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

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

%FIXME

% - qualified names

% - class intro rules;

% - class axioms;

\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 \emph{definitional}

interface to type classes (cf.~\S\ref{sec:classes}).  Thus any object logic

may make use of this light-weight mechanism of abstract theories

\cite{Wenzel:1997:TPHOL}.  There is also a tutorial on using axiomatic type

classes in isabelle \cite{isabelle-axclass} that is part of the standard

Isabelle documentation.

\begin{rail}

  'axclass' classdecl (axmdecl prop comment? +)

  ;

  'instance' (nameref ('<' | subseteq) nameref | nameref '::' simplearity) comment?

  ;

\end{rail}

\begin{descr}

\item [$\AXCLASS~c \subseteq \vec c~axms$] 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

  $intro_classes$ to support instantiation proofs of this class.

\item [$\INSTANCE~c@1 \subseteq c@2$ and $\INSTANCE~t :: (\vec s)c$] setup a

  goal stating a class relation or type arity.  The proof would usually

  proceed by $intro_classes$, 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 [$intro_classes$] repeatedly expands all class introduction rules of

  this theory.  Note that this method usually needs not be named explicitly,

  as it is already included in the default proof step (of $\PROOFNAME$,

  $\BYNAME$, etc.).  In particular, instantiation of trivial (syntactic)

  classes may be performed by a single ``$\DDOT$'' proof step.

\end{descr}

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

\indexisarcmd{also}\indexisarcmd{finally}

\indexisarcmd{moreover}\indexisarcmd{ultimately}

\indexisarcmd{print-trans-rules}\indexisaratt{trans}

\begin{matharray}{rcl}

  \isarcmd{also} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{finally} & : & \isartrans{proof(state)}{proof(chain)} \\

  \isarcmd{moreover} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{ultimately} & : & \isartrans{proof(state)}{proof(chain)} \\

  \isarcmd{print_trans_rules}^* & : & \isarkeep{theory~|~proof} \\

  trans & : & \isaratt \\

\end{matharray}

Calculational proof is forward reasoning with implicit application of

transitivity rules (such those of $=$, $\leq$, $<$).  Isabelle/Isar maintains

an auxiliary register $calculation$\indexisarthm{calculation} for accumulating

results obtained by transitivity composed with the current result.  Command

$\ALSO$ updates $calculation$ involving $this$, while $\FINALLY$ exhibits the

final $calculation$ 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 proof of

$\HAVENAME$, $\SHOWNAME$ etc.  The $\MOREOVER$ and $\ULTIMATELY$ commands are

similar to $\ALSO$ and $\FINALLY$, but only collect further results in

$calculation$ without applying any rules yet.

Also note that the automatic term abbreviation ``$\dots$'' has its canonical

application with calculational proofs.  It 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.

\medskip

The Isar calculation proof commands may be defined as

follows:\footnote{Internal bookkeeping such as proper handling of

  block-structure has been suppressed.}

\begin{matharray}{rcl}

  \ALSO@0 & \equiv & \NOTE{calculation}{this} \\

  \ALSO@{n+1} & \equiv & \NOTE{calculation}{trans~[OF~calculation~this]} \\[0.5ex]

  \FINALLY & \equiv & \ALSO~\FROM{calculation} \\

  \MOREOVER & \equiv & \NOTE{calculation}{calculation~this} \\

  \ULTIMATELY & \equiv & \MOREOVER~\FROM{calculation} \\

\end{matharray}

\begin{rail}

  ('also' | 'finally') transrules? comment?

  ;

  ('moreover' | 'ultimately') comment?

  ;

  'trans' (() | 'add' | 'del')

  ;

  transrules: '(' thmrefs ')' interest?

  ;

\end{rail}

\begin{descr}

\item [$\ALSO~(\vec a)$] maintains the auxiliary $calculation$ register as

  follows.  The first occurrence of $\ALSO$ in some calculational thread

  initializes $calculation$ by $this$. Any subsequent $\ALSO$ on the same

  level of block-structure updates $calculation$ by some transitivity rule

  applied to $calculation$ and $this$ (in that order).  Transitivity rules are

  picked from the current context, unless alternative rules are given as

  explicit arguments.

\item [$\FINALLY~(\vec a)$] 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 $\ALSO~\FROM{calculation}$.  Note that

  ``$\FINALLY~\SHOW{}{\Var{thesis}}~\DOT$'' and

  ``$\FINALLY~\HAVE{}{\phi}~\DOT$'' are typical idioms for concluding

  calculational proofs.

\item [$\MOREOVER$ and $\ULTIMATELY$] are analogous to $\ALSO$ and $\FINALLY$,

  but collect results only, without applying rules.

\item [$\isarkeyword{print_trans_rules}$] prints the list of transitivity

  rules declared in the current context.

\item [$trans$] declares theorems as transitivity rules.

\end{descr}

\section{Named local contexts (cases)}\label{sec:cases}

\indexisarcmd{case}\indexisarcmd{print-cases}

\indexisaratt{case-names}\indexisaratt{params}\indexisaratt{consumes}

\begin{matharray}{rcl}

  \isarcmd{case} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{print_cases}^* & : & \isarkeep{proof} \\

  case_names & : & \isaratt \\

  params & : & \isaratt \\

  consumes & : & \isaratt \\

\end{matharray}

Basically, Isar proof contexts are built up explicitly using commands like

$\FIXNAME$, $\ASSUMENAME$ etc.\ (see \S\ref{sec:proof-context}).  In typical

verification tasks this can become hard to manage, though.  In particular, a

large number of local contexts may emerge from case analysis or induction over

inductive sets and types.

\medskip

The $\CASENAME$ command provides a shorthand to refer to certain parts of

logical context symbolically.  Proof methods may provide an environment of

named ``cases'' of the form $c\colon \vec x, \vec \phi$.  Then the effect of

$\CASE{c}$ is exactly the same as $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$.

It is important to note that $\CASENAME$ does \emph{not} provide any means to

peek at the current goal state, which is treated as strictly non-observable in

Isar!  Instead, the cases considered here usually emerge in a canonical way

from certain pieces of specification that appear in the theory somewhere else

(e.g.\ in an inductive definition, or recursive function).  See also

\S\ref{sec:induct-method} for more details of how this works in HOL.

\medskip

Named cases may be exhibited in the current proof context only if both the

proof method and the rules involved support this.  Case names and parameters

of basic rules may be declared by hand as well, by using appropriate

attributes.  Thus variant versions of rules that have been derived manually

may be used in advanced case analysis later.

\railalias{casenames}{case\_names}

\railterm{casenames}

\begin{rail}

  'case' nameref attributes?

  ;

  casenames (name + )

  ;

  'params' ((name * ) + 'and')

  ;

  'consumes' nat?

  ;

\end{rail}

%FIXME bug in rail

\begin{descr}

\item [$\CASE{c}$] invokes a named local context $c\colon \vec x, \vec \phi$,

  as provided by an appropriate proof method (such as $cases$ and $induct$ in

  Isabelle/HOL, see \S\ref{sec:induct-method}).  The command $\CASE{c}$

  abbreviates $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$.

\item [$\isarkeyword{print_cases}$] prints all local contexts of the current

  state, using Isar proof language notation.  This is a diagnostic command;

  $undo$ does not apply.

\item [$case_names~\vec c$] declares names for the local contexts of premises

  of some theorem; $\vec c$ refers to the \emph{suffix} of the list of

  premises.

\item [$params~\vec p@1 \dots \vec p@n$] renames the innermost parameters of

  premises $1, \dots, n$ of some theorem.  An empty list of names may be given

  to skip positions, leaving the present parameters unchanged.

  Note that the default usage of case rules does \emph{not} directly expose

  parameters to the proof context (see also \S\ref{sec:induct-method-proper}).

\item [$consumes~n$] declares the number of ``major premises'' of a rule,

  i.e.\ the number of facts to be consumed when it is applied by an

  appropriate proof method (cf.\ \S\ref{sec:induct-method}).  The default

  value of $consumes$ is $n = 1$, which is appropriate for the usual kind of

  cases and induction rules for inductive sets (cf.\ \S\ref{sec:inductive}).

  Rules without any $consumes$ declaration given are treated as if

  $consumes~0$ had been specified.

  Note that explicit $consumes$ declarations are only rarely needed; this is

  already taken care of automatically by the higher-level $cases$ and $induct$

  declarations, see also \S\ref{sec:induct-att}.

\end{descr}

\section{Generalized existence}\label{sec:obtain}

\indexisarcmd{obtain}

\begin{matharray}{rcl}

  \isarcmd{obtain} & : & \isartrans{proof(state)}{proof(prove)} \\

\end{matharray}

Generalized existence means that additional elements with certain properties

may introduced in the current context.  Technically, the $\OBTAINNAME$

language element is like a declaration of $\FIXNAME$ and $\ASSUMENAME$ (see

also see \S\ref{sec:proof-context}), together with a soundness proof of its

additional claim.  According to the nature of existential reasoning,

assumptions get eliminated from any result exported from the context later,

provided that the corresponding parameters do \emph{not} occur in the

conclusion.

\begin{rail}

  'obtain' (vars + 'and') comment? \\ 'where' (assm comment? + 'and')

  ;

\end{rail}

$\OBTAINNAME$ is defined as a derived Isar command as follows, where $\vec b$

shall refer to (optional) facts indicated for forward chaining.

\begin{matharray}{l}

  \langle facts~\vec b\rangle \\

  \OBTAIN{\vec x}{a}{\vec \phi}~~\langle proof\rangle \equiv {} \\[1ex]

  \quad \BG \\

  \qquad \FIX{thesis} \\

  \qquad \ASSUME{that~[simp, intro]}{\All{\vec x} \vec\phi \Imp thesis} \\

  \qquad \FROM{\vec b}~\HAVE{}{thesis}~~\langle proof\rangle \\

  \quad \EN \\

  \quad \FIX{\vec x}~\ASSUMENAME^\ast~a\colon~\vec\phi \\

\end{matharray}

Typically, the soundness proof is relatively straight-forward, often just by

canonical automated tools such as $\BY{simp}$ (see \S\ref{sec:simp}) or

$\BY{blast}$ (see \S\ref{sec:classical-auto}).  Accordingly, the ``$that$''

reduction above is declared as simplification and introduction rule.

\medskip

In a sense, $\OBTAINNAME$ represents at the level of Isar proofs what would be

meta-logical existential quantifiers and conjunctions.  This concept has a

broad range of useful applications, ranging from plain elimination (or even

introduction) of object-level existentials and conjunctions, to elimination

over results of symbolic evaluation of recursive definitions, for example.

Also note that $\OBTAINNAME$ without parameters acts much like $\HAVENAME$,

where the result is treated as an assumption.

\section{Miscellaneous methods and attributes}\label{sec:misc-methods}

\indexisarmeth{unfold}\indexisarmeth{fold}\indexisarmeth{insert}

\indexisarmeth{erule}\indexisarmeth{drule}\indexisarmeth{frule}

\indexisarmeth{fail}\indexisarmeth{succeed}

\begin{matharray}{rcl}

  unfold & : & \isarmeth \\

  fold & : & \isarmeth \\

  insert & : & \isarmeth \\[0.5ex]

  erule^* & : & \isarmeth \\

  drule^* & : & \isarmeth \\

  frule^* & : & \isarmeth \\[0.5ex]

  succeed & : & \isarmeth \\

  fail & : & \isarmeth \\

\end{matharray}

\begin{rail}

  ('fold' | 'unfold' | 'insert') thmrefs

  ;

  ('erule' | 'drule' | 'frule') ('('nat')')? thmrefs

  ;

\end{rail}

\begin{descr}

\item [$unfold~\vec a$ and $fold~\vec a$] expand and fold back again the given

  meta-level definitions throughout all goals; any facts provided are inserted

  into the goal and subject to rewriting as well.

\item [$insert~\vec a$] inserts theorems as facts into all goals of the proof

  state.  Note that current facts indicated for forward chaining are ignored.

\item [$erule~\vec a$, $drule~\vec a$, and $frule~\vec a$] are similar to the

  basic $rule$ method (see \S\ref{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.

  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 basic $rule$ method, with

  forward chaining of current facts.

\item [$succeed$] yields a single (unchanged) result; it is the identity of

  the ``\texttt{,}'' method combinator (cf.\ \S\ref{sec:syn-meth}).

\item [$fail$] yields an empty result sequence; it is the identity of the

  ``\texttt{|}'' method combinator (cf.\ \S\ref{sec:syn-meth}).

\end{descr}

\indexisaratt{tagged}\indexisaratt{untagged}

\indexisaratt{THEN}\indexisaratt{COMP}

\indexisaratt{where}\indexisaratt{unfolded}\indexisaratt{folded}

\indexisaratt{standard}\indexisaratt{elim-format}

\indexisaratt{no-vars}\indexisaratt{exported}

\begin{matharray}{rcl}

  tagged & : & \isaratt \\

  untagged & : & \isaratt \\[0.5ex]

  THEN & : & \isaratt \\

  COMP & : & \isaratt \\[0.5ex]

  where & : & \isaratt \\[0.5ex]

  unfolded & : & \isaratt \\

  folded & : & \isaratt \\[0.5ex]

  standard & : & \isaratt \\

  elim_format & : & \isaratt \\

  no_vars^* & : & \isaratt \\

  exported^* & : & \isaratt \\

\end{matharray}

\begin{rail}

  'tagged' (nameref+)

  ;

  'untagged' name

  ;

  ('THEN' | 'COMP') ('[' nat ']')? thmref

  ;

  'where' (name '=' term * 'and')

  ;

  ('unfolded' | 'folded') thmrefs

  ;

\end{rail}

\begin{descr}

\item [$tagged~name~args$ and $untagged~name$] add and remove $tags$ of some

  theorem.  Tags may be any list of strings that serve as comment for some

  tools (e.g.\ $\LEMMANAME$ causes the tag ``$lemma$'' to be added to the

  result).  The first string is considered the tag name, the rest its

  arguments.  Note that untag removes any tags of the same name.

\item [$THEN~n~a$ and $COMP~n~a$] compose rules.  $THEN$ resolves with the

  $n$-th premise of $a$; the $COMP$ version skips the automatic lifting

  process that is normally intended (cf.\ \texttt{RS} and \texttt{COMP} in

  \cite[\S5]{isabelle-ref}).

\item [$where~\vec x = \vec t$] perform named instantiation of schematic

  variables occurring in a theorem.  Unlike instantiation tactics such as

  $rule_tac$ (see \S\ref{sec:tactic-commands}), actual schematic variables

  have to be specified (e.g.\ $\Var{x@3}$).

\item [$unfolded~\vec a$ and $folded~\vec a$] expand and fold back again the

  given meta-level definitions throughout a rule.

\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 [$elim_format$] turns a destruction rule into elimination rule format;

  see also the ML function \texttt{make\_elim} (see \cite{isabelle-ref}).

\item [$no_vars$] replaces schematic variables by free ones; this is mainly

  for tuning output of pretty printed theorems.

\item [$exported$] lifts a local result out of the current proof context,

  generalizing all fixed variables and discharging all assumptions.  Note that

  proper incremental export is already done as part of the basic Isar

  machinery.  This attribute is mainly for experimentation.

\end{descr}

\section{Tactic emulations}\label{sec:tactics}

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 are read and type-checked according to a subgoal of

  the current dynamic goal state, rather than the static proof context!  In

  particular, locally fixed variables and term abbreviations may not be

  included in the term specifications.  Thus schematic variables are left to

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

\end{warn}

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

named $foo_tac$.

\indexisarmeth{rule-tac}\indexisarmeth{erule-tac}

\indexisarmeth{drule-tac}\indexisarmeth{frule-tac}

\indexisarmeth{cut-tac}\indexisarmeth{thin-tac}

\indexisarmeth{subgoal-tac}\indexisarmeth{rename-tac}

\indexisarmeth{rotate-tac}\indexisarmeth{tactic}

\begin{matharray}{rcl}

  rule_tac^* & : & \isarmeth \\

  erule_tac^* & : & \isarmeth \\

  drule_tac^* & : & \isarmeth \\

  frule_tac^* & : & \isarmeth \\

  cut_tac^* & : & \isarmeth \\

  thin_tac^* & : & \isarmeth \\

  subgoal_tac^* & : & \isarmeth \\

  rename_tac^* & : & \isarmeth \\

  rotate_tac^* & : & \isarmeth \\

  tactic^* & : & \isarmeth \\

\end{matharray}

\railalias{ruletac}{rule\_tac}

\railterm{ruletac}

\railalias{eruletac}{erule\_tac}

\railterm{eruletac}

\railalias{druletac}{drule\_tac}

\railterm{druletac}

\railalias{fruletac}{frule\_tac}

\railterm{fruletac}

\railalias{cuttac}{cut\_tac}

\railterm{cuttac}

\railalias{thintac}{thin\_tac}

\railterm{thintac}

\railalias{subgoaltac}{subgoal\_tac}

\railterm{subgoaltac}

\railalias{renametac}{rename\_tac}

\railterm{renametac}

\railalias{rotatetac}{rotate\_tac}

\railterm{rotatetac}

\begin{rail}

  ( ruletac | eruletac | druletac | fruletac | cuttac | thintac ) goalspec?

  ( insts thmref | thmrefs )

  ;

  subgoaltac goalspec? (prop +)

  ;

  renametac goalspec? (name +)

  ;

  rotatetac goalspec? int?

  ;

  'tactic' text

  ;

  insts: ((name '=' term) + 'and') 'in'

  ;

\end{rail}

\begin{descr}

\item [$rule_tac$ etc.] do resolution of rules with explicit instantiation.

  This works the same way as the ML tactics \texttt{res_inst_tac} etc. (see

  \cite[\S3]{isabelle-ref}).

  Note that multiple rules may be only given there is no instantiation.  Then

  $rule_tac$ is the same as \texttt{resolve_tac} in ML (see

  \cite[\S3]{isabelle-ref}).

\item [$cut_tac$] inserts facts into the proof state as assumption of a

  subgoal, see also \texttt{cut_facts_tac} in \cite[\S3]{isabelle-ref}.  Note

  that the scope of schmatic variables is spread over the main goal statement.

  Instantiations may be given as well, see also ML tactic

  \texttt{cut_inst_tac} in \cite[\S3]{isabelle-ref}.

\item [$thin_tac~\phi$] deletes the specified assumption from a subgoal; note

  that $\phi$ may contain schematic variables.  See also \texttt{thin_tac} in

  \cite[\S3]{isabelle-ref}.

\item [$subgoal_tac~\phi$] adds $\phi$ as an assumption to a subgoal.  See

  also \texttt{subgoal_tac} and \texttt{subgoals_tac} in

  \cite[\S3]{isabelle-ref}.

\item [$rename_tac~\vec x$] renames parameters of a goal according to the list

  $\vec x$, which refers to the \emph{suffix} of variables.

\item [$rotate_tac~n$] rotates the assumptions of a goal by $n$ positions:

  from right to left if $n$ is positive, and from left to right if $n$ is

  negative; the default value is $1$.  See also \texttt{rotate_tac} in

  \cite[\S3]{isabelle-ref}.

\item [$tactic~text$] produces a proof method from any ML text of type

  \texttt{tactic}.  Apart from the usual ML environment and the current

  implicit theory context, the ML code may refer to the following locally

  bound values:

%%FIXME ttbox produces too much trailing space (why?)

{\footnotesize\begin{verbatim}

val ctxt  : Proof.context

val facts : thm list

val thm   : string -> thm

val thms  : string -> thm list

\end{verbatim}}

  Here \texttt{ctxt} refers to the current proof context, \texttt{facts}

  indicates any current facts for forward-chaining, and

  \texttt{thm}~/~\texttt{thms} retrieve named facts (including global

  theorems) from the context.

\end{descr}

\section{The Simplifier}\label{sec:simplifier}

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

\indexisarmeth{simp}\indexisarmeth{simp-all}

\begin{matharray}{rcl}

  simp & : & \isarmeth \\

  simp_all & : & \isarmeth \\

\end{matharray}

\railalias{simpall}{simp\_all}

\railterm{simpall}

\railalias{noasm}{no\_asm}

\railterm{noasm}

\railalias{noasmsimp}{no\_asm\_simp}

\railterm{noasmsimp}

\railalias{noasmuse}{no\_asm\_use}

\railterm{noasmuse}

\indexouternonterm{simpmod}

\begin{rail}

  ('simp' | simpall) ('!' ?) opt? (simpmod * )

  ;

  opt: '(' (noasm | noasmsimp | noasmuse) ')'

  ;

  simpmod: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') |

    'split' (() | 'add' | 'del')) ':' thmrefs

  ;

\end{rail}

\begin{descr}

\item [$simp$] invokes Isabelle's 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 [$simp_all$] is similar to $simp$, but acts on all goals.

\end{descr}

By default, the Simplifier methods are based on \texttt{asm_full_simp_tac}

internally \cite[\S10]{isabelle-ref}, which means that assumptions are both

simplified as well as used in simplifying the conclusion.  In structured

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

of the actual goal are involved, additional facts may inserted via explicit

forward-chaining (using $\THEN$, $\FROMNAME$ etc.).  The full context of

assumptions is only included if the ``$!$'' (bang) argument is given, which

should be used with some care, though.

Additional Simplifier options may be specified to tune the behavior even

further: $(no_asm)$ means assumptions are ignored completely (cf.\

\texttt{simp_tac}), $(no_asm_simp)$ means assumptions are used in the

simplification of the conclusion but are not themselves simplified (cf.\

\texttt{asm_simp_tac}), and $(no_asm_use)$ means assumptions are simplified

but are not used in the simplification of each other or the conclusion (cf.

\texttt{full_simp_tac}).

\medskip

The Splitter package is usually configured to work as part of the Simplifier.

The effect of repeatedly applying \texttt{split_tac} can be simulated by

$(simp~only\colon~split\colon~\vec a)$.  There is also a separate $split$

method available for single-step case splitting, see \S\ref{sec:basic-eq}.

\subsection{Declaring rules}

\indexisarcmd{print-simpset}

\indexisaratt{simp}\indexisaratt{split}\indexisaratt{cong}

\begin{matharray}{rcl}

  print_simpset^* & : & \isarkeep{theory~|~proof} \\

  simp & : & \isaratt \\

  cong & : & \isaratt \\

  split & : & \isaratt \\

\end{matharray}

\begin{rail}

  ('simp' | 'cong' | 'split') (() | 'add' | 'del')

  ;

\end{rail}

\begin{descr}

\item [$print_simpset$] prints the collection of rules declared to the

  Simplifier, which is also known as ``simpset'' internally

  \cite{isabelle-ref}.  This is a diagnostic command; $undo$ does not apply.

\item [$simp$] declares simplification rules.

\item [$cong$] declares congruence rules.

\item [$split$] declares case split rules.

\end{descr}

\subsection{Forward simplification}

\indexisaratt{simplified}

\begin{matharray}{rcl}

  simplified & : & \isaratt \\

\end{matharray}

\begin{rail}

  'simplified' opt?

  ;

  opt: '(' (noasm | noasmsimp | noasmuse) ')'

  ;

\end{rail}

\begin{descr}

\item [$simplified$] causes a theorem to be simplified according to the

  current Simplifier context (there are no separate arguments for declaring

  additional rules).  By default the result is fully simplified, including

  assumptions and conclusion.  The options $no_asm$ etc.\ restrict the

  Simplifier in the same way as the for the $simp$ method (see

  \S\ref{sec:simp}).

  The $simplified$ operation should be used only very rarely, usually for

  experimentation only.

\end{descr}

\section{Basic equational reasoning}\label{sec:basic-eq}

\indexisarmeth{subst}\indexisarmeth{hypsubst}\indexisarmeth{split}\indexisaratt{symmetric}

\begin{matharray}{rcl}

  subst & : & \isarmeth \\

  hypsubst^* & : & \isarmeth \\

  split & : & \isarmeth \\

  symmetric & : & \isaratt \\

\end{matharray}

\begin{rail}

  'subst' thmref

  ;

  'split' ('(' 'asm' ')')? thmrefs

  ;

\end{rail}

These methods and attributes provide basic facilities for equational reasoning

that are intended for specialized applications only.  Normally, single step

reasoning would be performed by calculation (see \S\ref{sec:calculation}),

while the Simplifier is the canonical tool for automated normalization (see

\S\ref{sec:simplifier}).

\begin{descr}

\item [$subst~thm$] performs a single substitution step using rule $thm$,

  which may be either a meta or object equality.

\item [$hypsubst$] performs substitution using some assumption.

\item [$split~thms$] performs single-step case splitting using rules $thms$.

  By default, splitting is performed in the conclusion of a goal; the $asm$

  option indicates to operate on assumptions instead.

  Note that the $simp$ method already involves repeated application of split

  rules as declared in the current context (see \S\ref{sec:simp}).

\item [$symmetric$] applies the symmetry rule of meta or object equality.

\end{descr}

\section{The Classical Reasoner}\label{sec:classical}

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

\indexisarmeth{rule}\indexisarmeth{intro}

\indexisarmeth{elim}\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 [$rule$] as offered by the classical reasoner is a refinement over the

  primitive one (see \S\ref{sec:pure-meth-att}).  In 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}).

  This is made the default method for basic proof steps, such as $\PROOFNAME$

  and ``$\DDOT$'' (two dots), see also \S\ref{sec:proof-steps} and

  \S\ref{sec:pure-meth-att}.

\item [$intro$ and $elim$] repeatedly refine some goal by intro- or

  elim-resolution, after having inserted any facts.  Omitting the arguments

  refers to any suitable rules declared in 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 actual sub-proof.

\item [$contradiction$] solves some goal by contradiction, deriving any result

  from both $\neg A$ and $A$.  Facts, which are guaranteed to participate, may

  appear in either order.

\end{descr}

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

\indexisarmeth{blast}\indexisarmeth{fast}\indexisarmeth{slow}

\indexisarmeth{best}\indexisarmeth{safe}\indexisarmeth{clarify}

\begin{matharray}{rcl}

  blast & : & \isarmeth \\

  fast & : & \isarmeth \\

  slow & : & \isarmeth \\

  best & : & \isarmeth \\

  safe & : & \isarmeth \\

  clarify & : & \isarmeth \\

\end{matharray}

\indexouternonterm{clamod}

\begin{rail}

  'blast' ('!' ?) nat? (clamod * )

  ;

  ('fast' | 'slow' | 'best' | 'safe' | 'clarify') ('!' ?) (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 [$fast$, $slow$, $best$, $safe$, and $clarify$] refer to the generic

  classical reasoner.  See \texttt{fast_tac}, \texttt{slow_tac},

  \texttt{best_tac}, \texttt{safe_tac}, and \texttt{clarify_tac} in

  \cite[\S11]{isabelle-ref} for more information.

\end{descr}

Any of above methods support additional modifiers of the context of classical

rules.  Their semantics is analogous to the attributes given in

\S\ref{sec:classical-mod}.  Facts provided by forward chaining are

inserted\footnote{These methods usually cannot make proper use of actual rules

  inserted that way, though.} into the goal before doing the search.  The

``!''~argument causes the full context of assumptions to be included as well.

This is slightly less hazardous than for the Simplifier (see

\S\ref{sec:simp}).

\subsection{Combined automated methods}\label{sec:clasimp}

\indexisarmeth{auto}\indexisarmeth{force}\indexisarmeth{clarsimp}

\indexisarmeth{fastsimp}\indexisarmeth{slowsimp}\indexisarmeth{bestsimp}

\begin{matharray}{rcl}

  auto & : & \isarmeth \\

  force & : & \isarmeth \\

  clarsimp & : & \isarmeth \\

  fastsimp & : & \isarmeth \\

  slowsimp & : & \isarmeth \\

  bestsimp & : & \isarmeth \\

\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{descr}

\item [$auto$, $force$, $clarsimp$, $fastsimp$, $slowsimp$, and $bestsimp$]

  provide access to Isabelle's combined simplification and classical reasoning

  tactics.  These correspond to \texttt{auto_tac}, \texttt{force_tac},

  \texttt{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 \S\ref{sec:simp} and

  \S\ref{sec:classical-auto}.  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 ``!''~argument causes the full context of assumptions to be

  included as well.

\end{descr}

\subsection{Declaring rules}\label{sec:classical-mod}

\indexisarcmd{print-claset}

\indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest}

\indexisaratt{iff}\indexisaratt{rule}

\begin{matharray}{rcl}

  print_claset^* & : & \isarkeep{theory~|~proof} \\

  intro & : & \isaratt \\

  elim & : & \isaratt \\

  dest & : & \isaratt \\

  rule & : & \isaratt \\

  iff & : & \isaratt \\

\end{matharray}

\begin{rail}

  ('intro' | 'elim' | 'dest') ('!' | () | '?')

  ;

  'rule' 'del'

  ;

  'iff' (((() | 'add') '?'?) | 'del')

  ;

\end{rail}

\begin{descr}

\item [$print_claset$] prints the collection of rules declared to the

  Classical Reasoner, which is also known as ``simpset'' internally

  \cite{isabelle-ref}.  This is a diagnostic command; $undo$ does not apply.

\item [$intro$, $elim$, and $dest$] declare introduction, elimination, and

  destruction rules, respectively.  By default, rules are considered as

  \emph{unsafe} (i.e.\ not applied blindly without backtracking), while a

  single ``!'' classifies as \emph{safe}, and ``?'' as \emph{extra} (i.e.\ not

  applied in the search-oriented automated methods, but only in single-step

  methods such as $rule$).

\item [$rule~del$] deletes introduction, elimination, or destruction rules from

  the context.

\item [$iff$] declares a (possibly conditional) ``safe'' rule to the context in

  several ways.   The rule is declared as a rewrite rule to the Simplifier. 

  Furthermore, it is 

  declared in several ways (depending on its structure) to the Classical 

  Reasoner for aggressive use, which would normally be indicated by ``!'').

  If the rule is an equivalence, the two corresponding implications are 

  declared as introduction and destruction rules. Otherwise, a warning is issued

  and if the rule is an inequality, the corresponding negation elimination rule

  is declared, else the rule itself is declared as an introduction rule.

  The ``?'' version of $iff$ declares ``extra'' Classical Reasoner rules only,

  and omits the Simplifier declaration.  Thus the declaration does not have

  any effect on automated proof tools, but only on simple methods such as

  $rule$ (see \S\ref{sec:misc-methods}).

\end{descr}

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