author wenzelm
Sun, 29 Jan 2012 22:12:25 +0100
changeset 46278 408e3acfdbb9
parent 46277 aea65ff733b4
permissions -rw-r--r--
updated hint about asm_rl;

theory Tactic
imports Base

chapter {* Tactical reasoning *}

text {* Tactical reasoning works by refining an initial claim in a
  backwards fashion, until a solved form is reached.  A @{text "goal"}
  consists of several subgoals that need to be solved in order to
  achieve the main statement; zero subgoals means that the proof may
  be finished.  A @{text "tactic"} is a refinement operation that maps
  a goal to a lazy sequence of potential successors.  A @{text
  "tactical"} is a combinator for composing tactics.  *}

section {* Goals \label{sec:tactical-goals} *}

text {*
  Isabelle/Pure represents a goal as a theorem stating that the
  subgoals imply the main goal: @{text "A\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^sub>n \<Longrightarrow>
  C"}.  The outermost goal structure is that of a Horn Clause: i.e.\
  an iterated implication without any quantifiers\footnote{Recall that
  outermost @{text "\<And>x. \<phi>[x]"} is always represented via schematic
  variables in the body: @{text "\<phi>[?x]"}.  These variables may get
  instantiated during the course of reasoning.}.  For @{text "n = 0"}
  a goal is called ``solved''.

  The structure of each subgoal @{text "A\<^sub>i"} is that of a
  general Hereditary Harrop Formula @{text "\<And>x\<^sub>1 \<dots>
  \<And>x\<^sub>k. H\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> H\<^sub>m \<Longrightarrow> B"}.  Here @{text
  "x\<^sub>1, \<dots>, x\<^sub>k"} are goal parameters, i.e.\
  arbitrary-but-fixed entities of certain types, and @{text
  "H\<^sub>1, \<dots>, H\<^sub>m"} are goal hypotheses, i.e.\ facts that may
  be assumed locally.  Together, this forms the goal context of the
  conclusion @{text B} to be established.  The goal hypotheses may be
  again arbitrary Hereditary Harrop Formulas, although the level of
  nesting rarely exceeds 1--2 in practice.

  The main conclusion @{text C} is internally marked as a protected
  proposition, which is represented explicitly by the notation @{text
  "#C"} here.  This ensures that the decomposition into subgoals and
  main conclusion is well-defined for arbitrarily structured claims.

  \medskip Basic goal management is performed via the following
  Isabelle/Pure rules:

  \infer[@{text "(init)"}]{@{text "C \<Longrightarrow> #C"}}{} \qquad
  \infer[@{text "(finish)"}]{@{text "C"}}{@{text "#C"}}

  \medskip The following low-level variants admit general reasoning
  with protected propositions:

  \infer[@{text "(protect)"}]{@{text "#C"}}{@{text "C"}} \qquad
  \infer[@{text "(conclude)"}]{@{text "A\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^sub>n \<Longrightarrow> C"}}{@{text "A\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^sub>n \<Longrightarrow> #C"}}

text %mlref {*
  @{index_ML Goal.init: "cterm -> thm"} \\
  @{index_ML Goal.finish: "Proof.context -> thm -> thm"} \\
  @{index_ML Goal.protect: "thm -> thm"} \\
  @{index_ML Goal.conclude: "thm -> thm"} \\


  \item @{ML "Goal.init"}~@{text C} initializes a tactical goal from
  the well-formed proposition @{text C}.

  \item @{ML "Goal.finish"}~@{text "ctxt thm"} checks whether theorem
  @{text "thm"} is a solved goal (no subgoals), and concludes the
  result by removing the goal protection.  The context is only
  required for printing error messages.

  \item @{ML "Goal.protect"}~@{text "thm"} protects the full statement
  of theorem @{text "thm"}.

  \item @{ML "Goal.conclude"}~@{text "thm"} removes the goal
  protection, even if there are pending subgoals.


section {* Tactics\label{sec:tactics} *}

text {* A @{text "tactic"} is a function @{text "goal \<rightarrow> goal\<^sup>*\<^sup>*"} that
  maps a given goal state (represented as a theorem, cf.\
  \secref{sec:tactical-goals}) to a lazy sequence of potential
  successor states.  The underlying sequence implementation is lazy
  both in head and tail, and is purely functional in \emph{not}
  supporting memoing.\footnote{The lack of memoing and the strict
  nature of SML requires some care when working with low-level
  sequence operations, to avoid duplicate or premature evaluation of
  results.  It also means that modified runtime behavior, such as
  timeout, is very hard to achieve for general tactics.}

  An \emph{empty result sequence} means that the tactic has failed: in
  a compound tactic expression other tactics might be tried instead,
  or the whole refinement step might fail outright, producing a
  toplevel error message in the end.  When implementing tactics from
  scratch, one should take care to observe the basic protocol of
  mapping regular error conditions to an empty result; only serious
  faults should emerge as exceptions.

  By enumerating \emph{multiple results}, a tactic can easily express
  the potential outcome of an internal search process.  There are also
  combinators for building proof tools that involve search
  systematically, see also \secref{sec:tacticals}.

  \medskip As explained before, a goal state essentially consists of a
  list of subgoals that imply the main goal (conclusion).  Tactics may
  operate on all subgoals or on a particularly specified subgoal, but
  must not change the main conclusion (apart from instantiating
  schematic goal variables).

  Tactics with explicit \emph{subgoal addressing} are of the form
  @{text "int \<rightarrow> tactic"} and may be applied to a particular subgoal
  (counting from 1).  If the subgoal number is out of range, the
  tactic should fail with an empty result sequence, but must not raise
  an exception!

  Operating on a particular subgoal means to replace it by an interval
  of zero or more subgoals in the same place; other subgoals must not
  be affected, apart from instantiating schematic variables ranging
  over the whole goal state.

  A common pattern of composing tactics with subgoal addressing is to
  try the first one, and then the second one only if the subgoal has
  not been solved yet.  Special care is required here to avoid bumping
  into unrelated subgoals that happen to come after the original
  subgoal.  Assuming that there is only a single initial subgoal is a
  very common error when implementing tactics!

  Tactics with internal subgoal addressing should expose the subgoal
  index as @{text "int"} argument in full generality; a hardwired
  subgoal 1 is not acceptable.
  \medskip The main well-formedness conditions for proper tactics are
  summarized as follows.


  \item General tactic failure is indicated by an empty result, only
  serious faults may produce an exception.

  \item The main conclusion must not be changed, apart from
  instantiating schematic variables.

  \item A tactic operates either uniformly on all subgoals, or
  specifically on a selected subgoal (without bumping into unrelated

  \item Range errors in subgoal addressing produce an empty result.


  Some of these conditions are checked by higher-level goal
  infrastructure (\secref{sec:struct-goals}); others are not checked
  explicitly, and violating them merely results in ill-behaved tactics
  experienced by the user (e.g.\ tactics that insist in being
  applicable only to singleton goals, or prevent composition via
  standard tacticals such as @{ML REPEAT}).

text %mlref {*
  @{index_ML_type tactic: "thm -> thm Seq.seq"} \\
  @{index_ML no_tac: tactic} \\
  @{index_ML all_tac: tactic} \\
  @{index_ML print_tac: "string -> tactic"} \\[1ex]
  @{index_ML PRIMITIVE: "(thm -> thm) -> tactic"} \\[1ex]
  @{index_ML SUBGOAL: "(term * int -> tactic) -> int -> tactic"} \\
  @{index_ML CSUBGOAL: "(cterm * int -> tactic) -> int -> tactic"} \\


  \item Type @{ML_type tactic} represents tactics.  The
  well-formedness conditions described above need to be observed.  See
  also @{file "~~/src/Pure/General/seq.ML"} for the underlying
  implementation of lazy sequences.

  \item Type @{ML_type "int -> tactic"} represents tactics with
  explicit subgoal addressing, with well-formedness conditions as
  described above.

  \item @{ML no_tac} is a tactic that always fails, returning the
  empty sequence.

  \item @{ML all_tac} is a tactic that always succeeds, returning a
  singleton sequence with unchanged goal state.

  \item @{ML print_tac}~@{text "message"} is like @{ML all_tac}, but
  prints a message together with the goal state on the tracing

  \item @{ML PRIMITIVE}~@{text rule} turns a primitive inference rule
  into a tactic with unique result.  Exception @{ML THM} is considered
  a regular tactic failure and produces an empty result; other
  exceptions are passed through.

  \item @{ML SUBGOAL}~@{text "(fn (subgoal, i) => tactic)"} is the
  most basic form to produce a tactic with subgoal addressing.  The
  given abstraction over the subgoal term and subgoal number allows to
  peek at the relevant information of the full goal state.  The
  subgoal range is checked as required above.

  \item @{ML CSUBGOAL} is similar to @{ML SUBGOAL}, but passes the
  subgoal as @{ML_type cterm} instead of raw @{ML_type term}.  This
  avoids expensive re-certification in situations where the subgoal is
  used directly for primitive inferences.


subsection {* Resolution and assumption tactics \label{sec:resolve-assume-tac} *}

text {* \emph{Resolution} is the most basic mechanism for refining a
  subgoal using a theorem as object-level rule.
  \emph{Elim-resolution} is particularly suited for elimination rules:
  it resolves with a rule, proves its first premise by assumption, and
  finally deletes that assumption from any new subgoals.
  \emph{Destruct-resolution} is like elim-resolution, but the given
  destruction rules are first turned into canonical elimination
  format.  \emph{Forward-resolution} is like destruct-resolution, but
  without deleting the selected assumption.  The @{text "r/e/d/f"}
  naming convention is maintained for several different kinds of
  resolution rules and tactics.

  Assumption tactics close a subgoal by unifying some of its premises
  against its conclusion.

  \medskip All the tactics in this section operate on a subgoal
  designated by a positive integer.  Other subgoals might be affected
  indirectly, due to instantiation of schematic variables.

  There are various sources of non-determinism, the tactic result
  sequence enumerates all possibilities of the following choices (if


  \item selecting one of the rules given as argument to the tactic;

  \item selecting a subgoal premise to eliminate, unifying it against
  the first premise of the rule;

  \item unifying the conclusion of the subgoal to the conclusion of
  the rule.


  Recall that higher-order unification may produce multiple results
  that are enumerated here.

text %mlref {*
  @{index_ML resolve_tac: "thm list -> int -> tactic"} \\
  @{index_ML eresolve_tac: "thm list -> int -> tactic"} \\
  @{index_ML dresolve_tac: "thm list -> int -> tactic"} \\
  @{index_ML forward_tac: "thm list -> int -> tactic"} \\[1ex]
  @{index_ML assume_tac: "int -> tactic"} \\
  @{index_ML eq_assume_tac: "int -> tactic"} \\[1ex]
  @{index_ML match_tac: "thm list -> int -> tactic"} \\
  @{index_ML ematch_tac: "thm list -> int -> tactic"} \\
  @{index_ML dmatch_tac: "thm list -> int -> tactic"} \\


  \item @{ML resolve_tac}~@{text "thms i"} refines the goal state
  using the given theorems, which should normally be introduction
  rules.  The tactic resolves a rule's conclusion with subgoal @{text
  i}, replacing it by the corresponding versions of the rule's

  \item @{ML eresolve_tac}~@{text "thms i"} performs elim-resolution
  with the given theorems, which are normally be elimination rules.

  Note that @{ML "eresolve_tac [asm_rl]"} is equivalent to @{ML
  assume_tac}, which facilitates mixing of assumption steps with
  genuine eliminations.

  \item @{ML dresolve_tac}~@{text "thms i"} performs
  destruct-resolution with the given theorems, which should normally
  be destruction rules.  This replaces an assumption by the result of
  applying one of the rules.

  \item @{ML forward_tac} is like @{ML dresolve_tac} except that the
  selected assumption is not deleted.  It applies a rule to an
  assumption, adding the result as a new assumption.

  \item @{ML assume_tac}~@{text i} attempts to solve subgoal @{text i}
  by assumption (modulo higher-order unification).

  \item @{ML eq_assume_tac} is similar to @{ML assume_tac}, but checks
  only for immediate @{text "\<alpha>"}-convertibility instead of using
  unification.  It succeeds (with a unique next state) if one of the
  assumptions is equal to the subgoal's conclusion.  Since it does not
  instantiate variables, it cannot make other subgoals unprovable.

  \item @{ML match_tac}, @{ML ematch_tac}, and @{ML dmatch_tac} are
  similar to @{ML resolve_tac}, @{ML eresolve_tac}, and @{ML
  dresolve_tac}, respectively, but do not instantiate schematic
  variables in the goal state.

  Flexible subgoals are not updated at will, but are left alone.
  Strictly speaking, matching means to treat the unknowns in the goal
  state as constants; these tactics merely discard unifiers that would
  update the goal state.


subsection {* Explicit instantiation within a subgoal context *}

text {* The main resolution tactics (\secref{sec:resolve-assume-tac})
  use higher-order unification, which works well in many practical
  situations despite its daunting theoretical properties.
  Nonetheless, there are important problem classes where unguided
  higher-order unification is not so useful.  This typically involves
  rules like universal elimination, existential introduction, or
  equational substitution.  Here the unification problem involves
  fully flexible @{text "?P ?x"} schemes, which are hard to manage
  without further hints.

  By providing a (small) rigid term for @{text "?x"} explicitly, the
  remaining unification problem is to assign a (large) term to @{text
  "?P"}, according to the shape of the given subgoal.  This is
  sufficiently well-behaved in most practical situations.

  \medskip Isabelle provides separate versions of the standard @{text
  "r/e/d/f"} resolution tactics that allow to provide explicit
  instantiations of unknowns of the given rule, wrt.\ terms that refer
  to the implicit context of the selected subgoal.

  An instantiation consists of a list of pairs of the form @{text
  "(?x, t)"}, where @{text ?x} is a schematic variable occurring in
  the given rule, and @{text t} is a term from the current proof
  context, augmented by the local goal parameters of the selected
  subgoal; cf.\ the @{text "focus"} operation described in

  Entering the syntactic context of a subgoal is a brittle operation,
  because its exact form is somewhat accidental, and the choice of
  bound variable names depends on the presence of other local and
  global names.  Explicit renaming of subgoal parameters prior to
  explicit instantiation might help to achieve a bit more robustness.

  Type instantiations may be given as well, via pairs like @{text
  "(?'a, \<tau>)"}.  Type instantiations are distinguished from term
  instantiations by the syntactic form of the schematic variable.
  Types are instantiated before terms are.  Since term instantiation
  already performs simple type-inference, so explicit type
  instantiations are seldom necessary.

text %mlref {*
  @{index_ML res_inst_tac: "Proof.context -> (indexname * string) list -> thm -> int -> tactic"} \\
  @{index_ML eres_inst_tac: "Proof.context -> (indexname * string) list -> thm -> int -> tactic"} \\
  @{index_ML dres_inst_tac: "Proof.context -> (indexname * string) list -> thm -> int -> tactic"} \\
  @{index_ML forw_inst_tac: "Proof.context -> (indexname * string) list -> thm -> int -> tactic"} \\
  @{index_ML subgoal_tac: "Proof.context -> string -> int -> tactic"} \\
  @{index_ML thin_tac: "Proof.context -> string -> int -> tactic"} \\
  @{index_ML rename_tac: "string list -> int -> tactic"} \\


  \item @{ML res_inst_tac}~@{text "ctxt insts thm i"} instantiates the
  rule @{text thm} with the instantiations @{text insts}, as described
  above, and then performs resolution on subgoal @{text i}.
  \item @{ML eres_inst_tac} is like @{ML res_inst_tac}, but performs

  \item @{ML dres_inst_tac} is like @{ML res_inst_tac}, but performs

  \item @{ML forw_inst_tac} is like @{ML dres_inst_tac} except that
  the selected assumption is not deleted.

  \item @{ML subgoal_tac}~@{text "ctxt \<phi> i"} adds the proposition
  @{text "\<phi>"} as local premise to subgoal @{text "i"}, and poses the
  same as a new subgoal @{text "i + 1"} (in the original context).

  \item @{ML thin_tac}~@{text "ctxt \<phi> i"} deletes the specified
  premise from subgoal @{text i}.  Note that @{text \<phi>} may contain
  schematic variables, to abbreviate the intended proposition; the
  first matching subgoal premise will be deleted.  Removing useless
  premises from a subgoal increases its readability and can make
  search tactics run faster.

  \item @{ML rename_tac}~@{text "names i"} renames the innermost
  parameters of subgoal @{text i} according to the provided @{text
  names} (which need to be distinct indentifiers).


  For historical reasons, the above instantiation tactics take
  unparsed string arguments, which makes them hard to use in general
  ML code.  The slightly more advanced @{ML Subgoal.FOCUS} combinator
  of \secref{sec:struct-goals} allows to refer to internal goal
  structure with explicit context management.

subsection {* Rearranging goal states *}

text {* In rare situations there is a need to rearrange goal states:
  either the overall collection of subgoals, or the local structure of
  a subgoal.  Various administrative tactics allow to operate on the
  concrete presentation these conceptual sets of formulae. *}

text %mlref {*
  @{index_ML rotate_tac: "int -> int -> tactic"} \\
  @{index_ML distinct_subgoals_tac: tactic} \\
  @{index_ML flexflex_tac: tactic} \\


  \item @{ML rotate_tac}~@{text "n i"} rotates the premises of subgoal
  @{text i} by @{text n} positions: from right to left if @{text n} is
  positive, and from left to right if @{text n} is negative.

  \item @{ML distinct_subgoals_tac} removes duplicate subgoals from a
  proof state.  This is potentially inefficient.

  \item @{ML flexflex_tac} removes all flex-flex pairs from the proof
  state by applying the trivial unifier.  This drastic step loses
  information.  It is already part of the Isar infrastructure for
  facts resulting from goals, and rarely needs to be invoked manually.

  Flex-flex constraints arise from difficult cases of higher-order
  unification.  To prevent this, use @{ML res_inst_tac} to instantiate
  some variables in a rule.  Normally flex-flex constraints can be
  ignored; they often disappear as unknowns get instantiated.


section {* Tacticals \label{sec:tacticals} *}

text {* A \emph{tactical} is a functional combinator for building up
  complex tactics from simpler ones.  Common tacticals perform
  sequential composition, disjunctive choice, iteration, or goal
  addressing.  Various search strategies may be expressed via

subsection {* Combining tactics *}

text {* Sequential composition and alternative choices are the most
  basic ways to combine tactics, similarly to ``@{verbatim ","}'' and
  ``@{verbatim "|"}'' in Isar method notation.  This corresponds to
  @{ML_op "THEN"} and @{ML_op "ORELSE"} in ML, but there are further
  possibilities for fine-tuning alternation of tactics such as @{ML_op
  "APPEND"}.  Further details become visible in ML due to explicit
  subgoal addressing.

text %mlref {*
  @{index_ML_op "THEN": "tactic * tactic -> tactic"} \\
  @{index_ML_op "ORELSE": "tactic * tactic -> tactic"} \\
  @{index_ML_op "APPEND": "tactic * tactic -> tactic"} \\
  @{index_ML "EVERY": "tactic list -> tactic"} \\
  @{index_ML "FIRST": "tactic list -> tactic"} \\[0.5ex]

  @{index_ML_op "THEN'": "('a -> tactic) * ('a -> tactic) -> 'a -> tactic"} \\
  @{index_ML_op "ORELSE'": "('a -> tactic) * ('a -> tactic) -> 'a -> tactic"} \\
  @{index_ML_op "APPEND'": "('a -> tactic) * ('a -> tactic) -> 'a -> tactic"} \\
  @{index_ML "EVERY'": "('a -> tactic) list -> 'a -> tactic"} \\
  @{index_ML "FIRST'": "('a -> tactic) list -> 'a -> tactic"} \\


  \item @{text "tac\<^sub>1"}~@{ML_op THEN}~@{text "tac\<^sub>2"} is the sequential
  composition of @{text "tac\<^sub>1"} and @{text "tac\<^sub>2"}.  Applied to a goal
  state, it returns all states reachable in two steps by applying
  @{text "tac\<^sub>1"} followed by @{text "tac\<^sub>2"}.  First, it applies @{text
  "tac\<^sub>1"} to the goal state, getting a sequence of possible next
  states; then, it applies @{text "tac\<^sub>2"} to each of these and
  concatenates the results to produce again one flat sequence of

  \item @{text "tac\<^sub>1"}~@{ML_op ORELSE}~@{text "tac\<^sub>2"} makes a choice
  between @{text "tac\<^sub>1"} and @{text "tac\<^sub>2"}.  Applied to a state, it
  tries @{text "tac\<^sub>1"} and returns the result if non-empty; if @{text
  "tac\<^sub>1"} fails then it uses @{text "tac\<^sub>2"}.  This is a deterministic
  choice: if @{text "tac\<^sub>1"} succeeds then @{text "tac\<^sub>2"} is excluded
  from the result.

  \item @{text "tac\<^sub>1"}~@{ML_op APPEND}~@{text "tac\<^sub>2"} concatenates the
  possible results of @{text "tac\<^sub>1"} and @{text "tac\<^sub>2"}.  Unlike
  @{ML_op "ORELSE"} there is \emph{no commitment} to either tactic, so
  @{ML_op "APPEND"} helps to avoid incompleteness during search, at
  the cost of potential inefficiencies.

  \item @{ML EVERY}~@{text "[tac\<^sub>1, \<dots>, tac\<^sub>n]"} abbreviates @{text
  "tac\<^sub>1"}~@{ML_op THEN}~@{text "\<dots>"}~@{ML_op THEN}~@{text "tac\<^sub>n"}.
  Note that @{ML "EVERY []"} is the same as @{ML all_tac}: it always

  \item @{ML FIRST}~@{text "[tac\<^sub>1, \<dots>, tac\<^sub>n]"} abbreviates @{text
  "tac\<^sub>1"}~@{ML_op ORELSE}~@{text "\<dots>"}~@{ML_op "ORELSE"}~@{text
  "tac\<^sub>n"}.  Note that @{ML "FIRST []"} is the same as @{ML no_tac}: it
  always fails.

  \item @{ML_op "THEN'"} is the lifted version of @{ML_op "THEN"}, for
  tactics with explicit subgoal addressing.  So @{text
  "(tac\<^sub>1"}~@{ML_op THEN'}~@{text "tac\<^sub>2) i"} is the same as @{text
  "(tac\<^sub>1 i"}~@{ML_op THEN}~@{text "tac\<^sub>2 i)"}.

  The other primed tacticals work analogously.


subsection {* Repetition tacticals *}

text {* These tacticals provide further control over repetition of
  tactics, beyond the stylized forms of ``@{verbatim "?"}''  and
  ``@{verbatim "+"}'' in Isar method expressions. *}

text %mlref {*
  @{index_ML "TRY": "tactic -> tactic"} \\
  @{index_ML "REPEAT": "tactic -> tactic"} \\
  @{index_ML "REPEAT1": "tactic -> tactic"} \\
  @{index_ML "REPEAT_DETERM": "tactic -> tactic"} \\
  @{index_ML "REPEAT_DETERM_N": "int -> tactic -> tactic"} \\


  \item @{ML TRY}~@{text "tac"} applies @{text "tac"} to the goal
  state and returns the resulting sequence, if non-empty; otherwise it
  returns the original state.  Thus, it applies @{text "tac"} at most

  Note that for tactics with subgoal addressing, the combinator can be
  applied via functional composition: @{ML "TRY"}~@{ML_op o}~@{text
  "tac"}.  There is no need for @{verbatim TRY'}.

  \item @{ML REPEAT}~@{text "tac"} applies @{text "tac"} to the goal
  state and, recursively, to each element of the resulting sequence.
  The resulting sequence consists of those states that make @{text
  "tac"} fail.  Thus, it applies @{text "tac"} as many times as
  possible (including zero times), and allows backtracking over each
  invocation of @{text "tac"}.  @{ML REPEAT} is more general than @{ML
  REPEAT_DETERM}, but requires more space.

  \item @{ML REPEAT1}~@{text "tac"} is like @{ML REPEAT}~@{text "tac"}
  but it always applies @{text "tac"} at least once, failing if this
  is impossible.

  \item @{ML REPEAT_DETERM}~@{text "tac"} applies @{text "tac"} to the
  goal state and, recursively, to the head of the resulting sequence.
  It returns the first state to make @{text "tac"} fail.  It is
  deterministic, discarding alternative outcomes.

  \item @{ML REPEAT_DETERM_N}~@{text "n tac"} is like @{ML
  REPEAT_DETERM}~@{text "tac"} but the number of repetitions is bound
  by @{text "n"} (where @{ML "~1"} means @{text "\<infinity>"}).


text %mlex {* The basic tactics and tacticals considered above follow
  some algebraic laws:


  \item @{ML all_tac} is the identity element of the tactical @{ML_op

  \item @{ML no_tac} is the identity element of @{ML_op "ORELSE"} and
  @{ML_op "APPEND"}.  Also, it is a zero element for @{ML_op "THEN"},
  which means that @{text "tac"}~@{ML_op THEN}~@{ML no_tac} is
  equivalent to @{ML no_tac}.

  \item @{ML TRY} and @{ML REPEAT} can be expressed as (recursive)
  functions over more basic combinators (ignoring some internal
  implementation tricks):


ML {*
  fun TRY tac = tac ORELSE all_tac;
  fun REPEAT tac st = ((tac THEN REPEAT tac) ORELSE all_tac) st;

text {* If @{text "tac"} can return multiple outcomes then so can @{ML
  REPEAT}~@{text "tac"}.  @{ML REPEAT} uses @{ML_op "ORELSE"} and not
  @{ML_op "APPEND"}, it applies @{text "tac"} as many times as
  possible in each outcome.

  Note the explicit abstraction over the goal state in the ML
  definition of @{ML REPEAT}.  Recursive tacticals must be coded in
  this awkward fashion to avoid infinite recursion of eager functional
  evaluation in Standard ML.  The following attempt would make @{ML
  REPEAT}~@{text "tac"} loop:

ML {*
  (*BAD -- does not terminate!*)
  fun REPEAT tac = (tac THEN REPEAT tac) ORELSE all_tac;

subsection {* Applying tactics to subgoal ranges *}

text {* Tactics with explicit subgoal addressing
  @{ML_type "int -> tactic"} can be used together with tacticals that
  act like ``subgoal quantifiers'': guided by success of the body
  tactic a certain range of subgoals is covered.  Thus the body tactic
  is applied to \emph{all} subgoals, \emph{some} subgoal etc.

  Suppose that the goal state has @{text "n \<ge> 0"} subgoals.  Many of
  these tacticals address subgoal ranges counting downwards from
  @{text "n"} towards @{text "1"}.  This has the fortunate effect that
  newly emerging subgoals are concatenated in the result, without
  interfering each other.  Nonetheless, there might be situations
  where a different order is desired. *}

text %mlref {*
  @{index_ML ALLGOALS: "(int -> tactic) -> tactic"} \\
  @{index_ML SOMEGOAL: "(int -> tactic) -> tactic"} \\
  @{index_ML FIRSTGOAL: "(int -> tactic) -> tactic"} \\
  @{index_ML HEADGOAL: "(int -> tactic) -> tactic"} \\
  @{index_ML REPEAT_SOME: "(int -> tactic) -> tactic"} \\
  @{index_ML REPEAT_FIRST: "(int -> tactic) -> tactic"} \\
  @{index_ML RANGE: "(int -> tactic) list -> int -> tactic"} \\


  \item @{ML ALLGOALS}~@{text "tac"} is equivalent to @{text "tac
  n"}~@{ML_op THEN}~@{text "\<dots>"}~@{ML_op THEN}~@{text "tac 1"}.  It
  applies the @{text tac} to all the subgoals, counting downwards.

  \item @{ML SOMEGOAL}~@{text "tac"} is equivalent to @{text "tac
  n"}~@{ML_op ORELSE}~@{text "\<dots>"}~@{ML_op ORELSE}~@{text "tac 1"}.  It
  applies @{text "tac"} to one subgoal, counting downwards.

  \item @{ML FIRSTGOAL}~@{text "tac"} is equivalent to @{text "tac
  1"}~@{ML_op ORELSE}~@{text "\<dots>"}~@{ML_op ORELSE}~@{text "tac n"}.  It
  applies @{text "tac"} to one subgoal, counting upwards.

  \item @{ML HEADGOAL}~@{text "tac"} is equivalent to @{text "tac 1"}.
  It applies @{text "tac"} unconditionally to the first subgoal.

  \item @{ML REPEAT_SOME}~@{text "tac"} applies @{text "tac"} once or
  more to a subgoal, counting downwards.

  \item @{ML REPEAT_FIRST}~@{text "tac"} applies @{text "tac"} once or
  more to a subgoal, counting upwards.

  \item @{ML RANGE}~@{text "[tac\<^sub>1, \<dots>, tac\<^sub>k] i"} is equivalent to
  @{text "tac\<^sub>k (i + k - 1)"}~@{ML_op THEN}~@{text "\<dots>"}~@{ML_op
  THEN}~@{text "tac\<^sub>1 i"}.  It applies the given list of tactics to the
  corresponding range of subgoals, counting downwards.


subsection {* Control and search tacticals *}

text {* A predicate on theorems @{ML_type "thm -> bool"} can test
  whether a goal state enjoys some desirable property --- such as
  having no subgoals.  Tactics that search for satisfactory goal
  states are easy to express.  The main search procedures,
  depth-first, breadth-first and best-first, are provided as
  tacticals.  They generate the search tree by repeatedly applying a
  given tactic.  *}

text %mlref ""

subsubsection {* Filtering a tactic's results *}

text {*
  @{index_ML FILTER: "(thm -> bool) -> tactic -> tactic"} \\
  @{index_ML CHANGED: "tactic -> tactic"} \\


  \item @{ML FILTER}~@{text "sat tac"} applies @{text "tac"} to the
  goal state and returns a sequence consisting of those result goal
  states that are satisfactory in the sense of @{text "sat"}.

  \item @{ML CHANGED}~@{text "tac"} applies @{text "tac"} to the goal
  state and returns precisely those states that differ from the
  original state (according to @{ML Thm.eq_thm}).  Thus @{ML
  CHANGED}~@{text "tac"} always has some effect on the state.


subsubsection {* Depth-first search *}

text {*
  @{index_ML DEPTH_FIRST: "(thm -> bool) -> tactic -> tactic"} \\
  @{index_ML DEPTH_SOLVE: "tactic -> tactic"} \\
  @{index_ML DEPTH_SOLVE_1: "tactic -> tactic"} \\


  \item @{ML DEPTH_FIRST}~@{text "sat tac"} returns the goal state if
  @{text "sat"} returns true.  Otherwise it applies @{text "tac"},
  then recursively searches from each element of the resulting
  sequence.  The code uses a stack for efficiency, in effect applying
  @{text "tac"}~@{ML_op THEN}~@{ML DEPTH_FIRST}~@{text "sat tac"} to
  the state.

  \item @{ML DEPTH_SOLVE}@{text "tac"} uses @{ML DEPTH_FIRST} to
  search for states having no subgoals.

  \item @{ML DEPTH_SOLVE_1}~@{text "tac"} uses @{ML DEPTH_FIRST} to
  search for states having fewer subgoals than the given state.  Thus,
  it insists upon solving at least one subgoal.


subsubsection {* Other search strategies *}

text {*
  @{index_ML BREADTH_FIRST: "(thm -> bool) -> tactic -> tactic"} \\
  @{index_ML BEST_FIRST: "(thm -> bool) * (thm -> int) -> tactic -> tactic"} \\
  @{index_ML THEN_BEST_FIRST: "tactic -> (thm -> bool) * (thm -> int) -> tactic -> tactic"} \\

  These search strategies will find a solution if one exists.
  However, they do not enumerate all solutions; they terminate after
  the first satisfactory result from @{text "tac"}.


  \item @{ML BREADTH_FIRST}~@{text "sat tac"} uses breadth-first
  search to find states for which @{text "sat"} is true.  For most
  applications, it is too slow.

  \item @{ML BEST_FIRST}~@{text "(sat, dist) tac"} does a heuristic
  search, using @{text "dist"} to estimate the distance from a
  satisfactory state (in the sense of @{text "sat"}).  It maintains a
  list of states ordered by distance.  It applies @{text "tac"} to the
  head of this list; if the result contains any satisfactory states,
  then it returns them.  Otherwise, @{ML BEST_FIRST} adds the new
  states to the list, and continues.

  The distance function is typically @{ML size_of_thm}, which computes
  the size of the state.  The smaller the state, the fewer and simpler
  subgoals it has.

  \item @{ML THEN_BEST_FIRST}~@{text "tac\<^sub>0 (sat, dist) tac"} is like
  @{ML BEST_FIRST}, except that the priority queue initially contains
  the result of applying @{text "tac\<^sub>0"} to the goal state.  This
  tactical permits separate tactics for starting the search and
  continuing the search.


subsubsection {* Auxiliary tacticals for searching *}

text {*
  @{index_ML COND: "(thm -> bool) -> tactic -> tactic -> tactic"} \\
  @{index_ML IF_UNSOLVED: "tactic -> tactic"} \\
  @{index_ML SOLVE: "tactic -> tactic"} \\
  @{index_ML DETERM: "tactic -> tactic"} \\


  \item @{ML COND}~@{text "sat tac\<^sub>1 tac\<^sub>2"} applies @{text "tac\<^sub>1"} to
  the goal state if it satisfies predicate @{text "sat"}, and applies
  @{text "tac\<^sub>2"}.  It is a conditional tactical in that only one of
  @{text "tac\<^sub>1"} and @{text "tac\<^sub>2"} is applied to a goal state.
  However, both @{text "tac\<^sub>1"} and @{text "tac\<^sub>2"} are evaluated
  because ML uses eager evaluation.

  \item @{ML IF_UNSOLVED}~@{text "tac"} applies @{text "tac"} to the
  goal state if it has any subgoals, and simply returns the goal state
  otherwise.  Many common tactics, such as @{ML resolve_tac}, fail if
  applied to a goal state that has no subgoals.

  \item @{ML SOLVE}~@{text "tac"} applies @{text "tac"} to the goal
  state and then fails iff there are subgoals left.

  \item @{ML DETERM}~@{text "tac"} applies @{text "tac"} to the goal
  state and returns the head of the resulting sequence.  @{ML DETERM}
  limits the search space by making its argument deterministic.


subsubsection {* Predicates and functions useful for searching *}

text {*
  @{index_ML has_fewer_prems: "int -> thm -> bool"} \\
  @{index_ML Thm.eq_thm: "thm * thm -> bool"} \\
  @{index_ML Thm.eq_thm_prop: "thm * thm -> bool"} \\
  @{index_ML size_of_thm: "thm -> int"} \\


  \item @{ML has_fewer_prems}~@{text "n thm"} reports whether @{text
  "thm"} has fewer than @{text "n"} premises.

  \item @{ML Thm.eq_thm}~@{text "(thm\<^sub>1, thm\<^sub>2)"} reports whether @{text
  "thm\<^sub>1"} and @{text "thm\<^sub>2"} are equal.  Both theorems must have
  compatible background theories.  Both theorems must have the same
  conclusions, the same set of hypotheses, and the same set of sort
  hypotheses.  Names of bound variables are ignored as usual.

  \item @{ML Thm.eq_thm_prop}~@{text "(thm\<^sub>1, thm\<^sub>2)"} reports whether
  the propositions of @{text "thm\<^sub>1"} and @{text "thm\<^sub>2"} are equal.
  Names of bound variables are ignored.

  \item @{ML size_of_thm}~@{text "thm"} computes the size of @{text
  "thm"}, namely the number of variables, constants and abstractions
  in its conclusion.  It may serve as a distance function for