%

\begin{isabellebody}%

\def\isabellecontext{Generic}%

%

\isadelimtheory

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\endisadelimtheory

%

\isatagtheory

\isacommand{theory}\isamarkupfalse%

\ Generic\isanewline

\isakeyword{imports}\ Base\ Main\isanewline

\isakeyword{begin}%

\endisatagtheory

{\isafoldtheory}%

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\isadelimtheory

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\endisadelimtheory

%

\isamarkupchapter{Generic tools and packages \label{ch:gen-tools}%

}

\isamarkuptrue%

%

\isamarkupsection{Configuration options \label{sec:config}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

Isabelle/Pure maintains a record of named configuration

  options within the theory or proof context, with values of type

  \verb|bool|, \verb|int|, \verb|real|, or \verb|string|.  Tools may declare options in ML, and then refer to these

  values (relative to the context).  Thus global reference variables

  are easily avoided.  The user may change the value of a

  configuration option by means of an associated attribute of the same

  name.  This form of context declaration works particularly well with

  commands such as \hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}} or \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}} like

  this:%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{declare}\isamarkupfalse%

\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5B}{\isacharbrackleft}}show{\isaliteral{5F}{\isacharunderscore}}main{\isaliteral{5F}{\isacharunderscore}}goal\ {\isaliteral{3D}{\isacharequal}}\ false{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5D}{\isacharbrackright}}\isanewline

\isanewline

\isacommand{notepad}\isamarkupfalse%

\isanewline

\isakeyword{begin}\isanewline

%

\isadelimproof

\ \ %

\endisadelimproof

%

\isatagproof

\isacommand{note}\isamarkupfalse%

\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5B}{\isacharbrackleft}}show{\isaliteral{5F}{\isacharunderscore}}main{\isaliteral{5F}{\isacharunderscore}}goal\ {\isaliteral{3D}{\isacharequal}}\ true{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5D}{\isacharbrackright}}%

\endisatagproof

{\isafoldproof}%

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\isadelimproof

\isanewline

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\endisadelimproof

\isacommand{end}\isamarkupfalse%

%

\begin{isamarkuptext}%

For historical reasons, some tools cannot take the full proof

  context into account and merely refer to the background theory.

  This is accommodated by configuration options being declared as

  ``global'', which may not be changed within a local context.

  \begin{matharray}{rcll}

    \indexdef{}{command}{print\_configs}\hypertarget{command.print-configs}{\hyperlink{command.print-configs}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}configs}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{6}{}

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]

\rail@bar

\rail@term{\isa{true}}[]

\rail@nextbar{2}

\rail@term{\isa{false}}[]

\rail@nextbar{3}

\rail@nont{\hyperlink{syntax.int}{\mbox{\isa{int}}}}[]

\rail@nextbar{4}

\rail@nont{\hyperlink{syntax.float}{\mbox{\isa{float}}}}[]

\rail@nextbar{5}

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@endbar

\rail@endbar

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.print-configs}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}configs}}}} prints the available configuration

  options, with names, types, and current values.

  \item \isa{{\isaliteral{22}{\isachardoublequote}}name\ {\isaliteral{3D}{\isacharequal}}\ value{\isaliteral{22}{\isachardoublequote}}} as an attribute expression modifies the

  named option, with the syntax of the value depending on the option's

  type.  For \verb|bool| the default value is \isa{true}.  Any

  attempt to change a global option in a local context is ignored.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Basic proof tools%

}

\isamarkuptrue%

%

\isamarkupsubsection{Miscellaneous methods and attributes \label{sec:misc-meth-att}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{method}{unfold}\hypertarget{method.unfold}{\hyperlink{method.unfold}{\mbox{\isa{unfold}}}} & : & \isa{method} \\

    \indexdef{}{method}{fold}\hypertarget{method.fold}{\hyperlink{method.fold}{\mbox{\isa{fold}}}} & : & \isa{method} \\

    \indexdef{}{method}{insert}\hypertarget{method.insert}{\hyperlink{method.insert}{\mbox{\isa{insert}}}} & : & \isa{method} \\[0.5ex]

    \indexdef{}{method}{erule}\hypertarget{method.erule}{\hyperlink{method.erule}{\mbox{\isa{erule}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{drule}\hypertarget{method.drule}{\hyperlink{method.drule}{\mbox{\isa{drule}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{frule}\hypertarget{method.frule}{\hyperlink{method.frule}{\mbox{\isa{frule}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{intro}\hypertarget{method.intro}{\hyperlink{method.intro}{\mbox{\isa{intro}}}} & : & \isa{method} \\

    \indexdef{}{method}{elim}\hypertarget{method.elim}{\hyperlink{method.elim}{\mbox{\isa{elim}}}} & : & \isa{method} \\

    \indexdef{}{method}{succeed}\hypertarget{method.succeed}{\hyperlink{method.succeed}{\mbox{\isa{succeed}}}} & : & \isa{method} \\

    \indexdef{}{method}{fail}\hypertarget{method.fail}{\hyperlink{method.fail}{\mbox{\isa{fail}}}} & : & \isa{method} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{3}{}

\rail@bar

\rail@term{\hyperlink{method.fold}{\mbox{\isa{fold}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{method.unfold}{\mbox{\isa{unfold}}}}[]

\rail@nextbar{2}

\rail@term{\hyperlink{method.insert}{\mbox{\isa{insert}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@end

\rail@begin{3}{}

\rail@bar

\rail@term{\hyperlink{method.erule}{\mbox{\isa{erule}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{method.drule}{\mbox{\isa{drule}}}}[]

\rail@nextbar{2}

\rail@term{\hyperlink{method.frule}{\mbox{\isa{frule}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@end

\rail@begin{2}{}

\rail@bar

\rail@term{\hyperlink{method.intro}{\mbox{\isa{intro}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{method.elim}{\mbox{\isa{elim}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@endbar

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{method.unfold}{\mbox{\isa{unfold}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} and \hyperlink{method.fold}{\mbox{\isa{fold}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} expand (or fold back) the given definitions throughout

  all goals; any chained facts provided are inserted into the goal and

  subject to rewriting as well.

  \item \hyperlink{method.insert}{\mbox{\isa{insert}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} inserts theorems as facts

  into all goals of the proof state.  Note that current facts

  indicated for forward chaining are ignored.

  \item \hyperlink{method.erule}{\mbox{\isa{erule}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}, \hyperlink{method.drule}{\mbox{\isa{drule}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}, and \hyperlink{method.frule}{\mbox{\isa{frule}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} are similar to the basic \hyperlink{method.rule}{\mbox{\isa{rule}}}

  method (see \secref{sec:pure-meth-att}), but apply rules by

  elim-resolution, destruct-resolution, and forward-resolution,

  respectively \cite{isabelle-implementation}.  The optional natural

  number argument (default 0) specifies additional assumption steps to

  be performed here.

  Note that these methods are improper ones, mainly serving for

  experimentation and tactic script emulation.  Different modes of

  basic rule application are usually expressed in Isar at the proof

  language level, rather than via implicit proof state manipulations.

  For example, a proper single-step elimination would be done using

  the plain \hyperlink{method.rule}{\mbox{\isa{rule}}} method, with forward chaining of current

  facts.

  \item \hyperlink{method.intro}{\mbox{\isa{intro}}} and \hyperlink{method.elim}{\mbox{\isa{elim}}} repeatedly refine some goal

  by intro- or elim-resolution, after having inserted any chained

  facts.  Exactly the rules given as arguments are taken into account;

  this allows fine-tuned decomposition of a proof problem, in contrast

  to common automated tools.

  \item \hyperlink{method.succeed}{\mbox{\isa{succeed}}} yields a single (unchanged) result; it is

  the identity of the ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{22}{\isachardoublequote}}}'' method combinator (cf.\

  \secref{sec:proof-meth}).

  \item \hyperlink{method.fail}{\mbox{\isa{fail}}} yields an empty result sequence; it is the

  identity of the ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}}'' method combinator (cf.\

  \secref{sec:proof-meth}).

  \end{description}

  \begin{matharray}{rcl}

    \indexdef{}{attribute}{tagged}\hypertarget{attribute.tagged}{\hyperlink{attribute.tagged}{\mbox{\isa{tagged}}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{untagged}\hypertarget{attribute.untagged}{\hyperlink{attribute.untagged}{\mbox{\isa{untagged}}}} & : & \isa{attribute} \\[0.5ex]

    \indexdef{}{attribute}{THEN}\hypertarget{attribute.THEN}{\hyperlink{attribute.THEN}{\mbox{\isa{THEN}}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{COMP}\hypertarget{attribute.COMP}{\hyperlink{attribute.COMP}{\mbox{\isa{COMP}}}} & : & \isa{attribute} \\[0.5ex]

    \indexdef{}{attribute}{unfolded}\hypertarget{attribute.unfolded}{\hyperlink{attribute.unfolded}{\mbox{\isa{unfolded}}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{folded}\hypertarget{attribute.folded}{\hyperlink{attribute.folded}{\mbox{\isa{folded}}}} & : & \isa{attribute} \\[0.5ex]

    \indexdef{}{attribute}{rotated}\hypertarget{attribute.rotated}{\hyperlink{attribute.rotated}{\mbox{\isa{rotated}}}} & : & \isa{attribute} \\

    \indexdef{Pure}{attribute}{elim\_format}\hypertarget{attribute.Pure.elim-format}{\hyperlink{attribute.Pure.elim-format}{\mbox{\isa{elim{\isaliteral{5F}{\isacharunderscore}}format}}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{standard}\hypertarget{attribute.standard}{\hyperlink{attribute.standard}{\mbox{\isa{standard}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{no\_vars}\hypertarget{attribute.no-vars}{\hyperlink{attribute.no-vars}{\mbox{\isa{no{\isaliteral{5F}{\isacharunderscore}}vars}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{attribute} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{1}{}

\rail@term{\hyperlink{attribute.tagged}{\mbox{\isa{tagged}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@end

\rail@begin{1}{}

\rail@term{\hyperlink{attribute.untagged}{\mbox{\isa{untagged}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@end

\rail@begin{2}{}

\rail@bar

\rail@term{\hyperlink{attribute.THEN}{\mbox{\isa{THEN}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{attribute.COMP}{\mbox{\isa{COMP}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]

\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]

\rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.thmref}{\mbox{\isa{thmref}}}}[]

\rail@end

\rail@begin{2}{}

\rail@bar

\rail@term{\hyperlink{attribute.unfolded}{\mbox{\isa{unfolded}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{attribute.folded}{\mbox{\isa{folded}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{attribute.rotated}{\mbox{\isa{rotated}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.int}{\mbox{\isa{int}}}}[]

\rail@endbar

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{attribute.tagged}{\mbox{\isa{tagged}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name\ value{\isaliteral{22}{\isachardoublequote}}} and \hyperlink{attribute.untagged}{\mbox{\isa{untagged}}}~\isa{name} add and remove \emph{tags} of some theorem.

  Tags may be any list of string pairs that serve as formal comment.

  The first string is considered the tag name, the second its value.

  Note that \hyperlink{attribute.untagged}{\mbox{\isa{untagged}}} removes any tags of the same name.

  \item \hyperlink{attribute.THEN}{\mbox{\isa{THEN}}}~\isa{a} and \hyperlink{attribute.COMP}{\mbox{\isa{COMP}}}~\isa{a}

  compose rules by resolution.  \hyperlink{attribute.THEN}{\mbox{\isa{THEN}}} resolves with the

  first premise of \isa{a} (an alternative position may be also

  specified); the \hyperlink{attribute.COMP}{\mbox{\isa{COMP}}} version skips the automatic

  lifting process that is normally intended (cf.\ \verb|op RS| and

  \verb|op COMP| in \cite{isabelle-implementation}).

  \item \hyperlink{attribute.unfolded}{\mbox{\isa{unfolded}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} and \hyperlink{attribute.folded}{\mbox{\isa{folded}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} expand and fold back again the given

  definitions throughout a rule.

  \item \hyperlink{attribute.rotated}{\mbox{\isa{rotated}}}~\isa{n} rotate the premises of a

  theorem by \isa{n} (default 1).

  \item \hyperlink{attribute.Pure.elim-format}{\mbox{\isa{elim{\isaliteral{5F}{\isacharunderscore}}format}}} turns a destruction rule into

  elimination rule format, by resolving with the rule \isa{{\isaliteral{22}{\isachardoublequote}}PROP\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}PROP\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ PROP\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ PROP\ B{\isaliteral{22}{\isachardoublequote}}}.

  Note that the Classical Reasoner (\secref{sec:classical}) provides

  its own version of this operation.

  \item \hyperlink{attribute.standard}{\mbox{\isa{standard}}} puts a theorem into the standard form of

  object-rules at the outermost theory level.  Note that this

  operation violates the local proof context (including active

  locales).

  \item \hyperlink{attribute.no-vars}{\mbox{\isa{no{\isaliteral{5F}{\isacharunderscore}}vars}}} replaces schematic variables by free

  ones; this is mainly for tuning output of pretty printed theorems.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Low-level equational reasoning%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{method}{subst}\hypertarget{method.subst}{\hyperlink{method.subst}{\mbox{\isa{subst}}}} & : & \isa{method} \\

    \indexdef{}{method}{hypsubst}\hypertarget{method.hypsubst}{\hyperlink{method.hypsubst}{\mbox{\isa{hypsubst}}}} & : & \isa{method} \\

    \indexdef{}{method}{split}\hypertarget{method.split}{\hyperlink{method.split}{\mbox{\isa{split}}}} & : & \isa{method} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{6}{}

\rail@term{\hyperlink{method.subst}{\mbox{\isa{subst}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@term{\isa{asm}}[]

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@endbar

\rail@cr{3}

\rail@bar

\rail@nextbar{4}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]

\rail@nextplus{5}

\rail@endplus

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.thmref}{\mbox{\isa{thmref}}}}[]

\rail@end

\rail@begin{1}{}

\rail@term{\hyperlink{method.split}{\mbox{\isa{split}}}}[]

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@end

\end{railoutput}

  These methods provide low-level facilities for equational reasoning

  that are intended for specialized applications only.  Normally,

  single step calculations would be performed in a structured text

  (see also \secref{sec:calculation}), while the Simplifier methods

  provide the canonical way for automated normalization (see

  \secref{sec:simplifier}).

  \begin{description}

  \item \hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{eq} performs a single substitution step

  using rule \isa{eq}, which may be either a meta or object

  equality.

  \item \hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}asm{\isaliteral{29}{\isacharparenright}}\ eq{\isaliteral{22}{\isachardoublequote}}} substitutes in an

  assumption.

  \item \hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ j{\isaliteral{29}{\isacharparenright}}\ eq{\isaliteral{22}{\isachardoublequote}}} performs several

  substitutions in the conclusion. The numbers \isa{i} to \isa{j}

  indicate the positions to substitute at.  Positions are ordered from

  the top of the term tree moving down from left to right. For

  example, in \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} there are three positions

  where commutativity of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{22}{\isachardoublequote}}} is applicable: 1 refers to \isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{22}{\isachardoublequote}}}, 2 to the whole term, and 3 to \isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{22}{\isachardoublequote}}}.

  If the positions in the list \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ j{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} are non-overlapping

  (e.g.\ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isadigit{3}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} in \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}) you may

  assume all substitutions are performed simultaneously.  Otherwise

  the behaviour of \isa{subst} is not specified.

  \item \hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}asm{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ j{\isaliteral{29}{\isacharparenright}}\ eq{\isaliteral{22}{\isachardoublequote}}} performs the

  substitutions in the assumptions. The positions refer to the

  assumptions in order from left to right.  For example, given in a

  goal of the form \isa{{\isaliteral{22}{\isachardoublequote}}P\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}, position 1 of

  commutativity of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{22}{\isachardoublequote}}} is the subterm \isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{22}{\isachardoublequote}}} and

  position 2 is the subterm \isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{22}{\isachardoublequote}}}.

  \item \hyperlink{method.hypsubst}{\mbox{\isa{hypsubst}}} performs substitution using some

  assumption; this only works for equations of the form \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{22}{\isachardoublequote}}} where \isa{x} is a free or bound variable.

  \item \hyperlink{method.split}{\mbox{\isa{split}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} performs single-step case

  splitting using the given rules.  Splitting is performed in the

  conclusion or some assumption of the subgoal, depending of the

  structure of the rule.

  Note that the \hyperlink{method.simp}{\mbox{\isa{simp}}} method already involves repeated

  application of split rules as declared in the current context, using

  \hyperlink{attribute.split}{\mbox{\isa{split}}}, for example.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Further tactic emulations \label{sec:tactics}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

The following improper proof methods emulate traditional tactics.

  These admit direct access to the goal state, which is normally

  considered harmful!  In particular, this may involve both numbered

  goal addressing (default 1), and dynamic instantiation within the

  scope of some subgoal.

  \begin{warn}

    Dynamic instantiations refer to universally quantified parameters

    of a subgoal (the dynamic context) rather than fixed variables and

    term abbreviations of a (static) Isar context.

  \end{warn}

  Tactic emulation methods, unlike their ML counterparts, admit

  simultaneous instantiation from both dynamic and static contexts.

  If names occur in both contexts goal parameters hide locally fixed

  variables.  Likewise, schematic variables refer to term

  abbreviations, if present in the static context.  Otherwise the

  schematic variable is interpreted as a schematic variable and left

  to be solved by unification with certain parts of the subgoal.

  Note that the tactic emulation proof methods in Isabelle/Isar are

  consistently named \isa{foo{\isaliteral{5F}{\isacharunderscore}}tac}.  Note also that variable names

  occurring on left hand sides of instantiations must be preceded by a

  question mark if they coincide with a keyword or contain dots.  This

  is consistent with the attribute \hyperlink{attribute.where}{\mbox{\isa{where}}} (see

  \secref{sec:pure-meth-att}).

  \begin{matharray}{rcl}

    \indexdef{}{method}{rule\_tac}\hypertarget{method.rule-tac}{\hyperlink{method.rule-tac}{\mbox{\isa{rule{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{erule\_tac}\hypertarget{method.erule-tac}{\hyperlink{method.erule-tac}{\mbox{\isa{erule{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{drule\_tac}\hypertarget{method.drule-tac}{\hyperlink{method.drule-tac}{\mbox{\isa{drule{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{frule\_tac}\hypertarget{method.frule-tac}{\hyperlink{method.frule-tac}{\mbox{\isa{frule{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{cut\_tac}\hypertarget{method.cut-tac}{\hyperlink{method.cut-tac}{\mbox{\isa{cut{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{thin\_tac}\hypertarget{method.thin-tac}{\hyperlink{method.thin-tac}{\mbox{\isa{thin{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{subgoal\_tac}\hypertarget{method.subgoal-tac}{\hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{rename\_tac}\hypertarget{method.rename-tac}{\hyperlink{method.rename-tac}{\mbox{\isa{rename{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{rotate\_tac}\hypertarget{method.rotate-tac}{\hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{tactic}\hypertarget{method.tactic}{\hyperlink{method.tactic}{\mbox{\isa{tactic}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

    \indexdef{}{method}{raw\_tactic}\hypertarget{method.raw-tactic}{\hyperlink{method.raw-tactic}{\mbox{\isa{raw{\isaliteral{5F}{\isacharunderscore}}tactic}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{9}{}

\rail@bar

\rail@term{\hyperlink{method.rule-tac}{\mbox{\isa{rule{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{method.erule-tac}{\mbox{\isa{erule{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@nextbar{2}

\rail@term{\hyperlink{method.drule-tac}{\mbox{\isa{drule{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@nextbar{3}

\rail@term{\hyperlink{method.frule-tac}{\mbox{\isa{frule{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@nextbar{4}

\rail@term{\hyperlink{method.cut-tac}{\mbox{\isa{cut{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@nextbar{5}

\rail@term{\hyperlink{method.thin-tac}{\mbox{\isa{thin{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]

\rail@endbar

\rail@cr{7}

\rail@bar

\rail@nont{\isa{dynamic{\isaliteral{5F}{\isacharunderscore}}insts}}[]

\rail@term{\isa{\isakeyword{in}}}[]

\rail@nont{\hyperlink{syntax.thmref}{\mbox{\isa{thmref}}}}[]

\rail@nextbar{8}

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@endbar

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]

\rail@endbar

\rail@plus

\rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]

\rail@nextplus{1}

\rail@endplus

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{method.rename-tac}{\mbox{\isa{rename{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]

\rail@endbar

\rail@plus

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nextplus{1}

\rail@endplus

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.int}{\mbox{\isa{int}}}}[]

\rail@endbar

\rail@end

\rail@begin{2}{}

\rail@bar

\rail@term{\hyperlink{method.tactic}{\mbox{\isa{tactic}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{method.raw-tactic}{\mbox{\isa{raw{\isaliteral{5F}{\isacharunderscore}}tactic}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[]

\rail@end

\rail@begin{2}{\isa{dynamic{\isaliteral{5F}{\isacharunderscore}}insts}}

\rail@plus

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]

\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]

\rail@nextplus{1}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\end{railoutput}

\begin{description}

  \item \hyperlink{method.rule-tac}{\mbox{\isa{rule{\isaliteral{5F}{\isacharunderscore}}tac}}} etc. do resolution of rules with explicit

  instantiation.  This works the same way as the ML tactics \verb|res_inst_tac| etc. (see \cite{isabelle-implementation})

  Multiple rules may be only given if there is no instantiation; then

  \hyperlink{method.rule-tac}{\mbox{\isa{rule{\isaliteral{5F}{\isacharunderscore}}tac}}} is the same as \verb|resolve_tac| in ML (see

  \cite{isabelle-implementation}).

  \item \hyperlink{method.cut-tac}{\mbox{\isa{cut{\isaliteral{5F}{\isacharunderscore}}tac}}} inserts facts into the proof state as

  assumption of a subgoal, see also \verb|Tactic.cut_facts_tac| in

  \cite{isabelle-implementation}.  Note that the scope of schematic

  variables is spread over the main goal statement.  Instantiations

  may be given as well, see also ML tactic \verb|cut_inst_tac| in

  \cite{isabelle-implementation}.

  \item \hyperlink{method.thin-tac}{\mbox{\isa{thin{\isaliteral{5F}{\isacharunderscore}}tac}}}~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} deletes the specified assumption

  from a subgoal; note that \isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} may contain schematic variables.

  See also \verb|thin_tac| in \cite{isabelle-implementation}.

  \item \hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isaliteral{5F}{\isacharunderscore}}tac}}}~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} adds \isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} as an

  assumption to a subgoal.  See also \verb|subgoal_tac| and \verb|subgoals_tac| in \cite{isabelle-implementation}.

  \item \hyperlink{method.rename-tac}{\mbox{\isa{rename{\isaliteral{5F}{\isacharunderscore}}tac}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} renames parameters of a

  goal according to the list \isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}, which refers to the

  \emph{suffix} of variables.

  \item \hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\isaliteral{5F}{\isacharunderscore}}tac}}}~\isa{n} rotates the assumptions of a

  goal by \isa{n} positions: from right to left if \isa{n} is

  positive, and from left to right if \isa{n} is negative; the

  default value is 1.  See also \verb|rotate_tac| in

  \cite{isabelle-implementation}.

  \item \hyperlink{method.tactic}{\mbox{\isa{tactic}}}~\isa{{\isaliteral{22}{\isachardoublequote}}text{\isaliteral{22}{\isachardoublequote}}} produces a proof method from

  any ML text of type \verb|tactic|.  Apart from the usual ML

  environment and the current proof context, the ML code may refer to

  the locally bound values \verb|facts|, which indicates any

  current facts used for forward-chaining.

  \item \hyperlink{method.raw-tactic}{\mbox{\isa{raw{\isaliteral{5F}{\isacharunderscore}}tactic}}} is similar to \hyperlink{method.tactic}{\mbox{\isa{tactic}}}, but

  presents the goal state in its raw internal form, where simultaneous

  subgoals appear as conjunction of the logical framework instead of

  the usual split into several subgoals.  While feature this is useful

  for debugging of complex method definitions, it should not never

  appear in production theories.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{The Simplifier \label{sec:simplifier}%

}

\isamarkuptrue%

%

\isamarkupsubsection{Simplification methods%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{method}{simp}\hypertarget{method.simp}{\hyperlink{method.simp}{\mbox{\isa{simp}}}} & : & \isa{method} \\

    \indexdef{}{method}{simp\_all}\hypertarget{method.simp-all}{\hyperlink{method.simp-all}{\mbox{\isa{simp{\isaliteral{5F}{\isacharunderscore}}all}}}} & : & \isa{method} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{2}{}

\rail@bar

\rail@term{\hyperlink{method.simp}{\mbox{\isa{simp}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{method.simp-all}{\mbox{\isa{simp{\isaliteral{5F}{\isacharunderscore}}all}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\isa{opt}}[]

\rail@endbar

\rail@plus

\rail@nextplus{1}

\rail@cnont{\hyperlink{syntax.simpmod}{\mbox{\isa{simpmod}}}}[]

\rail@endplus

\rail@end

\rail@begin{4}{\isa{opt}}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@bar

\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}asm}}[]

\rail@nextbar{1}

\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{5F}{\isacharunderscore}}simp}}[]

\rail@nextbar{2}

\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{5F}{\isacharunderscore}}use}}[]

\rail@nextbar{3}

\rail@term{\isa{asm{\isaliteral{5F}{\isacharunderscore}}lr}}[]

\rail@endbar

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@end

\rail@begin{9}{\indexdef{}{syntax}{simpmod}\hypertarget{syntax.simpmod}{\hyperlink{syntax.simpmod}{\mbox{\isa{simpmod}}}}}

\rail@bar

\rail@term{\isa{add}}[]

\rail@nextbar{1}

\rail@term{\isa{del}}[]

\rail@nextbar{2}

\rail@term{\isa{only}}[]

\rail@nextbar{3}

\rail@term{\isa{cong}}[]

\rail@bar

\rail@nextbar{4}

\rail@term{\isa{add}}[]

\rail@nextbar{5}

\rail@term{\isa{del}}[]

\rail@endbar

\rail@nextbar{6}

\rail@term{\isa{split}}[]

\rail@bar

\rail@nextbar{7}

\rail@term{\isa{add}}[]

\rail@nextbar{8}

\rail@term{\isa{del}}[]

\rail@endbar

\rail@endbar

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{method.simp}{\mbox{\isa{simp}}} invokes the Simplifier, after declaring

  additional rules according to the arguments given.  Note that the

  \isa{only} modifier first removes all other rewrite rules,

  congruences, and looper tactics (including splits), and then behaves

  like \isa{add}.

  \medskip The \isa{cong} modifiers add or delete Simplifier

  congruence rules (see also \cite{isabelle-ref}), the default is to

  add.

  \medskip The \isa{split} modifiers add or delete rules for the

  Splitter (see also \cite{isabelle-ref}), the default is to add.

  This works only if the Simplifier method has been properly setup to

  include the Splitter (all major object logics such HOL, HOLCF, FOL,

  ZF do this already).

  \item \hyperlink{method.simp-all}{\mbox{\isa{simp{\isaliteral{5F}{\isacharunderscore}}all}}} is similar to \hyperlink{method.simp}{\mbox{\isa{simp}}}, but acts on

  all goals (backwards from the last to the first one).

  \end{description}

  By default the Simplifier methods take local assumptions fully into

  account, using equational assumptions in the subsequent

  normalization process, or simplifying assumptions themselves (cf.\

  \verb|asm_full_simp_tac| in \cite{isabelle-ref}).  In structured

  proofs this is usually quite well behaved in practice: just the

  local premises of the actual goal are involved, additional 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.).

  Additional Simplifier options may be specified to tune the behavior

  further (mostly for unstructured scripts with many accidental local

  facts): ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' means assumptions are ignored

  completely (cf.\ \verb|simp_tac|), ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' means

  assumptions are used in the simplification of the conclusion but are

  not themselves simplified (cf.\ \verb|asm_simp_tac|), and ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{5F}{\isacharunderscore}}use{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' means assumptions are simplified but are not used

  in the simplification of each other or the conclusion (cf.\ \verb|full_simp_tac|).  For compatibility reasons, there is also an option

  ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}asm{\isaliteral{5F}{\isacharunderscore}}lr{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'', which means that an assumption is only used

  for simplifying assumptions which are to the right of it (cf.\ \verb|asm_lr_simp_tac|).

  The configuration option \isa{{\isaliteral{22}{\isachardoublequote}}depth{\isaliteral{5F}{\isacharunderscore}}limit{\isaliteral{22}{\isachardoublequote}}} limits the number of

  recursive invocations of the simplifier during conditional

  rewriting.

  \medskip The Splitter package is usually configured to work as part

  of the Simplifier.  The effect of repeatedly applying \verb|split_tac| can be simulated by ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{3A}{\isacharcolon}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}''.  There is also a separate \isa{split}

  method available for single-step case splitting.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Declaring rules%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{command}{print\_simpset}\hypertarget{command.print-simpset}{\hyperlink{command.print-simpset}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}simpset}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{attribute}{simp}\hypertarget{attribute.simp}{\hyperlink{attribute.simp}{\mbox{\isa{simp}}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{cong}\hypertarget{attribute.cong}{\hyperlink{attribute.cong}{\mbox{\isa{cong}}}} & : & \isa{attribute} \\

    \indexdef{}{attribute}{split}\hypertarget{attribute.split}{\hyperlink{attribute.split}{\mbox{\isa{split}}}} & : & \isa{attribute} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{3}{}

\rail@bar

\rail@term{\hyperlink{attribute.simp}{\mbox{\isa{simp}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{attribute.cong}{\mbox{\isa{cong}}}}[]

\rail@nextbar{2}

\rail@term{\hyperlink{attribute.split}{\mbox{\isa{split}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{add}}[]

\rail@nextbar{2}

\rail@term{\isa{del}}[]

\rail@endbar

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.print-simpset}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}simpset}}}} prints the collection of rules

  declared to the Simplifier, which is also known as ``simpset''

  internally \cite{isabelle-ref}.

  \item \hyperlink{attribute.simp}{\mbox{\isa{simp}}} declares simplification rules.

  \item \hyperlink{attribute.cong}{\mbox{\isa{cong}}} declares congruence rules.

  \item \hyperlink{attribute.split}{\mbox{\isa{split}}} declares case split rules.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Simplification procedures%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

Simplification procedures are ML functions that produce proven

  rewrite rules on demand.  They are associated with higher-order

  patterns that approximate the left-hand sides of equations.  The

  Simplifier first matches the current redex against one of the LHS

  patterns; if this succeeds, the corresponding ML function is

  invoked, passing the Simplifier context and redex term.  Thus rules

  may be specifically fashioned for particular situations, resulting

  in a more powerful mechanism than term rewriting by a fixed set of

  rules.

  Any successful result needs to be a (possibly conditional) rewrite

  rule \isa{{\isaliteral{22}{\isachardoublequote}}t\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ u{\isaliteral{22}{\isachardoublequote}}} that is applicable to the current redex.  The

  rule will be applied just as any ordinary rewrite rule.  It is

  expected to be already in \emph{internal form}, bypassing the

  automatic preprocessing of object-level equivalences.

  \begin{matharray}{rcl}

    \indexdef{}{command}{simproc\_setup}\hypertarget{command.simproc-setup}{\hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isaliteral{5F}{\isacharunderscore}}setup}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    simproc & : & \isa{attribute} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{6}{}

\rail@term{\hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isaliteral{5F}{\isacharunderscore}}setup}}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]

\rail@nextplus{1}

\rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]

\rail@endplus

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]

\rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[]

\rail@cr{3}

\rail@bar

\rail@nextbar{4}

\rail@term{\isa{\isakeyword{identifier}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@nextplus{5}

\rail@endplus

\rail@endbar

\rail@end

\rail@begin{3}{}

\rail@term{\hyperlink{attribute.simproc}{\mbox{\isa{simproc}}}}[]

\rail@bar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{add}}[]

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@endbar

\rail@nextbar{2}

\rail@term{\isa{del}}[]

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@endbar

\rail@plus

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nextplus{1}

\rail@endplus

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isaliteral{5F}{\isacharunderscore}}setup}}}} defines a named simplification

  procedure that is invoked by the Simplifier whenever any of the

  given term patterns match the current redex.  The implementation,

  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

  supposed to be some proven rewrite rule \isa{{\isaliteral{22}{\isachardoublequote}}r\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ r{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequote}}} (or a

  generalized version), or \verb|NONE| to indicate failure.  The

  \verb|simpset| argument holds the full context of the current

  Simplifier invocation, including the actual Isar proof context.  The

  \verb|morphism| informs about the difference of the original

  compilation context wrt.\ the one of the actual application later

  on.  The optional \hyperlink{keyword.identifier}{\mbox{\isa{\isakeyword{identifier}}}} specifies theorems that

  represent the logical content of the abstract theory of this

  simproc.

  Morphisms and identifiers are only relevant for simprocs that are

  defined within a local target context, e.g.\ in a locale.

  \item \isa{{\isaliteral{22}{\isachardoublequote}}simproc\ add{\isaliteral{3A}{\isacharcolon}}\ name{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}simproc\ del{\isaliteral{3A}{\isacharcolon}}\ name{\isaliteral{22}{\isachardoublequote}}}

  add or delete named simprocs to the current Simplifier context.  The

  default is to add a simproc.  Note that \hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isaliteral{5F}{\isacharunderscore}}setup}}}}

  already adds the new simproc to the subsequent context.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsubsection{Example%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

The following simplification procedure for \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}u{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}unit{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}} in HOL performs fine-grained

  control over rule application, beyond higher-order pattern matching.

  Declaring \isa{unit{\isaliteral{5F}{\isacharunderscore}}eq} as \hyperlink{attribute.simp}{\mbox{\isa{simp}}} directly would make

  the simplifier loop!  Note that a version of this simplification

  procedure is already active in Isabelle/HOL.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isadelimML

%

\endisadelimML

%

\isatagML

\isacommand{simproc{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse%

\ unit\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}unit{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline

\ \ fn\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ fn\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ fn\ ct\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline

\ \ \ \ if\ HOLogic{\isaliteral{2E}{\isachardot}}is{\isaliteral{5F}{\isacharunderscore}}unit\ {\isaliteral{28}{\isacharparenleft}}term{\isaliteral{5F}{\isacharunderscore}}of\ ct{\isaliteral{29}{\isacharparenright}}\ then\ NONE\isanewline

\ \ \ \ else\ SOME\ {\isaliteral{28}{\isacharparenleft}}mk{\isaliteral{5F}{\isacharunderscore}}meta{\isaliteral{5F}{\isacharunderscore}}eq\ %

\isaantiq

thm\ unit{\isaliteral{5F}{\isacharunderscore}}eq{}%

\endisaantiq

{\isaliteral{29}{\isacharparenright}}\isanewline

{\isaliteral{2A7D}{\isacharverbatimclose}}%

\endisatagML

{\isafoldML}%

%

\isadelimML

%

\endisadelimML

%

\begin{isamarkuptext}%

Since the Simplifier applies simplification procedures

  frequently, it is important to make the failure check in ML

  reasonably fast.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Forward simplification%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{attribute}{simplified}\hypertarget{attribute.simplified}{\hyperlink{attribute.simplified}{\mbox{\isa{simplified}}}} & : & \isa{attribute} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{2}{}

\rail@term{\hyperlink{attribute.simplified}{\mbox{\isa{simplified}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\isa{opt}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@endbar

\rail@end

\rail@begin{3}{\isa{opt}}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@bar

\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}asm}}[]

\rail@nextbar{1}

\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{5F}{\isacharunderscore}}simp}}[]

\rail@nextbar{2}

\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}asm{\isaliteral{5F}{\isacharunderscore}}use}}[]

\rail@endbar

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{attribute.simplified}{\mbox{\isa{simplified}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} causes a theorem to

  be simplified, either by exactly the specified rules \isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}, or the implicit Simplifier context if no arguments are given.

  The result is fully simplified by default, including assumptions and

  conclusion; the options \isa{no{\isaliteral{5F}{\isacharunderscore}}asm} etc.\ tune the Simplifier in

  the same way as the for the \isa{simp} method.

  Note that forward simplification restricts the simplifier to its

  most basic operation of term rewriting; solver and looper tactics

  \cite{isabelle-ref} are \emph{not} involved here.  The \isa{simplified} attribute should be only rarely required under normal

  circumstances.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{The Classical Reasoner \label{sec:classical}%

}

\isamarkuptrue%

%

\isamarkupsubsection{Basic concepts%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

Although Isabelle is generic, many users will be working in

  some extension of classical first-order logic.  Isabelle/ZF is built

  upon theory FOL, while Isabelle/HOL conceptually contains

  first-order logic as a fragment.  Theorem-proving in predicate logic

  is undecidable, but many automated strategies have been developed to

  assist in this task.

  Isabelle's classical reasoner is a generic package that accepts

  certain information about a logic and delivers a suite of automatic

  proof tools, based on rules that are classified and declared in the

  context.  These proof procedures are slow and simplistic compared

  with high-end automated theorem provers, but they can save

  considerable time and effort in practice.  They can prove theorems

  such as Pelletier's \cite{pelletier86} problems 40 and 41 in a few

  milliseconds (including full proof reconstruction):%

\end{isamarkuptext}%

\isamarkuptrue%

\isacommand{lemma}\isamarkupfalse%

\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ F\ x\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ F\ x\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ F\ z\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ F\ z\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline

%

\isadelimproof

\ \ %

\endisadelimproof

%

\isatagproof

\isacommand{by}\isamarkupfalse%

\ blast%

\endisatagproof

{\isafoldproof}%

%

\isadelimproof

\isanewline

%

\endisadelimproof

\isanewline

\isacommand{lemma}\isamarkupfalse%

\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ f\ x\ z\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ f\ x\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline

%

\isadelimproof

\ \ %

\endisadelimproof

%

\isatagproof

\isacommand{by}\isamarkupfalse%

\ blast%

\endisatagproof

{\isafoldproof}%

%