doc-src/IsarImplementation/Thy/document/Isar.tex

clarified singleton_fixate: intersection with supersort is identity, only replace actual type inference parameters;
tuned reject_other, after_infer_fixate;

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\isamarkupchapter{Isar language elements%
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\begin{isamarkuptext}%
The Isar proof language (see also
  \cite[\S2]{isabelle-isar-ref}) consists of three main categories of
  language elements as follows.

  \begin{enumerate}

  \item Proof \emph{commands} define the primary language of
  transactions of the underlying Isar/VM interpreter.  Typical
  examples are \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}, \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}, \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}, and \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}.

  Composing proof commands according to the rules of the Isar/VM leads
  to expressions of structured proof text, such that both the machine
  and the human reader can give it a meaning as formal reasoning.

  \item Proof \emph{methods} define a secondary language of mixed
  forward-backward refinement steps involving facts and goals.
  Typical examples are \hyperlink{method.rule}{\mbox{\isa{rule}}}, \hyperlink{method.unfold}{\mbox{\isa{unfold}}}, and \hyperlink{method.simp}{\mbox{\isa{simp}}}.

  Methods can occur in certain well-defined parts of the Isar proof
  language, say as arguments to \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}, \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}},
  or \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}.

  \item \emph{Attributes} define a tertiary language of small
  annotations to theorems being defined or referenced.  Attributes can
  modify both the context and the theorem.

  Typical examples are \hyperlink{attribute.intro}{\mbox{\isa{intro}}} (which affects the context),
  and \hyperlink{attribute.symmetric}{\mbox{\isa{symmetric}}} (which affects the theorem).

  \end{enumerate}%
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\isamarkupsection{Proof commands%
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\begin{isamarkuptext}%
A \emph{proof command} is state transition of the Isar/VM
  proof interpreter.

  In principle, Isar proof commands could be defined in user-space as
  well.  The system is built like that in the first place: one part of
  the commands are primitive, the other part is defined as derived
  elements.  Adding to the genuine structured proof language requires
  profound understanding of the Isar/VM machinery, though, so this is
  beyond the scope of this manual.

  What can be done realistically is to define some diagnostic commands
  that inspect the general state of the Isar/VM, and report some
  feedback to the user.  Typically this involves checking of the
  linguistic \emph{mode} of a proof state, or peeking at the pending
  goals (if available).

  Another common application is to define a toplevel command that
  poses a problem to the user as Isar proof state and processes the
  final result relatively to the context.  Thus a proof can be
  incorporated into the context of some user-space tool, without
  modifying the Isar proof language itself.%
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\begin{mldecls}
  \indexdef{}{ML type}{Proof.state}\verb|type Proof.state| \\
  \indexdef{}{ML}{Proof.assert\_forward}\verb|Proof.assert_forward: Proof.state -> Proof.state| \\
  \indexdef{}{ML}{Proof.assert\_chain}\verb|Proof.assert_chain: Proof.state -> Proof.state| \\
  \indexdef{}{ML}{Proof.assert\_backward}\verb|Proof.assert_backward: Proof.state -> Proof.state| \\
  \indexdef{}{ML}{Proof.simple\_goal}\verb|Proof.simple_goal: Proof.state -> {context: Proof.context, goal: thm}| \\
  \indexdef{}{ML}{Proof.goal}\verb|Proof.goal: Proof.state ->|\isasep\isanewline%
\verb|  {context: Proof.context, facts: thm list, goal: thm}| \\
  \indexdef{}{ML}{Proof.raw\_goal}\verb|Proof.raw_goal: Proof.state ->|\isasep\isanewline%
\verb|  {context: Proof.context, facts: thm list, goal: thm}| \\
  \indexdef{}{ML}{Proof.theorem}\verb|Proof.theorem: Method.text option ->|\isasep\isanewline%
\verb|  (thm list list -> Proof.context -> Proof.context) ->|\isasep\isanewline%
\verb|  (term * term list) list list -> Proof.context -> Proof.state| \\
  \end{mldecls}

  \begin{description}

  \item Type \verb|Proof.state| represents Isar proof states.
  This is a block-structured configuration with proof context,
  linguistic mode, and optional goal.  The latter consists of goal
  context, goal facts (``\isa{using}''), and tactical goal state
  (see \secref{sec:tactical-goals}).

  The general idea is that the facts shall contribute to the
  refinement of some parts of the tactical goal --- how exactly is
  defined by the proof method that is applied in that situation.

  \item \verb|Proof.assert_forward|, \verb|Proof.assert_chain|, \verb|Proof.assert_backward| are partial identity functions that fail
  unless a certain linguistic mode is active, namely ``\isa{proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}}'', ``\isa{proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}}'', ``\isa{proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}}'', respectively (using the terminology of
  \cite{isabelle-isar-ref}).

  It is advisable study the implementations of existing proof commands
  for suitable modes to be asserted.

  \item \verb|Proof.simple_goal|~\isa{state} returns the structured
  Isar goal (if available) in the form seen by ``simple'' methods
  (like \hyperlink{method.simp}{\mbox{\isa{simp}}} or \hyperlink{method.blast}{\mbox{\isa{blast}}}).  The Isar goal facts are
  already inserted as premises into the subgoals, which are presented
  individually as in \verb|Proof.goal|.

  \item \verb|Proof.goal|~\isa{state} returns the structured Isar
  goal (if available) in the form seen by regular methods (like
  \hyperlink{method.rule}{\mbox{\isa{rule}}}).  The auxiliary internal encoding of Pure
  conjunctions is split into individual subgoals as usual.

  \item \verb|Proof.raw_goal|~\isa{state} returns the structured
  Isar goal (if available) in the raw internal form seen by ``raw''
  methods (like \hyperlink{method.induct}{\mbox{\isa{induct}}}).  This form is rarely appropriate
  for dignostic tools; \verb|Proof.simple_goal| or \verb|Proof.goal|
  should be used in most situations.

  \item \verb|Proof.theorem|~\isa{before{\isaliteral{5F}{\isacharunderscore}}qed\ after{\isaliteral{5F}{\isacharunderscore}}qed\ statement\ ctxt}
  initializes a toplevel Isar proof state within a given context.

  The optional \isa{before{\isaliteral{5F}{\isacharunderscore}}qed} method is applied at the end of
  the proof, just before extracting the result (this feature is rarely
  used).

  The \isa{after{\isaliteral{5F}{\isacharunderscore}}qed} continuation receives the extracted result
  in order to apply it to the final context in a suitable way (e.g.\
  storing named facts).  Note that at this generic level the target
  context is specified as \verb|Proof.context|, but the usual
  wrapping of toplevel proofs into command transactions will provide a
  \verb|local_theory| here (\chref{ch:local-theory}).  This
  affects the way how results are stored.

  The \isa{statement} is given as a nested list of terms, each
  associated with optional \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} patterns as usual in the
  Isar source language.  The original nested list structure over terms
  is turned into one over theorems when \isa{after{\isaliteral{5F}{\isacharunderscore}}qed} is
  invoked.

  \end{description}%
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\begin{matharray}{rcl}
  \indexdef{}{ML antiquotation}{Isar.goal}\hypertarget{ML antiquotation.Isar.goal}{\hyperlink{ML antiquotation.Isar.goal}{\mbox{\isa{Isar{\isaliteral{2E}{\isachardot}}goal}}}} & : & \isa{ML{\isaliteral{5F}{\isacharunderscore}}antiquotation} \\
  \end{matharray}

  \begin{description}

  \item \isa{{\isaliteral{40}{\isacharat}}{\isaliteral{7B}{\isacharbraceleft}}Isar{\isaliteral{2E}{\isachardot}}goal{\isaliteral{7D}{\isacharbraceright}}} refers to the regular goal state (if
  available) of the current proof state managed by the Isar toplevel
  --- as abstract value.

  This only works for diagnostic ML commands, such as \hyperlink{command.ML-val}{\mbox{\isa{\isacommand{ML{\isaliteral{5F}{\isacharunderscore}}val}}}} or \hyperlink{command.ML-command}{\mbox{\isa{\isacommand{ML{\isaliteral{5F}{\isacharunderscore}}command}}}}.

  \end{description}%
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\begin{isamarkuptext}%
The following example peeks at a certain goal configuration.%
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\begin{isamarkuptext}%
Here we see 3 individual subgoals in the same way as regular
  proof methods would do.%
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\isamarkupsection{Proof methods%
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\begin{isamarkuptext}%
A \isa{method} is a function \isa{context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ thm\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ goal\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}cases\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ goal{\isaliteral{29}{\isacharparenright}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}} that operates on the full Isar goal
  configuration with context, goal facts, and tactical goal state and
  enumerates possible follow-up goal states, with the potential
  addition of named extensions of the proof context (\emph{cases}).
  The latter feature is rarely used.

  This means a proof method is like a structurally enhanced tactic
  (cf.\ \secref{sec:tactics}).  The well-formedness conditions for
  tactics need to hold for methods accordingly, with the following
  additions.

  \begin{itemize}

  \item Goal addressing is further limited either to operate either
  uniformly on \emph{all} subgoals, or specifically on the
  \emph{first} subgoal.

  Exception: old-style tactic emulations that are embedded into the
  method space, e.g.\ \hyperlink{method.rule-tac}{\mbox{\isa{rule{\isaliteral{5F}{\isacharunderscore}}tac}}}.

  \item A non-trivial method always needs to make progress: an
  identical follow-up goal state has to be avoided.\footnote{This
  enables the user to write method expressions like \isa{meth\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2B}{\isacharplus}}}
  without looping, while the trivial do-nothing case can be recovered
  via \isa{meth\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{3F}{\isacharquery}}}.}

  Exception: trivial stuttering steps, such as ``\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}'' or
  \hyperlink{method.succeed}{\mbox{\isa{succeed}}}.

  \item Goal facts passed to the method must not be ignored.  If there
  is no sensible use of facts outside the goal state, facts should be
  inserted into the subgoals that are addressed by the method.

  \end{itemize}

  \medskip Syntactically, the language of proof methods appears as
  arguments to Isar commands like \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}} or \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}.
  User-space additions are reasonably easy by plugging suitable
  method-valued parser functions into the framework, using the
  \hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}} command, for example.

  To get a better idea about the range of possibilities, consider the
  following Isar proof schemes.  This is the general form of
  structured proof text:

  \medskip
  \begin{tabular}{l}
  \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{props}~\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}} \\
  \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{28}{\isacharparenleft}}initial{\isaliteral{5F}{\isacharunderscore}}method{\isaliteral{29}{\isacharparenright}}} \\
  \quad\isa{body} \\
  \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isaliteral{28}{\isacharparenleft}}terminal{\isaliteral{5F}{\isacharunderscore}}method{\isaliteral{29}{\isacharparenright}}} \\
  \end{tabular}
  \medskip

  The goal configuration consists of \isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}} and
  \isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}} appended in that order, and various \isa{props} being claimed.  The \isa{initial{\isaliteral{5F}{\isacharunderscore}}method} is invoked
  with facts and goals together and refines the problem to something
  that is handled recursively in the proof \isa{body}.  The \isa{terminal{\isaliteral{5F}{\isacharunderscore}}method} has another chance to finish any remaining
  subgoals, but it does not see the facts of the initial step.

  \medskip This pattern illustrates unstructured proof scripts:

  \medskip
  \begin{tabular}{l}
  \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{props} \\
  \quad\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}}~\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}} \\
  \quad\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}} \\
  \quad\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{3}}}~\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{3}}} \\
  \quad\hyperlink{command.done}{\mbox{\isa{\isacommand{done}}}} \\
  \end{tabular}
  \medskip

  The \isa{method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}} operates on the original claim while
  using \isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}}.  Since the \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}} command
  structurally resets the facts, the \isa{method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}} will
  operate on the remaining goal state without facts.  The \isa{method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{3}}} will see again a collection of \isa{facts\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{3}}} that has been inserted into the script explicitly.

  \medskip Empirically, any Isar proof method can be categorized as
  follows.

  \begin{enumerate}

  \item \emph{Special method with cases} with named context additions
  associated with the follow-up goal state.

  Example: \hyperlink{method.induct}{\mbox{\isa{induct}}}, which is also a ``raw'' method since it
  operates on the internal representation of simultaneous claims as
  Pure conjunction (\secref{{sec:logic-aux}}), instead of separate
  subgoals (\secref{sec::tactical-goals}).

  \item \emph{Structured method} with strong emphasis on facts outside
  the goal state.

  Example: \hyperlink{method.rule}{\mbox{\isa{rule}}}, which captures the key ideas behind
  structured reasoning in Isar in purest form.

  \item \emph{Simple method} with weaker emphasis on facts, which are
  inserted into subgoals to emulate old-style tactical as
  ``premises''.

  Examples: \hyperlink{method.simp}{\mbox{\isa{simp}}}, \hyperlink{method.blast}{\mbox{\isa{blast}}}, \hyperlink{method.auto}{\mbox{\isa{auto}}}.

  \item \emph{Old-style tactic emulation} with detailed numeric goal
  addressing and explicit references to entities of the internal goal
  state (which are otherwise invisible from proper Isar proof text).
  The naming convention \isa{foo{\isaliteral{5F}{\isacharunderscore}}tac} makes this special
  non-standard status clear.

  Example: \hyperlink{method.rule-tac}{\mbox{\isa{rule{\isaliteral{5F}{\isacharunderscore}}tac}}}.

  \end{enumerate}

  When implementing proof methods, it is advisable to study existing
  implementations carefully and imitate the typical ``boiler plate''
  for context-sensitive parsing and further combinators to wrap-up
  tactic expressions as methods.\footnote{Aliases or abbreviations of
  the standard method combinators should be avoided.  Note that from
  Isabelle99 until Isabelle2009 the system did provide various odd
  combinations of method wrappers that made user applications more
  complicated than necessary.}%
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\begin{mldecls}
  \indexdef{}{ML type}{Proof.method}\verb|type Proof.method| \\
  \indexdef{}{ML}{METHOD\_CASES}\verb|METHOD_CASES: (thm list -> cases_tactic) -> Proof.method| \\
  \indexdef{}{ML}{METHOD}\verb|METHOD: (thm list -> tactic) -> Proof.method| \\
  \indexdef{}{ML}{SIMPLE\_METHOD}\verb|SIMPLE_METHOD: tactic -> Proof.method| \\
  \indexdef{}{ML}{SIMPLE\_METHOD'}\verb|SIMPLE_METHOD': (int -> tactic) -> Proof.method| \\
  \indexdef{}{ML}{HEADGOAL}\verb|HEADGOAL: (int -> tactic) -> tactic| \\
  \indexdef{}{ML}{Method.insert\_tac}\verb|Method.insert_tac: thm list -> int -> tactic| \\
  \indexdef{}{ML}{Method.setup}\verb|Method.setup: binding -> (Proof.context -> Proof.method) context_parser ->|\isasep\isanewline%
\verb|  string -> theory -> theory| \\
  \end{mldecls}

  \begin{description}

  \item Type \verb|Proof.method| represents proof methods as
  abstract type.

  \item \verb|METHOD_CASES|~\isa{{\isaliteral{28}{\isacharparenleft}}fn\ facts\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ cases{\isaliteral{5F}{\isacharunderscore}}tactic{\isaliteral{29}{\isacharparenright}}} wraps
  \isa{cases{\isaliteral{5F}{\isacharunderscore}}tactic} depending on goal facts as proof method with
  cases; the goal context is passed via method syntax.

  \item \verb|METHOD|~\isa{{\isaliteral{28}{\isacharparenleft}}fn\ facts\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ tactic{\isaliteral{29}{\isacharparenright}}} wraps \isa{tactic} depending on goal facts as regular proof method; the goal
  context is passed via method syntax.

  \item \verb|SIMPLE_METHOD|~\isa{tactic} wraps a tactic that
  addresses all subgoals uniformly as simple proof method.  Goal facts
  are already inserted into all subgoals before \isa{tactic} is
  applied.

  \item \verb|SIMPLE_METHOD'|~\isa{tactic} wraps a tactic that
  addresses a specific subgoal as simple proof method.  Goal facts are
  already inserted into the first subgoal before \isa{tactic} is
  applied to the same.

  \item \verb|HEADGOAL|~\isa{tactic} applies \isa{tactic} to
  the first subgoal.  This is convenient to reproduce part the \verb|SIMPLE_METHOD'| wrapping within regular \verb|METHOD|, for example.

  \item \verb|Method.insert_tac|~\isa{facts\ i} inserts \isa{facts} into subgoal \isa{i}.  This is convenient to reproduce
  part of the \verb|SIMPLE_METHOD| or \verb|SIMPLE_METHOD'| wrapping
  within regular \verb|METHOD|, for example.

  \item \verb|Method.setup|~\isa{name\ parser\ description} provides
  the functionality of the Isar command \hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}} as ML
  function.

  \end{description}%
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\begin{isamarkuptext}%
See also \hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}} in
  \cite{isabelle-isar-ref} which includes some abstract examples.

  \medskip The following toy examples illustrate how the goal facts
  and state are passed to proof methods.  The pre-defined proof method
  called ``\hyperlink{method.tactic}{\mbox{\isa{tactic}}}'' wraps ML source of type \verb|tactic| (abstracted over \verb|facts|).  This allows immediate
  experimentation without parsing of concrete syntax.%
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\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \ \ \isacommand{apply}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}tactic\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ rtac\ %
\isaantiq
thm\ conjI{}%
\endisaantiq
\ {\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \ \ \ \isacommand{using}\isamarkupfalse%
\ a\ \isacommand{apply}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}tactic\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ resolve{\isaliteral{5F}{\isacharunderscore}}tac\ facts\ {\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \ \ \ \isacommand{using}\isamarkupfalse%
\ b\ \isacommand{apply}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}tactic\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ resolve{\isaliteral{5F}{\isacharunderscore}}tac\ facts\ {\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \ \ \ \isacommand{done}\isamarkupfalse%
\isanewline
\isanewline
\ \ \isacommand{have}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \ \ \isacommand{using}\isamarkupfalse%
\ a\ \isakeyword{and}\ b%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
%
\endisadelimproof
%
\isadelimML
\ \ \ \ %
\endisadelimML
%
\isatagML
\isacommand{ML{\isaliteral{5F}{\isacharunderscore}}val}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{40}{\isacharat}}{\isaliteral{7B}{\isacharbraceleft}}Isar{\isaliteral{2E}{\isachardot}}goal{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
\endisatagML
{\isafoldML}%
%
\isadelimML
\isanewline
%
\endisadelimML
%
\isadelimproof
\ \ \ \ %
\endisadelimproof
%
\isatagproof
\isacommand{apply}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}tactic\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ Method{\isaliteral{2E}{\isachardot}}insert{\isaliteral{5F}{\isacharunderscore}}tac\ facts\ {\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \ \ \ \isacommand{apply}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}tactic\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ {\isaliteral{28}{\isacharparenleft}}rtac\ %
\isaantiq
thm\ conjI{}%
\endisaantiq
\ THEN{\isaliteral{5F}{\isacharunderscore}}ALL{\isaliteral{5F}{\isacharunderscore}}NEW\ atac{\isaliteral{29}{\isacharparenright}}\ {\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \ \ \ \isacommand{done}\isamarkupfalse%
%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
%
\endisadelimproof
\isacommand{end}\isamarkupfalse%
%
\begin{isamarkuptext}%
\medskip The next example implements a method that simplifies
  the first subgoal by rewrite rules given as arguments.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse%
\ my{\isaliteral{5F}{\isacharunderscore}}simp\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline
\ \ Attrib{\isaliteral{2E}{\isachardot}}thms\ {\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}fn\ thms\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ fn\ ctxt\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline
\ \ \ \ SIMPLE{\isaliteral{5F}{\isacharunderscore}}METHOD{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}fn\ i\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline
\ \ \ \ \ \ CHANGED\ {\isaliteral{28}{\isacharparenleft}}asm{\isaliteral{5F}{\isacharunderscore}}full{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5F}{\isacharunderscore}}tac\isanewline
\ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}HOL{\isaliteral{5F}{\isacharunderscore}}basic{\isaliteral{5F}{\isacharunderscore}}ss\ addsimps\ thms{\isaliteral{29}{\isacharparenright}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
{\isaliteral{2A7D}{\isacharverbatimclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}rewrite\ subgoal\ by\ given\ rules{\isaliteral{22}{\isachardoublequoteclose}}%
\endisatagML
{\isafoldML}%
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\isadelimML
%
\endisadelimML
%
\begin{isamarkuptext}%
The concrete syntax wrapping of \hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}} always
  passes-through the proof context at the end of parsing, but it is
  not used in this example.

  The \verb|Attrib.thms| parser produces a list of theorems from the
  usual Isar syntax involving attribute expressions etc.\ (syntax
  category \hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}) \cite{isabelle-isar-ref}.  The resulting
  \verb|thms| are added to \verb|HOL_basic_ss| which already
  contains the basic Simplifier setup for HOL.

  The tactic \verb|asm_full_simp_tac| is the one that is also used in
  method \hyperlink{method.simp}{\mbox{\isa{simp}}} by default.  The extra wrapping by the \verb|CHANGED| tactical ensures progress of simplification: identical goal
  states are filtered out explicitly to make the raw tactic conform to
  standard Isar method behaviour.

  \medskip Method \hyperlink{method.my-simp}{\mbox{\isa{my{\isaliteral{5F}{\isacharunderscore}}simp}}} can be used in Isar proofs like
  this:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{notepad}\isamarkupfalse%
\isanewline
\isakeyword{begin}\isanewline
%
\isadelimproof
\ \ %
\endisadelimproof
%
\isatagproof
\isacommand{fix}\isamarkupfalse%
\ a\ b\ c\isanewline
\ \ \isacommand{assume}\isamarkupfalse%
\ a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \isacommand{assume}\isamarkupfalse%
\ b{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}b\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \isacommand{have}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}my{\isaliteral{5F}{\isacharunderscore}}simp\ a\ b{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
%
\endisadelimproof
\isacommand{end}\isamarkupfalse%
%
\begin{isamarkuptext}%
Here is a similar method that operates on all subgoals,
  instead of just the first one.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse%
\ my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5F}{\isacharunderscore}}all\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline
\ \ Attrib{\isaliteral{2E}{\isachardot}}thms\ {\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}fn\ thms\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ fn\ ctxt\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline
\ \ \ \ SIMPLE{\isaliteral{5F}{\isacharunderscore}}METHOD\isanewline
\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}CHANGED\isanewline
\ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}ALLGOALS\ {\isaliteral{28}{\isacharparenleft}}asm{\isaliteral{5F}{\isacharunderscore}}full{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5F}{\isacharunderscore}}tac\isanewline
\ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}HOL{\isaliteral{5F}{\isacharunderscore}}basic{\isaliteral{5F}{\isacharunderscore}}ss\ addsimps\ thms{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
{\isaliteral{2A7D}{\isacharverbatimclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}rewrite\ all\ subgoals\ by\ given\ rules{\isaliteral{22}{\isachardoublequoteclose}}%
\endisatagML
{\isafoldML}%
%
\isadelimML
%
\endisadelimML
\isanewline
\isanewline
\isacommand{notepad}\isamarkupfalse%
\isanewline
\isakeyword{begin}\isanewline
%
\isadelimproof
\ \ %
\endisadelimproof
%
\isatagproof
\isacommand{fix}\isamarkupfalse%
\ a\ b\ c\isanewline
\ \ \isacommand{assume}\isamarkupfalse%
\ a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \isacommand{assume}\isamarkupfalse%
\ b{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}b\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \isacommand{have}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}c\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
\ {\isaliteral{28}{\isacharparenleft}}my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5F}{\isacharunderscore}}all\ a\ b{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
%
\isadelimproof
\isanewline
%
\endisadelimproof
\isacommand{end}\isamarkupfalse%
%
\begin{isamarkuptext}%
\medskip Apart from explicit arguments, common proof methods
  typically work with a default configuration provided by the context.
  As a shortcut to rule management we use a cheap solution via functor
  \verb|Named_Thms| (see also \verb|~~/src/Pure/Tools/named_thms.ML|).%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
\isacommand{ML}\isamarkupfalse%
\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline
\ \ structure\ My{\isaliteral{5F}{\isacharunderscore}}Simps\ {\isaliteral{3D}{\isacharequal}}\isanewline
\ \ \ \ Named{\isaliteral{5F}{\isacharunderscore}}Thms\isanewline
\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}val\ name\ {\isaliteral{3D}{\isacharequal}}\ %
\isaantiq
binding\ my{\isaliteral{5F}{\isacharunderscore}}simp{}%
\endisaantiq
\ val\ description\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}my{\isaliteral{5F}{\isacharunderscore}}simp\ rule{\isaliteral{22}{\isachardoublequote}}{\isaliteral{29}{\isacharparenright}}\isanewline
{\isaliteral{2A7D}{\isacharverbatimclose}}\isanewline
\isacommand{setup}\isamarkupfalse%
\ My{\isaliteral{5F}{\isacharunderscore}}Simps{\isaliteral{2E}{\isachardot}}setup%
\endisatagML
{\isafoldML}%
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\isadelimML
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\endisadelimML
%
\begin{isamarkuptext}%
This provides ML access to a list of theorems in canonical
  declaration order via \verb|My_Simps.get|.  The user can add or
  delete rules via the attribute \hyperlink{attribute.my-simp}{\mbox{\isa{my{\isaliteral{5F}{\isacharunderscore}}simp}}}.  The actual
  proof method is now defined as before, but we append the explicit
  arguments and the rules from the context.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimML
%
\endisadelimML
%
\isatagML
\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse%
\ my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline
\ \ Attrib{\isaliteral{2E}{\isachardot}}thms\ {\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}fn\ thms\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ fn\ ctxt\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline
\ \ \ \ SIMPLE{\isaliteral{5F}{\isacharunderscore}}METHOD{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}fn\ i\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline
\ \ \ \ \ \ CHANGED\ {\isaliteral{28}{\isacharparenleft}}asm{\isaliteral{5F}{\isacharunderscore}}full{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5F}{\isacharunderscore}}tac\isanewline
\ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}HOL{\isaliteral{5F}{\isacharunderscore}}basic{\isaliteral{5F}{\isacharunderscore}}ss\ addsimps\ {\isaliteral{28}{\isacharparenleft}}thms\ {\isaliteral{40}{\isacharat}}\ My{\isaliteral{5F}{\isacharunderscore}}Simps{\isaliteral{2E}{\isachardot}}get\ ctxt{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
{\isaliteral{2A7D}{\isacharverbatimclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}rewrite\ subgoal\ by\ given\ rules\ and\ my{\isaliteral{5F}{\isacharunderscore}}simp\ rules\ from\ the\ context{\isaliteral{22}{\isachardoublequoteclose}}%
\endisatagML
{\isafoldML}%
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\isadelimML
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\endisadelimML
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\begin{isamarkuptext}%
\medskip Method \hyperlink{method.my-simp'}{\mbox{\isa{my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{27}{\isacharprime}}}}} can be used in Isar proofs
  like this:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{notepad}\isamarkupfalse%
\isanewline
\isakeyword{begin}\isanewline
%
\isadelimproof
\ \ %
\endisadelimproof
%
\isatagproof
\isacommand{fix}\isamarkupfalse%
\ a\ b\ c\isanewline
\ \ \isacommand{assume}\isamarkupfalse%
\ {\isaliteral{5B}{\isacharbrackleft}}my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \isacommand{assume}\isamarkupfalse%
\ {\isaliteral{5B}{\isacharbrackleft}}my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}b\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\ \ \isacommand{have}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
\ my{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{27}{\isacharprime}}%
\endisatagproof
{\isafoldproof}%
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\isadelimproof
\isanewline
%
\endisadelimproof
\isacommand{end}\isamarkupfalse%
%
\begin{isamarkuptext}%
\medskip The \hyperlink{method.my-simp}{\mbox{\isa{my{\isaliteral{5F}{\isacharunderscore}}simp}}} variants defined above are
  ``simple'' methods, i.e.\ the goal facts are merely inserted as goal
  premises by the \verb|SIMPLE_METHOD'| or \verb|SIMPLE_METHOD| wrapper.
  For proof methods that are similar to the standard collection of
  \hyperlink{method.simp}{\mbox{\isa{simp}}}, \hyperlink{method.blast}{\mbox{\isa{blast}}}, \hyperlink{method.fast}{\mbox{\isa{fast}}}, \hyperlink{method.auto}{\mbox{\isa{auto}}}
  there is little more that can be done.

  Note that using the primary goal facts in the same manner as the
  method arguments obtained via concrete syntax or the context does
  not meet the requirement of ``strong emphasis on facts'' of regular
  proof methods, because rewrite rules as used above can be easily
  ignored.  A proof text ``\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{foo}~\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{my{\isaliteral{5F}{\isacharunderscore}}simp}'' where \isa{foo} is not used would
  deceive the reader.

  \medskip The technical treatment of rules from the context requires
  further attention.  Above we rebuild a fresh \verb|simpset| from
  the arguments and \emph{all} rules retrieved from the context on
  every invocation of the method.  This does not scale to really large
  collections of rules, which easily emerges in the context of a big
  theory library, for example.

  This is an inherent limitation of the simplistic rule management via
  functor \verb|Named_Thms|, because it lacks tool-specific
  storage and retrieval.  More realistic applications require
  efficient index-structures that organize theorems in a customized
  manner, such as a discrimination net that is indexed by the
  left-hand sides of rewrite rules.  For variations on the Simplifier,
  re-use of the existing type \verb|simpset| is adequate, but
  scalability would require it be maintained statically within the
  context data, not dynamically on each tool invocation.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Attributes \label{sec:attributes}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
An \emph{attribute} is a function \isa{context\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ thm\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ context\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ thm}, which means both a (generic) context and a theorem
  can be modified simultaneously.  In practice this mixed form is very
  rare, instead attributes are presented either as \emph{declaration
  attribute:} \isa{thm\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ context} or \emph{rule
  attribute:} \isa{context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ thm\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ thm}.

  Attributes can have additional arguments via concrete syntax.  There
  is a collection of context-sensitive parsers for various logical
  entities (types, terms, theorems).  These already take care of
  applying morphisms to the arguments when attribute expressions are
  moved into a different context (see also \secref{sec:morphisms}).

  When implementing declaration attributes, it is important to operate
  exactly on the variant of the generic context that is provided by
  the system, which is either global theory context or local proof
  context.  In particular, the background theory of a local context
  must not be modified in this situation!%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimmlref
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\endisadelimmlref
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\isatagmlref
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\begin{isamarkuptext}%
\begin{mldecls}
  \indexdef{}{ML type}{attribute}\verb|type attribute| \\
  \indexdef{}{ML}{Thm.rule\_attribute}\verb|Thm.rule_attribute: (Context.generic -> thm -> thm) -> attribute| \\
  \indexdef{}{ML}{Thm.declaration\_attribute}\verb|Thm.declaration_attribute: |\isasep\isanewline%
\verb|  (thm -> Context.generic -> Context.generic) -> attribute| \\
  \indexdef{}{ML}{Attrib.setup}\verb|Attrib.setup: binding -> attribute context_parser ->|\isasep\isanewline%
\verb|  string -> theory -> theory| \\
  \end{mldecls}

  \begin{description}

  \item Type \verb|attribute| represents attributes as concrete
  type alias.

  \item \verb|Thm.rule_attribute|~\isa{{\isaliteral{28}{\isacharparenleft}}fn\ context\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ rule{\isaliteral{29}{\isacharparenright}}} wraps
  a context-dependent rule (mapping on \verb|thm|) as attribute.

  \item \verb|Thm.declaration_attribute|~\isa{{\isaliteral{28}{\isacharparenleft}}fn\ thm\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ decl{\isaliteral{29}{\isacharparenright}}}
  wraps a theorem-dependent declaration (mapping on \verb|Context.generic|) as attribute.

  \item \verb|Attrib.setup|~\isa{name\ parser\ description} provides
  the functionality of the Isar command \hyperlink{command.attribute-setup}{\mbox{\isa{\isacommand{attribute{\isaliteral{5F}{\isacharunderscore}}setup}}}} as
  ML function.

  \end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
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\endisatagmlref
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\isadelimmlex
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\isatagmlex
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\begin{isamarkuptext}%
See also \hyperlink{command.attribute-setup}{\mbox{\isa{\isacommand{attribute{\isaliteral{5F}{\isacharunderscore}}setup}}}} in
  \cite{isabelle-isar-ref} which includes some abstract examples.%
\end{isamarkuptext}%
\isamarkuptrue%
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\endisatagmlex
{\isafoldmlex}%
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\isadelimmlex
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\endisadelimmlex
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\isadelimtheory
%
\endisadelimtheory
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\isatagtheory
\isacommand{end}\isamarkupfalse%
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
\isanewline
\end{isabellebody}%
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