diff -r 3a1aef73b2b2 -r aea5d7fa7ef5 doc-src/IsarImplementation/Thy/tactic.thy --- a/doc-src/IsarImplementation/Thy/tactic.thy Wed Mar 04 11:05:02 2009 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,420 +0,0 @@ - -(* $Id$ *) - -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 \ \ \ A\<^sub>n \ C"}. The outermost goal - structure is that of a Horn Clause\glossary{Horn Clause}{An iterated - implication @{text "A\<^sub>1 \ \ \ A\<^sub>n \ 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 "\"} and @{text "\"} - connectives).}: i.e.\ an iterated implication without any - quantifiers\footnote{Recall that outermost @{text "\x. \[x]"} is - always represented via schematic variables in the body: @{text - "\[?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 "\x\<^sub>1 \ \x\<^sub>k. H\<^sub>1 \ \ \ H\<^sub>m \ B"} in - normal form. Here @{text "x\<^sub>1, \, x\<^sub>k"} are goal parameters, i.e.\ - arbitrary-but-fixed entities of certain types, and @{text "H\<^sub>1, \, - 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: - - \[ - \infer[@{text "(init)"}]{@{text "C \ #C"}}{} \qquad - \infer[@{text "(finish)"}]{@{text "C"}}{@{text "#C"}} - \] - - \medskip The following low-level variants admit general reasoning - with protected propositions: - - \[ - \infer[@{text "(protect)"}]{@{text "#C"}}{@{text "C"}} \qquad - \infer[@{text "(conclude)"}]{@{text "A\<^sub>1 \ \ \ A\<^sub>n \ C"}}{@{text "A\<^sub>1 \ \ \ A\<^sub>n \ #C"}} - \] -*} - -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 \ 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 \ 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 "\"}-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, \)"}. 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 -