   536
  chaining is from earlier facts together with the current ones.
7167
wenzelm
parents: 7141
diff changeset
   537
\end{descr}
wenzelm
parents: 7141
diff changeset
   538
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   539
Note that the internal register of \emph{current facts} may be referred as
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   540
theorem list $facts$.\indexisarreg{facts}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   541
7167
wenzelm
parents: 7141
diff changeset
   542
wenzelm
parents: 7141
diff changeset
   543
\subsection{Goal statements}
wenzelm
parents: 7141
diff changeset
   544
wenzelm
parents: 7141
diff changeset
   545
\indexisarcmd{theorem}\indexisarcmd{lemma}
wenzelm
parents: 7141
diff changeset
   546
\indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus}
wenzelm
parents: 7141
diff changeset
   547
\begin{matharray}{rcl}
wenzelm
parents: 7141
diff changeset
   548
  \isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\
wenzelm
parents: 7141
diff changeset
   549
  \isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\
wenzelm
parents: 7141
diff changeset
   550
  \isarcmd{have} & : & \isartrans{proof(state)}{proof(prove)} \\
wenzelm
parents: 7141
diff changeset
   551
  \isarcmd{show} & : & \isartrans{proof(state)}{proof(prove)} \\
wenzelm
parents: 7141
diff changeset
   552
  \isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\
wenzelm
parents: 7141
diff changeset
   553
  \isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\
wenzelm
parents: 7141
diff changeset
   554
\end{matharray}
wenzelm
parents: 7141
diff changeset
   555
7175
wenzelm
parents: 7167
diff changeset
   556
Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$
wenzelm
parents: 7167
diff changeset
   557
and $\LEMMANAME$.  New local goals may be claimed within proof mode: four
wenzelm
parents: 7167
diff changeset
   558
variants are available, indicating whether the result is meant to solve some
wenzelm
parents: 7167
diff changeset
   559
pending goal and whether forward chaining is employed.
wenzelm
parents: 7167
diff changeset
   560
7167
wenzelm
parents: 7141
diff changeset
   561
\begin{rail}
wenzelm
parents: 7141
diff changeset
   562
  ('theorem' | 'lemma') goal
wenzelm
parents: 7141
diff changeset
   563
  ;
wenzelm
parents: 7141
diff changeset
   564
  ('have' | 'show' | 'hence' | 'thus') goal
wenzelm
parents: 7141
diff changeset
   565
  ;
wenzelm
parents: 7141
diff changeset
   566
wenzelm
parents: 7141
diff changeset
   567
  goal: thmdecl? proppat comment?
wenzelm
parents: 7141
diff changeset
   568
  ;
wenzelm
parents: 7141
diff changeset
   569
\end{rail}
wenzelm
parents: 7141
diff changeset
   570
wenzelm
parents: 7141
diff changeset
   571
\begin{descr}
wenzelm
parents: 7141
diff changeset
   572
\item [$\THEOREM{name}{\phi}$] enters proof mode with $\phi$ as main goal,
7175
wenzelm
parents: 7167
diff changeset
   573
  eventually resulting in some theorem $\turn \phi$, which will be stored in
wenzelm
parents: 7167
diff changeset
   574
  the theory.
7167
wenzelm
parents: 7141
diff changeset
   575
\item [$\LEMMANAME$] is similar to $\THEOREMNAME$, but tags the result as
wenzelm
parents: 7141
diff changeset
   576
  ``lemma''.
wenzelm
parents: 7141
diff changeset
   577
\item [$\HAVE{name}{\phi}$] claims a local goal, eventually resulting in a
wenzelm
parents: 7141
diff changeset
   578
  theorem with the current assumption context as hypotheses.
7175
wenzelm
parents: 7167
diff changeset
   579
\item [$\SHOW{name}{\phi}$] is similar to $\HAVE{name}{\phi}$, but solves some
wenzelm
parents: 7167
diff changeset
   580
  pending goal with the result \emph{exported} into the corresponding context.
7167
wenzelm
parents: 7141
diff changeset
   581
\item [$\HENCE{name}{\phi}$] abbreviates $\THEN~\HAVE{name}{\phi}$, i.e.\ 
wenzelm
parents: 7141
diff changeset
   582
  claims a local goal to be proven by forward chaining the current facts.
wenzelm
parents: 7141
diff changeset
   583
\item [$\THUS{name}{\phi}$] abbreviates $\THEN~\SHOW{name}{\phi}$.
wenzelm
parents: 7141
diff changeset
   584
\end{descr}
wenzelm
parents: 7141
diff changeset
   585
wenzelm
parents: 7141
diff changeset
   586
wenzelm
parents: 7141
diff changeset
   587
\subsection{Initial and terminal proof steps}\label{sec:proof-steps}
wenzelm
parents: 7141
diff changeset
   588
7175
wenzelm
parents: 7167
diff changeset
   589
\indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by}
wenzelm
parents: 7167
diff changeset
   590
\indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry}
wenzelm
parents: 7167
diff changeset
   591
\begin{matharray}{rcl}
wenzelm
parents: 7167
diff changeset
   592
  \isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\
wenzelm
parents: 7167
diff changeset
   593
  \isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\
wenzelm
parents: 7167
diff changeset
   594
  \isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm
parents: 7167
diff changeset
   595
  \isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm
parents: 7167
diff changeset
   596
  \isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm
parents: 7167
diff changeset
   597
  \isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
wenzelm
parents: 7167
diff changeset
   598
\end{matharray}
wenzelm
parents: 7167
diff changeset
   599
7167
wenzelm
parents: 7141
diff changeset
   600
Arbitrary goal refinements via tactics is considered harmful.  Consequently
wenzelm
parents: 7141
diff changeset
   601
the Isar framework admits proof methods to be invoked in two places only.
wenzelm
parents: 7141
diff changeset
   602
\begin{enumerate}
7175
wenzelm
parents: 7167
diff changeset
   603
\item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated
wenzelm
parents: 7167
diff changeset
   604
  intermediate goal to a number of sub-goals that are to be solved later.
wenzelm
parents: 7167
diff changeset
   605
  Facts are passed to $m@1$ for forward chaining if so indicated by
wenzelm
parents: 7167
diff changeset
   606
  $proof(chain)$ mode.
7167
wenzelm
parents: 7141
diff changeset
   607
  
7175
wenzelm
parents: 7167
diff changeset
   608
\item A \emph{terminal} conclusion step $\QED{m@2}$ solves any remaining
7167
wenzelm
parents: 7141
diff changeset
   609
  pending goals completely.  No facts are passed to $m@2$.
wenzelm
parents: 7141
diff changeset
   610
\end{enumerate}
wenzelm
parents: 7141
diff changeset
   611
wenzelm
parents: 7141
diff changeset
   612
The only other proper way to affect pending goals is by $\SHOWNAME$, which
wenzelm
parents: 7141
diff changeset
   613
involves an explicit statement of what is solved.
wenzelm
parents: 7141
diff changeset
   614
7175
wenzelm
parents: 7167
diff changeset
   615
\medskip
wenzelm
parents: 7167
diff changeset
   616
7167
wenzelm
parents: 7141
diff changeset
   617
Also note that initial proof methods should either solve the goal completely,
wenzelm
parents: 7141
diff changeset
   618
or constitute some well-understood deterministic reduction to new sub-goals.
wenzelm
parents: 7141
diff changeset
   619
Arbitrary automatic proof tools that are prone leave a large number of badly
wenzelm
parents: 7141
diff changeset
   620
structured sub-goals are no help in continuing the proof document in any
7175
wenzelm
parents: 7167
diff changeset
   621
intelligible way.  A much better technique would be to $\SHOWNAME$ some
wenzelm
parents: 7167
diff changeset
   622
non-trivial reduction as an explicit rule, which is solved completely by some
wenzelm
parents: 7167
diff changeset
   623
automated method, and then applied to some pending goal.
7167
wenzelm
parents: 7141
diff changeset
   624
7175
wenzelm
parents: 7167
diff changeset
   625
\medskip
wenzelm
parents: 7167
diff changeset
   626
wenzelm
parents: 7167
diff changeset
   627
Unless given explicitly by the user, the default initial method is
wenzelm
parents: 7167
diff changeset
   628
``$default$'', which is usually set up to apply a single standard elimination
wenzelm
parents: 7167
diff changeset
   629
or introduction rule according to the topmost symbol involved.  The default
wenzelm
parents: 7167
diff changeset
   630
terminal method is ``$finish$''; it solves all goals by assumption.
7167
wenzelm
parents: 7141
diff changeset
   631
wenzelm
parents: 7141
diff changeset
   632
\begin{rail}
wenzelm
parents: 7141
diff changeset
   633
  'proof' interest? meth? comment?
wenzelm
parents: 7141
diff changeset
   634
  ;
wenzelm
parents: 7141
diff changeset
   635
  'qed' meth? comment?
wenzelm
parents: 7141
diff changeset
   636
  ;
wenzelm
parents: 7141
diff changeset
   637
  'by' meth meth? comment?
wenzelm
parents: 7141
diff changeset
   638
  ;
wenzelm
parents: 7141
diff changeset
   639
  ('.' | '..' | 'sorry') comment?
wenzelm
parents: 7141
diff changeset
   640
  ;
wenzelm
parents: 7141
diff changeset
   641
wenzelm
parents: 7141
diff changeset
   642
  meth: method interest?
wenzelm
parents: 7141
diff changeset
   643
  ;
wenzelm
parents: 7141
diff changeset
   644
\end{rail}
wenzelm
parents: 7141
diff changeset
   645
wenzelm
parents: 7141
diff changeset
   646
\begin{descr}
7175
wenzelm
parents: 7167
diff changeset
   647
\item [$\PROOF{m@1}$] refines the pending goal by proof method $m@1$; facts
wenzelm
parents: 7167
diff changeset
   648
  for forward chaining are passed if so indicated by $proof(chain)$.
wenzelm
parents: 7167
diff changeset
   649
\item [$\QED{m@2}$] refines any remaining goals by proof method $m@1$ and
7167
wenzelm
parents: 7141
diff changeset
   650
  concludes the sub-proof.  If the goal had been $\SHOWNAME$, some pending
wenzelm
parents: 7141
diff changeset
   651
  sub-goal is solved as well by the rule resulting from the result exported to
7175
wenzelm
parents: 7167
diff changeset
   652
  the enclosing goal context.  Thus $\QEDNAME$ may fail for two reasons:
wenzelm
parents: 7167
diff changeset
   653
  either $m@2$ fails to solve all remaining goals completely, or the resulting
wenzelm
parents: 7167
diff changeset
   654
  rule does not resolve with any enclosing goal.  Debugging such a situation
wenzelm
parents: 7167
diff changeset
   655
  might involve temporarily changing $\SHOWNAME$ into $\HAVENAME$, or
wenzelm
parents: 7167
diff changeset
   656
  weakening the local context by replacing $\ASSUMENAME$ by $\PRESUMENAME$.
wenzelm
parents: 7167
diff changeset
   657
\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}; it abbreviates
wenzelm
parents: 7167
diff changeset
   658
  $\PROOF{m@1}~\QED{m@2}$, with automatic backtracking across both methods.
wenzelm
parents: 7167
diff changeset
   659
  Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done by simply
wenzelm
parents: 7167
diff changeset
   660
  expanding the abbreviation by hand; note that $\PROOF{m@1}$ is usually
wenzelm
parents: 7167
diff changeset
   661
  sufficient to see what is going wrong.
wenzelm
parents: 7167
diff changeset
   662
\item [$\DDOT$] is a \emph{default proof}; it abbreviates $\BY{default}$.
wenzelm
parents: 7167
diff changeset
   663
\item [$\DOT$] is a \emph{trivial proof}, it abbreviates $\BY{-}$, where
wenzelm
parents: 7167
diff changeset
   664
  method ``$-$'' does nothing except inserting any facts into the proof state.
7167
wenzelm
parents: 7141
diff changeset
   665
\item [$\isarkeyword{sorry}$] is a \emph{fake proof}; provided that
wenzelm
parents: 7141
diff changeset
   666
  \texttt{quick_and_dirty} is enabled, $\isarkeyword{sorry}$ pretends to solve
wenzelm
parents: 7141
diff changeset
   667
  the goal without much ado.  Of course, the result is a fake theorem only,
7175
wenzelm
parents: 7167
diff changeset
   668
  involving some oracle in its internal derivation object (this is indicated
wenzelm
parents: 7167
diff changeset
   669
  as $[!]$ in the printed result.  The main application of
7167
wenzelm
parents: 7141
diff changeset
   670
  $\isarkeyword{sorry}$ is to support top-down proof development.
wenzelm
parents: 7141
diff changeset
   671
\end{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   672
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   673
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   674
\subsection{Improper proof steps}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   675
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   676
The following commands emulate unstructured tactic scripts to some extent.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   677
While these are anathema for writing proper Isar proof documents, they might
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   678
come in handy for exploring and debugging.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   679
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   680
\indexisarcmd{apply}\indexisarcmd{then-apply}\indexisarcmd{back}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   681
\begin{matharray}{rcl}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   682
  \isarcmd{apply}^* & : & \isartrans{proof}{proof} \\
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   683
  \isarcmd{then_apply}^* & : & \isartrans{proof}{proof} \\
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   684
  \isarcmd{back}^* & : & \isartrans{proof}{proof} \\
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   685
\end{matharray}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   686
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   687
\railalias{thenapply}{then\_apply}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   688
\railterm{thenapply}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   689
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   690
\begin{rail}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   691
  'apply' method
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   692
  ;
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   693
  thenapply method
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   694
  ;
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   695
  'back'
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   696
  ;
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   697
\end{rail}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   698
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   699
\begin{descr}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   700
\item [$\isarkeyword{apply}~m$] applies proof method $m$ in the
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   701
  plain-old-tactic sense.  Facts for forward chaining are ignored.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   702
\item [$\isarkeyword{then_apply}~m$] is similar to $\isarkeyword{apply}$, but
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   703
  observes the goal's facts.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   704
\item [$\isarkeyword{back}$] does back-tracking over the result sequence of
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   705
  the last proof command.  Basically, any proof command may return multiple
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   706
  results.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   707
\end{descr}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   708
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   709
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   710
\subsection{Term abbreviations}\label{sec:term-abbrev}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   711
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   712
\indexisarcmd{let}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   713
\begin{matharray}{rcl}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   714
  \isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   715
  \isarkeyword{is} & : & syntax \\
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   716
\end{matharray}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   717
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   718
Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements,
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   719
or by annotating assumptions or goal statements ($\ASSUMENAME$, $\SHOWNAME$
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   720
etc.) with a list of patterns $\IS{p@1 \dots p@n}$.  In both cases,
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   721
higher-order matching is applied to bind extra-logical text
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   722
variables\index{text variables}, which may be either of the form $\VVar{x}$
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   723
(token class \railtoken{textvar}, see \S\ref{sec:lex-syntax}) or nameless
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   724
dummies ``\verb,_,'' (underscore).\index{dummy variables} Note that in the
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   725
$\LETNAME$ form the patterns occur on the left-hand side, while the $\ISNAME$
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   726
patterns are in postfix position.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   727
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   728
Note that term abbreviations are quite different from actual local definitions
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   729
as introduced via $\DEFNAME$ (see \S\ref{sec:proof-context}).  The latter are
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   730
visible within the logic as actual equations, while abbreviations disappear
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   731
during the input process just after type checking.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   732
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   733
\begin{rail}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   734
  'let' ((term + 'as') '=' term comment? + 'and')
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   735
  ;  
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   736
\end{rail}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   737
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   738
The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   739
\railnonterm{proppat} (see \S\ref{sec:term-pats}).
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   740
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   741
\begin{descr}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   742
\item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   743
  by simultaneous higher-order matching against terms $\vec t$.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   744
\item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   745
  preceding statement.  Also note that $\ISNAME$ is not a separate command,
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   746
  but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.).
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   747
\end{descr}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   748
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   749
Furthermore, a few automatic term abbreviations\index{automatic abbreviation}
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   750
for goals and facts are available.  For any open goal, $\VVar{thesis}$ refers
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   751
to its object-logic statement, $\VVar{thesis_prop}$ to the full proposition
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   752
(which may be a rule), and $\VVar{thesis_concl}$ to its (atomic) conclusion.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   753
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   754
Facts (i.e.\ assumptions and finished goals) that have an application $f(x)$
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   755
as object-logic statement get $x$ bound to the special text variable
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   756
``$\dots$'' (three dots).  The canonical application of this feature are
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   757
calculational proofs, see \S\ref{sec:calculation}.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   758
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   759
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   760
\subsection{Block structure}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   761
7167
wenzelm
parents: 7141
diff changeset
   762
While Isar is inherently block-structured, opening and closing blocks is
wenzelm
parents: 7141
diff changeset
   763
mostly handled rather casually, with little explicit user-intervention.  Any
wenzelm
parents: 7141
diff changeset
   764
local goal statement automatically opens \emph{two} blocks, which are closed
wenzelm
parents: 7141
diff changeset
   765
again when concluding the sub-proof (by $\QEDNAME$ etc.).  Sections of
wenzelm
parents: 7141
diff changeset
   766
different context within a sub-proof are typically switched via
wenzelm
parents: 7141
diff changeset
   767
$\isarkeyword{next}$, which is just a single block-close followed by
wenzelm
parents: 7141
diff changeset
   768
block-open again.  Thus the effect of $\isarkeyword{next}$ is to reset the
wenzelm
parents: 7141
diff changeset
   769
proof context to that of the head of the sub-proof.  Note that there is no
7175
wenzelm
parents: 7167
diff changeset
   770
goal focus involved here!
7167
wenzelm
parents: 7141
diff changeset
   771
7175
wenzelm
parents: 7167
diff changeset
   772
For slightly more advanced applications, there are explicit block parentheses
wenzelm
parents: 7167
diff changeset
   773
as well.  These typically achieve a strong forward style of reasoning.
7167
wenzelm
parents: 7141
diff changeset
   774
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   775
\indexisarcmd{next}\indexisarcmd{\{\{}\indexisarcmd{\}\}}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   776
\begin{matharray}{rcl}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   777
  \isarcmd{next} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   778
  \isarcmd{\{\{} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   779
  \isarcmd{\}\}} & : & \isartrans{proof(state)}{proof(state)} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   780
\end{matharray}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   781
7167
wenzelm
parents: 7141
diff changeset
   782
\begin{descr}
wenzelm
parents: 7141
diff changeset
   783
\item [$\isarkeyword{next}$] switches to a fresh block within a sub-proof,
wenzelm
parents: 7141
diff changeset
   784
  resetting the context to the initial one.
wenzelm
parents: 7141
diff changeset
   785
\item [$\isarkeyword{\{\{}$ and $\isarkeyword{\}\}}$] explicitly open and
wenzelm
parents: 7141
diff changeset
   786
  close blocks.  Any current facts pass through $\isarkeyword{\{\{}$
wenzelm
parents: 7141
diff changeset
   787
  unchanged, while $\isarkeyword{\}\}}$ causes them to be \emph{exported} into
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   788
  the enclosing context.  Thus fixed variables are generalised, assumptions
7167
wenzelm
parents: 7141
diff changeset
   789
  discharged, and local definitions eliminated.
wenzelm
parents: 7141
diff changeset
   790
\end{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   791
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   792
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   793
\section{Other commands}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   794
7315
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   795
The following commands are not part of the actual proper or improper
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   796
Isabelle/Isar syntax, but assist interactive development, for example.  Also
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   797
note that $undo$ does not apply here, since the theory or proof configuration
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   798
is not changed.
76a39a3784b5 checkpoint;
wenzelm
parents: 7175
diff changeset
   799
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   800
\subsection{Diagnostics}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   801
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   802
\indexisarcmd{typ}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{thm}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   803
\begin{matharray}{rcl}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   804
  \isarcmd{typ} & : & \isarkeep{theory~|~proof} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   805
  \isarcmd{term} & : & \isarkeep{theory~|~proof} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   806
  \isarcmd{prop} & : & \isarkeep{theory~|~proof} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   807
  \isarcmd{thm} & : & \isarkeep{theory~|~proof} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   808
\end{matharray}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   809
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   810
\begin{rail}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   811
  'typ' type
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   812
  ;
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   813
  'term' term
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   814
  ;
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   815
  'prop' prop
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   816
  ;
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   817
  'thm' thmrefs
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   818
  ;
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   819
\end{rail}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   820
7167
wenzelm
parents: 7141
diff changeset
   821
\begin{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   822
\item [$\isarkeyword{typ}~\tau$, $\isarkeyword{term}~t$,
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   823
  $\isarkeyword{prop}~\phi$] read and print types / terms / propositions
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   824
  according to the current theory or proof context.
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   825
\item [$\isarkeyword{thm}~thms$] retrieves lists of theorems from the current
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   826
  theory or proof context.  Note that any attributes included in the theorem
7175
wenzelm
parents: 7167
diff changeset
   827
  specifications are applied to a temporary context derived from the current
wenzelm
parents: 7167
diff changeset
   828
  theory or proof; the result is discarded.
7167
wenzelm
parents: 7141
diff changeset
   829
\end{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   830
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   831
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   832
\subsection{System operations}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   833
7167
wenzelm
parents: 7141
diff changeset
   834
\indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{use-thy}\indexisarcmd{use-thy-only}
wenzelm
parents: 7141
diff changeset
   835
\indexisarcmd{update-thy}\indexisarcmd{update-thy-only}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   836
\begin{matharray}{rcl}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   837
  \isarcmd{cd} & : & \isarkeep{\cdot} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   838
  \isarcmd{pwd} & : & \isarkeep{\cdot} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   839
  \isarcmd{use_thy} & : & \isarkeep{\cdot} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   840
  \isarcmd{use_thy_only} & : & \isarkeep{\cdot} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   841
  \isarcmd{update_thy} & : & \isarkeep{\cdot} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   842
  \isarcmd{update_thy_only} & : & \isarkeep{\cdot} \\
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   843
\end{matharray}
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   844
7167
wenzelm
parents: 7141
diff changeset
   845
\begin{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   846
\item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   847
  process.
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   848
\item [$\isarkeyword{pwd}~$] prints the current working directory.
7175
wenzelm
parents: 7167
diff changeset
   849
\item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$,
wenzelm
parents: 7167
diff changeset
   850
  $\isarkeyword{update_thy}$, and $\isarkeyword{update_thy_only}$] load some
wenzelm
parents: 7167
diff changeset
   851
  theory given as $name$ argument.  These commands are exactly the same as the
wenzelm
parents: 7167
diff changeset
   852
  corresponding ML functions (see also \cite[\S1 and \S6]{isabelle-ref}).
wenzelm
parents: 7167
diff changeset
   853
  Note that both the ML and Isar versions of these commands may load new- and
wenzelm
parents: 7167
diff changeset
   854
  old-style theories alike.
7167
wenzelm
parents: 7141
diff changeset
   855
\end{descr}
7134
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   856
320b412e5800 more stuff;
wenzelm
parents: 7046
diff changeset
   857
7046
9f755ff43cff skeleton only;
wenzelm
parents:
diff changeset
   858
%%% Local Variables: 
9f755ff43cff skeleton only;
wenzelm
parents:
diff changeset
   859
%%% mode: latex
9f755ff43cff skeleton only;
wenzelm
parents:
diff changeset
   860
%%% TeX-master: "isar-ref"
9f755ff43cff skeleton only;
wenzelm
parents:
diff changeset
   861
%%% End: