theory Generic

 imports Main

 begin

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

 section {* Configuration options *}

 text {*

   Isabelle/Pure maintains a record of named configuration options

   within the theory or proof context, with values of type @{ML_type

   bool}, @{ML_type int}, or @{ML_type string}.  Tools may declare

   options in ML, and then refer to these values (relative to the

   context).  Thus global reference variables are easily avoided.  The

   user may change the value of a configuration option by means of an

   associated attribute of the same name.  This form of context

   declaration works particularly well with commands such as @{command

   "declare"} or @{command "using"}.

   For historical reasons, some tools cannot take the full proof

   context into account and merely refer to the background theory.

   This is accommodated by configuration options being declared as

   ``global'', which may not be changed within a local context.

   \begin{matharray}{rcll}

     @{command_def "print_configs"} & : & @{text "context \<rightarrow>"} \\

   \end{matharray}

   \begin{rail}

     name ('=' ('true' | 'false' | int | name))?

   \end{rail}

   \begin{description}

   \item @{command "print_configs"} prints the available configuration

   options, with names, types, and current values.

   \item @{text "name = value"} as an attribute expression modifies the

   named option, with the syntax of the value depending on the option's

   type.  For @{ML_type bool} the default value is @{text true}.  Any

   attempt to change a global option in a local context is ignored.

   \end{description}

 *}

 section {* Basic proof tools *}

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

 text {*

   \begin{matharray}{rcl}

     @{method_def unfold} & : & @{text method} \\

     @{method_def fold} & : & @{text method} \\

     @{method_def insert} & : & @{text method} \\[0.5ex]

     @{method_def erule}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def drule}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def frule}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def succeed} & : & @{text method} \\

     @{method_def fail} & : & @{text method} \\

   \end{matharray}

   \begin{rail}

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

     ;

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

     ;

   \end{rail}

   \begin{description}

   \item @{method unfold}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{method fold}~@{text

   "a\<^sub>1 \<dots> a\<^sub>n"} expand (or fold back) the given definitions throughout

   all goals; any chained facts provided are inserted into the goal and

   subject to rewriting as well.

   \item @{method insert}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} inserts theorems as facts

   into all goals of the proof state.  Note that current facts

   indicated for forward chaining are ignored.

   \item @{method erule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, @{method

   drule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, and @{method frule}~@{text

   "a\<^sub>1 \<dots> a\<^sub>n"} are similar to the basic @{method rule}

   method (see \secref{sec:pure-meth-att}), but apply rules by

   elim-resolution, destruct-resolution, and forward-resolution,

   respectively \cite{isabelle-implementation}.  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 @{method rule} method, with forward chaining of current

   facts.

   \item @{method succeed} yields a single (unchanged) result; it is

   the identity of the ``@{text ","}'' method combinator (cf.\

   \secref{sec:proof-meth}).

   \item @{method fail} yields an empty result sequence; it is the

   identity of the ``@{text "|"}'' method combinator (cf.\

   \secref{sec:proof-meth}).

   \end{description}

   \begin{matharray}{rcl}

     @{attribute_def tagged} & : & @{text attribute} \\

     @{attribute_def untagged} & : & @{text attribute} \\[0.5ex]

     @{attribute_def THEN} & : & @{text attribute} \\

     @{attribute_def COMP} & : & @{text attribute} \\[0.5ex]

     @{attribute_def unfolded} & : & @{text attribute} \\

     @{attribute_def folded} & : & @{text attribute} \\[0.5ex]

     @{attribute_def rotated} & : & @{text attribute} \\

     @{attribute_def (Pure) elim_format} & : & @{text attribute} \\

     @{attribute_def standard}@{text "\<^sup>*"} & : & @{text attribute} \\

     @{attribute_def no_vars}@{text "\<^sup>*"} & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

     'tagged' name name

     ;

     'untagged' name

     ;

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

     ;

     ('unfolded' | 'folded') thmrefs

     ;

     'rotated' ( int )?

   \end{rail}

   \begin{description}

   \item @{attribute tagged}~@{text "name value"} and @{attribute

   untagged}~@{text name} add and remove \emph{tags} of some theorem.

   Tags may be any list of string pairs that serve as formal comment.

   The first string is considered the tag name, the second its value.

   Note that @{attribute untagged} removes any tags of the same name.

   \item @{attribute THEN}~@{text a} and @{attribute COMP}~@{text a}

   compose rules by resolution.  @{attribute THEN} resolves with the

   first premise of @{text a} (an alternative position may be also

   specified); the @{attribute COMP} version skips the automatic

   lifting process that is normally intended (cf.\ @{ML "op RS"} and

   @{ML "op COMP"} in \cite{isabelle-implementation}).

   \item @{attribute unfolded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{attribute

   folded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} expand and fold back again the given

   definitions throughout a rule.

   \item @{attribute rotated}~@{text n} rotate the premises of a

   theorem by @{text n} (default 1).

   \item @{attribute (Pure) elim_format} turns a destruction rule into

   elimination rule format, by resolving with the rule @{prop "PROP A \<Longrightarrow>

   (PROP A \<Longrightarrow> PROP B) \<Longrightarrow> PROP B"}.

   Note that the Classical Reasoner (\secref{sec:classical}) provides

   its own version of this operation.

   \item @{attribute 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 @{attribute no_vars} replaces schematic variables by free

   ones; this is mainly for tuning output of pretty printed theorems.

   \end{description}

 *}

 subsection {* Low-level equational reasoning *}

 text {*

   \begin{matharray}{rcl}

     @{method_def subst} & : & @{text method} \\

     @{method_def hypsubst} & : & @{text method} \\

     @{method_def split} & : & @{text method} \\

   \end{matharray}

   \begin{rail}

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

     ;

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

     ;

   \end{rail}

   These methods provide low-level facilities for equational reasoning

   that are intended for specialized applications only.  Normally,

   single step calculations would be performed in a structured text

   (see also \secref{sec:calculation}), while the Simplifier methods

   provide the canonical way for automated normalization (see

   \secref{sec:simplifier}).

   \begin{description}

   \item @{method subst}~@{text eq} performs a single substitution step

   using rule @{text eq}, which may be either a meta or object

   equality.

   \item @{method subst}~@{text "(asm) eq"} substitutes in an

   assumption.

   \item @{method subst}~@{text "(i \<dots> j) eq"} performs several

   substitutions in the conclusion. The numbers @{text i} to @{text 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 @{text "(a + b) + (c + d)"} there are three positions

   where commutativity of @{text "+"} is applicable: 1 refers to @{text

   "a + b"}, 2 to the whole term, and 3 to @{text "c + d"}.

   If the positions in the list @{text "(i \<dots> j)"} are non-overlapping

   (e.g.\ @{text "(2 3)"} in @{text "(a + b) + (c + d)"}) you may

   assume all substitutions are performed simultaneously.  Otherwise

   the behaviour of @{text subst} is not specified.

   \item @{method subst}~@{text "(asm) (i \<dots> j) eq"} performs the

   substitutions in the assumptions. The positions refer to the

   assumptions in order from left to right.  For example, given in a

   goal of the form @{text "P (a + b) \<Longrightarrow> P (c + d) \<Longrightarrow> \<dots>"}, position 1 of

   commutativity of @{text "+"} is the subterm @{text "a + b"} and

   position 2 is the subterm @{text "c + d"}.

   \item @{method hypsubst} performs substitution using some

   assumption; this only works for equations of the form @{text "x =

   t"} where @{text x} is a free or bound variable.

   \item @{method split}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} performs single-step case

   splitting using the given rules.  By default, splitting is performed

   in the conclusion of a goal; the @{text "(asm)"} option indicates to

   operate on assumptions instead.

   Note that the @{method simp} method already involves repeated

   application of split rules as declared in the current context.

   \end{description}

 *}

 subsection {* Further tactic emulations \label{sec:tactics} *}

 text {*

   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 @{text 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 @{attribute "where"} (see

   \secref{sec:pure-meth-att}).

   \begin{matharray}{rcl}

     @{method_def rule_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def erule_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def drule_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def frule_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def cut_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def thin_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def subgoal_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def rename_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def rotate_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def tactic}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def raw_tactic}@{text "\<^sup>*"} & : & @{text method} \\

   \end{matharray}

   \begin{rail}

     ( 'rule\_tac' | 'erule\_tac' | 'drule\_tac' | 'frule\_tac' | 'cut\_tac' | 'thin\_tac' ) goalspec?

     ( insts thmref | thmrefs )

     ;

     'subgoal\_tac' goalspec? (prop +)

     ;

     'rename\_tac' goalspec? (name +)

     ;

     'rotate\_tac' goalspec? int?

     ;

     ('tactic' | 'raw_tactic') text

     ;

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

     ;

   \end{rail}

 \begin{description}

   \item @{method rule_tac} etc. do resolution of rules with explicit

   instantiation.  This works the same way as the ML tactics @{ML

   res_inst_tac} etc. (see \cite{isabelle-implementation})

   Multiple rules may be only given if there is no instantiation; then

   @{method rule_tac} is the same as @{ML resolve_tac} in ML (see

   \cite{isabelle-implementation}).

   \item @{method cut_tac} inserts facts into the proof state as

   assumption of a subgoal, see also @{ML Tactic.cut_facts_tac} in

   \cite{isabelle-implementation}.  Note that the scope of schematic

   variables is spread over the main goal statement.  Instantiations

   may be given as well, see also ML tactic @{ML cut_inst_tac} in

   \cite{isabelle-implementation}.

   \item @{method thin_tac}~@{text \<phi>} deletes the specified assumption

   from a subgoal; note that @{text \<phi>} may contain schematic variables.

   See also @{ML thin_tac} in \cite{isabelle-implementation}.

   \item @{method subgoal_tac}~@{text \<phi>} adds @{text \<phi>} as an

   assumption to a subgoal.  See also @{ML subgoal_tac} and @{ML

   subgoals_tac} in \cite{isabelle-implementation}.

   \item @{method rename_tac}~@{text "x\<^sub>1 \<dots> x\<^sub>n"} renames parameters of a

   goal according to the list @{text "x\<^sub>1, \<dots>, x\<^sub>n"}, which refers to the

   \emph{suffix} of variables.

   \item @{method rotate_tac}~@{text n} rotates the assumptions of a

   goal by @{text n} positions: from right to left if @{text n} is

   positive, and from left to right if @{text n} is negative; the

   default value is 1.  See also @{ML rotate_tac} in

   \cite{isabelle-implementation}.

   \item @{method tactic}~@{text "text"} produces a proof method from

   any ML text of type @{ML_type tactic}.  Apart from the usual ML

   environment and the current proof context, the ML code may refer to

   the locally bound values @{ML_text facts}, which indicates any

   current facts used for forward-chaining.

   \item @{method raw_tactic} is similar to @{method tactic}, but

   presents the goal state in its raw internal form, where simultaneous

   subgoals appear as conjunction of the logical framework instead of

   the usual split into several subgoals.  While feature this is useful

   for debugging of complex method definitions, it should not never

   appear in production theories.

   \end{description}

 *}

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

 subsection {* Simplification methods *}

 text {*

   \begin{matharray}{rcl}

     @{method_def simp} & : & @{text method} \\

     @{method_def simp_all} & : & @{text method} \\

   \end{matharray}

   \indexouternonterm{simpmod}

   \begin{rail}

     ('simp' | 'simp\_all') opt? (simpmod *)

     ;

     opt: '(' ('no\_asm' | 'no\_asm\_simp' | 'no\_asm\_use' | 'asm\_lr' ) ')'

     ;

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

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

     ;

   \end{rail}

   \begin{description}

   \item @{method simp} invokes the Simplifier, after declaring

   additional rules according to the arguments given.  Note that the

   \railtterm{only} modifier first removes all other rewrite rules,

   congruences, and looper tactics (including splits), and then behaves

   like \railtterm{add}.

   \medskip The \railtterm{cong} modifiers add or delete Simplifier

   congruence rules (see also \cite{isabelle-ref}), the default is to

   add.

   \medskip The \railtterm{split} modifiers add or delete rules for the

   Splitter (see also \cite{isabelle-ref}), the default is to add.

   This works only if the Simplifier method has been properly setup to

   include the Splitter (all major object logics such HOL, HOLCF, FOL,

   ZF do this already).

   \item @{method simp_all} is similar to @{method simp}, but acts on

   all goals (backwards from the last to the first one).

   \end{description}

   By default the Simplifier methods take local assumptions fully into

   account, using equational assumptions in the subsequent

   normalization process, or simplifying assumptions themselves (cf.\

   @{ML asm_full_simp_tac} in \cite{isabelle-ref}).  In structured

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

   local premises of the actual goal are involved, additional facts may

   be inserted via explicit forward-chaining (via @{command "then"},

   @{command "from"}, @{command "using"} etc.).

   Additional Simplifier options may be specified to tune the behavior

   further (mostly for unstructured scripts with many accidental local

   facts): ``@{text "(no_asm)"}'' means assumptions are ignored

   completely (cf.\ @{ML simp_tac}), ``@{text "(no_asm_simp)"}'' means

   assumptions are used in the simplification of the conclusion but are

   not themselves simplified (cf.\ @{ML asm_simp_tac}), and ``@{text

   "(no_asm_use)"}'' means assumptions are simplified but are not used

   in the simplification of each other or the conclusion (cf.\ @{ML

   full_simp_tac}).  For compatibility reasons, there is also an option

   ``@{text "(asm_lr)"}'', which means that an assumption is only used

   for simplifying assumptions which are to the right of it (cf.\ @{ML

   asm_lr_simp_tac}).

   The configuration option @{text "depth_limit"} limits the number of

   recursive invocations of the simplifier during conditional

   rewriting.

   \medskip The Splitter package is usually configured to work as part

   of the Simplifier.  The effect of repeatedly applying @{ML

   split_tac} can be simulated by ``@{text "(simp only: split:

   a\<^sub>1 \<dots> a\<^sub>n)"}''.  There is also a separate @{text split}

   method available for single-step case splitting.

 *}

 subsection {* Declaring rules *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "print_simpset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{attribute_def simp} & : & @{text attribute} \\

     @{attribute_def cong} & : & @{text attribute} \\

     @{attribute_def split} & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

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

     ;

   \end{rail}

   \begin{description}

   \item @{command "print_simpset"} prints the collection of rules

   declared to the Simplifier, which is also known as ``simpset''

   internally \cite{isabelle-ref}.

   \item @{attribute simp} declares simplification rules.

   \item @{attribute cong} declares congruence rules.

   \item @{attribute split} declares case split rules.

   \end{description}

 *}

 subsection {* Simplification procedures *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "simproc_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     simproc & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

     'simproc\_setup' name '(' (term + '|') ')' '=' text \\ ('identifier' (nameref+))?

     ;

     'simproc' (('add' ':')? | 'del' ':') (name+)

     ;

   \end{rail}

   \begin{description}

   \item @{command "simproc_setup"} defines a named simplification

   procedure that is invoked by the Simplifier whenever any of the

   given term patterns match the current redex.  The implementation,

   which is provided as ML source text, needs to be of type @{ML_type

   "morphism -> simpset -> cterm -> thm option"}, where the @{ML_type

   cterm} represents the current redex @{text r} and the result is

   supposed to be some proven rewrite rule @{text "r \<equiv> r'"} (or a

   generalized version), or @{ML NONE} to indicate failure.  The

   @{ML_type simpset} argument holds the full context of the current

   Simplifier invocation, including the actual Isar proof context.  The

   @{ML_type morphism} informs about the difference of the original

   compilation context wrt.\ the one of the actual application later

   on.  The optional @{keyword "identifier"} specifies theorems that

   represent the logical content of the abstract theory of this

   simproc.

   Morphisms and identifiers are only relevant for simprocs that are

   defined within a local target context, e.g.\ in a locale.

   \item @{text "simproc add: name"} and @{text "simproc del: name"}

   add or delete named simprocs to the current Simplifier context.  The

   default is to add a simproc.  Note that @{command "simproc_setup"}

   already adds the new simproc to the subsequent context.

   \end{description}

 *}

 subsection {* Forward simplification *}

 text {*

   \begin{matharray}{rcl}

     @{attribute_def simplified} & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

     'simplified' opt? thmrefs?

     ;

     opt: '(' ('no\_asm' | 'no\_asm\_simp' | 'no\_asm\_use') ')'

     ;

   \end{rail}

   \begin{description}

   \item @{attribute simplified}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} causes a theorem to

   be simplified, either by exactly the specified rules @{text "a\<^sub>1, \<dots>,

   a\<^sub>n"}, or the implicit Simplifier context if no arguments are given.

   The result is fully simplified by default, including assumptions and

   conclusion; the options @{text no_asm} etc.\ tune the Simplifier in

   the same way as the for the @{text 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 @{text

   simplified} attribute should be only rarely required under normal

   circumstances.

   \end{description}

 *}

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

 subsection {* Basic methods *}

 text {*

   \begin{matharray}{rcl}

     @{method_def rule} & : & @{text method} \\

     @{method_def contradiction} & : & @{text method} \\

     @{method_def intro} & : & @{text method} \\

     @{method_def elim} & : & @{text method} \\

   \end{matharray}

   \begin{rail}

     ('rule' | 'intro' | 'elim') thmrefs?

     ;

   \end{rail}

   \begin{description}

   \item @{method rule} as offered by the Classical Reasoner is a

   refinement over the primitive one (see \secref{sec:pure-meth-att}).

   Both versions essentially work the same, but the classical version

   observes the classical rule context in addition to that of

   Isabelle/Pure.

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

   classical rules (even if these would qualify as intuitionistic

   ones), but only few declarations to the rule context of

   Isabelle/Pure (\secref{sec:pure-meth-att}).

   \item @{method contradiction} solves some goal by contradiction,

   deriving any result from both @{text "\<not> A"} and @{text A}.  Chained

   facts, which are guaranteed to participate, may appear in either

   order.

   \item @{method intro} and @{method elim} repeatedly refine some goal

   by intro- or elim-resolution, after having inserted any chained

   facts.  Exactly the rules given as arguments are taken into account;

   this allows fine-tuned decomposition of a proof problem, in contrast

   to common automated tools.

   \end{description}

 *}

 subsection {* Automated methods *}

 text {*

   \begin{matharray}{rcl}

     @{method_def blast} & : & @{text method} \\

     @{method_def fast} & : & @{text method} \\

     @{method_def slow} & : & @{text method} \\

     @{method_def best} & : & @{text method} \\

     @{method_def safe} & : & @{text method} \\

     @{method_def clarify} & : & @{text method} \\

   \end{matharray}

   \indexouternonterm{clamod}

   \begin{rail}

     'blast' nat? (clamod *)

     ;

     ('fast' | 'slow' | 'best' | 'safe' | 'clarify') (clamod *)

     ;

     clamod: (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' thmrefs

     ;

   \end{rail}

   \begin{description}

   \item @{method blast} refers to the classical tableau prover (see

   @{ML blast_tac} in \cite{isabelle-ref}).  The optional argument

   specifies a user-supplied search bound (default 20).

   \item @{method fast}, @{method slow}, @{method best}, @{method

   safe}, and @{method clarify} refer to the generic classical

   reasoner.  See @{ML fast_tac}, @{ML slow_tac}, @{ML best_tac}, @{ML

   safe_tac}, and @{ML clarify_tac} in \cite{isabelle-ref} for more

   information.

   \end{description}

   Any of the above methods support additional modifiers of the context

   of classical rules.  Their semantics is analogous to the attributes

   given before.  Facts provided by forward chaining are inserted into

   the goal before commencing proof search.

 *}

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

 text {*

   \begin{matharray}{rcl}

     @{method_def auto} & : & @{text method} \\

     @{method_def force} & : & @{text method} \\

     @{method_def clarsimp} & : & @{text method} \\

     @{method_def fastsimp} & : & @{text method} \\

     @{method_def slowsimp} & : & @{text method} \\

     @{method_def bestsimp} & : & @{text method} \\

   \end{matharray}

   \indexouternonterm{clasimpmod}

   \begin{rail}

     'auto' (nat nat)? (clasimpmod *)

     ;

     ('force' | 'clarsimp' | 'fastsimp' | 'slowsimp' | 'bestsimp') (clasimpmod *)

     ;

     clasimpmod: ('simp' (() | 'add' | 'del' | 'only') |

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

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

       (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' thmrefs

   \end{rail}

   \begin{description}

   \item @{method auto}, @{method force}, @{method clarsimp}, @{method

   fastsimp}, @{method slowsimp}, and @{method bestsimp} provide access

   to Isabelle's combined simplification and classical reasoning

   tactics.  These correspond to @{ML auto_tac}, @{ML force_tac}, @{ML

   clarsimp_tac}, and Classical Reasoner tactics with the Simplifier

   added as wrapper, see \cite{isabelle-ref} for more information.  The

   modifier arguments correspond to those given in

   \secref{sec:simplifier} and \secref{sec:classical}.  Just note that

   the ones related to the Simplifier are prefixed by \railtterm{simp}

   here.

   Facts provided by forward chaining are inserted into the goal before

   doing the search.

   \end{description}

 *}

 subsection {* Declaring rules *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "print_claset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{attribute_def intro} & : & @{text attribute} \\

     @{attribute_def elim} & : & @{text attribute} \\

     @{attribute_def dest} & : & @{text attribute} \\

     @{attribute_def rule} & : & @{text attribute} \\

     @{attribute_def iff} & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

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

     ;

     'rule' 'del'

     ;

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

     ;

   \end{rail}

   \begin{description}

   \item @{command "print_claset"} prints the collection of rules

   declared to the Classical Reasoner, which is also known as

   ``claset'' internally \cite{isabelle-ref}.

   \item @{attribute intro}, @{attribute elim}, and @{attribute dest}

   declare introduction, elimination, and destruction rules,

   respectively.  By default, rules are considered as \emph{unsafe}

   (i.e.\ not applied blindly without backtracking), while ``@{text

   "!"}'' classifies as \emph{safe}.  Rule declarations marked by

   ``@{text "?"}'' coincide with those of Isabelle/Pure, cf.\

   \secref{sec:pure-meth-att} (i.e.\ are only applied in single steps

   of the @{method rule} method).  The optional natural number

   specifies an explicit weight argument, which is ignored by automated

   tools, but determines the search order of single rule steps.

   \item @{attribute rule}~@{text del} deletes introduction,

   elimination, or destruction rules from the context.

   \item @{attribute 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 @{text

   "\<not>"}-elimination internally).

   The ``@{text "?"}'' version of @{attribute iff} declares rules to

   the Isabelle/Pure context only, and omits the Simplifier

   declaration.

   \end{description}

 *}

 subsection {* Classical operations *}

 text {*

   \begin{matharray}{rcl}

     @{attribute_def swapped} & : & @{text attribute} \\

   \end{matharray}

   \begin{description}

   \item @{attribute swapped} turns an introduction rule into an

   elimination, by resolving with the classical swap principle @{text

   "(\<not> B \<Longrightarrow> A) \<Longrightarrow> (\<not> A \<Longrightarrow> B)"}.

   \end{description}

 *}

 section {* Object-logic setup \label{sec:object-logic} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "judgment"} & : & @{text "theory \<rightarrow> theory"} \\

     @{method_def atomize} & : & @{text method} \\

     @{attribute_def atomize} & : & @{text attribute} \\

     @{attribute_def rule_format} & : & @{text attribute} \\

     @{attribute_def rulify} & : & @{text attribute} \\

   \end{matharray}

   The very starting point for any Isabelle object-logic is a ``truth

   judgment'' that links object-level statements to the meta-logic

   (with its minimal language of @{text prop} that covers universal

   quantification @{text "\<And>"} and implication @{text "\<Longrightarrow>"}).

   Common object-logics are sufficiently expressive to internalize rule

   statements over @{text "\<And>"} and @{text "\<Longrightarrow>"} within their own

   language.  This is useful in certain situations where a rule needs

   to be viewed as an atomic statement from the meta-level perspective,

   e.g.\ @{text "\<And>x. x \<in> A \<Longrightarrow> P x"} versus @{text "\<forall>x \<in> A. P x"}.

   From the following language elements, only the @{method atomize}

   method and @{attribute rule_format} attribute are occasionally

   required by end-users, the rest is for those who need to setup their

   own object-logic.  In the latter case existing formulations of

   Isabelle/FOL or Isabelle/HOL may be taken as realistic examples.

   Generic tools may refer to the information provided by object-logic

   declarations internally.

   \begin{rail}

     'judgment' constdecl

     ;

     'atomize' ('(' 'full' ')')?

     ;

     'rule\_format' ('(' 'noasm' ')')?

     ;

   \end{rail}

   \begin{description}

   \item @{command "judgment"}~@{text "c :: \<sigma> (mx)"} declares constant

   @{text c} as the truth judgment of the current object-logic.  Its

   type @{text \<sigma>} should specify a coercion of the category of

   object-level propositions to @{text prop} of the Pure meta-logic;

   the mixfix annotation @{text "(mx)"} would typically just link the

   object language (internally of syntactic category @{text logic})

   with that of @{text prop}.  Only one @{command "judgment"}

   declaration may be given in any theory development.

   \item @{method atomize} (as a method) rewrites any non-atomic

   premises of a sub-goal, using the meta-level equations declared via

   @{attribute atomize} (as an attribute) beforehand.  As a result,

   heavily nested goals become amenable to fundamental operations such

   as resolution (cf.\ the @{method rule} method).  Giving the ``@{text

   "(full)"}'' option here means to turn the whole subgoal into an

   object-statement (if possible), including the outermost parameters

   and assumptions as well.

   A typical collection of @{attribute atomize} rules for a particular

   object-logic would provide an internalization for each of the

   connectives of @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"}.

   Meta-level conjunction should be covered as well (this is

   particularly important for locales, see \secref{sec:locale}).

   \item @{attribute rule_format} rewrites a theorem by the equalities

   declared as @{attribute rulify} rules in the current object-logic.

   By default, the result is fully normalized, including assumptions

   and conclusions at any depth.  The @{text "(no_asm)"} option

   restricts the transformation to the conclusion of a rule.

   In common object-logics (HOL, FOL, ZF), the effect of @{attribute

   rule_format} is to replace (bounded) universal quantification

   (@{text "\<forall>"}) and implication (@{text "\<longrightarrow>"}) by the corresponding

   rule statements over @{text "\<And>"} and @{text "\<Longrightarrow>"}.

   \end{description}

 *}

 end