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%
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\begin{isabellebody}%
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\def\isabellecontext{Generic}%
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%
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\isadelimtheory
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\isanewline
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\isanewline
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%
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\endisadelimtheory
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%
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ Generic\isanewline
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\isakeyword{imports}\ Main\isanewline
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\isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupchapter{Generic tools and packages \label{ch:gen-tools}%
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}
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\isamarkuptrue%
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%
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\isamarkupsection{Configuration options%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Isabelle/Pure maintains a record of named configuration options
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within the theory or proof context, with values of type \verb|bool|, \verb|int|, or \verb|string|. Tools may declare
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options in ML, and then refer to these values (relative to the
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context). Thus global reference variables are easily avoided. The
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user may change the value of a configuration option by means of an
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associated attribute of the same name. This form of context
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declaration works particularly well with commands such as \hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}} or \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}.
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For historical reasons, some tools cannot take the full proof
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context into account and merely refer to the background theory.
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This is accommodated by configuration options being declared as
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``global'', which may not be changed within a local context.
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\begin{matharray}{rcll}
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\indexdef{}{command}{print\_configs}\hypertarget{command.print-configs}{\hyperlink{command.print-configs}{\mbox{\isa{\isacommand{print{\isacharunderscore}configs}}}}} & : & \isarkeep{theory~|~proof} \\
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\end{matharray}
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\begin{rail}
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name ('=' ('true' | 'false' | int | name))?
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\end{rail}
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\begin{descr}
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\item [\hyperlink{command.print-configs}{\mbox{\isa{\isacommand{print{\isacharunderscore}configs}}}}] prints the available
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configuration options, with names, types, and current values.
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\item [\isa{{\isachardoublequote}name\ {\isacharequal}\ value{\isachardoublequote}}] as an attribute expression modifies
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the named option, with the syntax of the value depending on the
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option's type. For \verb|bool| the default value is \isa{true}. Any attempt to change a global option in a local context is
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ignored.
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\end{descr}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Basic proof tools%
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}
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\isamarkuptrue%
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%
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\isamarkupsubsection{Miscellaneous methods and attributes \label{sec:misc-meth-att}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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\indexdef{}{method}{unfold}\hypertarget{method.unfold}{\hyperlink{method.unfold}{\mbox{\isa{unfold}}}} & : & \isarmeth \\
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\indexdef{}{method}{fold}\hypertarget{method.fold}{\hyperlink{method.fold}{\mbox{\isa{fold}}}} & : & \isarmeth \\
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\indexdef{}{method}{insert}\hypertarget{method.insert}{\hyperlink{method.insert}{\mbox{\isa{insert}}}} & : & \isarmeth \\[0.5ex]
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\indexdef{}{method}{erule}\hypertarget{method.erule}{\hyperlink{method.erule}{\mbox{\isa{erule}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
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\indexdef{}{method}{drule}\hypertarget{method.drule}{\hyperlink{method.drule}{\mbox{\isa{drule}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
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\indexdef{}{method}{frule}\hypertarget{method.frule}{\hyperlink{method.frule}{\mbox{\isa{frule}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
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\indexdef{}{method}{succeed}\hypertarget{method.succeed}{\hyperlink{method.succeed}{\mbox{\isa{succeed}}}} & : & \isarmeth \\
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\indexdef{}{method}{fail}\hypertarget{method.fail}{\hyperlink{method.fail}{\mbox{\isa{fail}}}} & : & \isarmeth \\
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\end{matharray}
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\begin{rail}
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('fold' | 'unfold' | 'insert') thmrefs
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;
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('erule' | 'drule' | 'frule') ('('nat')')? thmrefs
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;
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\end{rail}
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\begin{descr}
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\item [\hyperlink{method.unfold}{\mbox{\isa{unfold}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} and \hyperlink{method.fold}{\mbox{\isa{fold}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] expand (or fold back) the
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given definitions throughout all goals; any chained facts provided
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are inserted into the goal and subject to rewriting as well.
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\item [\hyperlink{method.insert}{\mbox{\isa{insert}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] inserts
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theorems as facts into all goals of the proof state. Note that
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current facts indicated for forward chaining are ignored.
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\item [\hyperlink{method.erule}{\mbox{\isa{erule}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}, \hyperlink{method.drule}{\mbox{\isa{drule}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}, and \hyperlink{method.frule}{\mbox{\isa{frule}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] are similar to the basic \hyperlink{method.rule}{\mbox{\isa{rule}}}
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method (see \secref{sec:pure-meth-att}), but apply rules by
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elim-resolution, destruct-resolution, and forward-resolution,
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respectively \cite{isabelle-ref}. The optional natural number
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argument (default 0) specifies additional assumption steps to be
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performed here.
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Note that these methods are improper ones, mainly serving for
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experimentation and tactic script emulation. Different modes of
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basic rule application are usually expressed in Isar at the proof
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language level, rather than via implicit proof state manipulations.
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For example, a proper single-step elimination would be done using
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the plain \hyperlink{method.rule}{\mbox{\isa{rule}}} method, with forward chaining of current
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facts.
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\item [\hyperlink{method.succeed}{\mbox{\isa{succeed}}}] yields a single (unchanged) result; it is
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the identity of the ``\isa{{\isachardoublequote}{\isacharcomma}{\isachardoublequote}}'' method combinator (cf.\
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\secref{sec:syn-meth}).
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\item [\hyperlink{method.fail}{\mbox{\isa{fail}}}] yields an empty result sequence; it is the
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identity of the ``\isa{{\isachardoublequote}{\isacharbar}{\isachardoublequote}}'' method combinator (cf.\
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\secref{sec:syn-meth}).
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\end{descr}
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\begin{matharray}{rcl}
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\indexdef{}{attribute}{tagged}\hypertarget{attribute.tagged}{\hyperlink{attribute.tagged}{\mbox{\isa{tagged}}}} & : & \isaratt \\
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\indexdef{}{attribute}{untagged}\hypertarget{attribute.untagged}{\hyperlink{attribute.untagged}{\mbox{\isa{untagged}}}} & : & \isaratt \\[0.5ex]
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\indexdef{}{attribute}{THEN}\hypertarget{attribute.THEN}{\hyperlink{attribute.THEN}{\mbox{\isa{THEN}}}} & : & \isaratt \\
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\indexdef{}{attribute}{COMP}\hypertarget{attribute.COMP}{\hyperlink{attribute.COMP}{\mbox{\isa{COMP}}}} & : & \isaratt \\[0.5ex]
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\indexdef{}{attribute}{unfolded}\hypertarget{attribute.unfolded}{\hyperlink{attribute.unfolded}{\mbox{\isa{unfolded}}}} & : & \isaratt \\
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\indexdef{}{attribute}{folded}\hypertarget{attribute.folded}{\hyperlink{attribute.folded}{\mbox{\isa{folded}}}} & : & \isaratt \\[0.5ex]
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\indexdef{}{attribute}{rotated}\hypertarget{attribute.rotated}{\hyperlink{attribute.rotated}{\mbox{\isa{rotated}}}} & : & \isaratt \\
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\indexdef{Pure}{attribute}{elim\_format}\hypertarget{attribute.Pure.elim-format}{\hyperlink{attribute.Pure.elim-format}{\mbox{\isa{elim{\isacharunderscore}format}}}} & : & \isaratt \\
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\indexdef{}{attribute}{standard}\hypertarget{attribute.standard}{\hyperlink{attribute.standard}{\mbox{\isa{standard}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
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\indexdef{}{attribute}{no\_vars}\hypertarget{attribute.no-vars}{\hyperlink{attribute.no-vars}{\mbox{\isa{no{\isacharunderscore}vars}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
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\end{matharray}
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\begin{rail}
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'tagged' nameref
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;
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'untagged' name
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;
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('THEN' | 'COMP') ('[' nat ']')? thmref
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;
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('unfolded' | 'folded') thmrefs
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;
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'rotated' ( int )?
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\end{rail}
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\begin{descr}
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\item [\hyperlink{attribute.tagged}{\mbox{\isa{tagged}}}~\isa{{\isachardoublequote}name\ arg{\isachardoublequote}} and \hyperlink{attribute.untagged}{\mbox{\isa{untagged}}}~\isa{name}] add and remove \emph{tags} of some theorem.
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Tags may be any list of string pairs that serve as formal comment.
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The first string is considered the tag name, the second its
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argument. Note that \hyperlink{attribute.untagged}{\mbox{\isa{untagged}}} removes any tags of the
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same name.
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\item [\hyperlink{attribute.THEN}{\mbox{\isa{THEN}}}~\isa{a} and \hyperlink{attribute.COMP}{\mbox{\isa{COMP}}}~\isa{a}]
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compose rules by resolution. \hyperlink{attribute.THEN}{\mbox{\isa{THEN}}} resolves with the
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first premise of \isa{a} (an alternative position may be also
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specified); the \hyperlink{attribute.COMP}{\mbox{\isa{COMP}}} version skips the automatic
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lifting process that is normally intended (cf.\ \verb|"op RS"| and
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\verb|"op COMP"| in \cite[\S5]{isabelle-ref}).
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\item [\hyperlink{attribute.unfolded}{\mbox{\isa{unfolded}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} and
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\hyperlink{attribute.folded}{\mbox{\isa{folded}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] expand and fold
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back again the given definitions throughout a rule.
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\item [\hyperlink{attribute.rotated}{\mbox{\isa{rotated}}}~\isa{n}] rotate the premises of a
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theorem by \isa{n} (default 1).
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\item [\hyperlink{attribute.Pure.elim-format}{\mbox{\isa{elim{\isacharunderscore}format}}}] turns a destruction rule
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into elimination rule format, by resolving with the rule \isa{{\isachardoublequote}PROP\ A\ {\isasymLongrightarrow}\ {\isacharparenleft}PROP\ A\ {\isasymLongrightarrow}\ PROP\ B{\isacharparenright}\ {\isasymLongrightarrow}\ PROP\ B{\isachardoublequote}}.
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Note that the Classical Reasoner (\secref{sec:classical}) provides
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its own version of this operation.
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\item [\hyperlink{attribute.standard}{\mbox{\isa{standard}}}] puts a theorem into the standard form
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of object-rules at the outermost theory level. Note that this
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operation violates the local proof context (including active
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locales).
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\item [\hyperlink{attribute.no-vars}{\mbox{\isa{no{\isacharunderscore}vars}}}] replaces schematic variables by free
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ones; this is mainly for tuning output of pretty printed theorems.
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\end{descr}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Low-level equational reasoning%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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\indexdef{}{method}{subst}\hypertarget{method.subst}{\hyperlink{method.subst}{\mbox{\isa{subst}}}} & : & \isarmeth \\
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\indexdef{}{method}{hypsubst}\hypertarget{method.hypsubst}{\hyperlink{method.hypsubst}{\mbox{\isa{hypsubst}}}} & : & \isarmeth \\
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\indexdef{}{method}{split}\hypertarget{method.split}{\hyperlink{method.split}{\mbox{\isa{split}}}} & : & \isarmeth \\
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\end{matharray}
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\begin{rail}
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'subst' ('(' 'asm' ')')? ('(' (nat+) ')')? thmref
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;
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'split' ('(' 'asm' ')')? thmrefs
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;
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\end{rail}
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These methods provide low-level facilities for equational reasoning
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that are intended for specialized applications only. Normally,
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single step calculations would be performed in a structured text
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(see also \secref{sec:calculation}), while the Simplifier methods
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provide the canonical way for automated normalization (see
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\secref{sec:simplifier}).
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\begin{descr}
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\item [\hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{eq}] performs a single substitution
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step using rule \isa{eq}, which may be either a meta or object
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equality.
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\item [\hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharparenright}\ eq{\isachardoublequote}}] substitutes in an
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assumption.
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\item [\hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isachardoublequote}{\isacharparenleft}i\ {\isasymdots}\ j{\isacharparenright}\ eq{\isachardoublequote}}] performs several
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substitutions in the conclusion. The numbers \isa{i} to \isa{j}
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indicate the positions to substitute at. Positions are ordered from
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the top of the term tree moving down from left to right. For
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example, in \isa{{\isachardoublequote}{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharplus}\ {\isacharparenleft}c\ {\isacharplus}\ d{\isacharparenright}{\isachardoublequote}} there are three positions
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where commutativity of \isa{{\isachardoublequote}{\isacharplus}{\isachardoublequote}} is applicable: 1 refers to
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\isa{{\isachardoublequote}a\ {\isacharplus}\ b{\isachardoublequote}}, 2 to the whole term, and 3 to \isa{{\isachardoublequote}c\ {\isacharplus}\ d{\isachardoublequote}}.
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If the positions in the list \isa{{\isachardoublequote}{\isacharparenleft}i\ {\isasymdots}\ j{\isacharparenright}{\isachardoublequote}} are non-overlapping
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(e.g.\ \isa{{\isachardoublequote}{\isacharparenleft}{\isadigit{2}}\ {\isadigit{3}}{\isacharparenright}{\isachardoublequote}} in \isa{{\isachardoublequote}{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharplus}\ {\isacharparenleft}c\ {\isacharplus}\ d{\isacharparenright}{\isachardoublequote}}) you may
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assume all substitutions are performed simultaneously. Otherwise
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the behaviour of \isa{subst} is not specified.
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\item [\hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharparenright}\ {\isacharparenleft}i\ {\isasymdots}\ j{\isacharparenright}\ eq{\isachardoublequote}}] performs the
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substitutions in the assumptions. The positions refer to the
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assumptions in order from left to right. For example, given in a
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goal of the form \isa{{\isachardoublequote}P\ {\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}c\ {\isacharplus}\ d{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}{\isachardoublequote}}, position 1 of
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commutativity of \isa{{\isachardoublequote}{\isacharplus}{\isachardoublequote}} is the subterm \isa{{\isachardoublequote}a\ {\isacharplus}\ b{\isachardoublequote}} and
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position 2 is the subterm \isa{{\isachardoublequote}c\ {\isacharplus}\ d{\isachardoublequote}}.
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\item [\hyperlink{method.hypsubst}{\mbox{\isa{hypsubst}}}] performs substitution using some
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assumption; this only works for equations of the form \isa{{\isachardoublequote}x\ {\isacharequal}\ t{\isachardoublequote}} where \isa{x} is a free or bound variable.
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\item [\hyperlink{method.split}{\mbox{\isa{split}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] performs
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single-step case splitting using the given rules. By default,
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splitting is performed in the conclusion of a goal; the \isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharparenright}{\isachardoublequote}} option indicates to operate on assumptions instead.
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Note that the \hyperlink{method.simp}{\mbox{\isa{simp}}} method already involves repeated
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application of split rules as declared in the current context.
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\end{descr}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Further tactic emulations \label{sec:tactics}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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The following improper proof methods emulate traditional tactics.
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These admit direct access to the goal state, which is normally
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considered harmful! In particular, this may involve both numbered
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goal addressing (default 1), and dynamic instantiation within the
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scope of some subgoal.
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\begin{warn}
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Dynamic instantiations refer to universally quantified parameters
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of a subgoal (the dynamic context) rather than fixed variables and
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term abbreviations of a (static) Isar context.
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\end{warn}
|
|
278 |
|
|
279 |
Tactic emulation methods, unlike their ML counterparts, admit
|
|
280 |
simultaneous instantiation from both dynamic and static contexts.
|
|
281 |
If names occur in both contexts goal parameters hide locally fixed
|
|
282 |
variables. Likewise, schematic variables refer to term
|
|
283 |
abbreviations, if present in the static context. Otherwise the
|
|
284 |
schematic variable is interpreted as a schematic variable and left
|
|
285 |
to be solved by unification with certain parts of the subgoal.
|
|
286 |
|
|
287 |
Note that the tactic emulation proof methods in Isabelle/Isar are
|
|
288 |
consistently named \isa{foo{\isacharunderscore}tac}. Note also that variable names
|
|
289 |
occurring on left hand sides of instantiations must be preceded by a
|
|
290 |
question mark if they coincide with a keyword or contain dots. This
|
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|
291 |
is consistent with the attribute \hyperlink{attribute.where}{\mbox{\isa{where}}} (see
|
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|
292 |
\secref{sec:pure-meth-att}).
|
|
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|
|
294 |
\begin{matharray}{rcl}
|
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|
295 |
\indexdef{}{method}{rule\_tac}\hypertarget{method.rule-tac}{\hyperlink{method.rule-tac}{\mbox{\isa{rule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
296 |
\indexdef{}{method}{erule\_tac}\hypertarget{method.erule-tac}{\hyperlink{method.erule-tac}{\mbox{\isa{erule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
297 |
\indexdef{}{method}{drule\_tac}\hypertarget{method.drule-tac}{\hyperlink{method.drule-tac}{\mbox{\isa{drule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
298 |
\indexdef{}{method}{frule\_tac}\hypertarget{method.frule-tac}{\hyperlink{method.frule-tac}{\mbox{\isa{frule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
299 |
\indexdef{}{method}{cut\_tac}\hypertarget{method.cut-tac}{\hyperlink{method.cut-tac}{\mbox{\isa{cut{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
300 |
\indexdef{}{method}{thin\_tac}\hypertarget{method.thin-tac}{\hyperlink{method.thin-tac}{\mbox{\isa{thin{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
301 |
\indexdef{}{method}{subgoal\_tac}\hypertarget{method.subgoal-tac}{\hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
302 |
\indexdef{}{method}{rename\_tac}\hypertarget{method.rename-tac}{\hyperlink{method.rename-tac}{\mbox{\isa{rename{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
|
303 |
\indexdef{}{method}{rotate\_tac}\hypertarget{method.rotate-tac}{\hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
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|
304 |
\indexdef{}{method}{tactic}\hypertarget{method.tactic}{\hyperlink{method.tactic}{\mbox{\isa{tactic}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
|
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|
305 |
\end{matharray}
|
|
306 |
|
|
307 |
\begin{rail}
|
|
308 |
( 'rule\_tac' | 'erule\_tac' | 'drule\_tac' | 'frule\_tac' | 'cut\_tac' | 'thin\_tac' ) goalspec?
|
|
309 |
( insts thmref | thmrefs )
|
|
310 |
;
|
|
311 |
'subgoal\_tac' goalspec? (prop +)
|
|
312 |
;
|
|
313 |
'rename\_tac' goalspec? (name +)
|
|
314 |
;
|
|
315 |
'rotate\_tac' goalspec? int?
|
|
316 |
;
|
|
317 |
'tactic' text
|
|
318 |
;
|
|
319 |
|
|
320 |
insts: ((name '=' term) + 'and') 'in'
|
|
321 |
;
|
|
322 |
\end{rail}
|
|
323 |
|
|
324 |
\begin{descr}
|
|
325 |
|
26907
|
326 |
\item [\hyperlink{method.rule-tac}{\mbox{\isa{rule{\isacharunderscore}tac}}} etc.] do resolution of rules with explicit
|
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|
327 |
instantiation. This works the same way as the ML tactics \verb|res_inst_tac| etc. (see \cite[\S3]{isabelle-ref}).
|
|
328 |
|
|
329 |
Multiple rules may be only given if there is no instantiation; then
|
26907
|
330 |
\hyperlink{method.rule-tac}{\mbox{\isa{rule{\isacharunderscore}tac}}} is the same as \verb|resolve_tac| in ML (see
|
26782
|
331 |
\cite[\S3]{isabelle-ref}).
|
|
332 |
|
26907
|
333 |
\item [\hyperlink{method.cut-tac}{\mbox{\isa{cut{\isacharunderscore}tac}}}] inserts facts into the proof state as
|
26782
|
334 |
assumption of a subgoal, see also \verb|cut_facts_tac| in
|
|
335 |
\cite[\S3]{isabelle-ref}. Note that the scope of schematic
|
|
336 |
variables is spread over the main goal statement. Instantiations
|
|
337 |
may be given as well, see also ML tactic \verb|cut_inst_tac| in
|
|
338 |
\cite[\S3]{isabelle-ref}.
|
|
339 |
|
26907
|
340 |
\item [\hyperlink{method.thin-tac}{\mbox{\isa{thin{\isacharunderscore}tac}}}~\isa{{\isasymphi}}] deletes the specified
|
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|
341 |
assumption from a subgoal; note that \isa{{\isasymphi}} may contain schematic
|
|
342 |
variables. See also \verb|thin_tac| in \cite[\S3]{isabelle-ref}.
|
|
343 |
|
26907
|
344 |
\item [\hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isacharunderscore}tac}}}~\isa{{\isasymphi}}] adds \isa{{\isasymphi}} as an
|
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|
345 |
assumption to a subgoal. See also \verb|subgoal_tac| and \verb|subgoals_tac| in \cite[\S3]{isabelle-ref}.
|
|
346 |
|
26907
|
347 |
\item [\hyperlink{method.rename-tac}{\mbox{\isa{rename{\isacharunderscore}tac}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n{\isachardoublequote}}] renames
|
26842
|
348 |
parameters of a goal according to the list \isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub n{\isachardoublequote}}, which refers to the \emph{suffix} of variables.
|
26782
|
349 |
|
26907
|
350 |
\item [\hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\isacharunderscore}tac}}}~\isa{n}] rotates the assumptions of a
|
26782
|
351 |
goal by \isa{n} positions: from right to left if \isa{n} is
|
|
352 |
positive, and from left to right if \isa{n} is negative; the
|
|
353 |
default value is 1. See also \verb|rotate_tac| in
|
|
354 |
\cite[\S3]{isabelle-ref}.
|
|
355 |
|
26902
|
356 |
\item [\hyperlink{method.tactic}{\mbox{\isa{tactic}}}~\isa{{\isachardoublequote}text{\isachardoublequote}}] produces a proof method from
|
26782
|
357 |
any ML text of type \verb|tactic|. Apart from the usual ML
|
|
358 |
environment and the current implicit theory context, the ML code may
|
|
359 |
refer to the following locally bound values:
|
|
360 |
|
|
361 |
%FIXME check
|
|
362 |
{\footnotesize\begin{verbatim}
|
|
363 |
val ctxt : Proof.context
|
|
364 |
val facts : thm list
|
|
365 |
val thm : string -> thm
|
|
366 |
val thms : string -> thm list
|
|
367 |
\end{verbatim}}
|
|
368 |
|
|
369 |
Here \verb|ctxt| refers to the current proof context, \verb|facts| indicates any current facts for forward-chaining, and \verb|thm|~/~\verb|thms| retrieve named facts (including global theorems)
|
|
370 |
from the context.
|
|
371 |
|
|
372 |
\end{descr}%
|
|
373 |
\end{isamarkuptext}%
|
|
374 |
\isamarkuptrue%
|
|
375 |
%
|
27042
|
376 |
\isamarkupsection{The Simplifier \label{sec:simplifier}%
|
26782
|
377 |
}
|
|
378 |
\isamarkuptrue%
|
|
379 |
%
|
27042
|
380 |
\isamarkupsubsection{Simplification methods%
|
26782
|
381 |
}
|
|
382 |
\isamarkuptrue%
|
|
383 |
%
|
|
384 |
\begin{isamarkuptext}%
|
|
385 |
\begin{matharray}{rcl}
|
26902
|
386 |
\indexdef{}{method}{simp}\hypertarget{method.simp}{\hyperlink{method.simp}{\mbox{\isa{simp}}}} & : & \isarmeth \\
|
26907
|
387 |
\indexdef{}{method}{simp\_all}\hypertarget{method.simp-all}{\hyperlink{method.simp-all}{\mbox{\isa{simp{\isacharunderscore}all}}}} & : & \isarmeth \\
|
26782
|
388 |
\end{matharray}
|
|
389 |
|
|
390 |
\indexouternonterm{simpmod}
|
|
391 |
\begin{rail}
|
|
392 |
('simp' | 'simp\_all') ('!' ?) opt? (simpmod *)
|
|
393 |
;
|
|
394 |
|
27094
|
395 |
opt: '(' ('no\_asm' | 'no\_asm\_simp' | 'no\_asm\_use' | 'asm\_lr' ) ')'
|
26782
|
396 |
;
|
|
397 |
simpmod: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') |
|
|
398 |
'split' (() | 'add' | 'del')) ':' thmrefs
|
|
399 |
;
|
|
400 |
\end{rail}
|
|
401 |
|
|
402 |
\begin{descr}
|
|
403 |
|
26902
|
404 |
\item [\hyperlink{method.simp}{\mbox{\isa{simp}}}] invokes the Simplifier, after declaring
|
26782
|
405 |
additional rules according to the arguments given. Note that the
|
|
406 |
\railtterm{only} modifier first removes all other rewrite rules,
|
|
407 |
congruences, and looper tactics (including splits), and then behaves
|
|
408 |
like \railtterm{add}.
|
|
409 |
|
|
410 |
\medskip The \railtterm{cong} modifiers add or delete Simplifier
|
|
411 |
congruence rules (see also \cite{isabelle-ref}), the default is to
|
|
412 |
add.
|
|
413 |
|
|
414 |
\medskip The \railtterm{split} modifiers add or delete rules for the
|
|
415 |
Splitter (see also \cite{isabelle-ref}), the default is to add.
|
|
416 |
This works only if the Simplifier method has been properly setup to
|
|
417 |
include the Splitter (all major object logics such HOL, HOLCF, FOL,
|
|
418 |
ZF do this already).
|
|
419 |
|
26907
|
420 |
\item [\hyperlink{method.simp-all}{\mbox{\isa{simp{\isacharunderscore}all}}}] is similar to \hyperlink{method.simp}{\mbox{\isa{simp}}}, but acts on
|
26782
|
421 |
all goals (backwards from the last to the first one).
|
|
422 |
|
|
423 |
\end{descr}
|
|
424 |
|
|
425 |
By default the Simplifier methods take local assumptions fully into
|
|
426 |
account, using equational assumptions in the subsequent
|
|
427 |
normalization process, or simplifying assumptions themselves (cf.\
|
|
428 |
\verb|asm_full_simp_tac| in \cite[\S10]{isabelle-ref}). In
|
|
429 |
structured proofs this is usually quite well behaved in practice:
|
|
430 |
just the local premises of the actual goal are involved, additional
|
26902
|
431 |
facts may be inserted via explicit forward-chaining (via \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}, \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}, \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}} etc.). The full
|
26842
|
432 |
context of premises is only included if the ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' (bang)
|
26782
|
433 |
argument is given, which should be used with some care, though.
|
|
434 |
|
|
435 |
Additional Simplifier options may be specified to tune the behavior
|
|
436 |
further (mostly for unstructured scripts with many accidental local
|
26842
|
437 |
facts): ``\isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isachardoublequote}}'' means assumptions are ignored
|
|
438 |
completely (cf.\ \verb|simp_tac|), ``\isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharunderscore}simp{\isacharparenright}{\isachardoublequote}}'' means
|
26782
|
439 |
assumptions are used in the simplification of the conclusion but are
|
26842
|
440 |
not themselves simplified (cf.\ \verb|asm_simp_tac|), and ``\isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharunderscore}use{\isacharparenright}{\isachardoublequote}}'' means assumptions are simplified but are not used
|
26782
|
441 |
in the simplification of each other or the conclusion (cf.\ \verb|full_simp_tac|). For compatibility reasons, there is also an option
|
26842
|
442 |
``\isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharunderscore}lr{\isacharparenright}{\isachardoublequote}}'', which means that an assumption is only used
|
26782
|
443 |
for simplifying assumptions which are to the right of it (cf.\ \verb|asm_lr_simp_tac|).
|
|
444 |
|
27094
|
445 |
The configuration option \isa{{\isachardoublequote}depth{\isacharunderscore}limit{\isachardoublequote}} limits the number of
|
26782
|
446 |
recursive invocations of the simplifier during conditional
|
|
447 |
rewriting.
|
|
448 |
|
|
449 |
\medskip The Splitter package is usually configured to work as part
|
26842
|
450 |
of the Simplifier. The effect of repeatedly applying \verb|split_tac| can be simulated by ``\isa{{\isachardoublequote}{\isacharparenleft}simp\ only{\isacharcolon}\ split{\isacharcolon}\ a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharparenright}{\isachardoublequote}}''. There is also a separate \isa{split}
|
26782
|
451 |
method available for single-step case splitting.%
|
|
452 |
\end{isamarkuptext}%
|
|
453 |
\isamarkuptrue%
|
|
454 |
%
|
27042
|
455 |
\isamarkupsubsection{Declaring rules%
|
26782
|
456 |
}
|
|
457 |
\isamarkuptrue%
|
|
458 |
%
|
|
459 |
\begin{isamarkuptext}%
|
|
460 |
\begin{matharray}{rcl}
|
26907
|
461 |
\indexdef{}{command}{print\_simpset}\hypertarget{command.print-simpset}{\hyperlink{command.print-simpset}{\mbox{\isa{\isacommand{print{\isacharunderscore}simpset}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
|
26902
|
462 |
\indexdef{}{attribute}{simp}\hypertarget{attribute.simp}{\hyperlink{attribute.simp}{\mbox{\isa{simp}}}} & : & \isaratt \\
|
|
463 |
\indexdef{}{attribute}{cong}\hypertarget{attribute.cong}{\hyperlink{attribute.cong}{\mbox{\isa{cong}}}} & : & \isaratt \\
|
|
464 |
\indexdef{}{attribute}{split}\hypertarget{attribute.split}{\hyperlink{attribute.split}{\mbox{\isa{split}}}} & : & \isaratt \\
|
26782
|
465 |
\end{matharray}
|
|
466 |
|
|
467 |
\begin{rail}
|
|
468 |
('simp' | 'cong' | 'split') (() | 'add' | 'del')
|
|
469 |
;
|
|
470 |
\end{rail}
|
|
471 |
|
|
472 |
\begin{descr}
|
|
473 |
|
26907
|
474 |
\item [\hyperlink{command.print-simpset}{\mbox{\isa{\isacommand{print{\isacharunderscore}simpset}}}}] prints the collection of rules
|
26782
|
475 |
declared to the Simplifier, which is also known as ``simpset''
|
|
476 |
internally \cite{isabelle-ref}.
|
|
477 |
|
26902
|
478 |
\item [\hyperlink{attribute.simp}{\mbox{\isa{simp}}}] declares simplification rules.
|
26782
|
479 |
|
26902
|
480 |
\item [\hyperlink{attribute.cong}{\mbox{\isa{cong}}}] declares congruence rules.
|
26782
|
481 |
|
26902
|
482 |
\item [\hyperlink{attribute.split}{\mbox{\isa{split}}}] declares case split rules.
|
26782
|
483 |
|
|
484 |
\end{descr}%
|
|
485 |
\end{isamarkuptext}%
|
|
486 |
\isamarkuptrue%
|
|
487 |
%
|
27042
|
488 |
\isamarkupsubsection{Simplification procedures%
|
26782
|
489 |
}
|
|
490 |
\isamarkuptrue%
|
|
491 |
%
|
|
492 |
\begin{isamarkuptext}%
|
|
493 |
\begin{matharray}{rcl}
|
26907
|
494 |
\indexdef{}{command}{simproc\_setup}\hypertarget{command.simproc-setup}{\hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isacharunderscore}setup}}}}} & : & \isarkeep{local{\dsh}theory} \\
|
26782
|
495 |
simproc & : & \isaratt \\
|
|
496 |
\end{matharray}
|
|
497 |
|
|
498 |
\begin{rail}
|
|
499 |
'simproc\_setup' name '(' (term + '|') ')' '=' text \\ ('identifier' (nameref+))?
|
|
500 |
;
|
|
501 |
|
|
502 |
'simproc' (('add' ':')? | 'del' ':') (name+)
|
|
503 |
;
|
|
504 |
\end{rail}
|
|
505 |
|
|
506 |
\begin{descr}
|
|
507 |
|
26907
|
508 |
\item [\hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isacharunderscore}setup}}}}] defines a named simplification
|
26782
|
509 |
procedure that is invoked by the Simplifier whenever any of the
|
|
510 |
given term patterns match the current redex. The implementation,
|
26842
|
511 |
which is provided as ML source text, needs to be of type \verb|"morphism -> simpset -> cterm -> thm option"|, where the \verb|cterm| represents the current redex \isa{r} and the result is
|
|
512 |
supposed to be some proven rewrite rule \isa{{\isachardoublequote}r\ {\isasymequiv}\ r{\isacharprime}{\isachardoublequote}} (or a
|
26782
|
513 |
generalized version), or \verb|NONE| to indicate failure. The
|
|
514 |
\verb|simpset| argument holds the full context of the current
|
|
515 |
Simplifier invocation, including the actual Isar proof context. The
|
|
516 |
\verb|morphism| informs about the difference of the original
|
|
517 |
compilation context wrt.\ the one of the actual application later
|
26902
|
518 |
on. The optional \hyperlink{keyword.identifier}{\mbox{\isa{\isakeyword{identifier}}}} specifies theorems that
|
26782
|
519 |
represent the logical content of the abstract theory of this
|
|
520 |
simproc.
|
|
521 |
|
|
522 |
Morphisms and identifiers are only relevant for simprocs that are
|
|
523 |
defined within a local target context, e.g.\ in a locale.
|
|
524 |
|
26842
|
525 |
\item [\isa{{\isachardoublequote}simproc\ add{\isacharcolon}\ name{\isachardoublequote}} and \isa{{\isachardoublequote}simproc\ del{\isacharcolon}\ name{\isachardoublequote}}]
|
26782
|
526 |
add or delete named simprocs to the current Simplifier context. The
|
26907
|
527 |
default is to add a simproc. Note that \hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isacharunderscore}setup}}}}
|
26782
|
528 |
already adds the new simproc to the subsequent context.
|
|
529 |
|
|
530 |
\end{descr}%
|
|
531 |
\end{isamarkuptext}%
|
|
532 |
\isamarkuptrue%
|
|
533 |
%
|
27042
|
534 |
\isamarkupsubsection{Forward simplification%
|
26782
|
535 |
}
|
|
536 |
\isamarkuptrue%
|
|
537 |
%
|
|
538 |
\begin{isamarkuptext}%
|
|
539 |
\begin{matharray}{rcl}
|
26902
|
540 |
\indexdef{}{attribute}{simplified}\hypertarget{attribute.simplified}{\hyperlink{attribute.simplified}{\mbox{\isa{simplified}}}} & : & \isaratt \\
|
26782
|
541 |
\end{matharray}
|
|
542 |
|
|
543 |
\begin{rail}
|
|
544 |
'simplified' opt? thmrefs?
|
|
545 |
;
|
|
546 |
|
26788
|
547 |
opt: '(' ('no\_asm' | 'no\_asm\_simp' | 'no\_asm\_use') ')'
|
26782
|
548 |
;
|
|
549 |
\end{rail}
|
|
550 |
|
|
551 |
\begin{descr}
|
|
552 |
|
26902
|
553 |
\item [\hyperlink{attribute.simplified}{\mbox{\isa{simplified}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}]
|
26782
|
554 |
causes a theorem to be simplified, either by exactly the specified
|
26842
|
555 |
rules \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ a\isactrlsub n{\isachardoublequote}}, or the implicit Simplifier
|
26782
|
556 |
context if no arguments are given. The result is fully simplified
|
|
557 |
by default, including assumptions and conclusion; the options \isa{no{\isacharunderscore}asm} etc.\ tune the Simplifier in the same way as the for the
|
|
558 |
\isa{simp} method.
|
|
559 |
|
|
560 |
Note that forward simplification restricts the simplifier to its
|
|
561 |
most basic operation of term rewriting; solver and looper tactics
|
|
562 |
\cite{isabelle-ref} are \emph{not} involved here. The \isa{simplified} attribute should be only rarely required under normal
|
|
563 |
circumstances.
|
|
564 |
|
|
565 |
\end{descr}%
|
|
566 |
\end{isamarkuptext}%
|
|
567 |
\isamarkuptrue%
|
|
568 |
%
|
27042
|
569 |
\isamarkupsection{The Classical Reasoner \label{sec:classical}%
|
26782
|
570 |
}
|
|
571 |
\isamarkuptrue%
|
|
572 |
%
|
27042
|
573 |
\isamarkupsubsection{Basic methods%
|
26782
|
574 |
}
|
|
575 |
\isamarkuptrue%
|
|
576 |
%
|
|
577 |
\begin{isamarkuptext}%
|
|
578 |
\begin{matharray}{rcl}
|
26902
|
579 |
\indexdef{}{method}{rule}\hypertarget{method.rule}{\hyperlink{method.rule}{\mbox{\isa{rule}}}} & : & \isarmeth \\
|
|
580 |
\indexdef{}{method}{contradiction}\hypertarget{method.contradiction}{\hyperlink{method.contradiction}{\mbox{\isa{contradiction}}}} & : & \isarmeth \\
|
|
581 |
\indexdef{}{method}{intro}\hypertarget{method.intro}{\hyperlink{method.intro}{\mbox{\isa{intro}}}} & : & \isarmeth \\
|
|
582 |
\indexdef{}{method}{elim}\hypertarget{method.elim}{\hyperlink{method.elim}{\mbox{\isa{elim}}}} & : & \isarmeth \\
|
26782
|
583 |
\end{matharray}
|
|
584 |
|
|
585 |
\begin{rail}
|
|
586 |
('rule' | 'intro' | 'elim') thmrefs?
|
|
587 |
;
|
|
588 |
\end{rail}
|
|
589 |
|
|
590 |
\begin{descr}
|
|
591 |
|
26902
|
592 |
\item [\hyperlink{method.rule}{\mbox{\isa{rule}}}] as offered by the Classical Reasoner is a
|
26782
|
593 |
refinement over the primitive one (see \secref{sec:pure-meth-att}).
|
|
594 |
Both versions essentially work the same, but the classical version
|
|
595 |
observes the classical rule context in addition to that of
|
|
596 |
Isabelle/Pure.
|
|
597 |
|
|
598 |
Common object logics (HOL, ZF, etc.) declare a rich collection of
|
|
599 |
classical rules (even if these would qualify as intuitionistic
|
|
600 |
ones), but only few declarations to the rule context of
|
|
601 |
Isabelle/Pure (\secref{sec:pure-meth-att}).
|
|
602 |
|
26902
|
603 |
\item [\hyperlink{method.contradiction}{\mbox{\isa{contradiction}}}] solves some goal by contradiction,
|
26842
|
604 |
deriving any result from both \isa{{\isachardoublequote}{\isasymnot}\ A{\isachardoublequote}} and \isa{A}. Chained
|
26782
|
605 |
facts, which are guaranteed to participate, may appear in either
|
|
606 |
order.
|
|
607 |
|
26902
|
608 |
\item [\hyperlink{method.intro}{\mbox{\isa{intro}}} and \hyperlink{method.elim}{\mbox{\isa{elim}}}] repeatedly refine some
|
|
609 |
goal by intro- or elim-resolution, after having inserted any chained
|
|
610 |
facts. Exactly the rules given as arguments are taken into account;
|
|
611 |
this allows fine-tuned decomposition of a proof problem, in contrast
|
|
612 |
to common automated tools.
|
26782
|
613 |
|
|
614 |
\end{descr}%
|
|
615 |
\end{isamarkuptext}%
|
|
616 |
\isamarkuptrue%
|
|
617 |
%
|
27042
|
618 |
\isamarkupsubsection{Automated methods%
|
26782
|
619 |
}
|
|
620 |
\isamarkuptrue%
|
|
621 |
%
|
|
622 |
\begin{isamarkuptext}%
|
|
623 |
\begin{matharray}{rcl}
|
26902
|
624 |
\indexdef{}{method}{blast}\hypertarget{method.blast}{\hyperlink{method.blast}{\mbox{\isa{blast}}}} & : & \isarmeth \\
|
|
625 |
\indexdef{}{method}{fast}\hypertarget{method.fast}{\hyperlink{method.fast}{\mbox{\isa{fast}}}} & : & \isarmeth \\
|
|
626 |
\indexdef{}{method}{slow}\hypertarget{method.slow}{\hyperlink{method.slow}{\mbox{\isa{slow}}}} & : & \isarmeth \\
|
|
627 |
\indexdef{}{method}{best}\hypertarget{method.best}{\hyperlink{method.best}{\mbox{\isa{best}}}} & : & \isarmeth \\
|
|
628 |
\indexdef{}{method}{safe}\hypertarget{method.safe}{\hyperlink{method.safe}{\mbox{\isa{safe}}}} & : & \isarmeth \\
|
|
629 |
\indexdef{}{method}{clarify}\hypertarget{method.clarify}{\hyperlink{method.clarify}{\mbox{\isa{clarify}}}} & : & \isarmeth \\
|
26782
|
630 |
\end{matharray}
|
|
631 |
|
|
632 |
\indexouternonterm{clamod}
|
|
633 |
\begin{rail}
|
|
634 |
'blast' ('!' ?) nat? (clamod *)
|
|
635 |
;
|
|
636 |
('fast' | 'slow' | 'best' | 'safe' | 'clarify') ('!' ?) (clamod *)
|
|
637 |
;
|
|
638 |
|
|
639 |
clamod: (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' thmrefs
|
|
640 |
;
|
|
641 |
\end{rail}
|
|
642 |
|
|
643 |
\begin{descr}
|
|
644 |
|
26902
|
645 |
\item [\hyperlink{method.blast}{\mbox{\isa{blast}}}] refers to the classical tableau prover (see
|
26782
|
646 |
\verb|blast_tac| in \cite[\S11]{isabelle-ref}). The optional
|
|
647 |
argument specifies a user-supplied search bound (default 20).
|
|
648 |
|
26902
|
649 |
\item [\hyperlink{method.fast}{\mbox{\isa{fast}}}, \hyperlink{method.slow}{\mbox{\isa{slow}}}, \hyperlink{method.best}{\mbox{\isa{best}}}, \hyperlink{method.safe}{\mbox{\isa{safe}}}, and \hyperlink{method.clarify}{\mbox{\isa{clarify}}}] refer to the generic classical
|
26782
|
650 |
reasoner. See \verb|fast_tac|, \verb|slow_tac|, \verb|best_tac|, \verb|safe_tac|, and \verb|clarify_tac| in \cite[\S11]{isabelle-ref} for
|
|
651 |
more information.
|
|
652 |
|
|
653 |
\end{descr}
|
|
654 |
|
|
655 |
Any of the above methods support additional modifiers of the context
|
|
656 |
of classical rules. Their semantics is analogous to the attributes
|
|
657 |
given before. Facts provided by forward chaining are inserted into
|
26842
|
658 |
the goal before commencing proof search. The ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}''~argument causes the full context of assumptions to be
|
26782
|
659 |
included as well.%
|
|
660 |
\end{isamarkuptext}%
|
|
661 |
\isamarkuptrue%
|
|
662 |
%
|
27042
|
663 |
\isamarkupsubsection{Combined automated methods \label{sec:clasimp}%
|
26782
|
664 |
}
|
|
665 |
\isamarkuptrue%
|
|
666 |
%
|
|
667 |
\begin{isamarkuptext}%
|
|
668 |
\begin{matharray}{rcl}
|
26902
|
669 |
\indexdef{}{method}{auto}\hypertarget{method.auto}{\hyperlink{method.auto}{\mbox{\isa{auto}}}} & : & \isarmeth \\
|
|
670 |
\indexdef{}{method}{force}\hypertarget{method.force}{\hyperlink{method.force}{\mbox{\isa{force}}}} & : & \isarmeth \\
|
|
671 |
\indexdef{}{method}{clarsimp}\hypertarget{method.clarsimp}{\hyperlink{method.clarsimp}{\mbox{\isa{clarsimp}}}} & : & \isarmeth \\
|
|
672 |
\indexdef{}{method}{fastsimp}\hypertarget{method.fastsimp}{\hyperlink{method.fastsimp}{\mbox{\isa{fastsimp}}}} & : & \isarmeth \\
|
|
673 |
\indexdef{}{method}{slowsimp}\hypertarget{method.slowsimp}{\hyperlink{method.slowsimp}{\mbox{\isa{slowsimp}}}} & : & \isarmeth \\
|
|
674 |
\indexdef{}{method}{bestsimp}\hypertarget{method.bestsimp}{\hyperlink{method.bestsimp}{\mbox{\isa{bestsimp}}}} & : & \isarmeth \\
|
26782
|
675 |
\end{matharray}
|
|
676 |
|
|
677 |
\indexouternonterm{clasimpmod}
|
|
678 |
\begin{rail}
|
|
679 |
'auto' '!'? (nat nat)? (clasimpmod *)
|
|
680 |
;
|
|
681 |
('force' | 'clarsimp' | 'fastsimp' | 'slowsimp' | 'bestsimp') '!'? (clasimpmod *)
|
|
682 |
;
|
|
683 |
|
|
684 |
clasimpmod: ('simp' (() | 'add' | 'del' | 'only') |
|
|
685 |
('cong' | 'split') (() | 'add' | 'del') |
|
|
686 |
'iff' (((() | 'add') '?'?) | 'del') |
|
|
687 |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' thmrefs
|
|
688 |
\end{rail}
|
|
689 |
|
|
690 |
\begin{descr}
|
|
691 |
|
26902
|
692 |
\item [\hyperlink{method.auto}{\mbox{\isa{auto}}}, \hyperlink{method.force}{\mbox{\isa{force}}}, \hyperlink{method.clarsimp}{\mbox{\isa{clarsimp}}}, \hyperlink{method.fastsimp}{\mbox{\isa{fastsimp}}}, \hyperlink{method.slowsimp}{\mbox{\isa{slowsimp}}}, and \hyperlink{method.bestsimp}{\mbox{\isa{bestsimp}}}] provide
|
26782
|
693 |
access to Isabelle's combined simplification and classical reasoning
|
|
694 |
tactics. These correspond to \verb|auto_tac|, \verb|force_tac|, \verb|clarsimp_tac|, and Classical Reasoner tactics with the Simplifier
|
|
695 |
added as wrapper, see \cite[\S11]{isabelle-ref} for more
|
|
696 |
information. The modifier arguments correspond to those given in
|
|
697 |
\secref{sec:simplifier} and \secref{sec:classical}. Just note that
|
|
698 |
the ones related to the Simplifier are prefixed by \railtterm{simp}
|
|
699 |
here.
|
|
700 |
|
|
701 |
Facts provided by forward chaining are inserted into the goal before
|
26842
|
702 |
doing the search. The ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' argument causes the full
|
26782
|
703 |
context of assumptions to be included as well.
|
|
704 |
|
|
705 |
\end{descr}%
|
|
706 |
\end{isamarkuptext}%
|
|
707 |
\isamarkuptrue%
|
|
708 |
%
|
27042
|
709 |
\isamarkupsubsection{Declaring rules%
|
26782
|
710 |
}
|
|
711 |
\isamarkuptrue%
|
|
712 |
%
|
|
713 |
\begin{isamarkuptext}%
|
|
714 |
\begin{matharray}{rcl}
|
26907
|
715 |
\indexdef{}{command}{print\_claset}\hypertarget{command.print-claset}{\hyperlink{command.print-claset}{\mbox{\isa{\isacommand{print{\isacharunderscore}claset}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
|
26902
|
716 |
\indexdef{}{attribute}{intro}\hypertarget{attribute.intro}{\hyperlink{attribute.intro}{\mbox{\isa{intro}}}} & : & \isaratt \\
|
|
717 |
\indexdef{}{attribute}{elim}\hypertarget{attribute.elim}{\hyperlink{attribute.elim}{\mbox{\isa{elim}}}} & : & \isaratt \\
|
|
718 |
\indexdef{}{attribute}{dest}\hypertarget{attribute.dest}{\hyperlink{attribute.dest}{\mbox{\isa{dest}}}} & : & \isaratt \\
|
|
719 |
\indexdef{}{attribute}{rule}\hypertarget{attribute.rule}{\hyperlink{attribute.rule}{\mbox{\isa{rule}}}} & : & \isaratt \\
|
|
720 |
\indexdef{}{attribute}{iff}\hypertarget{attribute.iff}{\hyperlink{attribute.iff}{\mbox{\isa{iff}}}} & : & \isaratt \\
|
26782
|
721 |
\end{matharray}
|
|
722 |
|
|
723 |
\begin{rail}
|
|
724 |
('intro' | 'elim' | 'dest') ('!' | () | '?') nat?
|
|
725 |
;
|
|
726 |
'rule' 'del'
|
|
727 |
;
|
|
728 |
'iff' (((() | 'add') '?'?) | 'del')
|
|
729 |
;
|
|
730 |
\end{rail}
|
|
731 |
|
|
732 |
\begin{descr}
|
|
733 |
|
26907
|
734 |
\item [\hyperlink{command.print-claset}{\mbox{\isa{\isacommand{print{\isacharunderscore}claset}}}}] prints the collection of rules
|
26782
|
735 |
declared to the Classical Reasoner, which is also known as
|
|
736 |
``claset'' internally \cite{isabelle-ref}.
|
|
737 |
|
26902
|
738 |
\item [\hyperlink{attribute.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.elim}{\mbox{\isa{elim}}}, and \hyperlink{attribute.dest}{\mbox{\isa{dest}}}]
|
26782
|
739 |
declare introduction, elimination, and destruction rules,
|
|
740 |
respectively. By default, rules are considered as \emph{unsafe}
|
26842
|
741 |
(i.e.\ not applied blindly without backtracking), while ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' classifies as \emph{safe}. Rule declarations marked by
|
|
742 |
``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' coincide with those of Isabelle/Pure, cf.\
|
26782
|
743 |
\secref{sec:pure-meth-att} (i.e.\ are only applied in single steps
|
26902
|
744 |
of the \hyperlink{method.rule}{\mbox{\isa{rule}}} method). The optional natural number
|
26782
|
745 |
specifies an explicit weight argument, which is ignored by automated
|
|
746 |
tools, but determines the search order of single rule steps.
|
|
747 |
|
26902
|
748 |
\item [\hyperlink{attribute.rule}{\mbox{\isa{rule}}}~\isa{del}] deletes introduction,
|
26782
|
749 |
elimination, or destruction rules from the context.
|
|
750 |
|
26902
|
751 |
\item [\hyperlink{attribute.iff}{\mbox{\isa{iff}}}] declares logical equivalences to the
|
26782
|
752 |
Simplifier and the Classical reasoner at the same time.
|
|
753 |
Non-conditional rules result in a ``safe'' introduction and
|
|
754 |
elimination pair; conditional ones are considered ``unsafe''. Rules
|
26842
|
755 |
with negative conclusion are automatically inverted (using \isa{{\isachardoublequote}{\isasymnot}{\isachardoublequote}}-elimination internally).
|
26782
|
756 |
|
26902
|
757 |
The ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' version of \hyperlink{attribute.iff}{\mbox{\isa{iff}}} declares rules to
|
26782
|
758 |
the Isabelle/Pure context only, and omits the Simplifier
|
|
759 |
declaration.
|
|
760 |
|
|
761 |
\end{descr}%
|
|
762 |
\end{isamarkuptext}%
|
|
763 |
\isamarkuptrue%
|
|
764 |
%
|
27042
|
765 |
\isamarkupsubsection{Classical operations%
|
26782
|
766 |
}
|
|
767 |
\isamarkuptrue%
|
|
768 |
%
|
|
769 |
\begin{isamarkuptext}%
|
|
770 |
\begin{matharray}{rcl}
|
26902
|
771 |
\indexdef{}{attribute}{swapped}\hypertarget{attribute.swapped}{\hyperlink{attribute.swapped}{\mbox{\isa{swapped}}}} & : & \isaratt \\
|
26782
|
772 |
\end{matharray}
|
|
773 |
|
|
774 |
\begin{descr}
|
|
775 |
|
26902
|
776 |
\item [\hyperlink{attribute.swapped}{\mbox{\isa{swapped}}}] turns an introduction rule into an
|
26842
|
777 |
elimination, by resolving with the classical swap principle \isa{{\isachardoublequote}{\isacharparenleft}{\isasymnot}\ B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymnot}\ A\ {\isasymLongrightarrow}\ B{\isacharparenright}{\isachardoublequote}}.
|
26782
|
778 |
|
|
779 |
\end{descr}%
|
|
780 |
\end{isamarkuptext}%
|
|
781 |
\isamarkuptrue%
|
|
782 |
%
|
27047
|
783 |
\isamarkupsection{Object-logic setup \label{sec:object-logic}%
|
26790
|
784 |
}
|
|
785 |
\isamarkuptrue%
|
|
786 |
%
|
|
787 |
\begin{isamarkuptext}%
|
|
788 |
\begin{matharray}{rcl}
|
26902
|
789 |
\indexdef{}{command}{judgment}\hypertarget{command.judgment}{\hyperlink{command.judgment}{\mbox{\isa{\isacommand{judgment}}}}} & : & \isartrans{theory}{theory} \\
|
|
790 |
\indexdef{}{method}{atomize}\hypertarget{method.atomize}{\hyperlink{method.atomize}{\mbox{\isa{atomize}}}} & : & \isarmeth \\
|
|
791 |
\indexdef{}{attribute}{atomize}\hypertarget{attribute.atomize}{\hyperlink{attribute.atomize}{\mbox{\isa{atomize}}}} & : & \isaratt \\
|
26907
|
792 |
\indexdef{}{attribute}{rule\_format}\hypertarget{attribute.rule-format}{\hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}}} & : & \isaratt \\
|
26902
|
793 |
\indexdef{}{attribute}{rulify}\hypertarget{attribute.rulify}{\hyperlink{attribute.rulify}{\mbox{\isa{rulify}}}} & : & \isaratt \\
|
26790
|
794 |
\end{matharray}
|
|
795 |
|
|
796 |
The very starting point for any Isabelle object-logic is a ``truth
|
|
797 |
judgment'' that links object-level statements to the meta-logic
|
|
798 |
(with its minimal language of \isa{prop} that covers universal
|
26842
|
799 |
quantification \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}} and implication \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}}).
|
26790
|
800 |
|
|
801 |
Common object-logics are sufficiently expressive to internalize rule
|
26842
|
802 |
statements over \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}} within their own
|
26790
|
803 |
language. This is useful in certain situations where a rule needs
|
|
804 |
to be viewed as an atomic statement from the meta-level perspective,
|
26842
|
805 |
e.g.\ \isa{{\isachardoublequote}{\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ P\ x{\isachardoublequote}} versus \isa{{\isachardoublequote}{\isasymforall}x\ {\isasymin}\ A{\isachardot}\ P\ x{\isachardoublequote}}.
|
26790
|
806 |
|
26902
|
807 |
From the following language elements, only the \hyperlink{method.atomize}{\mbox{\isa{atomize}}}
|
26907
|
808 |
method and \hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}} attribute are occasionally
|
26790
|
809 |
required by end-users, the rest is for those who need to setup their
|
|
810 |
own object-logic. In the latter case existing formulations of
|
|
811 |
Isabelle/FOL or Isabelle/HOL may be taken as realistic examples.
|
|
812 |
|
|
813 |
Generic tools may refer to the information provided by object-logic
|
|
814 |
declarations internally.
|
|
815 |
|
|
816 |
\begin{rail}
|
|
817 |
'judgment' constdecl
|
|
818 |
;
|
|
819 |
'atomize' ('(' 'full' ')')?
|
|
820 |
;
|
|
821 |
'rule\_format' ('(' 'noasm' ')')?
|
|
822 |
;
|
|
823 |
\end{rail}
|
|
824 |
|
|
825 |
\begin{descr}
|
|
826 |
|
26902
|
827 |
\item [\hyperlink{command.judgment}{\mbox{\isa{\isacommand{judgment}}}}~\isa{{\isachardoublequote}c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\ {\isacharparenleft}mx{\isacharparenright}{\isachardoublequote}}] declares
|
26790
|
828 |
constant \isa{c} as the truth judgment of the current
|
|
829 |
object-logic. Its type \isa{{\isasymsigma}} should specify a coercion of the
|
|
830 |
category of object-level propositions to \isa{prop} of the Pure
|
26842
|
831 |
meta-logic; the mixfix annotation \isa{{\isachardoublequote}{\isacharparenleft}mx{\isacharparenright}{\isachardoublequote}} would typically
|
26790
|
832 |
just link the object language (internally of syntactic category
|
26902
|
833 |
\isa{logic}) with that of \isa{prop}. Only one \hyperlink{command.judgment}{\mbox{\isa{\isacommand{judgment}}}} declaration may be given in any theory development.
|
26790
|
834 |
|
26902
|
835 |
\item [\hyperlink{method.atomize}{\mbox{\isa{atomize}}} (as a method)] rewrites any non-atomic
|
26790
|
836 |
premises of a sub-goal, using the meta-level equations declared via
|
26902
|
837 |
\hyperlink{attribute.atomize}{\mbox{\isa{atomize}}} (as an attribute) beforehand. As a result,
|
26790
|
838 |
heavily nested goals become amenable to fundamental operations such
|
26902
|
839 |
as resolution (cf.\ the \hyperlink{method.rule}{\mbox{\isa{rule}}} method). Giving the ``\isa{{\isachardoublequote}{\isacharparenleft}full{\isacharparenright}{\isachardoublequote}}'' option here means to turn the whole subgoal into an
|
26790
|
840 |
object-statement (if possible), including the outermost parameters
|
|
841 |
and assumptions as well.
|
|
842 |
|
26902
|
843 |
A typical collection of \hyperlink{attribute.atomize}{\mbox{\isa{atomize}}} rules for a particular
|
26790
|
844 |
object-logic would provide an internalization for each of the
|
26842
|
845 |
connectives of \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}}, \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}}, and \isa{{\isachardoublequote}{\isasymequiv}{\isachardoublequote}}.
|
26790
|
846 |
Meta-level conjunction should be covered as well (this is
|
|
847 |
particularly important for locales, see \secref{sec:locale}).
|
|
848 |
|
26907
|
849 |
\item [\hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}}] rewrites a theorem by the
|
26902
|
850 |
equalities declared as \hyperlink{attribute.rulify}{\mbox{\isa{rulify}}} rules in the current
|
26790
|
851 |
object-logic. By default, the result is fully normalized, including
|
26842
|
852 |
assumptions and conclusions at any depth. The \isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isachardoublequote}}
|
26790
|
853 |
option restricts the transformation to the conclusion of a rule.
|
|
854 |
|
26907
|
855 |
In common object-logics (HOL, FOL, ZF), the effect of \hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}} is to replace (bounded) universal quantification
|
26842
|
856 |
(\isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}}) and implication (\isa{{\isachardoublequote}{\isasymlongrightarrow}{\isachardoublequote}}) by the corresponding
|
|
857 |
rule statements over \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}}.
|
26790
|
858 |
|
|
859 |
\end{descr}%
|
|
860 |
\end{isamarkuptext}%
|
|
861 |
\isamarkuptrue%
|
|
862 |
%
|
26782
|
863 |
\isadelimtheory
|
|
864 |
%
|
|
865 |
\endisadelimtheory
|
|
866 |
%
|
|
867 |
\isatagtheory
|
|
868 |
\isacommand{end}\isamarkupfalse%
|
|
869 |
%
|
|
870 |
\endisatagtheory
|
|
871 |
{\isafoldtheory}%
|
|
872 |
%
|
|
873 |
\isadelimtheory
|
|
874 |
%
|
|
875 |
\endisadelimtheory
|
|
876 |
\isanewline
|
|
877 |
\end{isabellebody}%
|
|
878 |
%%% Local Variables:
|
|
879 |
%%% mode: latex
|
|
880 |
%%% TeX-master: "root"
|
|
881 |
%%% End:
|