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doc-src/Ref/tactic.tex

author | oheimb |

Wed, 12 Nov 1997 18:58:50 +0100 | |

changeset 4223 | f60e3d2c81d3 |

parent 3950 | e9d5bcae8351 |

child 4276 | a770eae2cdb0 |

permissions | -rw-r--r-- |

added thin_refl to hyp_subst_tac

%% $Id$ \chapter{Tactics} \label{tactics} \index{tactics|(} Tactics have type \mltydx{tactic}. This is just an abbreviation for functions from theorems to theorem sequences, where the theorems represent states of a backward proof. Tactics seldom need to be coded from scratch, as functions; instead they are expressed using basic tactics and tacticals. This chapter only presents the primitive tactics. Substantial proofs require the power of simplification (Chapter~\ref{chap:simplification}) and classical reasoning (Chapter~\ref{chap:classical}). \section{Resolution and assumption tactics} {\bf Resolution} is Isabelle's basic mechanism for refining a subgoal using a rule. {\bf Elim-resolution} is particularly suited for elimination rules, while {\bf destruct-resolution} is particularly suited for destruction rules. The {\tt r}, {\tt e}, {\tt d} naming convention is maintained for several different kinds of resolution tactics, as well as the shortcuts in the subgoal module. All the tactics in this section act on a subgoal designated by a positive integer~$i$. They fail (by returning the empty sequence) if~$i$ is out of range. \subsection{Resolution tactics} \index{resolution!tactics} \index{tactics!resolution|bold} \begin{ttbox} resolve_tac : thm list -> int -> tactic eresolve_tac : thm list -> int -> tactic dresolve_tac : thm list -> int -> tactic forward_tac : thm list -> int -> tactic \end{ttbox} These perform resolution on a list of theorems, $thms$, representing a list of object-rules. When generating next states, they take each of the rules in the order given. Each rule may yield several next states, or none: higher-order resolution may yield multiple resolvents. \begin{ttdescription} \item[\ttindexbold{resolve_tac} {\it thms} {\it i}] refines the proof state using the rules, which should normally be introduction rules. It resolves a rule's conclusion with subgoal~$i$ of the proof state. \item[\ttindexbold{eresolve_tac} {\it thms} {\it i}] \index{elim-resolution} performs elim-resolution with the rules, which should normally be elimination rules. It resolves with a rule, solves its first premise by assumption, and finally {\em deletes\/} that assumption from any new subgoals. \item[\ttindexbold{dresolve_tac} {\it thms} {\it i}] \index{forward proof}\index{destruct-resolution} performs destruct-resolution with the rules, which normally should be destruction rules. This replaces an assumption by the result of applying one of the rules. \item[\ttindexbold{forward_tac}]\index{forward proof} is like {\tt dresolve_tac} except that the selected assumption is not deleted. It applies a rule to an assumption, adding the result as a new assumption. \end{ttdescription} \subsection{Assumption tactics} \index{tactics!assumption|bold}\index{assumptions!tactics for} \begin{ttbox} assume_tac : int -> tactic eq_assume_tac : int -> tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{assume_tac} {\it i}] attempts to solve subgoal~$i$ by assumption. \item[\ttindexbold{eq_assume_tac}] is like {\tt assume_tac} but does not use unification. It succeeds (with a {\em unique\/} next state) if one of the assumptions is identical to the subgoal's conclusion. Since it does not instantiate variables, it cannot make other subgoals unprovable. It is intended to be called from proof strategies, not interactively. \end{ttdescription} \subsection{Matching tactics} \label{match_tac} \index{tactics!matching} \begin{ttbox} match_tac : thm list -> int -> tactic ematch_tac : thm list -> int -> tactic dmatch_tac : thm list -> int -> tactic \end{ttbox} These are just like the resolution tactics except that they never instantiate unknowns in the proof state. Flexible subgoals are not updated willy-nilly, but are left alone. Matching --- strictly speaking --- means treating the unknowns in the proof state as constants; these tactics merely discard unifiers that would update the proof state. \begin{ttdescription} \item[\ttindexbold{match_tac} {\it thms} {\it i}] refines the proof state using the rules, matching a rule's conclusion with subgoal~$i$ of the proof state. \item[\ttindexbold{ematch_tac}] is like {\tt match_tac}, but performs elim-resolution. \item[\ttindexbold{dmatch_tac}] is like {\tt match_tac}, but performs destruct-resolution. \end{ttdescription} \subsection{Resolution with instantiation} \label{res_inst_tac} \index{tactics!instantiation}\index{instantiation} \begin{ttbox} res_inst_tac : (string*string)list -> thm -> int -> tactic eres_inst_tac : (string*string)list -> thm -> int -> tactic dres_inst_tac : (string*string)list -> thm -> int -> tactic forw_inst_tac : (string*string)list -> thm -> int -> tactic \end{ttbox} These tactics are designed for applying rules such as substitution and induction, which cause difficulties for higher-order unification. The tactics accept explicit instantiations for unknowns in the rule --- typically, in the rule's conclusion. Each instantiation is a pair {\tt($v$,$e$)}, where $v$ is an unknown {\em without\/} its leading question mark! \begin{itemize} \item If $v$ is the type unknown {\tt'a}, then the rule must contain a type unknown \verb$?'a$ of some sort~$s$, and $e$ should be a type of sort $s$. \item If $v$ is the unknown {\tt P}, then the rule must contain an unknown \verb$?P$ of some type~$\tau$, and $e$ should be a term of some type~$\sigma$ such that $\tau$ and $\sigma$ are unifiable. If the unification of $\tau$ and $\sigma$ instantiates any type unknowns in $\tau$, these instantiations are recorded for application to the rule. \end{itemize} Types are instantiated before terms. Because type instantiations are inferred from term instantiations, explicit type instantiations are seldom necessary --- if \verb$?t$ has type \verb$?'a$, then the instantiation list \verb$[("'a","bool"),("t","True")]$ may be simplified to \verb$[("t","True")]$. Type unknowns in the proof state may cause failure because the tactics cannot instantiate them. The instantiation tactics act on a given subgoal. Terms in the instantiations are type-checked in the context of that subgoal --- in particular, they may refer to that subgoal's parameters. Any unknowns in the terms receive subscripts and are lifted over the parameters; thus, you may not refer to unknowns in the subgoal. \begin{ttdescription} \item[\ttindexbold{res_inst_tac} {\it insts} {\it thm} {\it i}] instantiates the rule {\it thm} with the instantiations {\it insts}, as described above, and then performs resolution on subgoal~$i$. Resolution typically causes further instantiations; you need not give explicit instantiations for every unknown in the rule. \item[\ttindexbold{eres_inst_tac}] is like {\tt res_inst_tac}, but performs elim-resolution. \item[\ttindexbold{dres_inst_tac}] is like {\tt res_inst_tac}, but performs destruct-resolution. \item[\ttindexbold{forw_inst_tac}] is like {\tt dres_inst_tac} except that the selected assumption is not deleted. It applies the instantiated rule to an assumption, adding the result as a new assumption. \end{ttdescription} \section{Other basic tactics} \subsection{Tactic shortcuts} \index{shortcuts!for tactics} \index{tactics!resolution}\index{tactics!assumption} \index{tactics!meta-rewriting} \begin{ttbox} rtac : thm -> int -> tactic etac : thm -> int -> tactic dtac : thm -> int -> tactic atac : int -> tactic ares_tac : thm list -> int -> tactic rewtac : thm -> tactic \end{ttbox} These abbreviate common uses of tactics. \begin{ttdescription} \item[\ttindexbold{rtac} {\it thm} {\it i}] abbreviates \hbox{\tt resolve_tac [{\it thm}] {\it i}}, doing resolution. \item[\ttindexbold{etac} {\it thm} {\it i}] abbreviates \hbox{\tt eresolve_tac [{\it thm}] {\it i}}, doing elim-resolution. \item[\ttindexbold{dtac} {\it thm} {\it i}] abbreviates \hbox{\tt dresolve_tac [{\it thm}] {\it i}}, doing destruct-resolution. \item[\ttindexbold{atac} {\it i}] abbreviates \hbox{\tt assume_tac {\it i}}, doing proof by assumption. \item[\ttindexbold{ares_tac} {\it thms} {\it i}] tries proof by assumption and resolution; it abbreviates \begin{ttbox} assume_tac {\it i} ORELSE resolve_tac {\it thms} {\it i} \end{ttbox} \item[\ttindexbold{rewtac} {\it def}] abbreviates \hbox{\tt rewrite_goals_tac [{\it def}]}, unfolding a definition. \end{ttdescription} \subsection{Inserting premises and facts}\label{cut_facts_tac} \index{tactics!for inserting facts}\index{assumptions!inserting} \begin{ttbox} cut_facts_tac : thm list -> int -> tactic cut_inst_tac : (string*string)list -> thm -> int -> tactic subgoal_tac : string -> int -> tactic subgoal_tacs : string list -> int -> tactic \end{ttbox} These tactics add assumptions to a subgoal. \begin{ttdescription} \item[\ttindexbold{cut_facts_tac} {\it thms} {\it i}] adds the {\it thms} as new assumptions to subgoal~$i$. Once they have been inserted as assumptions, they become subject to tactics such as {\tt eresolve_tac} and {\tt rewrite_goals_tac}. Only rules with no premises are inserted: Isabelle cannot use assumptions that contain $\Imp$ or~$\Forall$. Sometimes the theorems are premises of a rule being derived, returned by~{\tt goal}; instead of calling this tactic, you could state the goal with an outermost meta-quantifier. \item[\ttindexbold{cut_inst_tac} {\it insts} {\it thm} {\it i}] instantiates the {\it thm} with the instantiations {\it insts}, as described in \S\ref{res_inst_tac}. It adds the resulting theorem as a new assumption to subgoal~$i$. \item[\ttindexbold{subgoal_tac} {\it formula} {\it i}] adds the {\it formula} as a assumption to subgoal~$i$, and inserts the same {\it formula} as a new subgoal, $i+1$. \item[\ttindexbold{subgoals_tac} {\it formulae} {\it i}] uses {\tt subgoal_tac} to add the members of the list of {\it formulae} as assumptions to subgoal~$i$. \end{ttdescription} \subsection{``Putting off'' a subgoal} \begin{ttbox} defer_tac : int -> tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{defer_tac} {\it i}] moves subgoal~$i$ to the last position in the proof state. It can be useful when correcting a proof script: if the tactic given for subgoal~$i$ fails, calling {\tt defer_tac} instead will let you continue with the rest of the script. The tactic fails if subgoal~$i$ does not exist or if the proof state contains type unknowns. \end{ttdescription} \subsection{Definitions and meta-level rewriting} \index{tactics!meta-rewriting|bold}\index{meta-rewriting|bold} \index{definitions} Definitions in Isabelle have the form $t\equiv u$, where $t$ is typically a constant or a constant applied to a list of variables, for example $\it sqr(n)\equiv n\times n$. (Conditional definitions, $\phi\Imp t\equiv u$, are not supported.) {\bf Unfolding} the definition ${t\equiv u}$ means using it as a rewrite rule, replacing~$t$ by~$u$ throughout a theorem. {\bf Folding} $t\equiv u$ means replacing~$u$ by~$t$. Rewriting continues until no rewrites are applicable to any subterm. There are rules for unfolding and folding definitions; Isabelle does not do this automatically. The corresponding tactics rewrite the proof state, yielding a single next state. See also the {\tt goalw} command, which is the easiest way of handling definitions. \begin{ttbox} rewrite_goals_tac : thm list -> tactic rewrite_tac : thm list -> tactic fold_goals_tac : thm list -> tactic fold_tac : thm list -> tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{rewrite_goals_tac} {\it defs}] unfolds the {\it defs} throughout the subgoals of the proof state, while leaving the main goal unchanged. Use \ttindex{SELECT_GOAL} to restrict it to a particular subgoal. \item[\ttindexbold{rewrite_tac} {\it defs}] unfolds the {\it defs} throughout the proof state, including the main goal --- not normally desirable! \item[\ttindexbold{fold_goals_tac} {\it defs}] folds the {\it defs} throughout the subgoals of the proof state, while leaving the main goal unchanged. \item[\ttindexbold{fold_tac} {\it defs}] folds the {\it defs} throughout the proof state. \end{ttdescription} \subsection{Theorems useful with tactics} \index{theorems!of pure theory} \begin{ttbox} asm_rl: thm cut_rl: thm \end{ttbox} \begin{ttdescription} \item[\tdx{asm_rl}] is $\psi\Imp\psi$. Under elim-resolution it does proof by assumption, and \hbox{\tt eresolve_tac (asm_rl::{\it thms}) {\it i}} is equivalent to \begin{ttbox} assume_tac {\it i} ORELSE eresolve_tac {\it thms} {\it i} \end{ttbox} \item[\tdx{cut_rl}] is $\List{\psi\Imp\theta,\psi}\Imp\theta$. It is useful for inserting assumptions; it underlies {\tt forward_tac}, {\tt cut_facts_tac} and {\tt subgoal_tac}. \end{ttdescription} \section{Obscure tactics} \subsection{Renaming parameters in a goal} \index{parameters!renaming} \begin{ttbox} rename_tac : string -> int -> tactic rename_last_tac : string -> string list -> int -> tactic Logic.set_rename_prefix : string -> unit Logic.auto_rename : bool ref \hfill{\bf initially false} \end{ttbox} When creating a parameter, Isabelle chooses its name by matching variable names via the object-rule. Given the rule $(\forall I)$ formalized as $\left(\Forall x. P(x)\right) \Imp \forall x.P(x)$, Isabelle will note that the $\Forall$-bound variable in the premise has the same name as the $\forall$-bound variable in the conclusion. Sometimes there is insufficient information and Isabelle chooses an arbitrary name. The renaming tactics let you override Isabelle's choice. Because renaming parameters has no logical effect on the proof state, the {\tt by} command prints the message {\tt Warning:\ same as previous level}. Alternatively, you can suppress the naming mechanism described above and have Isabelle generate uniform names for parameters. These names have the form $p${\tt a}, $p${\tt b}, $p${\tt c},~\ldots, where $p$ is any desired prefix. They are ugly but predictable. \begin{ttdescription} \item[\ttindexbold{rename_tac} {\it str} {\it i}] interprets the string {\it str} as a series of blank-separated variable names, and uses them to rename the parameters of subgoal~$i$. The names must be distinct. If there are fewer names than parameters, then the tactic renames the innermost parameters and may modify the remaining ones to ensure that all the parameters are distinct. \item[\ttindexbold{rename_last_tac} {\it prefix} {\it suffixes} {\it i}] generates a list of names by attaching each of the {\it suffixes\/} to the {\it prefix}. It is intended for coding structural induction tactics, where several of the new parameters should have related names. \item[\ttindexbold{Logic.set_rename_prefix} {\it prefix};] sets the prefix for uniform renaming to~{\it prefix}. The default prefix is {\tt"k"}. \item[\ttindexbold{Logic.auto_rename} := true;] makes Isabelle generate uniform names for parameters. \end{ttdescription} \subsection{Manipulating assumptions} \index{assumptions!rotating} \begin{ttbox} thin_tac : string -> int -> tactic rotate_tac : int -> int -> tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{thin_tac} {\it formula} $i$] \index{assumptions!deleting} deletes the specified assumption from subgoal $i$. Often the assumption can be abbreviated, replacing subformul{\ae} by unknowns; the first matching assumption will be deleted. Removing useless assumptions from a subgoal increases its readability and can make search tactics run faster. \item[\ttindexbold{rotate_tac} $n$ $i$] \index{assumptions!rotating} rotates the assumptions of subgoal $i$ by $n$ positions: from right to left if $n$ is positive, and from left to right if $n$ is negative. This is sometimes necessary in connection with \ttindex{asm_full_simp_tac}, which processes assumptions from left to right. \end{ttdescription} \subsection{Tidying the proof state} \index{duplicate subgoals!removing} \index{parameters!removing unused} \index{flex-flex constraints} \begin{ttbox} distinct_subgoals_tac : tactic prune_params_tac : tactic flexflex_tac : tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{distinct_subgoals_tac}] removes duplicate subgoals from a proof state. (These arise especially in \ZF{}, where the subgoals are essentially type constraints.) \item[\ttindexbold{prune_params_tac}] removes unused parameters from all subgoals of the proof state. It works by rewriting with the theorem $(\Forall x. V)\equiv V$. This tactic can make the proof state more readable. It is used with \ttindex{rule_by_tactic} to simplify the resulting theorem. \item[\ttindexbold{flexflex_tac}] removes all flex-flex pairs from the proof state by applying the trivial unifier. This drastic step loses information, and should only be done as the last step of a proof. Flex-flex constraints arise from difficult cases of higher-order unification. To prevent this, use \ttindex{res_inst_tac} to instantiate some variables in a rule~(\S\ref{res_inst_tac}). Normally flex-flex constraints can be ignored; they often disappear as unknowns get instantiated. \end{ttdescription} \subsection{Composition: resolution without lifting} \index{tactics!for composition} \begin{ttbox} compose_tac: (bool * thm * int) -> int -> tactic \end{ttbox} {\bf Composing} two rules means resolving them without prior lifting or renaming of unknowns. This low-level operation, which underlies the resolution tactics, may occasionally be useful for special effects. A typical application is \ttindex{res_inst_tac}, which lifts and instantiates a rule, then passes the result to {\tt compose_tac}. \begin{ttdescription} \item[\ttindexbold{compose_tac} ($flag$, $rule$, $m$) $i$] refines subgoal~$i$ using $rule$, without lifting. The $rule$ is taken to have the form $\List{\psi@1; \ldots; \psi@m} \Imp \psi$, where $\psi$ need not be atomic; thus $m$ determines the number of new subgoals. If $flag$ is {\tt true} then it performs elim-resolution --- it solves the first premise of~$rule$ by assumption and deletes that assumption. \end{ttdescription} \section{Managing lots of rules} These operations are not intended for interactive use. They are concerned with the processing of large numbers of rules in automatic proof strategies. Higher-order resolution involving a long list of rules is slow. Filtering techniques can shorten the list of rules given to resolution, and can also detect whether a subgoal is too flexible, with too many rules applicable. \subsection{Combined resolution and elim-resolution} \label{biresolve_tac} \index{tactics!resolution} \begin{ttbox} biresolve_tac : (bool*thm)list -> int -> tactic bimatch_tac : (bool*thm)list -> int -> tactic subgoals_of_brl : bool*thm -> int lessb : (bool*thm) * (bool*thm) -> bool \end{ttbox} {\bf Bi-resolution} takes a list of $\it (flag,rule)$ pairs. For each pair, it applies resolution if the flag is~{\tt false} and elim-resolution if the flag is~{\tt true}. A single tactic call handles a mixture of introduction and elimination rules. \begin{ttdescription} \item[\ttindexbold{biresolve_tac} {\it brls} {\it i}] refines the proof state by resolution or elim-resolution on each rule, as indicated by its flag. It affects subgoal~$i$ of the proof state. \item[\ttindexbold{bimatch_tac}] is like {\tt biresolve_tac}, but performs matching: unknowns in the proof state are never updated (see~\S\ref{match_tac}). \item[\ttindexbold{subgoals_of_brl}({\it flag},{\it rule})] returns the number of new subgoals that bi-resolution would yield for the pair (if applied to a suitable subgoal). This is $n$ if the flag is {\tt false} and $n-1$ if the flag is {\tt true}, where $n$ is the number of premises of the rule. Elim-resolution yields one fewer subgoal than ordinary resolution because it solves the major premise by assumption. \item[\ttindexbold{lessb} ({\it brl1},{\it brl2})] returns the result of \begin{ttbox} subgoals_of_brl{\it brl1} < subgoals_of_brl{\it brl2} \end{ttbox} \end{ttdescription} Note that \hbox{\tt sort lessb {\it brls}} sorts a list of $\it (flag,rule)$ pairs by the number of new subgoals they will yield. Thus, those that yield the fewest subgoals should be tried first. \subsection{Discrimination nets for fast resolution}\label{filt_resolve_tac} \index{discrimination nets|bold} \index{tactics!resolution} \begin{ttbox} net_resolve_tac : thm list -> int -> tactic net_match_tac : thm list -> int -> tactic net_biresolve_tac: (bool*thm) list -> int -> tactic net_bimatch_tac : (bool*thm) list -> int -> tactic filt_resolve_tac : thm list -> int -> int -> tactic could_unify : term*term->bool filter_thms : (term*term->bool) -> int*term*thm list -> thm list \end{ttbox} The module {\tt Net} implements a discrimination net data structure for fast selection of rules \cite[Chapter 14]{charniak80}. A term is classified by the symbol list obtained by flattening it in preorder. The flattening takes account of function applications, constants, and free and bound variables; it identifies all unknowns and also regards \index{lambda abs@$\lambda$-abstractions} $\lambda$-abstractions as unknowns, since they could $\eta$-contract to anything. A discrimination net serves as a polymorphic dictionary indexed by terms. The module provides various functions for inserting and removing items from nets. It provides functions for returning all items whose term could match or unify with a target term. The matching and unification tests are overly lax (due to the identifications mentioned above) but they serve as useful filters. A net can store introduction rules indexed by their conclusion, and elimination rules indexed by their major premise. Isabelle provides several functions for `compiling' long lists of rules into fast resolution tactics. When supplied with a list of theorems, these functions build a discrimination net; the net is used when the tactic is applied to a goal. To avoid repeatedly constructing the nets, use currying: bind the resulting tactics to \ML{} identifiers. \begin{ttdescription} \item[\ttindexbold{net_resolve_tac} {\it thms}] builds a discrimination net to obtain the effect of a similar call to {\tt resolve_tac}. \item[\ttindexbold{net_match_tac} {\it thms}] builds a discrimination net to obtain the effect of a similar call to {\tt match_tac}. \item[\ttindexbold{net_biresolve_tac} {\it brls}] builds a discrimination net to obtain the effect of a similar call to {\tt biresolve_tac}. \item[\ttindexbold{net_bimatch_tac} {\it brls}] builds a discrimination net to obtain the effect of a similar call to {\tt bimatch_tac}. \item[\ttindexbold{filt_resolve_tac} {\it thms} {\it maxr} {\it i}] uses discrimination nets to extract the {\it thms} that are applicable to subgoal~$i$. If more than {\it maxr\/} theorems are applicable then the tactic fails. Otherwise it calls {\tt resolve_tac}. This tactic helps avoid runaway instantiation of unknowns, for example in type inference. \item[\ttindexbold{could_unify} ({\it t},{\it u})] returns {\tt false} if~$t$ and~$u$ are `obviously' non-unifiable, and otherwise returns~{\tt true}. It assumes all variables are distinct, reporting that {\tt ?a=?a} may unify with {\tt 0=1}. \item[\ttindexbold{filter_thms} $could\; (limit,prem,thms)$] returns the list of potentially resolvable rules (in {\it thms\/}) for the subgoal {\it prem}, using the predicate {\it could\/} to compare the conclusion of the subgoal with the conclusion of each rule. The resulting list is no longer than {\it limit}. \end{ttdescription} \section{Programming tools for proof strategies} Do not consider using the primitives discussed in this section unless you really need to code tactics from scratch. \subsection{Operations on type {\tt tactic}} \index{tactics!primitives for coding} A tactic maps theorems to sequences of theorems. The type constructor for sequences (lazy lists) is called \mltydx{Sequence.seq}. To simplify the types of tactics and tacticals, Isabelle defines a type abbreviation: \begin{ttbox} type tactic = thm -> thm Sequence.seq \end{ttbox} The following operations provide means for coding tactics in a clean style. \begin{ttbox} PRIMITIVE : (thm -> thm) -> tactic SUBGOAL : ((term*int) -> tactic) -> int -> tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{PRIMITIVE} $f$] packages the meta-rule~$f$ as a tactic that applies $f$ to the proof state and returns the result as a one-element sequence. If $f$ raises an exception, then the tactic's result is the empty sequence. \item[\ttindexbold{SUBGOAL} $f$ $i$] extracts subgoal~$i$ from the proof state as a term~$t$, and computes a tactic by calling~$f(t,i)$. It applies the resulting tactic to the same state. The tactic body is expressed using tactics and tacticals, but may peek at a particular subgoal: \begin{ttbox} SUBGOAL (fn (t,i) => {\it tactic-valued expression}) \end{ttbox} \end{ttdescription} \subsection{Tracing} \index{tactics!tracing} \index{tracing!of tactics} \begin{ttbox} pause_tac: tactic print_tac: tactic \end{ttbox} These tactics print tracing information when they are applied to a proof state. Their output may be difficult to interpret. Note that certain of the searching tacticals, such as {\tt REPEAT}, have built-in tracing options. \begin{ttdescription} \item[\ttindexbold{pause_tac}] prints {\footnotesize\tt** Press RETURN to continue:} and then reads a line from the terminal. If this line is blank then it returns the proof state unchanged; otherwise it fails (which may terminate a repetition). \item[\ttindexbold{print_tac}] returns the proof state unchanged, with the side effect of printing it at the terminal. \end{ttdescription} \section{Sequences} \index{sequences (lazy lists)|bold} The module {\tt Sequence} declares a type of lazy lists. It uses Isabelle's type \mltydx{option} to represent the possible presence (\ttindexbold{Some}) or absence (\ttindexbold{None}) of a value: \begin{ttbox} datatype 'a option = None | Some of 'a; \end{ttbox} For clarity, the module name {\tt Sequence} is omitted from the signature specifications below; for instance, {\tt null} appears instead of {\tt Sequence.null}. \subsection{Basic operations on sequences} \begin{ttbox} null : 'a seq seqof : (unit -> ('a * 'a seq) option) -> 'a seq single : 'a -> 'a seq pull : 'a seq -> ('a * 'a seq) option \end{ttbox} \begin{ttdescription} \item[Sequence.null] is the empty sequence. \item[\tt Sequence.seqof (fn()=> Some($x$,$s$))] constructs the sequence with head~$x$ and tail~$s$, neither of which is evaluated. \item[Sequence.single $x$] constructs the sequence containing the single element~$x$. \item[Sequence.pull $s$] returns {\tt None} if the sequence is empty and {\tt Some($x$,$s'$)} if the sequence has head~$x$ and tail~$s'$. Warning: calling \hbox{Sequence.pull $s$} again will {\it recompute\/} the value of~$x$; it is not stored! \end{ttdescription} \subsection{Converting between sequences and lists} \begin{ttbox} chop : int * 'a seq -> 'a list * 'a seq list_of_s : 'a seq -> 'a list s_of_list : 'a list -> 'a seq \end{ttbox} \begin{ttdescription} \item[Sequence.chop($n$,$s$)] returns the first~$n$ elements of~$s$ as a list, paired with the remaining elements of~$s$. If $s$ has fewer than~$n$ elements, then so will the list. \item[Sequence.list_of_s $s$] returns the elements of~$s$, which must be finite, as a list. \item[Sequence.s_of_list $l$] creates a sequence containing the elements of~$l$. \end{ttdescription} \subsection{Combining sequences} \begin{ttbox} append : 'a seq * 'a seq -> 'a seq interleave : 'a seq * 'a seq -> 'a seq flats : 'a seq seq -> 'a seq maps : ('a -> 'b) -> 'a seq -> 'b seq filters : ('a -> bool) -> 'a seq -> 'a seq \end{ttbox} \begin{ttdescription} \item[Sequence.append($s@1$,$s@2$)] concatenates $s@1$ to $s@2$. \item[Sequence.interleave($s@1$,$s@2$)] joins $s@1$ with $s@2$ by interleaving their elements. The result contains all the elements of the sequences, even if both are infinite. \item[Sequence.flats $ss$] concatenates a sequence of sequences. \item[Sequence.maps $f$ $s$] applies $f$ to every element of~$s=x@1,x@2,\ldots$, yielding the sequence $f(x@1),f(x@2),\ldots$. \item[Sequence.filters $p$ $s$] returns the sequence consisting of all elements~$x$ of~$s$ such that $p(x)$ is {\tt true}. \end{ttdescription} \index{tactics|)}