doc-src/IsarImplementation/Thy/tactic.thy

-
-(* $Id$ *)
-
-theory tactic imports base begin
-
-chapter {* Tactical reasoning *}
-
-text {*
-  Tactical reasoning works by refining the 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\glossary{Tactical goal}{A theorem of
-  \seeglossary{Horn Clause} form stating that a number of subgoals
-  imply the main conclusion, which is marked as a protected
-  proposition.} 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\glossary{Horn Clause}{An iterated
-  implication @{text "A\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^sub>n \<Longrightarrow> C"}, without any
-  outermost quantifiers.  Strictly speaking, propositions @{text
-  "A\<^sub>i"} need to be atomic in Horn Clauses, but Isabelle admits
-  arbitrary substructure here (nested @{text "\<Longrightarrow>"} and @{text "\<And>"}
-  connectives).}: 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"} in
-  normal form.  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\glossary{Protected proposition}{An arbitrarily
-  structured proposition @{text "C"} which is forced to appear as
-  atomic by wrapping it into a propositional identity operator;
-  notation @{text "#C"}.  Protecting a proposition prevents basic
-  inferences from entering into that structure for the time being.},
-  which is represented explicitly by the notation @{text "#C"}.  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 {*
-  \begin{mldecls}
-  @{index_ML Goal.init: "cterm -> thm"} \\
-  @{index_ML Goal.finish: "thm -> thm"} \\
-  @{index_ML Goal.protect: "thm -> thm"} \\
-  @{index_ML Goal.conclude: "thm -> thm"} \\
-  \end{mldecls}
-
-  \begin{description}
-
-  \item @{ML "Goal.init"}~@{text C} initializes a tactical goal from
-  the well-formed proposition @{text C}.
-
-  \item @{ML "Goal.finish"}~@{text "thm"} checks whether theorem
-  @{text "thm"} is a solved goal (no subgoals), and concludes the
-  result by removing the goal protection.
-
-  \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.
-
-  \end{description}
-*}
-
-
-section {* 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.}
-
-  An \emph{empty result sequence} means that the tactic has failed: in
-  a compound tactic expressions other tactics might be tried instead,
-  or the whole refinement step might fail outright, producing a
-  toplevel error message.  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 in \secref{sec:tactical-goals}, 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 inappropriate.
-  
-  \medskip The main well-formedness conditions for proper tactics are
-  summarized as follows.
-
-  \begin{itemize}
-
-  \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
-  subgoals).
-
-  \item Range errors in subgoal addressing produce an empty result.
-
-  \end{itemize}
-
-  Some of these conditions are checked by higher-level goal
-  infrastructure (\secref{sec:results}); 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 disallow composition with
-  basic tacticals).
-*}
-
-text %mlref {*
-  \begin{mldecls}
-  @{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"} \\
-  \end{mldecls}
-
-  \begin{description}
-
-  \item @{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 @{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
-  channel.
-
-  \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.
-
-  \end{description}
-*}
-
-
-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
-  applicable):
-
-  \begin{enumerate}
-
-  \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.
-
-  \end{enumerate}
-
-  Recall that higher-order unification may produce multiple results
-  that are enumerated here.
-*}
-
-text %mlref {*
-  \begin{mldecls}
-  @{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"} \\
-  \end{mldecls}
-
-  \begin{description}
-
-  \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
-  premises.
-
-  \item @{ML eresolve_tac}~@{text "thms i"} performs elim-resolution
-  with the given theorems, which should normally be elimination rules.
-
-  \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.
-
-  \end{description}
-*}
-
-
-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
-  \secref{sec:variables}.
-
-  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 type-inference as expected, explicit type
-  instantiations are seldom necessary.
-*}
-
-text %mlref {*
-  \begin{mldecls}
-  @{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"} \\[1ex]
-  @{index_ML rename_tac: "string list -> int -> tactic"} \\
-  \end{mldecls}
-
-  \begin{description}
-
-  \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
-  elim-resolution.
-
-  \item @{ML dres_inst_tac} is like @{ML res_inst_tac}, but performs
-  destruct-resolution.
-
-  \item @{ML forw_inst_tac} is like @{ML dres_inst_tac} except that
-  the selected assumption is not deleted.
-
-  \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).
-
-  \end{description}
-*}
-
-
-section {* Tacticals \label{sec:tacticals} *}
-
-text {*
-
-FIXME
-
-\glossary{Tactical}{A functional combinator for building up complex
-tactics from simpler ones.  Typical tactical perform sequential
-composition, disjunction (choice), iteration, or goal addressing.
-Various search strategies may be expressed via tacticals.}
-
-*}
-
-end
-