
\chapter{Generic Tools and Packages}\label{ch:gen-tools}

\section{Basic proof methods}\label{sec:pure-meth}

\indexisarmeth{fail}\indexisarmeth{succeed}\indexisarmeth{$-$}\indexisarmeth{assumption}
\indexisarmeth{finish}\indexisarmeth{fold}\indexisarmeth{unfold}
\indexisarmeth{rule}\indexisarmeth{erule}
\begin{matharray}{rcl}
  - & : & \isarmeth \\
  assumption & : & \isarmeth \\
  finish & : & \isarmeth \\[0.5ex]
  rule & : & \isarmeth \\
  erule^* & : & \isarmeth \\[0.5ex]
  fold & : & \isarmeth \\
  unfold & : & \isarmeth \\[0.5ex]
  succeed & : & \isarmeth \\
  fail & : & \isarmeth \\
\end{matharray}

\begin{rail}
  ('fold' | 'unfold' | 'rule' | 'erule') thmrefs
  ;
\end{rail}

\begin{descr}
\item [``$-$''] does nothing but insert the forward chaining facts as premises
  into the goal.  Note that command $\PROOFNAME$ without any method actually
  performs a single reduction step using the $rule$ method (see below); thus a
  plain \emph{do-nothing} proof step would be $\PROOF{-}$ rather than
  $\PROOFNAME$ alone.
\item [$assumption$] solves some goal by assumption, after inserting the
  goal's facts.
\item [$finish$] solves all remaining goals by assumption; this is the default
  terminal proof method for $\QEDNAME$, i.e.\ it usually does not have to be
  spelled out explicitly.
\item [$rule~thms$] applies some rule given as argument in backward manner;
  facts are used to reduce the rule before applying it to the goal.  Thus
  $rule$ without facts is plain \emph{introduction}, while with facts it
  becomes an \emph{elimination}.
  
  Note that the classical reasoner introduces another version of $rule$ that
  is able to pick appropriate rules automatically, whenever explicit $thms$
  are omitted (see \S\ref{sec:classical-basic}); that method is the default
  one for initial proof steps, such as $\PROOFNAME$ and ``$\DDOT$'' (two
  dots).
\item [$erule~thms$] is similar to $rule$, but applies rules by
  elim-resolution.  This is an improper method, mainly for experimentation and
  porting of old scripts.  Actual elimination proofs are usually done with
  $rule$ (single step, involving facts) or $elim$ (multiple steps, see
  \S\ref{sec:classical-basic}).
\item [$unfold~thms$ and $fold~thms$] expand and fold back again the given
  meta-level definitions throughout all goals; facts may not be involved.
\item [$succeed$] yields a single (unchanged) result; it is the identify of
  the ``\texttt{,}'' method combinator.
\item [$fail$] yields an empty result sequence; it is the identify of the
  ``\texttt{|}'' method combinator.
\end{descr}


\section{Miscellaneous attributes}

\indexisaratt{tag}\indexisaratt{untag}\indexisaratt{COMP}\indexisaratt{RS}
\indexisaratt{OF}\indexisaratt{where}\indexisaratt{of}\indexisaratt{standard}
\indexisaratt{elimify}\indexisaratt{transfer}\indexisaratt{export}
\begin{matharray}{rcl}
  tag & : & \isaratt \\
  untag & : & \isaratt \\[0.5ex]
  OF & : & \isaratt \\
  RS & : & \isaratt \\
  COMP & : & \isaratt \\[0.5ex]
  of & : & \isaratt \\
  where & : & \isaratt \\[0.5ex]
  standard & : & \isaratt \\
  elimify & : & \isaratt \\
  export^* & : & \isaratt \\
  transfer & : & \isaratt \\
\end{matharray}

\begin{rail}
  ('tag' | 'untag') (nameref+)
  ;
  'OF' thmrefs
  ;
  ('RS' | 'COMP') nat? thmref
  ;
  'of' (inst * ) ('concl' ':' (inst * ))?
  ;
  'where' (name '=' term * 'and')
  ;

  inst: underscore | term
  ;
\end{rail}

\begin{descr}
\item [$tag~tags$ and $untag~tags$] add and remove $tags$ to the theorem,
  respectively.  Tags may be any list of strings that serve as comment for
  some tools (e.g.\ $\LEMMANAME$ causes tag ``$lemma$'' to be added to the
  result).
\item [$OF~thms$, $RS~n~thm$, and $COMP~n~thm$] compose rules.  $OF$ applies
  $thms$ in parallel (cf.\ \texttt{MRS} in \cite[\S5]{isabelle-ref}, but note
  the reversed order).  Note that premises may be skipped by including $\_$
  (underscore) as argument.
  
  $RS$ resolves with the $n$-th premise of $thm$; $COMP$ is a version of $RS$
  that does not include the automatic lifting process that is normally
  intended (see also \texttt{RS} and \texttt{COMP} in
  \cite[\S5]{isabelle-ref}).
  
\item [$of~ts$ and $where~\vec x = \vec t$] perform positional and named
  instantiation, respectively.  The terms given in $of$ are substituted for
  any schematic variables occurring in a theorem from left to right;
  ``\texttt{_}'' (underscore) indicates to skip a position.
 
\item [$standard$] puts a theorem into the standard form of object-rules, just
  as the ML function \texttt{standard} (see \cite[\S5]{isabelle-ref}).
  
\item [$elimify$] turns an destruction rule into an elimination.
  
\item [$export$] lifts a local result out of the current proof context,
  generalizing all fixed variables and discharging all assumptions.  Note that
  (partial) export is usually done automatically behind the scenes.  This
  attribute is mainly for experimentation.
  
\item [$transfer$] promotes a theorem to the current theory context, which has
  to enclose the former one.  Normally, this is done automatically when rules
  are joined by inference.

\end{descr}


\section{Calculational proof}\label{sec:calculation}

\indexisarcmd{also}\indexisarcmd{finally}\indexisaratt{trans}
\begin{matharray}{rcl}
  \isarcmd{also} & : & \isartrans{proof(state)}{proof(state)} \\
  \isarcmd{finally} & : & \isartrans{proof(state)}{proof(chain)} \\
  trans & : & \isaratt \\
\end{matharray}

Calculational proof is forward reasoning with implicit application of
transitivity rules (such those of $=$, $\le$, $<$).  Isabelle/Isar maintains
an auxiliary register $calculation$\indexisarthm{calculation} for accumulating
results obtained by transitivity obtained together with the current facts.
Command $\ALSO$ updates $calculation$ from the most recent result, while
$\FINALLY$ exhibits the final result by forward chaining towards the next goal
statement.  Both commands require valid current facts, i.e.\ may occur only
after commands that produce theorems such as $\ASSUMENAME$, $\NOTENAME$, or
some finished $\HAVENAME$ or $\SHOWNAME$.

Also note that the automatic term abbreviation ``$\dots$'' has its canonical
application with calculational proofs.  It automatically refers to the
argument\footnote{The argument of a curried infix expression is its right-hand
  side.} of the preceding statement.

Isabelle/Isar calculations are implicitly subject to block structure in the
sense that new threads of calculational reasoning are commenced for any new
block (as opened by a local goal, for example).  This means that, apart from
being able to nest calculations, there is no separate \emph{begin-calculation}
command required.

\begin{rail}
  ('also' | 'finally') transrules? comment?
  ;
  'trans' (() | 'add' ':' | 'del' ':') thmrefs
  ;

  transrules: '(' thmrefs ')' interest?
  ;
\end{rail}

\begin{descr}
\item [$\ALSO~(thms)$] maintains the auxiliary $calculation$ register as
  follows.  The first occurrence of $\ALSO$ in some calculational thread
  initialises $calculation$ by $facts$. Any subsequent $\ALSO$ on the same
  level of block-structure updates $calculation$ by some transitivity rule
  applied to $calculation$ and $facts$ (in that order).  Transitivity rules
  are picked from the current context plus those given as $thms$ (the latter
  have precedence).
  
\item [$\FINALLY~(thms)$] maintaining $calculation$ in the same way as
  $\ALSO$, and concludes the current calculational thread.  The final result
  is exhibited as fact for forward chaining towards the next goal. Basically,
  $\FINALLY$ just abbreviates $\ALSO~\FROM{calculation}$.  A typical proof
  idiom is ``$\FINALLY~\SHOW{}{\VVar{thesis}}~\DOT$''.
  
\item [$trans$] maintains the set of transitivity rules of the theory or proof
  context, by adding or deleting theorems (the default is to add).
\end{descr}

See theory \texttt{HOL/Isar_examples/Group} for a simple application of
calculations for basic equational reasoning.
\texttt{HOL/Isar_examples/KnasterTarski} involves a few more advanced
calculational steps in combination with natural deduction.


\section{Axiomatic Type Classes}\label{sec:axclass}

\indexisarcmd{axclass}\indexisarcmd{instance}\indexisarmeth{intro-classes}
\begin{matharray}{rcl}
  \isarcmd{axclass} & : & \isartrans{theory}{theory} \\
  \isarcmd{instance} & : & \isartrans{theory}{proof(prove)} \\
  intro_classes & : & \isarmeth \\
\end{matharray}

Axiomatic type classes are provided by Isabelle/Pure as a purely
\emph{definitional} interface to type classes (cf.~\S\ref{sec:classes}).  Thus
any object logic may make use of this light-weight mechanism for abstract
theories.  See \cite{Wenzel:1997:TPHOL} for more information.  There is also a
tutorial on \emph{Using Axiomatic Type Classes in Isabelle} that is part of
the standard Isabelle documentation.
%FIXME cite

\begin{rail}
  'axclass' classdecl (axmdecl prop comment? +)
  ;
  'instance' (nameref '<' nameref | nameref '::' simplearity) comment?
  ;
\end{rail}

\begin{descr}
\item [$\isarkeyword{axclass}~c < \vec c~axms$] defines an axiomatic type
  class as the intersection of existing classes, with additional axioms
  holding.  Class axioms may not contain more than one type variable.  The
  class axioms (with implicit sort constraints added) are bound to the given
  names.  Furthermore a class introduction rule is generated, which is
  employed by method $intro_classes$ in support instantiation proofs of this
  class.
  
\item [$\isarkeyword{instance}~c@1 < c@2$ and $\isarkeyword{instance}~t ::
  (\vec s)c$] setup up a goal stating the class relation or type arity.  The
  proof would usually proceed by the $intro_classes$ method, and then
  establish the characteristic theorems of the type classes involved.  After
  finishing the proof the theory will be augmented by a type signature
  declaration corresponding to the resulting theorem.
\item [Method $intro_classes$] iteratively expands the class introduction
  rules
\end{descr}

See theory \texttt{HOL/Isar_examples/Group} for a simple example of using
axiomatic type classes, including instantiation proofs.


\section{The Simplifier}

\subsection{Simplification methods}\label{sec:simp}

\indexisarmeth{simp}\indexisarmeth{asm-simp}
\begin{matharray}{rcl}
  simp & : & \isarmeth \\
  asm_simp & : & \isarmeth \\
\end{matharray}

\railalias{asmsimp}{asm\_simp}
\railterm{asmsimp}

\begin{rail}
  ('simp' | asmsimp) (simpmod * )
  ;

  simpmod: ('add' | 'del' | 'only' | 'other') ':' thmrefs
  ;
\end{rail}

\begin{descr}
\item [Methods $simp$ and $asm_simp$] invoke Isabelle's simplifier, after
  modifying the context by adding or deleting given rules.  The
  \railtoken{only} modifier first removes all other rewrite rules and
  congruences, and then is like \railtoken{add}.  In contrast,
  \railtoken{other} ignores its arguments; nevertheless there may be
  side-effects on the context via attributes.  This provides a back door for
  arbitrary context manipulation.
  
  Both of these methods are based on \texttt{asm_full_simp_tac}, see
  \cite[\S10]{isabelle-ref}; $simp$ removes any exisiting premises of the
  goal, before inserting the goal facts; $asm_simp$ leaves the premises.
\end{descr}

\subsection{Modifying the context}

\indexisaratt{simp}
\begin{matharray}{rcl}
  simp & : & \isaratt \\
\end{matharray}

\begin{rail}
  'simp' (() | 'add' | 'del')
  ;
\end{rail}

\begin{descr}
\item [Attribute $simp$] adds or deletes rules from the theory or proof
  context (the default is to add).
\end{descr}


\subsection{Forward simplification}

\indexisaratt{simplify}\indexisaratt{asm-simplify}
\indexisaratt{full-simplify}\indexisaratt{asm-full-simplify}
\begin{matharray}{rcl}
  simplify & : & \isaratt \\
  asm_simplify & : & \isaratt \\
  full_simplify & : & \isaratt \\
  asm_full_simplify & : & \isaratt \\
\end{matharray}

These attributes provide forward rules for simplification, which should be
used only very rarely.  See the ML functions of the same name in
\cite[\S10]{isabelle-ref} for more information.


\section{The Classical Reasoner}

\subsection{Basic methods}\label{sec:classical-basic}

\indexisarmeth{rule}\indexisarmeth{default}\indexisarmeth{contradiction}
\begin{matharray}{rcl}
  rule & : & \isarmeth \\
  intro & : & \isarmeth \\
  elim & : & \isarmeth \\
  contradiction & : & \isarmeth \\
\end{matharray}

\begin{rail}
  ('rule' | 'intro' | 'elim') thmrefs
  ;
\end{rail}

\begin{descr}
\item [Method $rule$] as offered by the classical reasoner is a refinement
  over the primitive one (see \S\ref{sec:pure-meth}).  In the case that no
  rules are provided as arguments, it automatically determines elimination and
  introduction rules from the context (see also \S\ref{sec:classical-mod}).
  In that form it is the default method for basic proof steps, such as
  $\PROOFNAME$ and ``$\DDOT$'' (two dots).
  
\item [Methods $intro$ and $elim$] repeatedly refine some goal by intro- or
  elim-resolution, after having inserted the facts.  Omitting the arguments
  refers to any suitable rules from the context, otherwise only the explicitly
  given ones may be applied.  The latter form admits better control of what
  actually happens, thus it is very appropriate as an initial method for
  $\PROOFNAME$ that splits up certain connectives of the goal, before entering
  the sub-proof.

\item [Method $contradiction$] solves some goal by contradiction: both $A$ and
  $\neg A$ have to be present in the assumptions.
\end{descr}


\subsection{Automatic methods}\label{sec:classical-auto}

\indexisarmeth{blast}
\indexisarmeth{fast}\indexisarmeth{best}\indexisarmeth{slow}\indexisarmeth{slow-best}
\begin{matharray}{rcl}
 blast & : & \isarmeth \\
 fast & : & \isarmeth \\
 best & : & \isarmeth \\
 slow & : & \isarmeth \\
 slow_best & : & \isarmeth \\
\end{matharray}

\railalias{slowbest}{slow\_best}
\railterm{slowbest}

\begin{rail}
  'blast' nat? (clamod * )
  ;
  ('fast' | 'best' | 'slow' | slowbest) (clamod * )
  ;

  clamod: (('intro' | 'elim' | 'dest') (() | '!' | '!!') | 'del') ':' thmrefs
  ;
\end{rail}

\begin{descr}
\item [$blast$] refers to the classical tableau prover (see \texttt{blast_tac}
  in \cite[\S11]{isabelle-ref}).  The optional argument specifies a
  user-supplied search bound (default 20).
\item [$fast$, $best$, $slow$, $slow_best$] refer to the generic classical
  reasoner (see \cite[\S11]{isabelle-ref}, tactic \texttt{fast_tac} etc).
\end{descr}

Any of above methods support additional modifiers of the context of classical
rules.  There semantics is analogous to the attributes given in
\S\ref{sec:classical-mod}.


\subsection{Combined automatic methods}

\indexisarmeth{auto}\indexisarmeth{force}
\begin{matharray}{rcl}
  force & : & \isarmeth \\
  auto & : & \isarmeth \\
\end{matharray}

\begin{rail}
  ('force' | 'auto') (clasimpmod * )
  ;

  clasimpmod: ('simp' ('add' | 'del' | 'only') | other |
    (('intro' | 'elim' | 'dest') (() | '!' | '!!') | 'del')) ':' thmrefs
\end{rail}

\begin{descr}
\item [$force$ and $auto$] provide access to Isabelle's combined
  simplification and classical reasoning tactics.  See \texttt{force_tac} and
  \texttt{auto_tac} in \cite[\S11]{isabelle-ref} for more information.  The
  modifier arguments correspond to those given in \S\ref{sec:simp} and
  \S\ref{sec:classical-auto}.  Note that the ones related to the Simplifier
  are prefixed by \railtoken{simp} here.
\end{descr}

\subsection{Modifying the context}\label{sec:classical-mod}

\indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest}
\indexisaratt{iff}\indexisaratt{delrule}
\begin{matharray}{rcl}
  intro & : & \isaratt \\
  elim & : & \isaratt \\
  dest & : & \isaratt \\
  iff & : & \isaratt \\
  delrule & : & \isaratt \\
\end{matharray}

\begin{rail}
  ('intro' | 'elim' | 'dest') (() | '!' | '!!')
  ;
\end{rail}

\begin{descr}
\item [$intro$, $elim$, $dest$] add introduction, elimination, destruct rules,
  respectively.  By default, rules are considered as \emph{safe}, while a
  single ``!'' classifies as \emph{unsafe}, and ``!!'' as \emph{extra} (i.e.\ 
  not applied in the search-oriented automatic methods).
  
\item [$iff$] declares equations both as rewrite rules for the simplifier and
  classical reasoning rules.

\item [$delrule$] deletes introduction or elimination rules from the context.
  Note that destruction rules would have to be turned into elimination rules
  first, e.g.\ by using the $elimify$ attribute.
\end{descr}


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