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

 \section{Theory specification commands}

 \subsection{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 \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 +)

   ;

   'instance' (nameref ('<' | subseteq) nameref | nameref '::' arity)

   ;

 \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 (being bound as

   $c_class{\dtt}intro$); this rule is employed by method $intro_classes$ to

   support instantiation proofs of this class.

   The ``axioms'' are stored as theorems according to the given name

   specifications, adding the class name $c$ as name space prefix; the same

   facts are also stored collectively as $c_class{\dtt}axioms$.

 \item [$\INSTANCE~c@1 \subseteq c@2$ and $\INSTANCE~t :: (\vec s)s$] 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 $\PROOFNAME$ etc.).

   In particular, instantiation of trivial (syntactic) classes may be performed

   by a single ``$\DDOT$'' proof step.

 \end{descr}

 \subsection{Locales and local contexts}\label{sec:locale}

 Locales are named local contexts, consisting of a list of declaration elements

 that are modeled after the Isar proof context commands (cf.\

 \S\ref{sec:proof-context}).

 \subsubsection{Localized commands}

 Existing locales may be augmented later on by adding new facts.  Note that the

 actual context definition may not be changed!  Several theory commands that

 produce facts in some way are available in ``localized'' versions, referring

 to a named locale instead of the global theory context.

 \indexouternonterm{locale}

 \begin{rail}

   locale: '(' 'in' name ')'

   ;

 \end{rail}

 Emerging facts of localized commands are stored in two versions, both in the

 target locale and the theory (after export).  The latter view produces a

 qualified binding, using the locale name as a name space prefix.

 For example, ``$\LEMMAS~(\IN~loc)~a = \vec b$'' retrieves facts $\vec b$ from

 the locale context of $loc$ and augments its body by an appropriate

 ``$\isarkeyword{notes}$'' element (see below).  The exported view of $a$,

 after discharging the locale context, is stored as $loc{.}a$ within the global

 theory.  A localized goal ``$\LEMMANAME~(\IN~loc)~a:~\phi$'' works similarly,

 only that the fact emerges through the subsequent proof, which may refer to

 the full infrastructure of the locale context (covering local parameters with

 typing and concrete syntax, assumptions, definitions etc.).  Most notably,

 fact declarations of the locale are active during the proof as well (e.g.\ 

 local $simp$ rules).

 As a general principle, results exported from a locale context acquire

 additional premises according to the specification.  Usually this is only a

 single predicate according to the standard ``closed'' view of locale

 specifications.

 \subsubsection{Locale specifications}

 \indexisarcmd{locale}\indexisarcmd{print-locale}\indexisarcmd{print-locales}

 \begin{matharray}{rcl}

   \isarcmd{locale} & : & \isarkeep{theory} \\

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

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

 \end{matharray}

 \indexouternonterm{contextexpr}\indexouternonterm{contextelem}

 \railalias{printlocale}{print\_locale}

 \railterm{printlocale}

 \begin{rail}

   'locale' ('(open)')? name ('=' localeexpr)?

   ;

   printlocale '!'? localeexpr

   ;

   localeexpr: ((contextexpr '+' (contextelem+)) | contextexpr | (contextelem+))

   ;

   contextexpr: nameref | '(' contextexpr ')' |

   (contextexpr (name mixfix? +)) | (contextexpr + '+')

   ;

   contextelem: fixes | constrains | assumes | defines | notes | includes

   ;

   fixes: 'fixes' (name ('::' type)? structmixfix? + 'and')

   ;

   constrains: 'constrains' (name '::' type + 'and')

   ;

   assumes: 'assumes' (thmdecl? props + 'and')

   ;

   defines: 'defines' (thmdecl? prop proppat? + 'and')

   ;

   notes: 'notes' (thmdef? thmrefs + 'and')

   ;

   includes: 'includes' contextexpr

   ;

 \end{rail}

 \begin{descr}

 \item [$\LOCALE~loc~=~import~+~body$] defines a new locale $loc$ as a context

   consisting of a certain view of existing locales ($import$) plus some

   additional elements ($body$).  Both $import$ and $body$ are optional; the

   degenerate form $\LOCALE~loc$ defines an empty locale, which may still be

   useful to collect declarations of facts later on.  Type-inference on locale

   expressions automatically takes care of the most general typing that the

   combined context elements may acquire.

   The $import$ consists of a structured context expression, consisting of

   references to existing locales, renamed contexts, or merged contexts.

   Renaming uses positional notation: $c~\vec x$ means that (a prefix of) the

   fixed parameters of context $c$ are named according to $\vec x$; a

   ``\texttt{_}'' (underscore) \indexisarthm{_@\texttt{_}} means to skip that

   position.  Renaming by default deletes existing syntax.  Optionally,

   new syntax may by specified with a mixfix annotation.  Note that the

   special syntax declared with ``$(structure)$'' (see below) is

   neither deleted nor can it be changed.

   Merging proceeds from left-to-right, suppressing any duplicates stemming

   from different paths through the import hierarchy.

   The $body$ consists of basic context elements, further context expressions

   may be included as well.

   \begin{descr}

   \item [$\FIXES{~x::\tau~(mx)}$] declares a local parameter of type $\tau$

     and mixfix annotation $mx$ (both are optional).  The special syntax

     declaration ``$(structure)$'' means that $x$ may be referenced

     implicitly in this context.

   \item [$\CONSTRAINS{~x::\tau}$] introduces a type constraint $\tau$

     on the local parameter $x$.

   \item [$\ASSUMES{a}{\vec\phi}$] introduces local premises, similar to

     $\ASSUMENAME$ within a proof (cf.\ \S\ref{sec:proof-context}).

   \item [$\DEFINES{a}{x \equiv t}$] defines a previously declared parameter.

     This is close to $\DEFNAME$ within a proof (cf.\

     \S\ref{sec:proof-context}), but $\DEFINESNAME$ takes an equational

     proposition instead of variable-term pair.  The left-hand side of the

     equation may have additional arguments, e.g.\ ``$\DEFINES{}{f~\vec x

       \equiv t}$''.

   \item [$\NOTES{a}{\vec b}$] reconsiders facts within a local context.  Most

     notably, this may include arbitrary declarations in any attribute

     specifications included here, e.g.\ a local $simp$ rule.

   \item [$\INCLUDES{c}$] copies the specified context in a statically scoped

     manner.  Only available in the long goal format of \S\ref{sec:goals}.

     In contrast, the initial $import$ specification of a locale expression

     maintains a dynamic relation to the locales being referenced (benefiting

     from any later fact declarations in the obvious manner).

   \end{descr}

   Note that ``$\IS{p}$'' patterns given in the syntax of $\ASSUMESNAME$ and

   $\DEFINESNAME$ above are illegal in locale definitions.  In the long goal

   format of \S\ref{sec:goals}, term bindings may be included as expected,

   though.

   \medskip By default, locale specifications are ``closed up'' by turning the

   given text into a predicate definition $loc_axioms$ and deriving the

   original assumptions as local lemmas (modulo local definitions).  The

   predicate statement covers only the newly specified assumptions, omitting

   the content of included locale expressions.  The full cumulative view is

   only provided on export, involving another predicate $loc$ that refers to

   the complete specification text.

   In any case, the predicate arguments are those locale parameters that

   actually occur in the respective piece of text.  Also note that these

   predicates operate at the meta-level in theory, but the locale packages

   attempts to internalize statements according to the object-logic setup

   (e.g.\ replacing $\Forall$ by $\forall$, and $\Imp$ by $\imp$ in HOL; see

   also \S\ref{sec:object-logic}).  Separate introduction rules

   $loc_axioms.intro$ and $loc.intro$ are declared as well.

   The $(open)$ option of a locale specification prevents both the current

   $loc_axioms$ and cumulative $loc$ predicate constructions.  Predicates are

   also omitted for empty specification texts.

 \item [$\isarkeyword{print_locale}~import~+~body$] prints the specified locale

   expression in a flattened form.  The notable special case

   $\isarkeyword{print_locale}~loc$ just prints the contents of the named

   locale, but keep in mind that type-inference will normalize type variables

   according to the usual alphabetical order.  The command omits

   $\isarkeyword{notes}$ elements by default.  Use

   $\isarkeyword{print_locale}!$ to get them included.

 \item [$\isarkeyword{print_locales}$] prints the names of all locales of the

   current theory.

 \end{descr}

 \subsubsection{Interpretation of locales}

 Locale expressions (more precisely, \emph{context expressions}) may be

 instantiated, and the instantiated facts added to the current context.

 This requires a proof of the instantiated specification and is called

 \emph{locale interpretation}.  Interpretation is possible in theories

 and locales

 (command $\isarcmd{interpretation}$) and also in proof contexts

 ($\isarcmd{interpret}$).

 \indexisarcmd{interpretation}\indexisarcmd{interpret}

 \indexisarcmd{print-interps}

 \begin{matharray}{rcl}

   \isarcmd{interpretation} & : & \isartrans{theory}{proof(prove)} \\

   \isarcmd{interpret} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\

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

 \end{matharray}

 \indexouternonterm{interp}

 \railalias{printinterps}{print\_interps}

 \railterm{printinterps}

 \begin{rail}

   'interpretation' (interp | name ('<' | subseteq) contextexp)

   ;

   'interpret' interp

   ;

   printinterps '!'? name

   ;

   interp: thmdecl? contextexpr ('[' (inst+) ']')?

   ;

 \end{rail}

 \begin{descr}

 \item [$\isarcmd{interpretation}~expr~insts$]

   The first form of $\isarcmd{interpretation}$ interprets $expr$

   in the theory.  The instantiation is given as a list of

   terms $insts$ and is positional.

   All parameters must receive an instantiation term --- with the

   exception of defined parameters.  These are, if omitted, derived

   from the defining equation and other instantiations.  Use ``\_'' to

   omit an instantiation term.  Free variables are automatically

   generalized.

   The command generates proof obligations for the instantiated

   specifications (assumes and defines elements).  Once these are

   discharged by the user, instantiated facts are added to the theory in

   a post-processing phase.

   The command is aware of interpretations already active in the

   theory.  No proof obligations are generated for those, neither is

   post-processing applied to their facts.  This avoids duplication of

   interpreted facts, in particular.  Note that, in the case of a

   locale with import, parts of the interpretation may already be

   active.  The command will only generate proof obligations and add

   facts for new parts.

   The context expression may be preceded by a name and/or attributes.

   These take effect in the post-processing of facts.  The name is used

   to prefix fact names, for example to avoid accidental hiding of

   other facts.  Attributes are applied after attributes of the

   interpreted facts.

   Adding facts to locales has the

   effect of adding interpreted facts to the theory for all active

   interpretations also.  That is, interpretations dynamically

   participate in any facts added to locales.

 \item [$\isarcmd{interpretation}~name~\subseteq~expr$]

   This form of the command interprets $expr$ in the locale $name$.  It

   requires a proof that the specification of $name$ implies the

   specification of $expr$.  As in the localized version of the theorem

   command, the proof is in the context of $name$.  After the proof

   obligation has been dischared, the facts of $expr$

   become part of locale $name$ as \emph{derived} context elements and

   are available when the context $name$ is subsequently entered.

   Note that, like import, this is dynamic: facts added to a locale

   part of $expr$ after interpretation become also available in

   $name$.  Like facts

   of renamed context elements, facts obtained by interpretation may be

   accessed by prefixing with the parameter renaming (where the parameters

   are separated by `\_').

   Unlike interpretation in theories, instantiation is confined to the

   renaming of parameters, which may be specified as part of the context

   expression $expr$.  Using defined parameters in $name$ one may

   achieve an effect similar to instantiation, though.

   Only specification fragments of $expr$ that are not already part of

   $name$ (be it imported, derived or a derived fragment of the import)

   are considered by interpretation.  This enables circular

   interpretations.

   If interpretations of $name$ exist in the current theory, the

   command adds interpretations for $expr$ as well, with the same

   prefix and attributes, although only for fragments of $expr$ that

   are not interpreted in the theory already.

 \item [$\isarcmd{interpret}~expr~insts$]

   interprets $expr$ in the proof context and is otherwise similar to

   interpretation in theories.  Free variables in instantiations are not

   generalized, however.

 \item [$\isarcmd{print_interps}~loc$]

   prints the interpretations of a particular locale $loc$ that are

   active in the current context, either theory or proof context.  The

   exclamation point argument causes triggers printing of

   \emph{witness} theorems justifying interpretations.  These are

   normally omitted from the output.

 \end{descr}

 \begin{warn}

   Since attributes are applied to interpreted theorems, interpretation

   may modify the current simpset and claset.  Take this into

   account when choosing attributes for local theorems.

 \end{warn}

 \begin{warn}

   An interpretation in a theory may subsume previous interpretations.

   This happens if the same specification fragment is interpreted twice

   and the instantiation of the second interpretation is more general

   than the interpretation of the first.  A warning

   is issued, since it is likely that these could have been generalized

   in the first place.  The locale package does not attempt to remove

   subsumed interpretations.  This situation is normally harmless, but

   note that $blast$ gets confused by the presence of multiple axclass

   instances of a rule.

 \end{warn}

 \section{Derived proof schemes}

 \subsection{Generalized elimination}\label{sec:obtain}

 \indexisarcmd{obtain}\indexisarcmd{guess}

 \begin{matharray}{rcl}

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

   \isarcmd{guess}^* & : & \isartrans{proof(state)}{proof(prove)} \\

 \end{matharray}

 Generalized elimination means that additional elements with certain properties

 may be introduced in the current context, by virtue of a locally proven

 ``soundness statement''.  Technically speaking, 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') 'where' (props + 'and')

   ;

   'guess' (vars + '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 \HAVE{}{\All{thesis} (\All{\vec x} \vec\phi \Imp thesis) \Imp thesis} \\

   \quad \PROOF{succeed} \\

   \qquad \FIX{thesis} \\

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

   \qquad \THUS{}{thesis} \\

   \quad\qquad \APPLY{-} \\

   \quad\qquad \USING{\vec b}~~\langle proof\rangle \\

   \quad \QED{} \\

   \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 ``$\BY{simp}$'' or ``$\BY{blast}$''.

 Accordingly, the ``$that$'' reduction above is declared as simplification and

 introduction rule.

 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

 introduction) of object-level existential 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 a genuine assumption.

 \medskip

 The improper variant $\isarkeyword{guess}$ is similar to $\OBTAINNAME$, but

 derives the obtained statement from the course of reasoning!  The proof starts

 with a fixed goal $thesis$.  The subsequent proof may refine this to anything

 of the form like $\All{\vec x} \vec\phi \Imp thesis$, but must not introduce

 new subgoals.  The final goal state is then used as reduction rule for the

 obtain scheme described above.  Obtained parameters $\vec x$ are marked as

 internal by default, which prevents the proof context from being polluted by

 ad-hoc variables.  The variable names and type constraints given as arguments

 for $\isarkeyword{guess}$ specify a prefix of obtained parameters explicitly

 in the text.

 It is important to note that the facts introduced by $\OBTAINNAME$ and

 $\isarkeyword{guess}$ may not be polymorphic: any type-variables occurring

 here are fixed in the present context!

 \subsection{Calculational reasoning}\label{sec:calculation}

 \indexisarcmd{also}\indexisarcmd{finally}

 \indexisarcmd{moreover}\indexisarcmd{ultimately}

 \indexisarcmd{print-trans-rules}

 \indexisaratt{trans}\indexisaratt{sym}\indexisaratt{symmetric}

 \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 \\

   sym & : & \isaratt \\

   symmetric & : & \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 implicit term abbreviation ``$\dots$'' has its canonical

 application with calculational proofs.  It refers to the argument of the

 preceding statement. (The argument of a curried infix expression happens to be

 its right-hand side.)

 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{We

   suppress internal bookkeeping such as proper handling of block-structure.}

 \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') ('(' thmrefs ')')?

   ;

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

   ;

 \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 (for calculational commands $\ALSO$ and $\FINALLY$) and symmetry rules

   (for the $symmetric$ operation and single step elimination patters) of the

   current context.

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

 \item [$sym$] declares symmetry rules.

 \item [$symmetric$] resolves a theorem with some rule declared as $sym$ in the

   current context.  For example, ``$\ASSUME{[symmetric]}{x = y}$'' produces a

   swapped fact derived from that assumption.

   In structured proof texts it is often more appropriate to use an explicit

   single-step elimination proof, such as ``$\ASSUME{}{x = y}~\HENCE{}{y =

     x}~\DDOT$''.  The very same rules known to $symmetric$ are declared as

   $elim?$ as well.

 \end{descr}

 \section{Proof tools}

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

 \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 \\

   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 (or fold back again) the

   given meta-level definitions throughout all goals; any chained 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 here.

   Note that these methods are improper ones, mainly serving for

   experimentation and tactic script emulation.  Different modes of basic rule

   application are usually expressed in Isar at the proof language level,

   rather than via implicit proof state manipulations.  For example, a proper

   single-step elimination would be done using the plain $rule$ 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{unfolded}\indexisaratt{folded}

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

 \indexisaratt{no-vars}

 \begin{matharray}{rcl}

   tagged & : & \isaratt \\

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

   THEN & : & \isaratt \\

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

   unfolded & : & \isaratt \\

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

   elim_format & : & \isaratt \\

   standard^* & : & \isaratt \\

   no_vars^* & : & \isaratt \\

 \end{matharray}

 \begin{rail}

   'tagged' (nameref+)

   ;

   'untagged' name

   ;

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

   ;

   ('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 [$THEN~a$ and $COMP~a$] compose rules by resolution.  $THEN$ resolves

   with the first premise of $a$ (an alternative position may be also

   specified); the $COMP$ version skips the automatic lifting process that is

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

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

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

   given meta-level definitions throughout a rule.

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

   by resolving with the rule $\PROP A \Imp (\PROP A \Imp \PROP B) \Imp \PROP

   B$.

   Note that the Classical Reasoner (\S\ref{sec:classical}) provides its own

   version of this operation.

 \item [$standard$] puts a theorem into the standard form of object-rules at

   the outermost theory level.  Note that this operation violates the local

   proof context (including active locales).

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

   for tuning output of pretty printed theorems.

 \end{descr}

 \subsection{Further 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 refer to universally quantified parameters of

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

   abbreviations of a (static) Isar context.

 \end{warn}

 Tactic emulation methods, unlike their ML counterparts, admit

 simultaneous instantiation from both dynamic and static contexts.  If

 names occur in both contexts goal parameters hide locally fixed

 variables.  Likewise, schematic variables refer to term abbreviations,

 if present in the static context.  Otherwise the schematic variable is

 interpreted as a schematic variable and left to be solved by unification

 with certain parts of the subgoal.

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

 named $foo_tac$.  Note also that variable names occurring on left hand sides

 of instantiations must be preceded by a question mark if they coincide with

 a keyword or contain dots.

 This is consistent with the attribute $where$ (see \S\ref{sec:pure-meth-att}).

 \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}).

   Multiple rules may be only given if 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 schematic 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:

 {\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}

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

 \subsubsection{Simplification methods}

 \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}

 \railalias{asmlr}{asm\_lr}

 \railterm{asmlr}

 \indexouternonterm{simpmod}

 \begin{rail}

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

   ;

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

   ;

   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 (backwards from

   the last to the first one).

 \end{descr}

 By default the Simplifier methods take local assumptions fully into account,

 using equational assumptions in the subsequent normalization process, or

 simplifying assumptions themselves (cf.\ \texttt{asm_full_simp_tac} in

 \cite[\S10]{isabelle-ref}).  In structured proofs this is usually quite well

 behaved in practice: just the local premises of the actual goal are involved,

 additional facts may be inserted via explicit forward-chaining (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 further

 (mostly for unstructured scripts with many accidental local facts):

 ``$(no_asm)$'' 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}).

 For compatibility reasons, there is also an option ``$(asm_lr)$'',

 which means that an assumption is only used for simplifying assumptions

 which are to the right of it (cf.\ \texttt{asm_lr_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.

 \subsubsection{Declaring rules}

 \indexisarcmd{print-simpset}

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

 \begin{matharray}{rcl}

   \isarcmd{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 [$\isarcmd{print_simpset}$] 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}

 \subsubsection{Forward simplification}

 \indexisaratt{simplified}

 \begin{matharray}{rcl}

   simplified & : & \isaratt \\

 \end{matharray}

 \begin{rail}

   'simplified' opt? thmrefs?

   ;

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

   ;

 \end{rail}

 \begin{descr}

 \item [$simplified~\vec a$] causes a theorem to be simplified, either by

   exactly the specified rules $\vec a$, or the implicit Simplifier context if

   no arguments are given.  The result is fully simplified by default,

   including assumptions and conclusion; the options $no_asm$ etc.\ tune the

   Simplifier in the same way as the for the $simp$ method.

   Note that forward simplification restricts the simplifier to its most basic

   operation of term rewriting; solver and looper tactics \cite{isabelle-ref}

   are \emph{not} involved here.  The $simplified$ attribute should be only

   rarely required under normal circumstances.

 \end{descr}

 \subsubsection{Low-level equational reasoning}

 \indexisarmeth{subst}\indexisarmeth{hypsubst}\indexisarmeth{split}

 \begin{matharray}{rcl}

   subst^* & : & \isarmeth \\

   hypsubst^* & : & \isarmeth \\

   split^* & : & \isarmeth \\

 \end{matharray}

 \begin{rail}

   'subst' ('(' 'asm' ')')? ('(' (nat+) ')')? thmref

   ;

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

   ;

 \end{rail}

 These methods provide low-level facilities for equational reasoning that are

 intended for specialized applications only.  Normally, single step

 calculations would be performed in a structured text (see also

 \S\ref{sec:calculation}), while the Simplifier methods provide the canonical

 way for automated normalization (see \S\ref{sec:simplifier}).

 \begin{descr}

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

   may be either a meta or object equality.

 \item [$subst~(asm)~eq$] substitutes in an assumption.

 \item [$subst~(i \dots j)~eq$] performs several substitutions in the

 conclusion. The numbers $i$ to $j$ indicate the positions to substitute at.

 Positions are ordered from the top of the term tree moving down from left to

 right. For example, in $(a+b)+(c+d)$ there are three positions where

 commutativity of $+$ is applicable: 1 refers to the whole term, 2 to $a+b$

 and 3 to $c+d$. If the positions in the list $(i \dots j)$ are

 non-overlapping (e.g. $(2~3)$ in $(a+b)+(c+d)$) you may assume all

 substitutions are performed simultaneously. Otherwise the behaviour of

 $subst$ is not specified.

 \item [$subst~(asm)~(i \dots j)~eq$] performs the substitutions in the

 assumptions. Positions $1 \dots i@1$ refer

 to assumption 1, positions $i@1+1 \dots i@2$ to assumption 2, and so on.

 \item [$hypsubst$] performs substitution using some assumption; this only

   works for equations of the form $x = t$ where $x$ is a free or bound

   variable.

 \item [$split~\vec a$] 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.

 \end{descr}

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

 \subsubsection{Basic methods}

 \indexisarmeth{rule}\indexisarmeth{default}\indexisarmeth{contradiction}

 \indexisarmeth{intro}\indexisarmeth{elim}

 \begin{matharray}{rcl}

   rule & : & \isarmeth \\

   contradiction & : & \isarmeth \\

   intro & : & \isarmeth \\

   elim & : & \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}).  Both versions essentially

   work the same, but the classical version observes the classical rule context

   in addition to that of Isabelle/Pure.

   Common object logics (HOL, ZF, etc.) declare a rich collection of classical

   rules (even if these would qualify as intuitionistic ones), but only few

   declarations to the rule context of Isabelle/Pure

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

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

   from both $\neg A$ and $A$.  Chained facts, which are guaranteed to

   participate, may appear in either order.

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

   elim-resolution, after having inserted any chained facts.  Exactly the rules

   given as arguments are taken into account; this allows fine-tuned

   decomposition of a proof problem, in contrast to common automated tools.

 \end{descr}

 \subsubsection{Automated methods}

 \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 the above methods support additional modifiers of the context of

 classical rules.  Their semantics is analogous to the attributes given before.

 Facts provided by forward chaining are inserted into the goal before

 commencing proof search.  The ``!''~argument causes the full context of

 assumptions to be included as well.

 \subsubsection{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:simplifier} and

   \S\ref{sec:classical}.  Just note that the ones related to the Simplifier

   are prefixed by \railtterm{simp} here.

   Facts provided by forward chaining are inserted into the goal before doing

   the search.  The ``!''~argument causes the full context of assumptions to be

   included as well.

 \end{descr}

 \subsubsection{Declaring rules}

 \indexisarcmd{print-claset}

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

 \indexisaratt{iff}\indexisaratt{rule}

 \begin{matharray}{rcl}

   \isarcmd{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 [$\isarcmd{print_claset}$] 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}.  Rule declarations marked by ``?''

   coincide with those of Isabelle/Pure, cf.\ \S\ref{sec:pure-meth-att} (i.e.\

   are only applied in single steps of the $rule$ method).

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

   the context.

 \item [$iff$] declares logical equivalences to the Simplifier and the

   Classical reasoner at the same time.  Non-conditional rules result in a

   ``safe'' introduction and elimination pair; conditional ones are considered

   ``unsafe''.  Rules with negative conclusion are automatically inverted

   (using $\neg$ elimination internally).

   The ``?'' version of $iff$ declares rules to the Isabelle/Pure context only,

   and omits the Simplifier declaration.

 \end{descr}

 \subsubsection{Classical operations}

 \indexisaratt{elim-format}\indexisaratt{swapped}

 \begin{matharray}{rcl}

   elim_format & : & \isaratt \\

   swapped & : & \isaratt \\

 \end{matharray}

 \begin{descr}

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

   this operation is similar to the the intuitionistic version

   (\S\ref{sec:misc-meth-att}), but each premise of the resulting rule acquires

   an additional local fact of the negated main thesis; according to the

   classical principle $(\neg A \Imp A) \Imp A$.

 \item [$swapped$] turns an introduction rule into an elimination, by resolving

   with the classical swap principle $(\neg B \Imp A) \Imp (\neg A \Imp B)$.

 \end{descr}

 \subsection{Proof by cases and induction}\label{sec:cases-induct}

 \subsubsection{Rule contexts}

 \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 that of ``$\FIX{\vec x}~\ASSUME{c}{\vec\phi}$''.  Term

 bindings may be covered as well, such as $\Var{case}$ for the intended

 conclusion.

 Normally the ``terminology'' of a case value (i.e.\ the parameters $\vec x$)

 are marked as hidden.  Using the explicit form ``$\CASE{(c~\vec x)}$'' enables

 proof writers to choose their own names for the subsequent proof text.

 \medskip

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

 to peek at the current goal state, which is generally considered

 non-observable in Isar.  The text of the cases basically emerge from standard

 elimination or induction rules, which in turn are derived from previous theory

 specifications in a canonical way (say from $\isarkeyword{inductive}$

 definitions).

 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' (caseref | '(' caseref ((name | underscore) +) ')')

   ;

   caseref: nameref attributes?

   ;

   casenames (name +)

   ;

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

   ;

   'consumes' nat?

   ;

 \end{rail}

 \begin{descr}

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

   \vec \phi$, as provided by an appropriate proof method (such as $cases$ and

   $induct$, see \S\ref{sec:cases-induct-meth}).  The command ``$\CASE{(c~\vec

     x)}$'' abbreviates ``$\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:cases-induct-meth}).

 \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:cases-induct-meth}).  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:hol-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:cases-induct-att}.

 \end{descr}

 \subsubsection{Proof methods}\label{sec:cases-induct-meth}

 \indexisarmeth{cases}\indexisarmeth{induct}

 \begin{matharray}{rcl}

   cases & : & \isarmeth \\

   induct & : & \isarmeth \\

 \end{matharray}

 The $cases$ and $induct$ methods provide a uniform interface to case analysis

 and induction over datatypes, inductive sets, and recursive functions.  The

 corresponding rules may be specified and instantiated in a casual manner.

 Furthermore, these methods provide named local contexts that may be invoked

 via the $\CASENAME$ proof command within the subsequent proof text.  This

 accommodates compact proof texts even when reasoning about large

 specifications.

 \begin{rail}

   'cases' spec

   ;

   'induct' spec

   ;

   spec: open? args rule?

   ;

   open: '(' 'open' ')'

   ;

   args: (insts * 'and')

   ;

   rule: ('type' | 'set') ':' nameref | 'rule' ':' thmref

   ;

 \end{rail}

 \begin{descr}

 \item [$cases~insts~R$] applies method $rule$ with an appropriate case

   distinction theorem, instantiated to the subjects $insts$.  Symbolic case

   names are bound according to the rule's local contexts.

   The rule is determined as follows, according to the facts and arguments

   passed to the $cases$ method:

   \begin{matharray}{llll}

     \Text{facts}    &       & \Text{arguments} & \Text{rule} \\\hline

                     & cases &           & \Text{classical case split} \\

                     & cases & t         & \Text{datatype exhaustion (type of $t$)} \\

     \edrv a \in A   & cases & \dots     & \Text{inductive set elimination (of $A$)} \\

     \dots           & cases & \dots ~ R & \Text{explicit rule $R$} \\

   \end{matharray}

   Several instantiations may be given, referring to the \emph{suffix} of

   premises of the case rule; within each premise, the \emph{prefix} of

   variables is instantiated.  In most situations, only a single term needs to

   be specified; this refers to the first variable of the last premise (it is

   usually the same for all cases).

   The ``$(open)$'' option causes the parameters of the new local contexts to

   be exposed to the current proof context.  Thus local variables stemming from

   distant parts of the theory development may be introduced in an implicit

   manner, which can be quite confusing to the reader.  Furthermore, this

   option may cause unwanted hiding of existing local variables, resulting in

   less robust proof texts.

 \item [$induct~insts~R$] is analogous to the $cases$ method, but refers to

   induction rules, which are determined as follows:

   \begin{matharray}{llll}

     \Text{facts}    &        & \Text{arguments} & \Text{rule} \\\hline

                     & induct & P ~ x ~ \dots & \Text{datatype induction (type of $x$)} \\

     \edrv x \in A   & induct & \dots         & \Text{set induction (of $A$)} \\

     \dots           & induct & \dots ~ R     & \Text{explicit rule $R$} \\

   \end{matharray}

   Several instantiations may be given, each referring to some part of a mutual

   inductive definition or datatype --- only related partial induction rules

   may be used together, though.  Any of the lists of terms $P, x, \dots$

   refers to the \emph{suffix} of variables present in the induction rule.

   This enables the writer to specify only induction variables, or both

   predicates and variables, for example.

   The ``$(open)$'' option works the same way as for $cases$.

 \end{descr}

 Above methods produce named local contexts, as determined by the instantiated

 rule as specified in the text.  Beyond that, the $induct$ method guesses

 further instantiations from the goal specification itself.  Any persisting

 unresolved schematic variables of the resulting rule will render the the

 corresponding case invalid.  The term binding $\Var{case}$\indexisarvar{case}

 for the conclusion will be provided with each case, provided that term is

 fully specified.

 The $\isarkeyword{print_cases}$ command prints all named cases present in the

 current proof state.

 \medskip

 It is important to note that there is a fundamental difference of the $cases$

 and $induct$ methods in handling of non-atomic goal statements: $cases$ just

 applies a certain rule in backward fashion, splitting the result into new

 goals with the local contexts being augmented in a purely monotonic manner.

 In contrast, $induct$ passes the full goal statement through the

 ``recursive'' course involved in the induction.  Thus the original statement

 is basically replaced by separate copies, corresponding to the induction

 hypotheses and conclusion; the original goal context is no longer available.

 This behavior allows \emph{strengthened induction predicates} to be expressed

 concisely as meta-level rule statements, i.e.\ $\All{\vec x} \vec\phi \Imp

 \psi$ to indicate ``variable'' parameters $\vec x$ and ``recursive''

 assumptions $\vec\phi$. Note that ``$\isarcmd{case}~c$'' already performs

 ``$\FIX{\vec x}$''.  Also note that local definitions may be expressed as

 $\All{\vec x} n \equiv t[\vec x] \Imp \phi[n]$, with induction over $n$.

 In induction proofs, local assumptions introduced by cases are split into two

 different kinds: $hyps$ stemming from the rule and $prems$ from the goal

 statement.  This is reflected in the extracted cases accordingly, so invoking

 ``$\isarcmd{case}~c$'' will provide separate facts $c\mathord.hyps$ and

 $c\mathord.prems$, as well as fact $c$ to hold the all-inclusive list.

 \medskip

 Facts presented to either method are consumed according to the number of

 ``major premises'' of the rule involved (see also \S\ref{sec:cases-induct}),

 which is usually $0$ for plain cases and induction rules of datatypes etc.\

 and $1$ for rules of inductive sets and the like.  The remaining facts are

 inserted into the goal verbatim before the actual $cases$ or $induct$ rule is

 applied (thus facts may be even passed through an induction).

 \subsubsection{Declaring rules}\label{sec:cases-induct-att}

 \indexisarcmd{print-induct-rules}\indexisaratt{cases}\indexisaratt{induct}

 \begin{matharray}{rcl}

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

   cases & : & \isaratt \\

   induct & : & \isaratt \\

 \end{matharray}

 \begin{rail}

   'cases' spec

   ;

   'induct' spec

   ;

   spec: ('type' | 'set') ':' nameref

   ;

 \end{rail}

 \begin{descr}

 \item [$\isarkeyword{print_induct_rules}$] prints cases and induct rules for

   sets and types of the current context.

 \item [$cases$ and $induct$] (as attributes) augment the corresponding context

   of rules for reasoning about inductive sets and types, using the

   corresponding methods of the same name.  Certain definitional packages of

   object-logics usually declare emerging cases and induction rules as

   expected, so users rarely need to intervene.

   Manual rule declarations usually include the the $case_names$ and $ps$

   attributes to adjust names of cases and parameters of a rule (see

   \S\ref{sec:cases-induct}); the $consumes$ declaration is taken care of

   automatically: $consumes~0$ is specified for ``type'' rules and $consumes~1$

   for ``set'' rules.

 \end{descr}

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