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\begin{isabellebody}%
\def\isabellecontext{Generic}%
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\isadelimtheory
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\endisadelimtheory
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\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ Generic\isanewline
\isakeyword{imports}\ Main\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
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\isadelimtheory
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\endisadelimtheory
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\isamarkupchapter{Generic tools and packages \label{ch:gen-tools}%
}
\isamarkuptrue%
%
\isamarkupsection{Configuration options%
}
\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|, 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}}}}.
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{\isacharunderscore}configs}}}}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
name ('=' ('true' | 'false' | int | name))?
\end{rail}
\begin{description}
\item \hyperlink{command.print-configs}{\mbox{\isa{\isacommand{print{\isacharunderscore}configs}}}} prints the available configuration
options, with names, types, and current values.
\item \isa{{\isachardoublequote}name\ {\isacharequal}\ value{\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{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{drule}\hypertarget{method.drule}{\hyperlink{method.drule}{\mbox{\isa{drule}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{frule}\hypertarget{method.frule}{\hyperlink{method.frule}{\mbox{\isa{frule}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \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{rail}
('fold' | 'unfold' | 'insert') thmrefs
;
('erule' | 'drule' | 'frule') ('('nat')')? thmrefs
;
\end{rail}
\begin{description}
\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 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{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\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{{\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}}} method (see \secref{sec:pure-meth-att}), but
apply rules by elim-resolution, destruct-resolution, and
forward-resolution, respectively \cite{isabelle-ref}. 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.succeed}{\mbox{\isa{succeed}}} yields a single (unchanged) result; it is
the identity of the ``\isa{{\isachardoublequote}{\isacharcomma}{\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{{\isachardoublequote}{\isacharbar}{\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{\isacharunderscore}format}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{standard}\hypertarget{attribute.standard}{\hyperlink{attribute.standard}{\mbox{\isa{standard}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\
\indexdef{}{attribute}{no\_vars}\hypertarget{attribute.no-vars}{\hyperlink{attribute.no-vars}{\mbox{\isa{no{\isacharunderscore}vars}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\
\end{matharray}
\begin{rail}
'tagged' name name
;
'untagged' name
;
('THEN' | 'COMP') ('[' nat ']')? thmref
;
('unfolded' | 'folded') thmrefs
;
'rotated' ( int )?
\end{rail}
\begin{description}
\item \hyperlink{attribute.tagged}{\mbox{\isa{tagged}}}~\isa{{\isachardoublequote}name\ value{\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[\S5]{isabelle-ref}).
\item \hyperlink{attribute.unfolded}{\mbox{\isa{unfolded}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} and \hyperlink{attribute.folded}{\mbox{\isa{folded}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\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{\isacharunderscore}format}}} turns a destruction rule 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}}.
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{\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{rail}
'subst' ('(' 'asm' ')')? ('(' (nat+) ')')? thmref
;
'split' ('(' 'asm' ')')? thmrefs
;
\end{rail}
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{{\isachardoublequote}{\isacharparenleft}asm{\isacharparenright}\ eq{\isachardoublequote}} substitutes in an
assumption.
\item \hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isachardoublequote}{\isacharparenleft}i\ {\isasymdots}\ j{\isacharparenright}\ eq{\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{{\isachardoublequote}{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharplus}\ {\isacharparenleft}c\ {\isacharplus}\ d{\isacharparenright}{\isachardoublequote}} there are three positions
where commutativity of \isa{{\isachardoublequote}{\isacharplus}{\isachardoublequote}} is applicable: 1 refers to \isa{{\isachardoublequote}a\ {\isacharplus}\ b{\isachardoublequote}}, 2 to the whole term, and 3 to \isa{{\isachardoublequote}c\ {\isacharplus}\ d{\isachardoublequote}}.
If the positions in the list \isa{{\isachardoublequote}{\isacharparenleft}i\ {\isasymdots}\ j{\isacharparenright}{\isachardoublequote}} are non-overlapping
(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
assume all substitutions are performed simultaneously. Otherwise
the behaviour of \isa{subst} is not specified.
\item \hyperlink{method.subst}{\mbox{\isa{subst}}}~\isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharparenright}\ {\isacharparenleft}i\ {\isasymdots}\ j{\isacharparenright}\ eq{\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{{\isachardoublequote}P\ {\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}c\ {\isacharplus}\ d{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}{\isachardoublequote}}, position 1 of
commutativity of \isa{{\isachardoublequote}{\isacharplus}{\isachardoublequote}} is the subterm \isa{{\isachardoublequote}a\ {\isacharplus}\ b{\isachardoublequote}} and
position 2 is the subterm \isa{{\isachardoublequote}c\ {\isacharplus}\ d{\isachardoublequote}}.
\item \hyperlink{method.hypsubst}{\mbox{\isa{hypsubst}}} performs substitution using some
assumption; this only works for equations of the form \isa{{\isachardoublequote}x\ {\isacharequal}\ t{\isachardoublequote}} where \isa{x} is a free or bound variable.
\item \hyperlink{method.split}{\mbox{\isa{split}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} performs single-step case
splitting using the given rules. By default, splitting is performed
in the conclusion of a goal; the \isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharparenright}{\isachardoublequote}} option indicates to
operate on assumptions instead.
Note that the \hyperlink{method.simp}{\mbox{\isa{simp}}} method already involves repeated
application of split rules as declared in the current context.
\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{\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{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{erule\_tac}\hypertarget{method.erule-tac}{\hyperlink{method.erule-tac}{\mbox{\isa{erule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{drule\_tac}\hypertarget{method.drule-tac}{\hyperlink{method.drule-tac}{\mbox{\isa{drule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{frule\_tac}\hypertarget{method.frule-tac}{\hyperlink{method.frule-tac}{\mbox{\isa{frule{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{cut\_tac}\hypertarget{method.cut-tac}{\hyperlink{method.cut-tac}{\mbox{\isa{cut{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{thin\_tac}\hypertarget{method.thin-tac}{\hyperlink{method.thin-tac}{\mbox{\isa{thin{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{subgoal\_tac}\hypertarget{method.subgoal-tac}{\hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{rename\_tac}\hypertarget{method.rename-tac}{\hyperlink{method.rename-tac}{\mbox{\isa{rename{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{rotate\_tac}\hypertarget{method.rotate-tac}{\hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{tactic}\hypertarget{method.tactic}{\hyperlink{method.tactic}{\mbox{\isa{tactic}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\indexdef{}{method}{raw\_tactic}\hypertarget{method.raw-tactic}{\hyperlink{method.raw-tactic}{\mbox{\isa{raw{\isacharunderscore}tactic}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
\end{matharray}
\begin{rail}
( 'rule\_tac' | 'erule\_tac' | 'drule\_tac' | 'frule\_tac' | 'cut\_tac' | 'thin\_tac' ) goalspec?
( insts thmref | thmrefs )
;
'subgoal\_tac' goalspec? (prop +)
;
'rename\_tac' goalspec? (name +)
;
'rotate\_tac' goalspec? int?
;
('tactic' | 'raw_tactic') text
;
insts: ((name '=' term) + 'and') 'in'
;
\end{rail}
\begin{description}
\item \hyperlink{method.rule-tac}{\mbox{\isa{rule{\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[\S3]{isabelle-ref})
Multiple rules may be only given if there is no instantiation; then
\hyperlink{method.rule-tac}{\mbox{\isa{rule{\isacharunderscore}tac}}} is the same as \verb|resolve_tac| in ML (see
\cite[\S3]{isabelle-ref}).
\item \hyperlink{method.cut-tac}{\mbox{\isa{cut{\isacharunderscore}tac}}} inserts facts into the proof state as
assumption of a subgoal, see also \verb|Tactic.cut_facts_tac| in
\cite[\S3]{isabelle-ref}. 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[\S3]{isabelle-ref}.
\item \hyperlink{method.thin-tac}{\mbox{\isa{thin{\isacharunderscore}tac}}}~\isa{{\isasymphi}} deletes the specified assumption
from a subgoal; note that \isa{{\isasymphi}} may contain schematic variables.
See also \verb|thin_tac| in \cite[\S3]{isabelle-ref}.
\item \hyperlink{method.subgoal-tac}{\mbox{\isa{subgoal{\isacharunderscore}tac}}}~\isa{{\isasymphi}} adds \isa{{\isasymphi}} as an
assumption to a subgoal. See also \verb|subgoal_tac| and \verb|subgoals_tac| in \cite[\S3]{isabelle-ref}.
\item \hyperlink{method.rename-tac}{\mbox{\isa{rename{\isacharunderscore}tac}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n{\isachardoublequote}} renames 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.
\item \hyperlink{method.rotate-tac}{\mbox{\isa{rotate{\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[\S3]{isabelle-ref}.
\item \hyperlink{method.tactic}{\mbox{\isa{tactic}}}~\isa{{\isachardoublequote}text{\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{\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{\isacharunderscore}all}}}} & : & \isa{method} \\
\end{matharray}
\indexouternonterm{simpmod}
\begin{rail}
('simp' | 'simp\_all') ('!' ?) opt? (simpmod *)
;
opt: '(' ('no\_asm' | 'no\_asm\_simp' | 'no\_asm\_use' | 'asm\_lr' ) ')'
;
simpmod: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') |
'split' (() | 'add' | 'del')) ':' thmrefs
;
\end{rail}
\begin{description}
\item \hyperlink{method.simp}{\mbox{\isa{simp}}} invokes the Simplifier, after declaring
additional rules according to the arguments given. Note that the
\railtterm{only} modifier first removes all other rewrite rules,
congruences, and looper tactics (including splits), and then behaves
like \railtterm{add}.
\medskip The \railtterm{cong} modifiers add or delete Simplifier
congruence rules (see also \cite{isabelle-ref}), the default is to
add.
\medskip The \railtterm{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{\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[\S10]{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.). The full
context of premises is only included if the ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' (bang)
argument is given, which should be used with some care, though.
Additional Simplifier options may be specified to tune the behavior
further (mostly for unstructured scripts with many accidental local
facts): ``\isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isachardoublequote}}'' means assumptions are ignored
completely (cf.\ \verb|simp_tac|), ``\isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharunderscore}simp{\isacharparenright}{\isachardoublequote}}'' means
assumptions are used in the simplification of the conclusion but are
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
in the simplification of each other or the conclusion (cf.\ \verb|full_simp_tac|). For compatibility reasons, there is also an option
``\isa{{\isachardoublequote}{\isacharparenleft}asm{\isacharunderscore}lr{\isacharparenright}{\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{{\isachardoublequote}depth{\isacharunderscore}limit{\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{{\isachardoublequote}{\isacharparenleft}simp\ only{\isacharcolon}\ split{\isacharcolon}\ a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharparenright}{\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{\isacharunderscore}simpset}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\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{rail}
('simp' | 'cong' | 'split') (() | 'add' | 'del')
;
\end{rail}
\begin{description}
\item \hyperlink{command.print-simpset}{\mbox{\isa{\isacommand{print{\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}%
\begin{matharray}{rcl}
\indexdef{}{command}{simproc\_setup}\hypertarget{command.simproc-setup}{\hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\isacharunderscore}setup}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
simproc & : & \isa{attribute} \\
\end{matharray}
\begin{rail}
'simproc\_setup' name '(' (term + '|') ')' '=' text \\ ('identifier' (nameref+))?
;
'simproc' (('add' ':')? | 'del' ':') (name+)
;
\end{rail}
\begin{description}
\item \hyperlink{command.simproc-setup}{\mbox{\isa{\isacommand{simproc{\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{{\isachardoublequote}r\ {\isasymequiv}\ r{\isacharprime}{\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{{\isachardoublequote}simproc\ add{\isacharcolon}\ name{\isachardoublequote}} and \isa{{\isachardoublequote}simproc\ del{\isacharcolon}\ name{\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{\isacharunderscore}setup}}}}
already adds the new simproc to the subsequent context.
\end{description}%
\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{rail}
'simplified' opt? thmrefs?
;
opt: '(' ('no\_asm' | 'no\_asm\_simp' | 'no\_asm\_use') ')'
;
\end{rail}
\begin{description}
\item \hyperlink{attribute.simplified}{\mbox{\isa{simplified}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} causes a theorem to
be simplified, either by exactly the specified rules \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ a\isactrlsub n{\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{\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 methods%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{method}{rule}\hypertarget{method.rule}{\hyperlink{method.rule}{\mbox{\isa{rule}}}} & : & \isa{method} \\
\indexdef{}{method}{contradiction}\hypertarget{method.contradiction}{\hyperlink{method.contradiction}{\mbox{\isa{contradiction}}}} & : & \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} \\
\end{matharray}
\begin{rail}
('rule' | 'intro' | 'elim') thmrefs?
;
\end{rail}
\begin{description}
\item \hyperlink{method.rule}{\mbox{\isa{rule}}} as offered by the Classical Reasoner is a
refinement over the primitive one (see \secref{sec:pure-meth-att}).
Both versions essentially work the same, but the classical version
observes the classical rule context in addition to that of
Isabelle/Pure.
Common object logics (HOL, ZF, etc.) declare a rich collection of
classical rules (even if these would qualify as intuitionistic
ones), but only few declarations to the rule context of
Isabelle/Pure (\secref{sec:pure-meth-att}).
\item \hyperlink{method.contradiction}{\mbox{\isa{contradiction}}} solves some goal by contradiction,
deriving any result from both \isa{{\isachardoublequote}{\isasymnot}\ A{\isachardoublequote}} and \isa{A}. Chained
facts, which are guaranteed to participate, may appear in either
order.
\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.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Automated methods%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{method}{blast}\hypertarget{method.blast}{\hyperlink{method.blast}{\mbox{\isa{blast}}}} & : & \isa{method} \\
\indexdef{}{method}{fast}\hypertarget{method.fast}{\hyperlink{method.fast}{\mbox{\isa{fast}}}} & : & \isa{method} \\
\indexdef{}{method}{slow}\hypertarget{method.slow}{\hyperlink{method.slow}{\mbox{\isa{slow}}}} & : & \isa{method} \\
\indexdef{}{method}{best}\hypertarget{method.best}{\hyperlink{method.best}{\mbox{\isa{best}}}} & : & \isa{method} \\
\indexdef{}{method}{safe}\hypertarget{method.safe}{\hyperlink{method.safe}{\mbox{\isa{safe}}}} & : & \isa{method} \\
\indexdef{}{method}{clarify}\hypertarget{method.clarify}{\hyperlink{method.clarify}{\mbox{\isa{clarify}}}} & : & \isa{method} \\
\end{matharray}
\indexouternonterm{clamod}
\begin{rail}
'blast' ('!' ?) nat? (clamod *)
;
('fast' | 'slow' | 'best' | 'safe' | 'clarify') ('!' ?) (clamod *)
;
clamod: (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' thmrefs
;
\end{rail}
\begin{description}
\item \hyperlink{method.blast}{\mbox{\isa{blast}}} refers to the classical tableau prover (see
\verb|blast_tac| in \cite[\S11]{isabelle-ref}). The optional
argument specifies a user-supplied search bound (default 20).
\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
reasoner. See \verb|fast_tac|, \verb|slow_tac|, \verb|best_tac|, \verb|safe_tac|, and \verb|clarify_tac| in \cite[\S11]{isabelle-ref} for
more information.
\end{description}
Any of the above methods support additional modifiers of the context
of classical rules. Their semantics is analogous to the attributes
given before. Facts provided by forward chaining are inserted into
the goal before commencing proof search. The ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}''~argument causes the full context of assumptions to be
included as well.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Combined automated methods \label{sec:clasimp}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{method}{auto}\hypertarget{method.auto}{\hyperlink{method.auto}{\mbox{\isa{auto}}}} & : & \isa{method} \\
\indexdef{}{method}{force}\hypertarget{method.force}{\hyperlink{method.force}{\mbox{\isa{force}}}} & : & \isa{method} \\
\indexdef{}{method}{clarsimp}\hypertarget{method.clarsimp}{\hyperlink{method.clarsimp}{\mbox{\isa{clarsimp}}}} & : & \isa{method} \\
\indexdef{}{method}{fastsimp}\hypertarget{method.fastsimp}{\hyperlink{method.fastsimp}{\mbox{\isa{fastsimp}}}} & : & \isa{method} \\
\indexdef{}{method}{slowsimp}\hypertarget{method.slowsimp}{\hyperlink{method.slowsimp}{\mbox{\isa{slowsimp}}}} & : & \isa{method} \\
\indexdef{}{method}{bestsimp}\hypertarget{method.bestsimp}{\hyperlink{method.bestsimp}{\mbox{\isa{bestsimp}}}} & : & \isa{method} \\
\end{matharray}
\indexouternonterm{clasimpmod}
\begin{rail}
'auto' '!'? (nat nat)? (clasimpmod *)
;
('force' | 'clarsimp' | 'fastsimp' | 'slowsimp' | 'bestsimp') '!'? (clasimpmod *)
;
clasimpmod: ('simp' (() | 'add' | 'del' | 'only') |
('cong' | 'split') (() | 'add' | 'del') |
'iff' (((() | 'add') '?'?) | 'del') |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' thmrefs
\end{rail}
\begin{description}
\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 access
to Isabelle's combined simplification and classical reasoning
tactics. These correspond to \verb|auto_tac|, \verb|force_tac|, \verb|clarsimp_tac|, and Classical Reasoner tactics with the Simplifier
added as wrapper, see \cite[\S11]{isabelle-ref} for more
information. The modifier arguments correspond to those given in
\secref{sec:simplifier} and \secref{sec:classical}. Just note that
the ones related to the Simplifier are prefixed by \railtterm{simp}
here.
Facts provided by forward chaining are inserted into the goal before
doing the search. The ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' argument causes the full
context of assumptions to be included as well.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Declaring rules%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{command}{print\_claset}\hypertarget{command.print-claset}{\hyperlink{command.print-claset}{\mbox{\isa{\isacommand{print{\isacharunderscore}claset}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{attribute}{intro}\hypertarget{attribute.intro}{\hyperlink{attribute.intro}{\mbox{\isa{intro}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{elim}\hypertarget{attribute.elim}{\hyperlink{attribute.elim}{\mbox{\isa{elim}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{dest}\hypertarget{attribute.dest}{\hyperlink{attribute.dest}{\mbox{\isa{dest}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{rule}\hypertarget{attribute.rule}{\hyperlink{attribute.rule}{\mbox{\isa{rule}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{iff}\hypertarget{attribute.iff}{\hyperlink{attribute.iff}{\mbox{\isa{iff}}}} & : & \isa{attribute} \\
\end{matharray}
\begin{rail}
('intro' | 'elim' | 'dest') ('!' | () | '?') nat?
;
'rule' 'del'
;
'iff' (((() | 'add') '?'?) | 'del')
;
\end{rail}
\begin{description}
\item \hyperlink{command.print-claset}{\mbox{\isa{\isacommand{print{\isacharunderscore}claset}}}} prints the collection of rules
declared to the Classical Reasoner, which is also known as
``claset'' internally \cite{isabelle-ref}.
\item \hyperlink{attribute.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.elim}{\mbox{\isa{elim}}}, and \hyperlink{attribute.dest}{\mbox{\isa{dest}}}
declare introduction, elimination, and destruction rules,
respectively. By default, rules are considered as \emph{unsafe}
(i.e.\ not applied blindly without backtracking), while ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' classifies as \emph{safe}. Rule declarations marked by
``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' coincide with those of Isabelle/Pure, cf.\
\secref{sec:pure-meth-att} (i.e.\ are only applied in single steps
of the \hyperlink{method.rule}{\mbox{\isa{rule}}} method). The optional natural number
specifies an explicit weight argument, which is ignored by automated
tools, but determines the search order of single rule steps.
\item \hyperlink{attribute.rule}{\mbox{\isa{rule}}}~\isa{del} deletes introduction,
elimination, or destruction rules from the context.
\item \hyperlink{attribute.iff}{\mbox{\isa{iff}}} declares logical equivalences to the
Simplifier and the Classical reasoner at the same time.
Non-conditional rules result in a ``safe'' introduction and
elimination pair; conditional ones are considered ``unsafe''. Rules
with negative conclusion are automatically inverted (using \isa{{\isachardoublequote}{\isasymnot}{\isachardoublequote}}-elimination internally).
The ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' version of \hyperlink{attribute.iff}{\mbox{\isa{iff}}} declares rules to
the Isabelle/Pure context only, and omits the Simplifier
declaration.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Classical operations%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{attribute}{swapped}\hypertarget{attribute.swapped}{\hyperlink{attribute.swapped}{\mbox{\isa{swapped}}}} & : & \isa{attribute} \\
\end{matharray}
\begin{description}
\item \hyperlink{attribute.swapped}{\mbox{\isa{swapped}}} turns an introduction rule into an
elimination, by resolving with the classical swap principle \isa{{\isachardoublequote}{\isacharparenleft}{\isasymnot}\ B\ {\isasymLongrightarrow}\ A{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymnot}\ A\ {\isasymLongrightarrow}\ B{\isacharparenright}{\isachardoublequote}}.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Object-logic setup \label{sec:object-logic}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{command}{judgment}\hypertarget{command.judgment}{\hyperlink{command.judgment}{\mbox{\isa{\isacommand{judgment}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{method}{atomize}\hypertarget{method.atomize}{\hyperlink{method.atomize}{\mbox{\isa{atomize}}}} & : & \isa{method} \\
\indexdef{}{attribute}{atomize}\hypertarget{attribute.atomize}{\hyperlink{attribute.atomize}{\mbox{\isa{atomize}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{rule\_format}\hypertarget{attribute.rule-format}{\hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}}} & : & \isa{attribute} \\
\indexdef{}{attribute}{rulify}\hypertarget{attribute.rulify}{\hyperlink{attribute.rulify}{\mbox{\isa{rulify}}}} & : & \isa{attribute} \\
\end{matharray}
The very starting point for any Isabelle object-logic is a ``truth
judgment'' that links object-level statements to the meta-logic
(with its minimal language of \isa{prop} that covers universal
quantification \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}} and implication \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}}).
Common object-logics are sufficiently expressive to internalize rule
statements over \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}} within their own
language. This is useful in certain situations where a rule needs
to be viewed as an atomic statement from the meta-level perspective,
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}}.
From the following language elements, only the \hyperlink{method.atomize}{\mbox{\isa{atomize}}}
method and \hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}} attribute are occasionally
required by end-users, the rest is for those who need to setup their
own object-logic. In the latter case existing formulations of
Isabelle/FOL or Isabelle/HOL may be taken as realistic examples.
Generic tools may refer to the information provided by object-logic
declarations internally.
\begin{rail}
'judgment' constdecl
;
'atomize' ('(' 'full' ')')?
;
'rule\_format' ('(' 'noasm' ')')?
;
\end{rail}
\begin{description}
\item \hyperlink{command.judgment}{\mbox{\isa{\isacommand{judgment}}}}~\isa{{\isachardoublequote}c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\ {\isacharparenleft}mx{\isacharparenright}{\isachardoublequote}} declares constant
\isa{c} as the truth judgment of the current object-logic. Its
type \isa{{\isasymsigma}} should specify a coercion of the category of
object-level propositions to \isa{prop} of the Pure meta-logic;
the mixfix annotation \isa{{\isachardoublequote}{\isacharparenleft}mx{\isacharparenright}{\isachardoublequote}} would typically just link the
object language (internally of syntactic category \isa{logic})
with that of \isa{prop}. Only one \hyperlink{command.judgment}{\mbox{\isa{\isacommand{judgment}}}}
declaration may be given in any theory development.
\item \hyperlink{method.atomize}{\mbox{\isa{atomize}}} (as a method) rewrites any non-atomic
premises of a sub-goal, using the meta-level equations declared via
\hyperlink{attribute.atomize}{\mbox{\isa{atomize}}} (as an attribute) beforehand. As a result,
heavily nested goals become amenable to fundamental operations such
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
object-statement (if possible), including the outermost parameters
and assumptions as well.
A typical collection of \hyperlink{attribute.atomize}{\mbox{\isa{atomize}}} rules for a particular
object-logic would provide an internalization for each of the
connectives of \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}}, \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}}, and \isa{{\isachardoublequote}{\isasymequiv}{\isachardoublequote}}.
Meta-level conjunction should be covered as well (this is
particularly important for locales, see \secref{sec:locale}).
\item \hyperlink{attribute.rule-format}{\mbox{\isa{rule{\isacharunderscore}format}}} rewrites a theorem by the equalities
declared as \hyperlink{attribute.rulify}{\mbox{\isa{rulify}}} rules in the current object-logic.
By default, the result is fully normalized, including assumptions
and conclusions at any depth. The \isa{{\isachardoublequote}{\isacharparenleft}no{\isacharunderscore}asm{\isacharparenright}{\isachardoublequote}} option
restricts the transformation to the conclusion of a rule.
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
(\isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}}) and implication (\isa{{\isachardoublequote}{\isasymlongrightarrow}{\isachardoublequote}}) by the corresponding
rule statements over \isa{{\isachardoublequote}{\isasymAnd}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}}.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
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\isatagtheory
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\endisatagtheory
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\isanewline
\end{isabellebody}%
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