src/Doc/Isar_Ref/Generic.thy

isabelle update_cartouches -t;

theory Generic
imports Base Main
begin

chapter \<open>Generic tools and packages \label{ch:gen-tools}\<close>

section \<open>Configuration options \label{sec:config}\<close>

text \<open>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:
\<close>

(*<*)experiment begin(*>*)
declare [[show_main_goal = false]]

notepad
begin
  note [[show_main_goal = true]]
end
(*<*)end(*>*)

text \<open>
  \begin{matharray}{rcll}
    @{command_def "print_options"} & : & \<open>context \<rightarrow>\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{command print_options} ('!'?)
    ;
    @{syntax name} ('=' ('true' | 'false' | @{syntax int} | @{syntax float} | @{syntax name}))?
  \<close>}

  \<^descr> @{command "print_options"} prints the available configuration
  options, with names, types, and current values; the ``\<open>!\<close>'' option
  indicates extra verbosity.
  
  \<^descr> \<open>name = value\<close> 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 \<open>true\<close>.  Any
  attempt to change a global option in a local context is ignored.
\<close>


section \<open>Basic proof tools\<close>

subsection \<open>Miscellaneous methods and attributes \label{sec:misc-meth-att}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def unfold} & : & \<open>method\<close> \\
    @{method_def fold} & : & \<open>method\<close> \\
    @{method_def insert} & : & \<open>method\<close> \\[0.5ex]
    @{method_def erule}\<open>\<^sup>*\<close> & : & \<open>method\<close> \\
    @{method_def drule}\<open>\<^sup>*\<close> & : & \<open>method\<close> \\
    @{method_def frule}\<open>\<^sup>*\<close> & : & \<open>method\<close> \\
    @{method_def intro} & : & \<open>method\<close> \\
    @{method_def elim} & : & \<open>method\<close> \\
    @{method_def fail} & : & \<open>method\<close> \\
    @{method_def succeed} & : & \<open>method\<close> \\
    @{method_def sleep} & : & \<open>method\<close> \\
  \end{matharray}

  @{rail \<open>
    (@@{method fold} | @@{method unfold} | @@{method insert}) @{syntax thmrefs}
    ;
    (@@{method erule} | @@{method drule} | @@{method frule})
      ('(' @{syntax nat} ')')? @{syntax thmrefs}
    ;
    (@@{method intro} | @@{method elim}) @{syntax thmrefs}?
    ;
    @@{method sleep} @{syntax real}
  \<close>}

  \<^descr> @{method unfold}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> and @{method fold}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> 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.

  \<^descr> @{method insert}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> inserts theorems as facts
  into all goals of the proof state.  Note that current facts
  indicated for forward chaining are ignored.

  \<^descr> @{method erule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close>, @{method
  drule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close>, and @{method frule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> 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.

  \<^descr> @{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.

  \<^descr> @{method fail} yields an empty result sequence; it is the
  identity of the ``\<open>|\<close>'' method combinator (cf.\
  \secref{sec:proof-meth}).

  \<^descr> @{method succeed} yields a single (unchanged) result; it is
  the identity of the ``\<open>,\<close>'' method combinator (cf.\
  \secref{sec:proof-meth}).

  \<^descr> @{method sleep}~\<open>s\<close> succeeds after a real-time delay of \<open>s\<close> seconds. This is occasionally useful for demonstration and testing
  purposes.


  \begin{matharray}{rcl}
    @{attribute_def tagged} & : & \<open>attribute\<close> \\
    @{attribute_def untagged} & : & \<open>attribute\<close> \\[0.5ex]
    @{attribute_def THEN} & : & \<open>attribute\<close> \\
    @{attribute_def unfolded} & : & \<open>attribute\<close> \\
    @{attribute_def folded} & : & \<open>attribute\<close> \\
    @{attribute_def abs_def} & : & \<open>attribute\<close> \\[0.5ex]
    @{attribute_def rotated} & : & \<open>attribute\<close> \\
    @{attribute_def (Pure) elim_format} & : & \<open>attribute\<close> \\
    @{attribute_def no_vars}\<open>\<^sup>*\<close> & : & \<open>attribute\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{attribute tagged} @{syntax name} @{syntax name}
    ;
    @@{attribute untagged} @{syntax name}
    ;
    @@{attribute THEN} ('[' @{syntax nat} ']')? @{syntax thmref}
    ;
    (@@{attribute unfolded} | @@{attribute folded}) @{syntax thmrefs}
    ;
    @@{attribute rotated} @{syntax int}?
  \<close>}

  \<^descr> @{attribute tagged}~\<open>name value\<close> and @{attribute
  untagged}~\<open>name\<close> add and remove \<^emph>\<open>tags\<close> 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.

  \<^descr> @{attribute THEN}~\<open>a\<close> composes rules by resolution; it
  resolves with the first premise of \<open>a\<close> (an alternative
  position may be also specified).  See also @{ML_op "RS"} in
  @{cite "isabelle-implementation"}.
  
  \<^descr> @{attribute unfolded}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> and @{attribute
  folded}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> expand and fold back again the given
  definitions throughout a rule.

  \<^descr> @{attribute abs_def} turns an equation of the form @{prop "f x
  y \<equiv> t"} into @{prop "f \<equiv> \<lambda>x y. t"}, which ensures that @{method
  simp} or @{method unfold} steps always expand it.  This also works
  for object-logic equality.

  \<^descr> @{attribute rotated}~\<open>n\<close> rotate the premises of a
  theorem by \<open>n\<close> (default 1).

  \<^descr> @{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.

  \<^descr> @{attribute no_vars} replaces schematic variables by free
  ones; this is mainly for tuning output of pretty printed theorems.
\<close>


subsection \<open>Low-level equational reasoning\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def subst} & : & \<open>method\<close> \\
    @{method_def hypsubst} & : & \<open>method\<close> \\
    @{method_def split} & : & \<open>method\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{method subst} ('(' 'asm' ')')? \<newline> ('(' (@{syntax nat}+) ')')? @{syntax thmref}
    ;
    @@{method split} @{syntax thmrefs}
  \<close>}

  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}).

  \<^descr> @{method subst}~\<open>eq\<close> performs a single substitution step
  using rule \<open>eq\<close>, which may be either a meta or object
  equality.

  \<^descr> @{method subst}~\<open>(asm) eq\<close> substitutes in an
  assumption.

  \<^descr> @{method subst}~\<open>(i \<dots> j) eq\<close> performs several
  substitutions in the conclusion. The numbers \<open>i\<close> to \<open>j\<close>
  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 \<open>(a + b) + (c + d)\<close> there are three positions
  where commutativity of \<open>+\<close> is applicable: 1 refers to \<open>a + b\<close>, 2 to the whole term, and 3 to \<open>c + d\<close>.

  If the positions in the list \<open>(i \<dots> j)\<close> are non-overlapping
  (e.g.\ \<open>(2 3)\<close> in \<open>(a + b) + (c + d)\<close>) you may
  assume all substitutions are performed simultaneously.  Otherwise
  the behaviour of \<open>subst\<close> is not specified.

  \<^descr> @{method subst}~\<open>(asm) (i \<dots> j) eq\<close> 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 \<open>P (a + b) \<Longrightarrow> P (c + d) \<Longrightarrow> \<dots>\<close>, position 1 of
  commutativity of \<open>+\<close> is the subterm \<open>a + b\<close> and
  position 2 is the subterm \<open>c + d\<close>.

  \<^descr> @{method hypsubst} performs substitution using some
  assumption; this only works for equations of the form \<open>x =
  t\<close> where \<open>x\<close> is a free or bound variable.

  \<^descr> @{method split}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> performs single-step case
  splitting using the given rules.  Splitting is performed in the
  conclusion or some assumption of the subgoal, depending of the
  structure of the rule.
  
  Note that the @{method simp} method already involves repeated
  application of split rules as declared in the current context, using
  @{attribute split}, for example.
\<close>


section \<open>The Simplifier \label{sec:simplifier}\<close>

text \<open>The Simplifier performs conditional and unconditional
  rewriting and uses contextual information: rule declarations in the
  background theory or local proof context are taken into account, as
  well as chained facts and subgoal premises (``local assumptions'').
  There are several general hooks that allow to modify the
  simplification strategy, or incorporate other proof tools that solve
  sub-problems, produce rewrite rules on demand etc.

  The rewriting strategy is always strictly bottom up, except for
  congruence rules, which are applied while descending into a term.
  Conditions in conditional rewrite rules are solved recursively
  before the rewrite rule is applied.

  The default Simplifier setup of major object logics (HOL, HOLCF,
  FOL, ZF) makes the Simplifier ready for immediate use, without
  engaging into the internal structures.  Thus it serves as
  general-purpose proof tool with the main focus on equational
  reasoning, and a bit more than that.
\<close>


subsection \<open>Simplification methods \label{sec:simp-meth}\<close>

text \<open>
  \begin{tabular}{rcll}
    @{method_def simp} & : & \<open>method\<close> \\
    @{method_def simp_all} & : & \<open>method\<close> \\
    @{attribute_def simp_depth_limit} & : & \<open>attribute\<close> & default \<open>100\<close> \\
  \end{tabular}
  \<^medskip>

  @{rail \<open>
    (@@{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' | 'split' (() | 'add' | 'del') |
      'cong' (() | 'add' | 'del')) ':' @{syntax thmrefs}
  \<close>}

  \<^descr> @{method simp} invokes the Simplifier on the first subgoal,
  after inserting chained facts as additional goal premises; further
  rule declarations may be included via \<open>(simp add: facts)\<close>.
  The proof method fails if the subgoal remains unchanged after
  simplification.

  Note that the original goal premises and chained facts are subject
  to simplification themselves, while declarations via \<open>add\<close>/\<open>del\<close> merely follow the policies of the object-logic
  to extract rewrite rules from theorems, without further
  simplification.  This may lead to slightly different behavior in
  either case, which might be required precisely like that in some
  boundary situations to perform the intended simplification step!

  \<^medskip>
  The \<open>only\<close> modifier first removes all other rewrite
  rules, looper tactics (including split rules), congruence rules, and
  then behaves like \<open>add\<close>.  Implicit solvers remain, which means
  that trivial rules like reflexivity or introduction of \<open>True\<close> are available to solve the simplified subgoals, but also
  non-trivial tools like linear arithmetic in HOL.  The latter may
  lead to some surprise of the meaning of ``only'' in Isabelle/HOL
  compared to English!

  \<^medskip>
  The \<open>split\<close> modifiers add or delete rules for the
  Splitter (see also \secref{sec:simp-strategies} on the looper).
  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).

  There is also a separate @{method_ref split} method available for
  single-step case splitting.  The effect of repeatedly applying
  \<open>(split thms)\<close> can be imitated by ``\<open>(simp only:
  split: thms)\<close>''.

  \<^medskip>
  The \<open>cong\<close> modifiers add or delete Simplifier
  congruence rules (see also \secref{sec:simp-rules}); the default is
  to add.

  \<^descr> @{method simp_all} is similar to @{method simp}, but acts on
  all goals, working backwards from the last to the first one as usual
  in Isabelle.\footnote{The order is irrelevant for goals without
  schematic variables, so simplification might actually be performed
  in parallel here.}

  Chained facts are inserted into all subgoals, before the
  simplification process starts.  Further rule declarations are the
  same as for @{method simp}.

  The proof method fails if all subgoals remain unchanged after
  simplification.

  \<^descr> @{attribute simp_depth_limit} limits the number of recursive
  invocations of the Simplifier during conditional rewriting.


  By default the Simplifier methods above take local assumptions fully
  into account, using equational assumptions in the subsequent
  normalization process, or simplifying assumptions themselves.
  Further options allow to fine-tune the behavior of the Simplifier
  in this respect, corresponding to a variety of ML tactics as
  follows.\footnote{Unlike the corresponding Isar proof methods, the
  ML tactics do not insist in changing the goal state.}

  \begin{center}
  \small
  \begin{tabular}{|l|l|p{0.3\textwidth}|}
  \hline
  Isar method & ML tactic & behavior \\\hline

  \<open>(simp (no_asm))\<close> & @{ML simp_tac} & assumptions are ignored
  completely \\\hline

  \<open>(simp (no_asm_simp))\<close> & @{ML asm_simp_tac} & assumptions
  are used in the simplification of the conclusion but are not
  themselves simplified \\\hline

  \<open>(simp (no_asm_use))\<close> & @{ML full_simp_tac} & assumptions
  are simplified but are not used in the simplification of each other
  or the conclusion \\\hline

  \<open>(simp)\<close> & @{ML asm_full_simp_tac} & assumptions are used in
  the simplification of the conclusion and to simplify other
  assumptions \\\hline

  \<open>(simp (asm_lr))\<close> & @{ML asm_lr_simp_tac} & compatibility
  mode: an assumption is only used for simplifying assumptions which
  are to the right of it \\\hline

  \end{tabular}
  \end{center}
\<close>


subsubsection \<open>Examples\<close>

text \<open>We consider basic algebraic simplifications in Isabelle/HOL.
  The rather trivial goal @{prop "0 + (x + 0) = x + 0 + 0"} looks like
  a good candidate to be solved by a single call of @{method simp}:
\<close>

lemma "0 + (x + 0) = x + 0 + 0" apply simp? oops

text \<open>The above attempt \<^emph>\<open>fails\<close>, because @{term "0"} and @{term
  "op +"} in the HOL library are declared as generic type class
  operations, without stating any algebraic laws yet.  More specific
  types are required to get access to certain standard simplifications
  of the theory context, e.g.\ like this:\<close>

lemma fixes x :: nat shows "0 + (x + 0) = x + 0 + 0" by simp
lemma fixes x :: int shows "0 + (x + 0) = x + 0 + 0" by simp
lemma fixes x :: "'a :: monoid_add" shows "0 + (x + 0) = x + 0 + 0" by simp

text \<open>
  \<^medskip>
  In many cases, assumptions of a subgoal are also needed in
  the simplification process.  For example:
\<close>

lemma fixes x :: nat shows "x = 0 \<Longrightarrow> x + x = 0" by simp
lemma fixes x :: nat assumes "x = 0" shows "x + x = 0" apply simp oops
lemma fixes x :: nat assumes "x = 0" shows "x + x = 0" using assms by simp

text \<open>As seen above, local assumptions that shall contribute to
  simplification need to be part of the subgoal already, or indicated
  explicitly for use by the subsequent method invocation.  Both too
  little or too much information can make simplification fail, for
  different reasons.

  In the next example the malicious assumption @{prop "\<And>x::nat. f x =
  g (f (g x))"} does not contribute to solve the problem, but makes
  the default @{method simp} method loop: the rewrite rule \<open>f
  ?x \<equiv> g (f (g ?x))\<close> extracted from the assumption does not
  terminate.  The Simplifier notices certain simple forms of
  nontermination, but not this one.  The problem can be solved
  nonetheless, by ignoring assumptions via special options as
  explained before:
\<close>

lemma "(\<And>x::nat. f x = g (f (g x))) \<Longrightarrow> f 0 = f 0 + 0"
  by (simp (no_asm))

text \<open>The latter form is typical for long unstructured proof
  scripts, where the control over the goal content is limited.  In
  structured proofs it is usually better to avoid pushing too many
  facts into the goal state in the first place.  Assumptions in the
  Isar proof context do not intrude the reasoning if not used
  explicitly.  This is illustrated for a toplevel statement and a
  local proof body as follows:
\<close>

lemma
  assumes "\<And>x::nat. f x = g (f (g x))"
  shows "f 0 = f 0 + 0" by simp

notepad
begin
  assume "\<And>x::nat. f x = g (f (g x))"
  have "f 0 = f 0 + 0" by simp
end

text \<open>
  \<^medskip>
  Because assumptions may simplify each other, there
  can be very subtle cases of nontermination. For example, the regular
  @{method simp} method applied to @{prop "P (f x) \<Longrightarrow> y = x \<Longrightarrow> f x = f y
  \<Longrightarrow> Q"} gives rise to the infinite reduction sequence
  \[
  \<open>P (f x)\<close> \stackrel{\<open>f x \<equiv> f y\<close>}{\longmapsto}
  \<open>P (f y)\<close> \stackrel{\<open>y \<equiv> x\<close>}{\longmapsto}
  \<open>P (f x)\<close> \stackrel{\<open>f x \<equiv> f y\<close>}{\longmapsto} \cdots
  \]
  whereas applying the same to @{prop "y = x \<Longrightarrow> f x = f y \<Longrightarrow> P (f x) \<Longrightarrow>
  Q"} terminates (without solving the goal):
\<close>

lemma "y = x \<Longrightarrow> f x = f y \<Longrightarrow> P (f x) \<Longrightarrow> Q"
  apply simp
  oops

text \<open>See also \secref{sec:simp-trace} for options to enable
  Simplifier trace mode, which often helps to diagnose problems with
  rewrite systems.
\<close>


subsection \<open>Declaring rules \label{sec:simp-rules}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{attribute_def simp} & : & \<open>attribute\<close> \\
    @{attribute_def split} & : & \<open>attribute\<close> \\
    @{attribute_def cong} & : & \<open>attribute\<close> \\
    @{command_def "print_simpset"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
  \end{matharray}

  @{rail \<open>
    (@@{attribute simp} | @@{attribute split} | @@{attribute cong})
      (() | 'add' | 'del')
    ;
    @@{command print_simpset} ('!'?)
  \<close>}

  \<^descr> @{attribute simp} declares rewrite rules, by adding or
  deleting them from the simpset within the theory or proof context.
  Rewrite rules are theorems expressing some form of equality, for
  example:

  \<open>Suc ?m + ?n = ?m + Suc ?n\<close> \\
  \<open>?P \<and> ?P \<longleftrightarrow> ?P\<close> \\
  \<open>?A \<union> ?B \<equiv> {x. x \<in> ?A \<or> x \<in> ?B}\<close>

  \<^medskip>
  Conditional rewrites such as \<open>?m < ?n \<Longrightarrow> ?m div ?n = 0\<close> are
  also permitted; the conditions can be arbitrary formulas.

  \<^medskip>
  Internally, all rewrite rules are translated into Pure
  equalities, theorems with conclusion \<open>lhs \<equiv> rhs\<close>. The
  simpset contains a function for extracting equalities from arbitrary
  theorems, which is usually installed when the object-logic is
  configured initially. For example, \<open>\<not> ?x \<in> {}\<close> could be
  turned into \<open>?x \<in> {} \<equiv> False\<close>. Theorems that are declared as
  @{attribute simp} and local assumptions within a goal are treated
  uniformly in this respect.

  The Simplifier accepts the following formats for the \<open>lhs\<close>
  term:

    \<^enum> First-order patterns, considering the sublanguage of
    application of constant operators to variable operands, without
    \<open>\<lambda>\<close>-abstractions or functional variables.
    For example:

    \<open>(?x + ?y) + ?z \<equiv> ?x + (?y + ?z)\<close> \\
    \<open>f (f ?x ?y) ?z \<equiv> f ?x (f ?y ?z)\<close>

    \<^enum> Higher-order patterns in the sense of @{cite "nipkow-patterns"}.
    These are terms in \<open>\<beta>\<close>-normal form (this will always be the
    case unless you have done something strange) where each occurrence
    of an unknown is of the form \<open>?F x\<^sub>1 \<dots> x\<^sub>n\<close>, where the
    \<open>x\<^sub>i\<close> are distinct bound variables.

    For example, \<open>(\<forall>x. ?P x \<and> ?Q x) \<equiv> (\<forall>x. ?P x) \<and> (\<forall>x. ?Q x)\<close>
    or its symmetric form, since the \<open>rhs\<close> is also a
    higher-order pattern.

    \<^enum> Physical first-order patterns over raw \<open>\<lambda>\<close>-term
    structure without \<open>\<alpha>\<beta>\<eta>\<close>-equality; abstractions and bound
    variables are treated like quasi-constant term material.

    For example, the rule \<open>?f ?x \<in> range ?f = True\<close> rewrites the
    term \<open>g a \<in> range g\<close> to \<open>True\<close>, but will fail to
    match \<open>g (h b) \<in> range (\<lambda>x. g (h x))\<close>. However, offending
    subterms (in our case \<open>?f ?x\<close>, which is not a pattern) can
    be replaced by adding new variables and conditions like this: \<open>?y = ?f ?x \<Longrightarrow> ?y \<in> range ?f = True\<close> is acceptable as a conditional
    rewrite rule of the second category since conditions can be
    arbitrary terms.

  \<^descr> @{attribute split} declares case split rules.

  \<^descr> @{attribute cong} declares congruence rules to the Simplifier
  context.

  Congruence rules are equalities of the form @{text [display]
  "\<dots> \<Longrightarrow> f ?x\<^sub>1 \<dots> ?x\<^sub>n = f ?y\<^sub>1 \<dots> ?y\<^sub>n"}

  This controls the simplification of the arguments of \<open>f\<close>.  For
  example, some arguments can be simplified under additional
  assumptions: @{text [display] "?P\<^sub>1 \<longleftrightarrow> ?Q\<^sub>1 \<Longrightarrow> (?Q\<^sub>1 \<Longrightarrow> ?P\<^sub>2 \<longleftrightarrow> ?Q\<^sub>2) \<Longrightarrow>
  (?P\<^sub>1 \<longrightarrow> ?P\<^sub>2) \<longleftrightarrow> (?Q\<^sub>1 \<longrightarrow> ?Q\<^sub>2)"}

  Given this rule, the Simplifier assumes \<open>?Q\<^sub>1\<close> and extracts
  rewrite rules from it when simplifying \<open>?P\<^sub>2\<close>.  Such local
  assumptions are effective for rewriting formulae such as \<open>x =
  0 \<longrightarrow> y + x = y\<close>.

  %FIXME
  %The local assumptions are also provided as theorems to the solver;
  %see \secref{sec:simp-solver} below.

  \<^medskip>
  The following congruence rule for bounded quantifiers also
  supplies contextual information --- about the bound variable:
  @{text [display] "(?A = ?B) \<Longrightarrow> (\<And>x. x \<in> ?B \<Longrightarrow> ?P x \<longleftrightarrow> ?Q x) \<Longrightarrow>
    (\<forall>x \<in> ?A. ?P x) \<longleftrightarrow> (\<forall>x \<in> ?B. ?Q x)"}

  \<^medskip>
  This congruence rule for conditional expressions can
  supply contextual information for simplifying the arms:
  @{text [display] "?p = ?q \<Longrightarrow> (?q \<Longrightarrow> ?a = ?c) \<Longrightarrow> (\<not> ?q \<Longrightarrow> ?b = ?d) \<Longrightarrow>
    (if ?p then ?a else ?b) = (if ?q then ?c else ?d)"}

  A congruence rule can also \<^emph>\<open>prevent\<close> simplification of some
  arguments.  Here is an alternative congruence rule for conditional
  expressions that conforms to non-strict functional evaluation:
  @{text [display] "?p = ?q \<Longrightarrow> (if ?p then ?a else ?b) = (if ?q then ?a else ?b)"}

  Only the first argument is simplified; the others remain unchanged.
  This can make simplification much faster, but may require an extra
  case split over the condition \<open>?q\<close> to prove the goal.

  \<^descr> @{command "print_simpset"} prints the collection of rules declared
  to the Simplifier, which is also known as ``simpset'' internally; the
  ``\<open>!\<close>'' option indicates extra verbosity.

  For historical reasons, simpsets may occur independently from the
  current context, but are conceptually dependent on it.  When the
  Simplifier is invoked via one of its main entry points in the Isar
  source language (as proof method \secref{sec:simp-meth} or rule
  attribute \secref{sec:simp-meth}), its simpset is derived from the
  current proof context, and carries a back-reference to that for
  other tools that might get invoked internally (e.g.\ simplification
  procedures \secref{sec:simproc}).  A mismatch of the context of the
  simpset and the context of the problem being simplified may lead to
  unexpected results.


  The implicit simpset of the theory context is propagated
  monotonically through the theory hierarchy: forming a new theory,
  the union of the simpsets of its imports are taken as starting
  point.  Also note that definitional packages like @{command
  "datatype"}, @{command "primrec"}, @{command "fun"} routinely
  declare Simplifier rules to the target context, while plain
  @{command "definition"} is an exception in \<^emph>\<open>not\<close> declaring
  anything.

  \<^medskip>
  It is up the user to manipulate the current simpset further
  by explicitly adding or deleting theorems as simplification rules,
  or installing other tools via simplification procedures
  (\secref{sec:simproc}).  Good simpsets are hard to design.  Rules
  that obviously simplify, like \<open>?n + 0 \<equiv> ?n\<close> are good
  candidates for the implicit simpset, unless a special
  non-normalizing behavior of certain operations is intended.  More
  specific rules (such as distributive laws, which duplicate subterms)
  should be added only for specific proof steps.  Conversely,
  sometimes a rule needs to be deleted just for some part of a proof.
  The need of frequent additions or deletions may indicate a poorly
  designed simpset.

  \begin{warn}
  The union of simpsets from theory imports (as described above) is
  not always a good starting point for the new theory.  If some
  ancestors have deleted simplification rules because they are no
  longer wanted, while others have left those rules in, then the union
  will contain the unwanted rules, and thus have to be deleted again
  in the theory body.
  \end{warn}
\<close>


subsection \<open>Ordered rewriting with permutative rules\<close>

text \<open>A rewrite rule is \<^emph>\<open>permutative\<close> if the left-hand side and
  right-hand side are the equal up to renaming of variables.  The most
  common permutative rule is commutativity: \<open>?x + ?y = ?y +
  ?x\<close>.  Other examples include \<open>(?x - ?y) - ?z = (?x - ?z) -
  ?y\<close> in arithmetic and \<open>insert ?x (insert ?y ?A) = insert ?y
  (insert ?x ?A)\<close> for sets.  Such rules are common enough to merit
  special attention.

  Because ordinary rewriting loops given such rules, the Simplifier
  employs a special strategy, called \<^emph>\<open>ordered rewriting\<close>.
  Permutative rules are detected and only applied if the rewriting
  step decreases the redex wrt.\ a given term ordering.  For example,
  commutativity rewrites \<open>b + a\<close> to \<open>a + b\<close>, but then
  stops, because the redex cannot be decreased further in the sense of
  the term ordering.

  The default is lexicographic ordering of term structure, but this
  could be also changed locally for special applications via
  @{index_ML Simplifier.set_termless} in Isabelle/ML.

  \<^medskip>
  Permutative rewrite rules are declared to the Simplifier
  just like other rewrite rules.  Their special status is recognized
  automatically, and their application is guarded by the term ordering
  accordingly.\<close>


subsubsection \<open>Rewriting with AC operators\<close>

text \<open>Ordered rewriting is particularly effective in the case of
  associative-commutative operators.  (Associativity by itself is not
  permutative.)  When dealing with an AC-operator \<open>f\<close>, keep
  the following points in mind:

  \<^item> The associative law must always be oriented from left to
  right, namely \<open>f (f x y) z = f x (f y z)\<close>.  The opposite
  orientation, if used with commutativity, leads to looping in
  conjunction with the standard term order.

  \<^item> To complete your set of rewrite rules, you must add not just
  associativity (A) and commutativity (C) but also a derived rule
  \<^emph>\<open>left-commutativity\<close> (LC): \<open>f x (f y z) = f y (f x z)\<close>.


  Ordered rewriting with the combination of A, C, and LC sorts a term
  lexicographically --- the rewriting engine imitates bubble-sort.
\<close>

experiment
  fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infix "\<bullet>" 60)
  assumes assoc: "(x \<bullet> y) \<bullet> z = x \<bullet> (y \<bullet> z)"
  assumes commute: "x \<bullet> y = y \<bullet> x"
begin

lemma left_commute: "x \<bullet> (y \<bullet> z) = y \<bullet> (x \<bullet> z)"
proof -
  have "(x \<bullet> y) \<bullet> z = (y \<bullet> x) \<bullet> z" by (simp only: commute)
  then show ?thesis by (simp only: assoc)
qed

lemmas AC_rules = assoc commute left_commute

text \<open>Thus the Simplifier is able to establish equalities with
  arbitrary permutations of subterms, by normalizing to a common
  standard form.  For example:\<close>

lemma "(b \<bullet> c) \<bullet> a = xxx"
  apply (simp only: AC_rules)
  txt \<open>@{subgoals}\<close>
  oops

lemma "(b \<bullet> c) \<bullet> a = a \<bullet> (b \<bullet> c)" by (simp only: AC_rules)
lemma "(b \<bullet> c) \<bullet> a = c \<bullet> (b \<bullet> a)" by (simp only: AC_rules)
lemma "(b \<bullet> c) \<bullet> a = (c \<bullet> b) \<bullet> a" by (simp only: AC_rules)

end

text \<open>Martin and Nipkow @{cite "martin-nipkow"} discuss the theory and
  give many examples; other algebraic structures are amenable to
  ordered rewriting, such as Boolean rings.  The Boyer-Moore theorem
  prover @{cite bm88book} also employs ordered rewriting.
\<close>


subsubsection \<open>Re-orienting equalities\<close>

text \<open>Another application of ordered rewriting uses the derived rule
  @{thm [source] eq_commute}: @{thm [source = false] eq_commute} to
  reverse equations.

  This is occasionally useful to re-orient local assumptions according
  to the term ordering, when other built-in mechanisms of
  reorientation and mutual simplification fail to apply.\<close>


subsection \<open>Simplifier tracing and debugging \label{sec:simp-trace}\<close>

text \<open>
  \begin{tabular}{rcll}
    @{attribute_def simp_trace} & : & \<open>attribute\<close> & default \<open>false\<close> \\
    @{attribute_def simp_trace_depth_limit} & : & \<open>attribute\<close> & default \<open>1\<close> \\
    @{attribute_def simp_debug} & : & \<open>attribute\<close> & default \<open>false\<close> \\
    @{attribute_def simp_trace_new} & : & \<open>attribute\<close> \\
    @{attribute_def simp_break} & : & \<open>attribute\<close> \\
  \end{tabular}
  \<^medskip>

  @{rail \<open>
    @@{attribute simp_trace_new} ('interactive')? \<newline>
      ('mode' '=' ('full' | 'normal'))? \<newline>
      ('depth' '=' @{syntax nat})?
    ;

    @@{attribute simp_break} (@{syntax term}*)
  \<close>}

  These attributes and configurations options control various aspects of
  Simplifier tracing and debugging.

  \<^descr> @{attribute simp_trace} makes the Simplifier output internal
  operations.  This includes rewrite steps, but also bookkeeping like
  modifications of the simpset.

  \<^descr> @{attribute simp_trace_depth_limit} limits the effect of
  @{attribute simp_trace} to the given depth of recursive Simplifier
  invocations (when solving conditions of rewrite rules).

  \<^descr> @{attribute simp_debug} makes the Simplifier output some extra
  information about internal operations.  This includes any attempted
  invocation of simplification procedures.

  \<^descr> @{attribute simp_trace_new} controls Simplifier tracing within
  Isabelle/PIDE applications, notably Isabelle/jEdit @{cite "isabelle-jedit"}.
  This provides a hierarchical representation of the rewriting steps
  performed by the Simplifier.

  Users can configure the behaviour by specifying breakpoints, verbosity and
  enabling or disabling the interactive mode. In normal verbosity (the
  default), only rule applications matching a breakpoint will be shown to
  the user. In full verbosity, all rule applications will be logged.
  Interactive mode interrupts the normal flow of the Simplifier and defers
  the decision how to continue to the user via some GUI dialog.

  \<^descr> @{attribute simp_break} declares term or theorem breakpoints for
  @{attribute simp_trace_new} as described above. Term breakpoints are
  patterns which are checked for matches on the redex of a rule application.
  Theorem breakpoints trigger when the corresponding theorem is applied in a
  rewrite step. For example:
\<close>

(*<*)experiment begin(*>*)
declare conjI [simp_break]
declare [[simp_break "?x \<and> ?y"]]
(*<*)end(*>*)


subsection \<open>Simplification procedures \label{sec:simproc}\<close>

text \<open>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 \<open>t \<equiv> u\<close> 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>\<open>internal form\<close>, bypassing the
  automatic preprocessing of object-level equivalences.

  \begin{matharray}{rcl}
    @{command_def "simproc_setup"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
    simproc & : & \<open>attribute\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{command simproc_setup} @{syntax name} '(' (@{syntax term} + '|') ')' '='
      @{syntax text} \<newline> (@'identifier' (@{syntax nameref}+))?
    ;

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

  \<^descr> @{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 \<open>r\<close> and the result is
  supposed to be some proven rewrite rule \<open>r \<equiv> r'\<close> (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.

  \<^descr> \<open>simproc add: name\<close> and \<open>simproc del: name\<close>
  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.
\<close>


subsubsection \<open>Example\<close>

text \<open>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.\<close>

(*<*)experiment begin(*>*)
simproc_setup unit ("x::unit") =
  \<open>fn _ => fn _ => fn ct =>
    if HOLogic.is_unit (Thm.term_of ct) then NONE
    else SOME (mk_meta_eq @{thm unit_eq})\<close>
(*<*)end(*>*)

text \<open>Since the Simplifier applies simplification procedures
  frequently, it is important to make the failure check in ML
  reasonably fast.\<close>


subsection \<open>Configurable Simplifier strategies \label{sec:simp-strategies}\<close>

text \<open>The core term-rewriting engine of the Simplifier is normally
  used in combination with some add-on components that modify the
  strategy and allow to integrate other non-Simplifier proof tools.
  These may be reconfigured in ML as explained below.  Even if the
  default strategies of object-logics like Isabelle/HOL are used
  unchanged, it helps to understand how the standard Simplifier
  strategies work.\<close>


subsubsection \<open>The subgoaler\<close>

text \<open>
  \begin{mldecls}
  @{index_ML Simplifier.set_subgoaler: "(Proof.context -> int -> tactic) ->
  Proof.context -> Proof.context"} \\
  @{index_ML Simplifier.prems_of: "Proof.context -> thm list"} \\
  \end{mldecls}

  The subgoaler is the tactic used to solve subgoals arising out of
  conditional rewrite rules or congruence rules.  The default should
  be simplification itself.  In rare situations, this strategy may
  need to be changed.  For example, if the premise of a conditional
  rule is an instance of its conclusion, as in \<open>Suc ?m < ?n \<Longrightarrow>
  ?m < ?n\<close>, the default strategy could loop.  % FIXME !??

  \<^descr> @{ML Simplifier.set_subgoaler}~\<open>tac ctxt\<close> sets the
  subgoaler of the context to \<open>tac\<close>.  The tactic will
  be applied to the context of the running Simplifier instance.

  \<^descr> @{ML Simplifier.prems_of}~\<open>ctxt\<close> retrieves the current
  set of premises from the context.  This may be non-empty only if
  the Simplifier has been told to utilize local assumptions in the
  first place (cf.\ the options in \secref{sec:simp-meth}).


  As an example, consider the following alternative subgoaler:
\<close>

ML_val \<open>
  fun subgoaler_tac ctxt =
    assume_tac ctxt ORELSE'
    resolve_tac ctxt (Simplifier.prems_of ctxt) ORELSE'
    asm_simp_tac ctxt
\<close>

text \<open>This tactic first tries to solve the subgoal by assumption or
  by resolving with with one of the premises, calling simplification
  only if that fails.\<close>


subsubsection \<open>The solver\<close>

text \<open>
  \begin{mldecls}
  @{index_ML_type solver} \\
  @{index_ML Simplifier.mk_solver: "string ->
  (Proof.context -> int -> tactic) -> solver"} \\
  @{index_ML_op setSolver: "Proof.context * solver -> Proof.context"} \\
  @{index_ML_op addSolver: "Proof.context * solver -> Proof.context"} \\
  @{index_ML_op setSSolver: "Proof.context * solver -> Proof.context"} \\
  @{index_ML_op addSSolver: "Proof.context * solver -> Proof.context"} \\
  \end{mldecls}

  A solver is a tactic that attempts to solve a subgoal after
  simplification.  Its core functionality is to prove trivial subgoals
  such as @{prop "True"} and \<open>t = t\<close>, but object-logics might
  be more ambitious.  For example, Isabelle/HOL performs a restricted
  version of linear arithmetic here.

  Solvers are packaged up in abstract type @{ML_type solver}, with
  @{ML Simplifier.mk_solver} as the only operation to create a solver.

  \<^medskip>
  Rewriting does not instantiate unknowns.  For example,
  rewriting alone cannot prove \<open>a \<in> ?A\<close> since this requires
  instantiating \<open>?A\<close>.  The solver, however, is an arbitrary
  tactic and may instantiate unknowns as it pleases.  This is the only
  way the Simplifier can handle a conditional rewrite rule whose
  condition contains extra variables.  When a simplification tactic is
  to be combined with other provers, especially with the Classical
  Reasoner, it is important whether it can be considered safe or not.
  For this reason a simpset contains two solvers: safe and unsafe.

  The standard simplification strategy solely uses the unsafe solver,
  which is appropriate in most cases.  For special applications where
  the simplification process is not allowed to instantiate unknowns
  within the goal, simplification starts with the safe solver, but may
  still apply the ordinary unsafe one in nested simplifications for
  conditional rules or congruences. Note that in this way the overall
  tactic is not totally safe: it may instantiate unknowns that appear
  also in other subgoals.

  \<^descr> @{ML Simplifier.mk_solver}~\<open>name tac\<close> turns \<open>tac\<close> into a solver; the \<open>name\<close> is only attached as a
  comment and has no further significance.

  \<^descr> \<open>ctxt setSSolver solver\<close> installs \<open>solver\<close> as
  the safe solver of \<open>ctxt\<close>.

  \<^descr> \<open>ctxt addSSolver solver\<close> adds \<open>solver\<close> as an
  additional safe solver; it will be tried after the solvers which had
  already been present in \<open>ctxt\<close>.

  \<^descr> \<open>ctxt setSolver solver\<close> installs \<open>solver\<close> as the
  unsafe solver of \<open>ctxt\<close>.

  \<^descr> \<open>ctxt addSolver solver\<close> adds \<open>solver\<close> as an
  additional unsafe solver; it will be tried after the solvers which
  had already been present in \<open>ctxt\<close>.


  \<^medskip>
  The solver tactic is invoked with the context of the
  running Simplifier.  Further operations
  may be used to retrieve relevant information, such as the list of
  local Simplifier premises via @{ML Simplifier.prems_of} --- this
  list may be non-empty only if the Simplifier runs in a mode that
  utilizes local assumptions (see also \secref{sec:simp-meth}).  The
  solver is also presented the full goal including its assumptions in
  any case.  Thus it can use these (e.g.\ by calling @{ML
  assume_tac}), even if the Simplifier proper happens to ignore local
  premises at the moment.

  \<^medskip>
  As explained before, the subgoaler is also used to solve
  the premises of congruence rules.  These are usually of the form
  \<open>s = ?x\<close>, where \<open>s\<close> needs to be simplified and
  \<open>?x\<close> needs to be instantiated with the result.  Typically,
  the subgoaler will invoke the Simplifier at some point, which will
  eventually call the solver.  For this reason, solver tactics must be
  prepared to solve goals of the form \<open>t = ?x\<close>, usually by
  reflexivity.  In particular, reflexivity should be tried before any
  of the fancy automated proof tools.

  It may even happen that due to simplification the subgoal is no
  longer an equality.  For example, \<open>False \<longleftrightarrow> ?Q\<close> could be
  rewritten to \<open>\<not> ?Q\<close>.  To cover this case, the solver could
  try resolving with the theorem \<open>\<not> False\<close> of the
  object-logic.

  \<^medskip>
  \begin{warn}
  If a premise of a congruence rule cannot be proved, then the
  congruence is ignored.  This should only happen if the rule is
  \<^emph>\<open>conditional\<close> --- that is, contains premises not of the form
  \<open>t = ?x\<close>.  Otherwise it indicates that some congruence rule,
  or possibly the subgoaler or solver, is faulty.
  \end{warn}
\<close>


subsubsection \<open>The looper\<close>

text \<open>
  \begin{mldecls}
  @{index_ML_op setloop: "Proof.context *
  (Proof.context -> int -> tactic) -> Proof.context"} \\
  @{index_ML_op addloop: "Proof.context *
  (string * (Proof.context -> int -> tactic))
  -> Proof.context"} \\
  @{index_ML_op delloop: "Proof.context * string -> Proof.context"} \\
  @{index_ML Splitter.add_split: "thm -> Proof.context -> Proof.context"} \\
  @{index_ML Splitter.del_split: "thm -> Proof.context -> Proof.context"} \\
  \end{mldecls}

  The looper is a list of tactics that are applied after
  simplification, in case the solver failed to solve the simplified
  goal.  If the looper succeeds, the simplification process is started
  all over again.  Each of the subgoals generated by the looper is
  attacked in turn, in reverse order.

  A typical looper is \<^emph>\<open>case splitting\<close>: the expansion of a
  conditional.  Another possibility is to apply an elimination rule on
  the assumptions.  More adventurous loopers could start an induction.

  \<^descr> \<open>ctxt setloop tac\<close> installs \<open>tac\<close> as the only
  looper tactic of \<open>ctxt\<close>.

  \<^descr> \<open>ctxt addloop (name, tac)\<close> adds \<open>tac\<close> as an
  additional looper tactic with name \<open>name\<close>, which is
  significant for managing the collection of loopers.  The tactic will
  be tried after the looper tactics that had already been present in
  \<open>ctxt\<close>.

  \<^descr> \<open>ctxt delloop name\<close> deletes the looper tactic that was
  associated with \<open>name\<close> from \<open>ctxt\<close>.

  \<^descr> @{ML Splitter.add_split}~\<open>thm ctxt\<close> adds split tactics
  for \<open>thm\<close> as additional looper tactics of \<open>ctxt\<close>.

  \<^descr> @{ML Splitter.del_split}~\<open>thm ctxt\<close> deletes the split
  tactic corresponding to \<open>thm\<close> from the looper tactics of
  \<open>ctxt\<close>.


  The splitter replaces applications of a given function; the
  right-hand side of the replacement can be anything.  For example,
  here is a splitting rule for conditional expressions:

  @{text [display] "?P (if ?Q ?x ?y) \<longleftrightarrow> (?Q \<longrightarrow> ?P ?x) \<and> (\<not> ?Q \<longrightarrow> ?P ?y)"}

  Another example is the elimination operator for Cartesian products
  (which happens to be called @{const case_prod} in Isabelle/HOL:

  @{text [display] "?P (case_prod ?f ?p) \<longleftrightarrow> (\<forall>a b. ?p = (a, b) \<longrightarrow> ?P (f a b))"}

  For technical reasons, there is a distinction between case splitting
  in the conclusion and in the premises of a subgoal.  The former is
  done by @{ML Splitter.split_tac} with rules like @{thm [source]
  split_if} or @{thm [source] option.split}, which do not split the
  subgoal, while the latter is done by @{ML Splitter.split_asm_tac}
  with rules like @{thm [source] split_if_asm} or @{thm [source]
  option.split_asm}, which split the subgoal.  The function @{ML
  Splitter.add_split} automatically takes care of which tactic to
  call, analyzing the form of the rules given as argument; it is the
  same operation behind \<open>split\<close> attribute or method modifier
  syntax in the Isar source language.

  Case splits should be allowed only when necessary; they are
  expensive and hard to control.  Case-splitting on if-expressions in
  the conclusion is usually beneficial, so it is enabled by default in
  Isabelle/HOL and Isabelle/FOL/ZF.

  \begin{warn}
  With @{ML Splitter.split_asm_tac} as looper component, the
  Simplifier may split subgoals!  This might cause unexpected problems
  in tactic expressions that silently assume 0 or 1 subgoals after
  simplification.
  \end{warn}
\<close>


subsection \<open>Forward simplification \label{sec:simp-forward}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{attribute_def simplified} & : & \<open>attribute\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{attribute simplified} opt? @{syntax thmrefs}?
    ;

    opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use') ')'
  \<close>}

  \<^descr> @{attribute simplified}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> causes a theorem to
  be simplified, either by exactly the specified rules \<open>a\<^sub>1, \<dots>,
  a\<^sub>n\<close>, or the implicit Simplifier context if no arguments are given.
  The result is fully simplified by default, including assumptions and
  conclusion; the options \<open>no_asm\<close> etc.\ tune the Simplifier in
  the same way as the for the \<open>simp\<close> method.

  Note that forward simplification restricts the Simplifier to its
  most basic operation of term rewriting; solver and looper tactics
  (\secref{sec:simp-strategies}) are \<^emph>\<open>not\<close> involved here.  The
  @{attribute simplified} attribute should be only rarely required
  under normal circumstances.
\<close>


section \<open>The Classical Reasoner \label{sec:classical}\<close>

subsection \<open>Basic concepts\<close>

text \<open>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):\<close>

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 \<open>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.\<close>


subsubsection \<open>The sequent calculus\<close>

text \<open>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>\<open>sequent calculus\<close>, a generalization of natural deduction,
  is easier to automate.

  A \<^bold>\<open>sequent\<close> has the form \<open>\<Gamma> \<turnstile> \<Delta>\<close>, where \<open>\<Gamma>\<close>
  and \<open>\<Delta>\<close> are sets of formulae.\footnote{For first-order
  logic, sequents can equivalently be made from lists or multisets of
  formulae.} The sequent \<open>P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n\<close> is
  \<^bold>\<open>valid\<close> if \<open>P\<^sub>1 \<and> \<dots> \<and> P\<^sub>m\<close> implies \<open>Q\<^sub>1 \<or> \<dots> \<or>
  Q\<^sub>n\<close>.  Thus \<open>P\<^sub>1, \<dots>, P\<^sub>m\<close> represent assumptions, each of which
  is true, while \<open>Q\<^sub>1, \<dots>, Q\<^sub>n\<close> represent alternative goals.  A
  sequent is \<^bold>\<open>basic\<close> if its left and right sides have a common
  formula, as in \<open>P, Q \<turnstile> Q, R\<close>; basic sequents are trivially
  valid.

  Sequent rules are classified as \<^bold>\<open>right\<close> or \<^bold>\<open>left\<close>,
  indicating which side of the \<open>\<turnstile>\<close> 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 \<open>(\<longrightarrow>I)\<close>
  is the rule
  \[
  \infer[\<open>(\<longrightarrow>R)\<close>]{\<open>\<Gamma> \<turnstile> \<Delta>, P \<longrightarrow> Q\<close>}{\<open>P, \<Gamma> \<turnstile> \<Delta>, Q\<close>}
  \]
  Applying the rule backwards, this breaks down some implication on
  the right side of a sequent; \<open>\<Gamma>\<close> and \<open>\<Delta>\<close> stand for
  the sets of formulae that are unaffected by the inference.  The
  analogue of the pair \<open>(\<or>I1)\<close> and \<open>(\<or>I2)\<close> is the
  single rule
  \[
  \infer[\<open>(\<or>R)\<close>]{\<open>\<Gamma> \<turnstile> \<Delta>, P \<or> Q\<close>}{\<open>\<Gamma> \<turnstile> \<Delta>, P, Q\<close>}
  \]
  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 \<open>(P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)\<close>.  Working
  backwards, we reduce this formula to a basic sequent:
  \[
  \infer[\<open>(\<or>R)\<close>]{\<open>\<turnstile> (P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)\<close>}
    {\infer[\<open>(\<longrightarrow>R)\<close>]{\<open>\<turnstile> (P \<longrightarrow> Q), (Q \<longrightarrow> P)\<close>}
      {\infer[\<open>(\<longrightarrow>R)\<close>]{\<open>P \<turnstile> Q, (Q \<longrightarrow> P)\<close>}
        {\<open>P, Q \<turnstile> Q, P\<close>}}}
  \]

  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 \<open>Chapter 10\<close>
  "paulson-ml2"} for further discussion.\<close>


subsubsection \<open>Simulating sequents by natural deduction\<close>

text \<open>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 \<open>P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n\<close> by the Isabelle formula
  \<open>P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>2 \<Longrightarrow> ... \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> Q\<^sub>1\<close> where the order of
  the assumptions and the choice of \<open>Q\<^sub>1\<close> 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 \<open>(\<or>E)\<close>, \<open>(\<bottom>E)\<close> and \<open>(\<exists>E)\<close> 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 \<open>(\<and>E1, 2)\<close> and \<open>(\<forall>E)\<close>.
  But note that the rule \<open>(\<not>L)\<close> has no effect under our
  representation of sequents!
  \[
  \infer[\<open>(\<not>L)\<close>]{\<open>\<not> P, \<Gamma> \<turnstile> \<Delta>\<close>}{\<open>\<Gamma> \<turnstile> \<Delta>, P\<close>}
  \]

  What about reasoning on the right?  Introduction rules can only
  affect the formula in the conclusion, namely \<open>Q\<^sub>1\<close>.  The
  other right-side formulae are represented as negated assumptions,
  \<open>\<not> Q\<^sub>2, \<dots>, \<not> Q\<^sub>n\<close>.  In order to operate on one of these, it
  must first be exchanged with \<open>Q\<^sub>1\<close>.  Elim-resolution with the
  \<open>swap\<close> rule has this effect: \<open>\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R\<close>

  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 \<open>(swap)\<close>.  The swapped form of \<open>(\<and>I)\<close>, which might be called \<open>(\<not>\<and>E)\<close>, is
  @{text [display] "\<not> (P \<and> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (\<not> R \<Longrightarrow> Q) \<Longrightarrow> R"}

  Similarly, the swapped form of \<open>(\<longrightarrow>I)\<close> 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>"}
\<close>


subsubsection \<open>Extra rules for the sequent calculus\<close>

text \<open>As mentioned, destruction rules such as \<open>(\<and>E1, 2)\<close> and
  \<open>(\<forall>E)\<close> must be replaced by sequent-style elimination rules.
  In addition, we need rules to embody the classical equivalence
  between \<open>P \<longrightarrow> Q\<close> and \<open>\<not> P \<or> Q\<close>.  The introduction
  rules \<open>(\<or>I1, 2)\<close> are replaced by a rule that simulates
  \<open>(\<or>R)\<close>: @{text [display] "(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q"}

  The destruction rule \<open>(\<longrightarrow>E)\<close> 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, \<open>\<exists>x. P x\<close> is equivalent to \<open>\<not> (\<forall>x. \<not> P x)\<close>;
  the rules \<open>(\<exists>R)\<close> and \<open>(\<forall>L)\<close> are dual:
  \[
  \infer[\<open>(\<exists>R)\<close>]{\<open>\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x\<close>}{\<open>\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x, P t\<close>}
  \qquad
  \infer[\<open>(\<forall>L)\<close>]{\<open>\<forall>x. P x, \<Gamma> \<turnstile> \<Delta>\<close>}{\<open>P t, \<forall>x. P x, \<Gamma> \<turnstile> \<Delta>\<close>}
  \]
  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
  \<open>n > 1\<close>.  Natural examples of the multiple use of an
  existential formula are rare; a standard one is \<open>\<exists>x. \<forall>y. P x
  \<longrightarrow> P y\<close>.

  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 \<open>(\<exists>R)\<close>, \<open>(\<exists>L)\<close> and
  \<open>(\<forall>R)\<close> by \<open>(\<exists>I)\<close>, \<open>(\<exists>E)\<close> and \<open>(\<forall>I)\<close>,
  respectively, and put \<open>(\<forall>E)\<close> 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 \<open>(\<exists>I)\<close> 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.
\<close>


subsection \<open>Rule declarations\<close>

text \<open>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>\<open>safe\<close> or
  \<^emph>\<open>unsafe\<close>.  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 \<open>P \<Longrightarrow> P \<or> Q\<close> is unsafe because it reduces \<open>P
  \<or> Q\<close> to \<open>P\<close>, which might turn out as premature choice of an
  unprovable subgoal.  Any rule is unsafe whose premises contain new
  unknowns.  The elimination rule \<open>\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q\<close> is
  unsafe, since it is applied via elim-resolution, which discards the
  assumption \<open>\<forall>x. P x\<close> and replaces it by the weaker
  assumption \<open>P t\<close>.  The rule \<open>P t \<Longrightarrow> \<exists>x. P x\<close> is
  unsafe for similar reasons.  The quantifier duplication rule \<open>\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q\<close> is unsafe in a different sense:
  since it keeps the assumption \<open>\<forall>x. P x\<close>, 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 \<open>(\<or>E)\<close> 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"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
    @{attribute_def intro} & : & \<open>attribute\<close> \\
    @{attribute_def elim} & : & \<open>attribute\<close> \\
    @{attribute_def dest} & : & \<open>attribute\<close> \\
    @{attribute_def rule} & : & \<open>attribute\<close> \\
    @{attribute_def iff} & : & \<open>attribute\<close> \\
    @{attribute_def swapped} & : & \<open>attribute\<close> \\
  \end{matharray}

  @{rail \<open>
    (@@{attribute intro} | @@{attribute elim} | @@{attribute dest}) ('!' | () | '?') @{syntax nat}?
    ;
    @@{attribute rule} 'del'
    ;
    @@{attribute iff} (((() | 'add') '?'?) | 'del')
  \<close>}

  \<^descr> @{command "print_claset"} prints the collection of rules
  declared to the Classical Reasoner, i.e.\ the @{ML_type claset}
  within the context.

  \<^descr> @{attribute intro}, @{attribute elim}, and @{attribute dest}
  declare introduction, elimination, and destruction rules,
  respectively.  By default, rules are considered as \<^emph>\<open>unsafe\<close>
  (i.e.\ not applied blindly without backtracking), while ``\<open>!\<close>'' classifies as \<^emph>\<open>safe\<close>.  Rule declarations marked by
  ``\<open>?\<close>'' 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.

  \<^descr> @{attribute rule}~\<open>del\<close> deletes all occurrences of a
  rule from the classical context, regardless of its classification as
  introduction~/ elimination~/ destruction and safe~/ unsafe.

  \<^descr> @{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 \<open>\<not>\<close>-elimination
  internally).

  The ``\<open>?\<close>'' version of @{attribute iff} declares rules to
  the Isabelle/Pure context only, and omits the Simplifier
  declaration.

  \<^descr> @{attribute swapped} turns an introduction rule into an
  elimination, by resolving with the classical swap principle \<open>\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R\<close> in the second position.  This is mainly for
  illustrative purposes: the Classical Reasoner already swaps rules
  internally as explained above.
\<close>


subsection \<open>Structured methods\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def rule} & : & \<open>method\<close> \\
    @{method_def contradiction} & : & \<open>method\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{method rule} @{syntax thmrefs}?
  \<close>}

  \<^descr> @{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}).

  \<^descr> @{method contradiction} solves some goal by contradiction,
  deriving any result from both \<open>\<not> A\<close> and \<open>A\<close>.  Chained
  facts, which are guaranteed to participate, may appear in either
  order.
\<close>


subsection \<open>Fully automated methods\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def blast} & : & \<open>method\<close> \\
    @{method_def auto} & : & \<open>method\<close> \\
    @{method_def force} & : & \<open>method\<close> \\
    @{method_def fast} & : & \<open>method\<close> \\
    @{method_def slow} & : & \<open>method\<close> \\
    @{method_def best} & : & \<open>method\<close> \\
    @{method_def fastforce} & : & \<open>method\<close> \\
    @{method_def slowsimp} & : & \<open>method\<close> \\
    @{method_def bestsimp} & : & \<open>method\<close> \\
    @{method_def deepen} & : & \<open>method\<close> \\
  \end{matharray}

  @{rail \<open>
    @@{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 fastforce} | @@{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}
  \<close>}

  \<^descr> @{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.

    \<^item> It does not use the classical wrapper tacticals, such as the
    integration with the Simplifier of @{method fastforce}.

    \<^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
    @{verbatim [display] \<open>Function unknown's argument not a bound variable\<close>}

    \<^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.

  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,
  \<open>(blast lim)\<close> tries to prove the goal using a search bound
  of \<open>lim\<close>.  Sometimes a slow proof using @{method blast} can
  be made much faster by supplying the successful search bound to this
  proof method instead.

  \<^descr> @{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 \<open>(auto m n)\<close> refer to its
  builtin classical reasoning procedures: \<open>m\<close> (default 4) is for
  @{method blast}, which is tried first, and \<open>n\<close> (default 2) is
  for a slower but more general alternative that also takes wrappers
  into account.

  \<^descr> @{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.

  \<^descr> @{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.

  \<^descr> @{method fastforce}, @{method slowsimp}, @{method bestsimp}
  are like @{method fast}, @{method slow}, @{method best},
  respectively, but use the Simplifier as additional wrapper. The name
  @{method fastforce}, reflects the behaviour of this popular method
  better without requiring an understanding of its implementation.

  \<^descr> @{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.


  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 \<open>simp\<close> 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.
\<close>


subsection \<open>Partially automated methods\<close>

text \<open>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} & : & \<open>method\<close> \\
    @{method_def clarify} & : & \<open>method\<close> \\
    @{method_def clarsimp} & : & \<open>method\<close> \\
  \end{matharray}

  @{rail \<open>
    (@@{method safe} | @@{method clarify}) (@{syntax clamod} * )
    ;
    @@{method clarsimp} (@{syntax clasimpmod} * )
  \<close>}

  \<^descr> @{method safe} repeatedly performs safe steps on all subgoals.
  It is deterministic, with at most one outcome.

  \<^descr> @{method clarify} performs a series of safe steps without
  splitting subgoals; see also @{method clarify_step}.

  \<^descr> @{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.
\<close>


subsection \<open>Single-step tactics\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def safe_step} & : & \<open>method\<close> \\
    @{method_def inst_step} & : & \<open>method\<close> \\
    @{method_def step} & : & \<open>method\<close> \\
    @{method_def slow_step} & : & \<open>method\<close> \\
    @{method_def clarify_step} & : &  \<open>method\<close> \\
  \end{matharray}

  These are the primitive tactics behind the automated proof methods
  of the Classical Reasoner.  By calling them yourself, you can
  execute these procedures one step at a time.

  \<^descr> @{method safe_step} performs a safe step on the first subgoal.
  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.

  \<^descr> @{method inst_step} is like @{method safe_step}, but allows
  unknowns to be instantiated.

  \<^descr> @{method step} is the basic step of the proof procedure, it
  operates on the first subgoal.  The unsafe wrapper tacticals are
  applied to a tactic that tries @{method safe}, @{method inst_step},
  or applies an unsafe rule from the context.

  \<^descr> @{method slow_step} resembles @{method step}, but allows
  backtracking between using safe rules with instantiation (@{method
  inst_step}) and using unsafe rules.  The resulting search space is
  larger.

  \<^descr> @{method clarify_step} performs a safe step on the first
  subgoal; no splitting step is applied.  For example, the subgoal
  \<open>A \<and> B\<close> is left as a conjunction.  Proof by assumption,
  Modus Ponens, etc., may be performed provided they do not
  instantiate unknowns.  Assumptions of the form \<open>x = t\<close> may
  be eliminated.  The safe wrapper tactical is applied.
\<close>


subsection \<open>Modifying the search step\<close>

text \<open>
  \begin{mldecls}
    @{index_ML_type wrapper: "(int -> tactic) -> (int -> tactic)"} \\[0.5ex]
    @{index_ML_op addSWrapper: "Proof.context *
  (string * (Proof.context -> wrapper)) -> Proof.context"} \\
    @{index_ML_op addSbefore: "Proof.context *
  (string * (Proof.context -> int -> tactic)) -> Proof.context"} \\
    @{index_ML_op addSafter: "Proof.context *
  (string * (Proof.context -> int -> tactic)) -> Proof.context"} \\
    @{index_ML_op delSWrapper: "Proof.context * string -> Proof.context"} \\[0.5ex]
    @{index_ML_op addWrapper: "Proof.context *
  (string * (Proof.context -> wrapper)) -> Proof.context"} \\
    @{index_ML_op addbefore: "Proof.context *
  (string * (Proof.context -> int -> tactic)) -> Proof.context"} \\
    @{index_ML_op addafter: "Proof.context *
  (string * (Proof.context -> int -> tactic)) -> Proof.context"} \\
    @{index_ML_op delWrapper: "Proof.context * string -> Proof.context"} \\[0.5ex]
    @{index_ML addSss: "Proof.context -> Proof.context"} \\
    @{index_ML addss: "Proof.context -> Proof.context"} \\
  \end{mldecls}

  The proof strategy of the Classical Reasoner is simple.  Perform as
  many safe inferences as possible; or else, apply certain safe rules,
  allowing instantiation of unknowns; or else, apply an unsafe rule.
  The tactics also eliminate assumptions of the form \<open>x = t\<close>
  by substitution if they have been set up to do so.  They may perform
  a form of Modus Ponens: if there are assumptions \<open>P \<longrightarrow> Q\<close> and
  \<open>P\<close>, then replace \<open>P \<longrightarrow> Q\<close> by \<open>Q\<close>.

  The classical reasoning tools --- except @{method blast} --- allow
  to modify this basic proof strategy by applying two lists of
  arbitrary \<^emph>\<open>wrapper tacticals\<close> to it.  The first wrapper list,
  which is considered to contain safe wrappers only, affects @{method
  safe_step} and all the tactics that call it.  The second one, which
  may contain unsafe wrappers, affects the unsafe parts of @{method
  step}, @{method slow_step}, and the tactics that call them.  A
  wrapper transforms each step of the search, for example by
  attempting other tactics before or after the original step tactic.
  All members of a wrapper list are applied in turn to the respective
  step tactic.

  Initially the two wrapper lists are empty, which means no
  modification of the step tactics. Safe and unsafe wrappers are added
  to the context with the functions given below, supplying them with
  wrapper names.  These names may be used to selectively delete
  wrappers.

  \<^descr> \<open>ctxt addSWrapper (name, wrapper)\<close> adds a new wrapper,
  which should yield a safe tactic, to modify the existing safe step
  tactic.

  \<^descr> \<open>ctxt addSbefore (name, tac)\<close> adds the given tactic as a
  safe wrapper, such that it is tried \<^emph>\<open>before\<close> each safe step of
  the search.

  \<^descr> \<open>ctxt addSafter (name, tac)\<close> adds the given tactic as a
  safe wrapper, such that it is tried when a safe step of the search
  would fail.

  \<^descr> \<open>ctxt delSWrapper name\<close> deletes the safe wrapper with
  the given name.

  \<^descr> \<open>ctxt addWrapper (name, wrapper)\<close> adds a new wrapper to
  modify the existing (unsafe) step tactic.

  \<^descr> \<open>ctxt addbefore (name, tac)\<close> adds the given tactic as an
  unsafe wrapper, such that it its result is concatenated
  \<^emph>\<open>before\<close> the result of each unsafe step.

  \<^descr> \<open>ctxt addafter (name, tac)\<close> adds the given tactic as an
  unsafe wrapper, such that it its result is concatenated \<^emph>\<open>after\<close>
  the result of each unsafe step.

  \<^descr> \<open>ctxt delWrapper name\<close> deletes the unsafe wrapper with
  the given name.

  \<^descr> \<open>addSss\<close> adds the simpset of the context to its
  classical set. The assumptions and goal will be simplified, in a
  rather safe way, after each safe step of the search.

  \<^descr> \<open>addss\<close> adds the simpset of the context to its
  classical set. The assumptions and goal will be simplified, before
  the each unsafe step of the search.
\<close>


section \<open>Object-logic setup \label{sec:object-logic}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def "judgment"} & : & \<open>theory \<rightarrow> theory\<close> \\
    @{method_def atomize} & : & \<open>method\<close> \\
    @{attribute_def atomize} & : & \<open>attribute\<close> \\
    @{attribute_def rule_format} & : & \<open>attribute\<close> \\
    @{attribute_def rulify} & : & \<open>attribute\<close> \\
  \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 \<open>prop\<close> that covers universal
  quantification \<open>\<And>\<close> and implication \<open>\<Longrightarrow>\<close>).

  Common object-logics are sufficiently expressive to internalize rule
  statements over \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> 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.\ \<open>\<And>x. x \<in> A \<Longrightarrow> P x\<close> versus \<open>\<forall>x \<in> A. P x\<close>.

  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 \<open>
    @@{command judgment} @{syntax name} '::' @{syntax type} @{syntax mixfix}?
    ;
    @@{attribute atomize} ('(' 'full' ')')?
    ;
    @@{attribute rule_format} ('(' 'noasm' ')')?
  \<close>}

  \<^descr> @{command "judgment"}~\<open>c :: \<sigma> (mx)\<close> declares constant
  \<open>c\<close> as the truth judgment of the current object-logic.  Its
  type \<open>\<sigma>\<close> should specify a coercion of the category of
  object-level propositions to \<open>prop\<close> of the Pure meta-logic;
  the mixfix annotation \<open>(mx)\<close> would typically just link the
  object language (internally of syntactic category \<open>logic\<close>)
  with that of \<open>prop\<close>.  Only one @{command "judgment"}
  declaration may be given in any theory development.
  
  \<^descr> @{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 ``\<open>(full)\<close>'' 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 \<open>\<And>\<close>, \<open>\<Longrightarrow>\<close>, and \<open>\<equiv>\<close>.
  Meta-level conjunction should be covered as well (this is
  particularly important for locales, see \secref{sec:locale}).

  \<^descr> @{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 \<open>(no_asm)\<close> 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
  (\<open>\<forall>\<close>) and implication (\<open>\<longrightarrow>\<close>) by the corresponding
  rule statements over \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close>.
\<close>


section \<open>Tracing higher-order unification\<close>

text \<open>
  \begin{tabular}{rcll}
    @{attribute_def unify_trace_simp} & : & \<open>attribute\<close> & default \<open>false\<close> \\
    @{attribute_def unify_trace_types} & : & \<open>attribute\<close> & default \<open>false\<close> \\
    @{attribute_def unify_trace_bound} & : & \<open>attribute\<close> & default \<open>50\<close> \\
    @{attribute_def unify_search_bound} & : & \<open>attribute\<close> & default \<open>60\<close> \\
  \end{tabular}
  \<^medskip>

  Higher-order unification works well in most practical situations,
  but sometimes needs extra care to identify problems.  These tracing
  options may help.

  \<^descr> @{attribute unify_trace_simp} controls tracing of the
  simplification phase of higher-order unification.

  \<^descr> @{attribute unify_trace_types} controls warnings of
  incompleteness, when unification is not considering all possible
  instantiations of schematic type variables.

  \<^descr> @{attribute unify_trace_bound} determines the depth where
  unification starts to print tracing information once it reaches
  depth; 0 for full tracing.  At the default value, tracing
  information is almost never printed in practice.

  \<^descr> @{attribute unify_search_bound} prevents unification from
  searching past the given depth.  Because of this bound, higher-order
  unification cannot return an infinite sequence, though it can return
  an exponentially long one.  The search rarely approaches the default
  value in practice.  If the search is cut off, unification prints a
  warning ``Unification bound exceeded''.


  \begin{warn}
  Options for unification cannot be modified in a local context.  Only
  the global theory content is taken into account.
  \end{warn}
\<close>

end