theory Generic

 imports Base Main

 begin

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

 section {* Configuration options \label{sec:config} *}

 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}, @{ML_type real}, 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"} like

   this:

 *}

 declare [[show_main_goal = false]]

 notepad

 begin

   note [[show_main_goal = true]]

 end

 text {* 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}

   @{rail "

     @{syntax name} ('=' ('true' | 'false' | @{syntax int} | @{syntax float} | @{syntax name}))?

   "}

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

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

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

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

   \end{matharray}

   @{rail "

     (@@{method fold} | @@{method unfold} | @@{method insert}) @{syntax thmrefs}

     ;

     (@@{method erule} | @@{method drule} | @@{method frule})

       ('(' @{syntax nat} ')')? @{syntax thmrefs}

     ;

     (@@{method intro} | @@{method elim}) @{syntax thmrefs}?

   "}

   \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 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.

   \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}

   @{rail "

     @@{attribute tagged} @{syntax name} @{syntax name}

     ;

     @@{attribute untagged} @{syntax name}

     ;

     (@@{attribute THEN} | @@{attribute COMP}) ('[' @{syntax nat} ']')? @{syntax thmref}

     ;

     (@@{attribute unfolded} | @@{attribute folded}) @{syntax thmrefs}

     ;

     @@{attribute rotated} @{syntax int}?

   "}

   \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}

   @{rail "

     @@{method subst} ('(' 'asm' ')')? \\ ('(' (@{syntax nat}+) ')')? @{syntax thmref}

     ;

     @@{method split} ('(' 'asm' ')')? @{syntax thmrefs}

   "}

   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}

   @{rail "

     (@@{method rule_tac} | @@{method erule_tac} | @@{method drule_tac} |

       @@{method frule_tac} | @@{method cut_tac} | @@{method thin_tac}) @{syntax goal_spec}? \\

     ( dynamic_insts @'in' @{syntax thmref} | @{syntax thmrefs} )

     ;

     @@{method subgoal_tac} @{syntax goal_spec}? (@{syntax prop} +)

     ;

     @@{method rename_tac} @{syntax goal_spec}? (@{syntax name} +)

     ;

     @@{method rotate_tac} @{syntax goal_spec}? @{syntax int}?

     ;

     (@@{method tactic} | @@{method raw_tactic}) @{syntax text}

     ;

     dynamic_insts: ((@{syntax name} '=' @{syntax term}) + @'and')

   "}

 \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}

   @{rail "

     (@@{method simp} | @@{method simp_all}) opt? (@{syntax simpmod} * )

     ;

     opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use' | 'asm_lr' ) ')'

     ;

     @{syntax_def simpmod}: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') |

       'split' (() | 'add' | 'del')) ':' @{syntax thmrefs}

   "}

   \begin{description}

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

   additional rules according to the arguments given.  Note that the

   @{text only} modifier first removes all other rewrite rules,

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

   like @{text add}.

   \medskip The @{text cong} modifiers add or delete Simplifier

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

   add.

   \medskip The @{text 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}

   @{rail "

     (@@{attribute simp} | @@{attribute cong} | @@{attribute split}) (() | 'add' | 'del')

   "}

   \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 {* Simplification procedures are ML functions that produce proven

   rewrite rules on demand.  They are associated with higher-order

   patterns that approximate the left-hand sides of equations.  The

   Simplifier first matches the current redex against one of the LHS

   patterns; if this succeeds, the corresponding ML function is

   invoked, passing the Simplifier context and redex term.  Thus rules

   may be specifically fashioned for particular situations, resulting

   in a more powerful mechanism than term rewriting by a fixed set of

   rules.

   Any successful result needs to be a (possibly conditional) rewrite

   rule @{text "t \<equiv> u"} that is applicable to the current redex.  The

   rule will be applied just as any ordinary rewrite rule.  It is

   expected to be already in \emph{internal form}, bypassing the

   automatic preprocessing of object-level equivalences.

   \begin{matharray}{rcl}

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

     simproc & : & @{text attribute} \\

   \end{matharray}

   @{rail "

     @@{command simproc_setup} @{syntax name} '(' (@{syntax term} + '|') ')' '='

       @{syntax text} \\ (@'identifier' (@{syntax nameref}+))?

     ;

     @@{attribute simproc} (('add' ':')? | 'del' ':') (@{syntax name}+)

   "}

   \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}

 *}

 subsubsection {* Example *}

 text {* The following simplification procedure for @{thm

   [source=false, show_types] unit_eq} in HOL performs fine-grained

   control over rule application, beyond higher-order pattern matching.

   Declaring @{thm unit_eq} as @{attribute simp} directly would make

   the simplifier loop!  Note that a version of this simplification

   procedure is already active in Isabelle/HOL.  *}

 simproc_setup unit ("x::unit") = {*

   fn _ => fn _ => fn ct =>

     if HOLogic.is_unit (term_of ct) then NONE

     else SOME (mk_meta_eq @{thm unit_eq})

 *}

 text {* Since the Simplifier applies simplification procedures

   frequently, it is important to make the failure check in ML

   reasonably fast. *}

 subsection {* Forward simplification *}

 text {*

   \begin{matharray}{rcl}

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

   \end{matharray}

   @{rail "

     @@{attribute simplified} opt? @{syntax thmrefs}?

     ;

     opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use') ')'

   "}

   \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 concepts *}

 text {* Although Isabelle is generic, many users will be working in

   some extension of classical first-order logic.  Isabelle/ZF is built

   upon theory FOL, while Isabelle/HOL conceptually contains

   first-order logic as a fragment.  Theorem-proving in predicate logic

   is undecidable, but many automated strategies have been developed to

   assist in this task.

   Isabelle's classical reasoner is a generic package that accepts

   certain information about a logic and delivers a suite of automatic

   proof tools, based on rules that are classified and declared in the

   context.  These proof procedures are slow and simplistic compared

   with high-end automated theorem provers, but they can save

   considerable time and effort in practice.  They can prove theorems

   such as Pelletier's \cite{pelletier86} problems 40 and 41 in a few

   milliseconds (including full proof reconstruction): *}

 lemma "(\<exists>y. \<forall>x. F x y \<longleftrightarrow> F x x) \<longrightarrow> \<not> (\<forall>x. \<exists>y. \<forall>z. F z y \<longleftrightarrow> \<not> F z x)"

   by blast

 lemma "(\<forall>z. \<exists>y. \<forall>x. f x y \<longleftrightarrow> f x z \<and> \<not> f x x) \<longrightarrow> \<not> (\<exists>z. \<forall>x. f x z)"

   by blast

 text {* The proof tools are generic.  They are not restricted to

   first-order logic, and have been heavily used in the development of

   the Isabelle/HOL library and applications.  The tactics can be

   traced, and their components can be called directly; in this manner,

   any proof can be viewed interactively.  *}

 subsubsection {* The sequent calculus *}

 text {* Isabelle supports natural deduction, which is easy to use for

   interactive proof.  But natural deduction does not easily lend

   itself to automation, and has a bias towards intuitionism.  For

   certain proofs in classical logic, it can not be called natural.

   The \emph{sequent calculus}, a generalization of natural deduction,

   is easier to automate.

   A \textbf{sequent} has the form @{text "\<Gamma> \<turnstile> \<Delta>"}, where @{text "\<Gamma>"}

   and @{text "\<Delta>"} are sets of formulae.\footnote{For first-order

   logic, sequents can equivalently be made from lists or multisets of

   formulae.} The sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} is

   \textbf{valid} if @{text "P\<^sub>1 \<and> \<dots> \<and> P\<^sub>m"} implies @{text "Q\<^sub>1 \<or> \<dots> \<or>

   Q\<^sub>n"}.  Thus @{text "P\<^sub>1, \<dots>, P\<^sub>m"} represent assumptions, each of which

   is true, while @{text "Q\<^sub>1, \<dots>, Q\<^sub>n"} represent alternative goals.  A

   sequent is \textbf{basic} if its left and right sides have a common

   formula, as in @{text "P, Q \<turnstile> Q, R"}; basic sequents are trivially

   valid.

   Sequent rules are classified as \textbf{right} or \textbf{left},

   indicating which side of the @{text "\<turnstile>"} symbol they operate on.

   Rules that operate on the right side are analogous to natural

   deduction's introduction rules, and left rules are analogous to

   elimination rules.  The sequent calculus analogue of @{text "(\<longrightarrow>I)"}

   is the rule

   \[

   \infer[@{text "(\<longrightarrow>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<longrightarrow> Q"}}{@{text "P, \<Gamma> \<turnstile> \<Delta>, Q"}}

   \]

   Applying the rule backwards, this breaks down some implication on

   the right side of a sequent; @{text "\<Gamma>"} and @{text "\<Delta>"} stand for

   the sets of formulae that are unaffected by the inference.  The

   analogue of the pair @{text "(\<or>I1)"} and @{text "(\<or>I2)"} is the

   single rule

   \[

   \infer[@{text "(\<or>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<or> Q"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P, Q"}}

   \]

   This breaks down some disjunction on the right side, replacing it by

   both disjuncts.  Thus, the sequent calculus is a kind of

   multiple-conclusion logic.

   To illustrate the use of multiple formulae on the right, let us

   prove the classical theorem @{text "(P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}.  Working

   backwards, we reduce this formula to a basic sequent:

   \[

   \infer[@{text "(\<or>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}}

     {\infer[@{text "(\<longrightarrow>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q), (Q \<longrightarrow> P)"}}

       {\infer[@{text "(\<longrightarrow>R)"}]{@{text "P \<turnstile> Q, (Q \<longrightarrow> P)"}}

         {@{text "P, Q \<turnstile> Q, P"}}}}

   \]

   This example is typical of the sequent calculus: start with the

   desired theorem and apply rules backwards in a fairly arbitrary

   manner.  This yields a surprisingly effective proof procedure.

   Quantifiers add only few complications, since Isabelle handles

   parameters and schematic variables.  See \cite[Chapter

   10]{paulson-ml2} for further discussion.  *}

 subsubsection {* Simulating sequents by natural deduction *}

 text {* Isabelle can represent sequents directly, as in the

   object-logic LK.  But natural deduction is easier to work with, and

   most object-logics employ it.  Fortunately, we can simulate the

   sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} by the Isabelle formula

   @{text "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>2 \<Longrightarrow> ... \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> Q\<^sub>1"} where the order of

   the assumptions and the choice of @{text "Q\<^sub>1"} are arbitrary.

   Elim-resolution plays a key role in simulating sequent proofs.

   We can easily handle reasoning on the left.  Elim-resolution with

   the rules @{text "(\<or>E)"}, @{text "(\<bottom>E)"} and @{text "(\<exists>E)"} achieves

   a similar effect as the corresponding sequent rules.  For the other

   connectives, we use sequent-style elimination rules instead of

   destruction rules such as @{text "(\<and>E1, 2)"} and @{text "(\<forall>E)"}.

   But note that the rule @{text "(\<not>L)"} has no effect under our

   representation of sequents!

   \[

   \infer[@{text "(\<not>L)"}]{@{text "\<not> P, \<Gamma> \<turnstile> \<Delta>"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P"}}

   \]

   What about reasoning on the right?  Introduction rules can only

   affect the formula in the conclusion, namely @{text "Q\<^sub>1"}.  The

   other right-side formulae are represented as negated assumptions,

   @{text "\<not> Q\<^sub>2, \<dots>, \<not> Q\<^sub>n"}.  In order to operate on one of these, it

   must first be exchanged with @{text "Q\<^sub>1"}.  Elim-resolution with the

   @{text swap} rule has this effect: @{text "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"}

   To ensure that swaps occur only when necessary, each introduction

   rule is converted into a swapped form: it is resolved with the

   second premise of @{text "(swap)"}.  The swapped form of @{text

   "(\<and>I)"}, which might be called @{text "(\<not>\<and>E)"}, is

   @{text [display] "\<not> (P \<and> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (\<not> R \<Longrightarrow> Q) \<Longrightarrow> R"}

   Similarly, the swapped form of @{text "(\<longrightarrow>I)"} is

   @{text [display] "\<not> (P \<longrightarrow> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P \<Longrightarrow> Q) \<Longrightarrow> R"}

   Swapped introduction rules are applied using elim-resolution, which

   deletes the negated formula.  Our representation of sequents also

   requires the use of ordinary introduction rules.  If we had no

   regard for readability of intermediate goal states, we could treat

   the right side more uniformly by representing sequents as @{text

   [display] "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> \<bottom>"}

 *}

 subsubsection {* Extra rules for the sequent calculus *}

 text {* As mentioned, destruction rules such as @{text "(\<and>E1, 2)"} and

   @{text "(\<forall>E)"} must be replaced by sequent-style elimination rules.

   In addition, we need rules to embody the classical equivalence

   between @{text "P \<longrightarrow> Q"} and @{text "\<not> P \<or> Q"}.  The introduction

   rules @{text "(\<or>I1, 2)"} are replaced by a rule that simulates

   @{text "(\<or>R)"}: @{text [display] "(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q"}

   The destruction rule @{text "(\<longrightarrow>E)"} is replaced by @{text [display]

   "(P \<longrightarrow> Q) \<Longrightarrow> (\<not> P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"}

   Quantifier replication also requires special rules.  In classical

   logic, @{text "\<exists>x. P x"} is equivalent to @{text "\<not> (\<forall>x. \<not> P x)"};

   the rules @{text "(\<exists>R)"} and @{text "(\<forall>L)"} are dual:

   \[

   \infer[@{text "(\<exists>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x"}}{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x, P t"}}

   \qquad

   \infer[@{text "(\<forall>L)"}]{@{text "\<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}{@{text "P t, \<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}

   \]

   Thus both kinds of quantifier may be replicated.  Theorems requiring

   multiple uses of a universal formula are easy to invent; consider

   @{text [display] "(\<forall>x. P x \<longrightarrow> P (f x)) \<and> P a \<longrightarrow> P (f\<^sup>n a)"} for any

   @{text "n > 1"}.  Natural examples of the multiple use of an

   existential formula are rare; a standard one is @{text "\<exists>x. \<forall>y. P x

   \<longrightarrow> P y"}.

   Forgoing quantifier replication loses completeness, but gains

   decidability, since the search space becomes finite.  Many useful

   theorems can be proved without replication, and the search generally

   delivers its verdict in a reasonable time.  To adopt this approach,

   represent the sequent rules @{text "(\<exists>R)"}, @{text "(\<exists>L)"} and

   @{text "(\<forall>R)"} by @{text "(\<exists>I)"}, @{text "(\<exists>E)"} and @{text "(\<forall>I)"},

   respectively, and put @{text "(\<forall>E)"} into elimination form: @{text

   [display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"}

   Elim-resolution with this rule will delete the universal formula

   after a single use.  To replicate universal quantifiers, replace the

   rule by @{text [display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"}

   To replicate existential quantifiers, replace @{text "(\<exists>I)"} by

   @{text [display] "(\<not> (\<exists>x. P x) \<Longrightarrow> P t) \<Longrightarrow> \<exists>x. P x"}

   All introduction rules mentioned above are also useful in swapped

   form.

   Replication makes the search space infinite; we must apply the rules

   with care.  The classical reasoner distinguishes between safe and

   unsafe rules, applying the latter only when there is no alternative.

   Depth-first search may well go down a blind alley; best-first search

   is better behaved in an infinite search space.  However, quantifier

   replication is too expensive to prove any but the simplest theorems.

 *}

 subsection {* Rule declarations *}

 text {* The proof tools of the Classical Reasoner depend on

   collections of rules declared in the context, which are classified

   as introduction, elimination or destruction and as \emph{safe} or

   \emph{unsafe}.  In general, safe rules can be attempted blindly,

   while unsafe rules must be used with care.  A safe rule must never

   reduce a provable goal to an unprovable set of subgoals.

   The rule @{text "P \<Longrightarrow> P \<or> Q"} is unsafe because it reduces @{text "P

   \<or> Q"} to @{text "P"}, which might turn out as premature choice of an

   unprovable subgoal.  Any rule is unsafe whose premises contain new

   unknowns.  The elimination rule @{text "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"} is

   unsafe, since it is applied via elim-resolution, which discards the

   assumption @{text "\<forall>x. P x"} and replaces it by the weaker

   assumption @{text "P t"}.  The rule @{text "P t \<Longrightarrow> \<exists>x. P x"} is

   unsafe for similar reasons.  The quantifier duplication rule @{text

   "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"} is unsafe in a different sense:

   since it keeps the assumption @{text "\<forall>x. P x"}, it is prone to

   looping.  In classical first-order logic, all rules are safe except

   those mentioned above.

   The safe~/ unsafe distinction is vague, and may be regarded merely

   as a way of giving some rules priority over others.  One could argue

   that @{text "(\<or>E)"} is unsafe, because repeated application of it

   could generate exponentially many subgoals.  Induction rules are

   unsafe because inductive proofs are difficult to set up

   automatically.  Any inference is unsafe that instantiates an unknown

   in the proof state --- thus matching must be used, rather than

   unification.  Even proof by assumption is unsafe if it instantiates

   unknowns shared with other subgoals.

   \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} \\

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

   \end{matharray}

   @{rail "

     (@@{attribute intro} | @@{attribute elim} | @@{attribute dest}) ('!' | () | '?') @{syntax nat}?

     ;

     @@{attribute rule} 'del'

     ;

     @@{attribute iff} (((() | 'add') '?'?) | 'del')

   "}

   \begin{description}

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

   declared to the Classical Reasoner, i.e.\ the @{ML_type claset}

   within the context.

   \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 the

   automated reasoning tools, but determines the search order of single

   rule steps.

   Introduction rules are those that can be applied using ordinary

   resolution.  Their swapped forms are generated internally, which

   will be applied using elim-resolution.  Elimination rules are

   applied using elim-resolution.  Rules are sorted by the number of

   new subgoals they will yield; rules that generate the fewest

   subgoals will be tried first.  Otherwise, later declarations take

   precedence over earlier ones.

   Rules already present in the context with the same classification

   are ignored.  A warning is printed if the rule has already been

   added with some other classification, but the rule is added anyway

   as requested.

   \item @{attribute rule}~@{text del} deletes all occurrences of a

   rule from the classical context, regardless of its classification as

   introduction~/ elimination~/ destruction and safe~/ unsafe.

   \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.

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

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

   "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"} in the second position.  This is mainly for

   illustrative purposes: the Classical Reasoner already swaps rules

   internally as explained above.

   \end{description}

 *}

 subsection {* Structured methods *}

 text {*

   \begin{matharray}{rcl}

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

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

   \end{matharray}

   @{rail "

     @@{method rule} @{syntax thmrefs}?

   "}

   \begin{description}

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

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

   versions 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.

   \end{description}

 *}

 subsection {* Automated methods *}

 text {*

   \begin{matharray}{rcl}

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

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

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

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

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

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

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

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

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

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

   \end{matharray}

   @{rail "

     @@{method blast} @{syntax nat}? (@{syntax clamod} * )

     ;

     @@{method auto} (@{syntax nat} @{syntax nat})? (@{syntax clasimpmod} * )

     ;

     @@{method force} (@{syntax clasimpmod} * )

     ;

     (@@{method fast} | @@{method slow} | @@{method best}) (@{syntax clamod} * )

     ;

     (@@{method fastsimp} | @@{method slowsimp} | @@{method bestsimp})

       (@{syntax clasimpmod} * )

     ;

     @@{method deepen} (@{syntax nat} ?) (@{syntax clamod} * )

     ;

     @{syntax_def clamod}:

       (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' @{syntax thmrefs}

     ;

     @{syntax_def clasimpmod}: ('simp' (() | 'add' | 'del' | 'only') |

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

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

       (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' @{syntax thmrefs}

   "}

   \begin{description}

   \item @{method blast} is a separate classical tableau prover that

   uses the same classical rule declarations as explained before.

   Proof search is coded directly in ML using special data structures.

   A successful proof is then reconstructed using regular Isabelle

   inferences.  It is faster and more powerful than the other classical

   reasoning tools, but has major limitations too.

   \begin{itemize}

   \item It does not use the classical wrapper tacticals, such as the

   integration with the Simplifier of @{method fastsimp}.

   \item It does not perform higher-order unification, as needed by the

   rule @{thm [source=false] rangeI} in HOL.  There are often

   alternatives to such rules, for example @{thm [source=false]

   range_eqI}.

   \item Function variables may only be applied to parameters of the

   subgoal.  (This restriction arises because the prover does not use

   higher-order unification.)  If other function variables are present

   then the prover will fail with the message \texttt{Function Var's

   argument not a bound variable}.

   \item Its proof strategy is more general than @{method fast} but can

   be slower.  If @{method blast} fails or seems to be running forever,

   try @{method fast} and the other proof tools described below.

   \end{itemize}

   The optional integer argument specifies a bound for the number of

   unsafe steps used in a proof.  By default, @{method blast} starts

   with a bound of 0 and increases it successively to 20.  In contrast,

   @{text "(blast lim)"} tries to prove the goal using a search bound

   of @{text "lim"}.  Sometimes a slow proof using @{method blast} can

   be made much faster by supplying the successful search bound to this

   proof method instead.

   \item @{method auto} combines classical reasoning with

   simplification.  It is intended for situations where there are a lot

   of mostly trivial subgoals; it proves all the easy ones, leaving the

   ones it cannot prove.  Occasionally, attempting to prove the hard

   ones may take a long time.

   The optional depth arguments in @{text "(auto m n)"} refer to its

   builtin classical reasoning procedures: @{text m} (default 4) is for

   @{method blast}, which is tried first, and @{text n} (default 2) is

   for a slower but more general alternative that also takes wrappers

   into account.

   \item @{method force} is intended to prove the first subgoal

   completely, using many fancy proof tools and performing a rather

   exhaustive search.  As a result, proof attempts may take rather long

   or diverge easily.

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

   prove the first subgoal using sequent-style reasoning as explained

   before.  Unlike @{method blast}, they construct proofs directly in

   Isabelle.

   There is a difference in search strategy and back-tracking: @{method

   fast} uses depth-first search and @{method best} uses best-first

   search (guided by a heuristic function: normally the total size of

   the proof state).

   Method @{method slow} is like @{method fast}, but conducts a broader

   search: it may, when backtracking from a failed proof attempt, undo

   even the step of proving a subgoal by assumption.

   \item @{method fastsimp}, @{method slowsimp}, @{method bestsimp} are

   like @{method fast}, @{method slow}, @{method best}, respectively,

   but use the Simplifier as additional wrapper.

   \item @{method deepen} works by exhaustive search up to a certain

   depth.  The start depth is 4 (unless specified explicitly), and the

   depth is increased iteratively up to 10.  Unsafe rules are modified

   to preserve the formula they act on, so that it be used repeatedly.

   This method can prove more goals than @{method fast}, but is much

   slower, for example if the assumptions have many universal

   quantifiers.

   \end{description}

   Any of the above methods support additional modifiers of the context

   of classical (and simplifier) rules, but the ones related to the

   Simplifier are explicitly prefixed by @{text simp} here.  The

   semantics of these ad-hoc rule declarations is analogous to the

   attributes given before.  Facts provided by forward chaining are

   inserted into the goal before commencing proof search.

 *}

 subsection {* Semi-automated methods *}

 text {* These proof methods may help in situations when the

   fully-automated tools fail.  The result is a simpler subgoal that

   can be tackled by other means, such as by manual instantiation of

   quantifiers.

   \begin{matharray}{rcl}

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

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

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

   \end{matharray}

   @{rail "

     (@@{method safe} | @@{method clarify}) (@{syntax clamod} * )

     ;

     @@{method clarsimp} (@{syntax clasimpmod} * )

   "}

   \begin{description}

   \item @{method safe} repeatedly performs safe steps on all subgoals.

   It is deterministic, with at most one outcome.

   \item @{method clarify} performs a series of safe steps without

   splitting subgoals; see also @{ML clarify_step_tac}.

   \item @{method clarsimp} acts like @{method clarify}, but also does

   simplification.  Note that if the Simplifier context includes a

   splitter for the premises, the subgoal may still be split.

   \end{description}

 *}

 subsection {* Single-step tactics *}

 text {*

   \begin{matharray}{rcl}

     @{index_ML safe_step_tac: "Proof.context -> int -> tactic"} \\

     @{index_ML inst_step_tac: "Proof.context -> int -> tactic"} \\

     @{index_ML step_tac: "Proof.context -> int -> tactic"} \\

     @{index_ML slow_step_tac: "Proof.context -> int -> tactic"} \\

     @{index_ML clarify_step_tac: "Proof.context -> int -> tactic"} \\

   \end{matharray}

   These are the primitive tactics behind the (semi)automated proof

   methods of the Classical Reasoner.  By calling them yourself, you

   can execute these procedures one step at a time.

   \begin{description}

   \item @{ML safe_step_tac}~@{text "ctxt i"} performs a safe step on

   subgoal @{text i}.  The safe wrapper tacticals are applied to a

   tactic that may include proof by assumption or Modus Ponens (taking

   care not to instantiate unknowns), or substitution.

   \item @{ML inst_step_tac} is like @{ML safe_step_tac}, but allows

   unknowns to be instantiated.

   \item @{ML step_tac}~@{text "ctxt i"} is the basic step of the proof

   procedure.  The unsafe wrapper tacticals are applied to a tactic

   that tries @{ML safe_tac}, @{ML inst_step_tac}, or applies an unsafe

   rule from the context.

   \item @{ML slow_step_tac} resembles @{ML step_tac}, but allows

   backtracking between using safe rules with instantiation (@{ML

   inst_step_tac}) and using unsafe rules.  The resulting search space

   is larger.

   \item @{ML clarify_step_tac}~@{text "ctxt i"} performs a safe step

   on subgoal @{text i}.  No splitting step is applied; for example,

   the subgoal @{text "A \<and> B"} is left as a conjunction.  Proof by

   assumption, Modus Ponens, etc., may be performed provided they do

   not instantiate unknowns.  Assumptions of the form @{text "x = t"}

   may be eliminated.  The safe wrapper tactical is applied.

   \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.

   @{rail "

     @@{command judgment} @{syntax constdecl}

     ;

     @@{attribute atomize} ('(' 'full' ')')?

     ;

     @@{attribute rule_format} ('(' 'noasm' ')')?

   "}

   \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 (Pure) 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