doc-src/Ref/tactic.tex
author paulson
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%% $Id$
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\chapter{Tactics} \label{tactics}
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\index{tactics|(}
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Tactics have type \mltydx{tactic}.  They are essentially
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functions from theorems to theorem sequences, where the theorems represent
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states of a backward proof.  Tactics seldom need to be coded from scratch,
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as functions; instead they are expressed using basic tactics and tacticals.
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This chapter only presents the primitive tactics.  Substantial proofs require
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the power of simplification (Chapter~\ref{simp-chap}) and classical reasoning
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(Chapter~\ref{chap:classical}).
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\section{Resolution and assumption tactics}
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{\bf Resolution} is Isabelle's basic mechanism for refining a subgoal using
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a rule.  {\bf Elim-resolution} is particularly suited for elimination
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rules, while {\bf destruct-resolution} is particularly suited for
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destruction rules.  The {\tt r}, {\tt e}, {\tt d} naming convention is
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maintained for several different kinds of resolution tactics, as well as
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the shortcuts in the subgoal module.
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All the tactics in this section act on a subgoal designated by a positive
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integer~$i$.  They fail (by returning the empty sequence) if~$i$ is out of
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range.
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\subsection{Resolution tactics}
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\index{resolution!tactics}
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\index{tactics!resolution|bold}
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\begin{ttbox} 
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resolve_tac  : thm list -> int -> tactic
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eresolve_tac : thm list -> int -> tactic
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dresolve_tac : thm list -> int -> tactic
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forward_tac  : thm list -> int -> tactic 
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\end{ttbox}
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These perform resolution on a list of theorems, $thms$, representing a list
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of object-rules.  When generating next states, they take each of the rules
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in the order given.  Each rule may yield several next states, or none:
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higher-order resolution may yield multiple resolvents.
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\begin{ttdescription}
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\item[\ttindexbold{resolve_tac} {\it thms} {\it i}] 
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  refines the proof state using the rules, which should normally be
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  introduction rules.  It resolves a rule's conclusion with
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  subgoal~$i$ of the proof state.
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\item[\ttindexbold{eresolve_tac} {\it thms} {\it i}] 
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  \index{elim-resolution}
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  performs elim-resolution with the rules, which should normally be
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  elimination rules.  It resolves with a rule, solves its first premise by
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  assumption, and finally {\em deletes\/} that assumption from any new
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  subgoals.
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\item[\ttindexbold{dresolve_tac} {\it thms} {\it i}] 
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  \index{forward proof}\index{destruct-resolution}
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  performs destruct-resolution with the rules, which normally should
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  be destruction rules.  This replaces an assumption by the result of
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  applying one of the rules.
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\item[\ttindexbold{forward_tac}]\index{forward proof}
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  is like {\tt dresolve_tac} except that the selected assumption is not
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  deleted.  It applies a rule to an assumption, adding the result as a new
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  assumption.
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\end{ttdescription}
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\subsection{Assumption tactics}
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\index{tactics!assumption|bold}\index{assumptions!tactics for}
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\begin{ttbox} 
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assume_tac    : int -> tactic
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eq_assume_tac : int -> tactic
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\end{ttbox} 
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\begin{ttdescription}
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\item[\ttindexbold{assume_tac} {\it i}] 
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attempts to solve subgoal~$i$ by assumption.
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\item[\ttindexbold{eq_assume_tac}] 
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is like {\tt assume_tac} but does not use unification.  It succeeds (with a
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{\em unique\/} next state) if one of the assumptions is identical to the
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subgoal's conclusion.  Since it does not instantiate variables, it cannot
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make other subgoals unprovable.  It is intended to be called from proof
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strategies, not interactively.
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\end{ttdescription}
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\subsection{Matching tactics} \label{match_tac}
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\index{tactics!matching}
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\begin{ttbox} 
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match_tac  : thm list -> int -> tactic
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ematch_tac : thm list -> int -> tactic
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dmatch_tac : thm list -> int -> tactic
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\end{ttbox}
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These are just like the resolution tactics except that they never
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instantiate unknowns in the proof state.  Flexible subgoals are not updated
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willy-nilly, but are left alone.  Matching --- strictly speaking --- means
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treating the unknowns in the proof state as constants; these tactics merely
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discard unifiers that would update the proof state.
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\begin{ttdescription}
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\item[\ttindexbold{match_tac} {\it thms} {\it i}] 
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refines the proof state using the rules, matching a rule's
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conclusion with subgoal~$i$ of the proof state.
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\item[\ttindexbold{ematch_tac}] 
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is like {\tt match_tac}, but performs elim-resolution.
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\item[\ttindexbold{dmatch_tac}] 
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is like {\tt match_tac}, but performs destruct-resolution.
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\end{ttdescription}
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\subsection{Resolution with instantiation} \label{res_inst_tac}
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\index{tactics!instantiation}\index{instantiation}
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\begin{ttbox} 
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res_inst_tac  : (string*string)list -> thm -> int -> tactic
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eres_inst_tac : (string*string)list -> thm -> int -> tactic
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dres_inst_tac : (string*string)list -> thm -> int -> tactic
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forw_inst_tac : (string*string)list -> thm -> int -> tactic
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\end{ttbox}
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These tactics are designed for applying rules such as substitution and
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induction, which cause difficulties for higher-order unification.  The
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tactics accept explicit instantiations for unknowns in the rule ---
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typically, in the rule's conclusion.  Each instantiation is a pair
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{\tt($v$,$e$)}, where $v$ is an unknown {\em without\/} its leading
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question mark!
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\begin{itemize}
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\item If $v$ is the type unknown {\tt'a}, then
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the rule must contain a type unknown \verb$?'a$ of some
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sort~$s$, and $e$ should be a type of sort $s$.
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\item If $v$ is the unknown {\tt P}, then
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the rule must contain an unknown \verb$?P$ of some type~$\tau$,
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and $e$ should be a term of some type~$\sigma$ such that $\tau$ and
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$\sigma$ are unifiable.  If the unification of $\tau$ and $\sigma$
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instantiates any type unknowns in $\tau$, these instantiations
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are recorded for application to the rule.
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\end{itemize}
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Types are instantiated before terms.  Because type instantiations are
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inferred from term instantiations, explicit type instantiations are seldom
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necessary --- if \verb$?t$ has type \verb$?'a$, then the instantiation list
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\verb$[("'a","bool"),("t","True")]$ may be simplified to
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\verb$[("t","True")]$.  Type unknowns in the proof state may cause
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failure because the tactics cannot instantiate them.
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The instantiation tactics act on a given subgoal.  Terms in the
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instantiations are type-checked in the context of that subgoal --- in
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particular, they may refer to that subgoal's parameters.  Any unknowns in
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the terms receive subscripts and are lifted over the parameters; thus, you
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may not refer to unknowns in the subgoal.
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\begin{ttdescription}
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\item[\ttindexbold{res_inst_tac} {\it insts} {\it thm} {\it i}]
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instantiates the rule {\it thm} with the instantiations {\it insts}, as
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described above, and then performs resolution on subgoal~$i$.  Resolution
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typically causes further instantiations; you need not give explicit
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instantiations for every unknown in the rule.
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\item[\ttindexbold{eres_inst_tac}] 
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is like {\tt res_inst_tac}, but performs elim-resolution.
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\item[\ttindexbold{dres_inst_tac}] 
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is like {\tt res_inst_tac}, but performs destruct-resolution.
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\item[\ttindexbold{forw_inst_tac}] 
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is like {\tt dres_inst_tac} except that the selected assumption is not
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deleted.  It applies the instantiated rule to an assumption, adding the
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result as a new assumption.
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\end{ttdescription}
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\section{Other basic tactics}
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\subsection{Tactic shortcuts}
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\index{shortcuts!for tactics}
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\index{tactics!resolution}\index{tactics!assumption}
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\index{tactics!meta-rewriting}
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\begin{ttbox} 
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rtac     :      thm -> int -> tactic
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etac     :      thm -> int -> tactic
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dtac     :      thm -> int -> tactic
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atac     :             int -> tactic
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ares_tac : thm list -> int -> tactic
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rewtac   :      thm ->        tactic
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\end{ttbox}
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These abbreviate common uses of tactics.
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\begin{ttdescription}
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\item[\ttindexbold{rtac} {\it thm} {\it i}] 
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abbreviates \hbox{\tt resolve_tac [{\it thm}] {\it i}}, doing resolution.
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\item[\ttindexbold{etac} {\it thm} {\it i}] 
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abbreviates \hbox{\tt eresolve_tac [{\it thm}] {\it i}}, doing elim-resolution.
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\item[\ttindexbold{dtac} {\it thm} {\it i}] 
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abbreviates \hbox{\tt dresolve_tac [{\it thm}] {\it i}}, doing
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destruct-resolution.
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\item[\ttindexbold{atac} {\it i}] 
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abbreviates \hbox{\tt assume_tac {\it i}}, doing proof by assumption.
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\item[\ttindexbold{ares_tac} {\it thms} {\it i}] 
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tries proof by assumption and resolution; it abbreviates
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\begin{ttbox}
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assume_tac {\it i} ORELSE resolve_tac {\it thms} {\it i}
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\end{ttbox}
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\item[\ttindexbold{rewtac} {\it def}] 
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abbreviates \hbox{\tt rewrite_goals_tac [{\it def}]}, unfolding a definition.
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\end{ttdescription}
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\subsection{Inserting premises and facts}\label{cut_facts_tac}
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\index{tactics!for inserting facts}\index{assumptions!inserting}
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\begin{ttbox} 
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cut_facts_tac : thm list -> int -> tactic
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cut_inst_tac  : (string*string)list -> thm -> int -> tactic
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subgoal_tac   : string -> int -> tactic
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subgoal_tacs  : string list -> int -> tactic
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\end{ttbox}
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These tactics add assumptions to a subgoal.
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\begin{ttdescription}
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\item[\ttindexbold{cut_facts_tac} {\it thms} {\it i}] 
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  adds the {\it thms} as new assumptions to subgoal~$i$.  Once they have
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  been inserted as assumptions, they become subject to tactics such as {\tt
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    eresolve_tac} and {\tt rewrite_goals_tac}.  Only rules with no premises
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  are inserted: Isabelle cannot use assumptions that contain $\Imp$
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  or~$\Forall$.  Sometimes the theorems are premises of a rule being
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  derived, returned by~{\tt goal}; instead of calling this tactic, you
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  could state the goal with an outermost meta-quantifier.
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\item[\ttindexbold{cut_inst_tac} {\it insts} {\it thm} {\it i}]
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  instantiates the {\it thm} with the instantiations {\it insts}, as
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  described in \S\ref{res_inst_tac}.  It adds the resulting theorem as a
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  new assumption to subgoal~$i$. 
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\item[\ttindexbold{subgoal_tac} {\it formula} {\it i}] 
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adds the {\it formula} as a assumption to subgoal~$i$, and inserts the same
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{\it formula} as a new subgoal, $i+1$.
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\item[\ttindexbold{subgoals_tac} {\it formulae} {\it i}] 
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  uses {\tt subgoal_tac} to add the members of the list of {\it
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    formulae} as assumptions to subgoal~$i$. 
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\end{ttdescription}
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\subsection{``Putting off'' a subgoal}
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\begin{ttbox} 
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defer_tac : int -> tactic
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{defer_tac} {\it i}] 
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  moves subgoal~$i$ to the last position in the proof state.  It can be
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  useful when correcting a proof script: if the tactic given for subgoal~$i$
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  fails, calling {\tt defer_tac} instead will let you continue with the rest
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  of the script.
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  The tactic fails if subgoal~$i$ does not exist or if the proof state
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  contains type unknowns. 
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\end{ttdescription}
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\subsection{Definitions and meta-level rewriting}
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\index{tactics!meta-rewriting|bold}\index{meta-rewriting|bold}
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\index{definitions}
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Definitions in Isabelle have the form $t\equiv u$, where $t$ is typically a
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constant or a constant applied to a list of variables, for example $\it
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sqr(n)\equiv n\times n$.  (Conditional definitions, $\phi\Imp t\equiv u$,
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are not supported.)  {\bf Unfolding} the definition ${t\equiv u}$ means using
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it as a rewrite rule, replacing~$t$ by~$u$ throughout a theorem.  {\bf
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Folding} $t\equiv u$ means replacing~$u$ by~$t$.  Rewriting continues until
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no rewrites are applicable to any subterm.
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There are rules for unfolding and folding definitions; Isabelle does not do
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this automatically.  The corresponding tactics rewrite the proof state,
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yielding a single next state.  See also the {\tt goalw} command, which is the
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easiest way of handling definitions.
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\begin{ttbox} 
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rewrite_goals_tac : thm list -> tactic
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rewrite_tac       : thm list -> tactic
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fold_goals_tac    : thm list -> tactic
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fold_tac          : thm list -> tactic
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{rewrite_goals_tac} {\it defs}]  
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unfolds the {\it defs} throughout the subgoals of the proof state, while
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leaving the main goal unchanged.  Use \ttindex{SELECT_GOAL} to restrict it to a
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particular subgoal.
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\item[\ttindexbold{rewrite_tac} {\it defs}]  
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unfolds the {\it defs} throughout the proof state, including the main goal
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--- not normally desirable!
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\item[\ttindexbold{fold_goals_tac} {\it defs}]  
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folds the {\it defs} throughout the subgoals of the proof state, while
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leaving the main goal unchanged.
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\item[\ttindexbold{fold_tac} {\it defs}]  
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folds the {\it defs} throughout the proof state.
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\end{ttdescription}
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\subsection{Theorems useful with tactics}
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\index{theorems!of pure theory}
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\begin{ttbox} 
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asm_rl: thm 
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cut_rl: thm 
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\end{ttbox}
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\begin{ttdescription}
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\item[\tdx{asm_rl}] 
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is $\psi\Imp\psi$.  Under elim-resolution it does proof by assumption, and
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\hbox{\tt eresolve_tac (asm_rl::{\it thms}) {\it i}} is equivalent to
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\begin{ttbox} 
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assume_tac {\it i}  ORELSE  eresolve_tac {\it thms} {\it i}
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\end{ttbox}
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\item[\tdx{cut_rl}] 
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is $\List{\psi\Imp\theta,\psi}\Imp\theta$.  It is useful for inserting
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assumptions; it underlies {\tt forward_tac}, {\tt cut_facts_tac}
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and {\tt subgoal_tac}.
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\end{ttdescription}
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\section{Obscure tactics}
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\subsection{Rotating assumptions}
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\index{assumptions!rotating}
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\begin{ttbox} 
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rotate_tac : int -> int -> tactic
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{rotate_tac} $n$ $i$]  
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rotates the assumptions of subgoal $i$ by $n$ positions: from right to left,
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if $n$ is positive, and from left to right, if $n$ is negative. Sometimes
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necessary in connection with \ttindex{asm_full_simp_tac}.
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\end{ttdescription}
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\subsection{Tidying the proof state}
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\index{parameters!removing unused}
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\index{flex-flex constraints}
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\begin{ttbox} 
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prune_params_tac : tactic
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flexflex_tac     : tactic
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{prune_params_tac}]  
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  removes unused parameters from all subgoals of the proof state.  It works
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  by rewriting with the theorem $(\Forall x. V)\equiv V$.  This tactic can
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  make the proof state more readable.  It is used with
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  \ttindex{rule_by_tactic} to simplify the resulting theorem.
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\item[\ttindexbold{flexflex_tac}]  
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  removes all flex-flex pairs from the proof state by applying the trivial
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  unifier.  This drastic step loses information, and should only be done as
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  the last step of a proof.
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  Flex-flex constraints arise from difficult cases of higher-order
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  unification.  To prevent this, use \ttindex{res_inst_tac} to instantiate
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  some variables in a rule~(\S\ref{res_inst_tac}).  Normally flex-flex
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  constraints can be ignored; they often disappear as unknowns get
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  instantiated.
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\end{ttdescription}
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\subsection{Renaming parameters in a goal} \index{parameters!renaming}
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\begin{ttbox} 
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rename_tac        : string -> int -> tactic
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rename_last_tac   : string -> string list -> int -> tactic
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Logic.set_rename_prefix : string -> unit
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Logic.auto_rename       : bool ref      \hfill{\bf initially false}
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\end{ttbox}
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When creating a parameter, Isabelle chooses its name by matching variable
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names via the object-rule.  Given the rule $(\forall I)$ formalized as
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$\left(\Forall x. P(x)\right) \Imp \forall x.P(x)$, Isabelle will note that
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the $\Forall$-bound variable in the premise has the same name as the
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$\forall$-bound variable in the conclusion.  
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Sometimes there is insufficient information and Isabelle chooses an
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arbitrary name.  The renaming tactics let you override Isabelle's choice.
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Because renaming parameters has no logical effect on the proof state, the
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{\tt by} command prints the message {\tt Warning:\ same as previous
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level}.
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Alternatively, you can suppress the naming mechanism described above and
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have Isabelle generate uniform names for parameters.  These names have the
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form $p${\tt a}, $p${\tt b}, $p${\tt c},~\ldots, where $p$ is any desired
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prefix.  They are ugly but predictable.
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   381
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\begin{ttdescription}
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\item[\ttindexbold{rename_tac} {\it str} {\it i}] 
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interprets the string {\it str} as a series of blank-separated variable
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   385
names, and uses them to rename the parameters of subgoal~$i$.  The names
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   386
must be distinct.  If there are fewer names than parameters, then the
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tactic renames the innermost parameters and may modify the remaining ones
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   388
to ensure that all the parameters are distinct.
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   389
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\item[\ttindexbold{rename_last_tac} {\it prefix} {\it suffixes} {\it i}] 
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generates a list of names by attaching each of the {\it suffixes\/} to the 
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   392
{\it prefix}.  It is intended for coding structural induction tactics,
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   393
where several of the new parameters should have related names.
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   394
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\item[\ttindexbold{Logic.set_rename_prefix} {\it prefix};] 
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sets the prefix for uniform renaming to~{\it prefix}.  The default prefix
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   397
is {\tt"k"}.
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   398
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\item[\ttindexbold{Logic.auto_rename} := true;] 
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makes Isabelle generate uniform names for parameters. 
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\end{ttdescription}
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\subsection{Composition: resolution without lifting}
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\index{tactics!for composition}
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\begin{ttbox}
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compose_tac: (bool * thm * int) -> int -> tactic
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\end{ttbox}
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{\bf Composing} two rules means resolving them without prior lifting or
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renaming of unknowns.  This low-level operation, which underlies the
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   411
resolution tactics, may occasionally be useful for special effects.
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   412
A typical application is \ttindex{res_inst_tac}, which lifts and instantiates a
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   413
rule, then passes the result to {\tt compose_tac}.
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diff changeset
   414
\begin{ttdescription}
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\item[\ttindexbold{compose_tac} ($flag$, $rule$, $m$) $i$] 
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   416
refines subgoal~$i$ using $rule$, without lifting.  The $rule$ is taken to
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have the form $\List{\psi@1; \ldots; \psi@m} \Imp \psi$, where $\psi$ need
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diff changeset
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not be atomic; thus $m$ determines the number of new subgoals.  If
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$flag$ is {\tt true} then it performs elim-resolution --- it solves the
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first premise of~$rule$ by assumption and deletes that assumption.
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\end{ttdescription}
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diff changeset
   422
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   423
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\section{Managing lots of rules}
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These operations are not intended for interactive use.  They are concerned
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diff changeset
   426
with the processing of large numbers of rules in automatic proof
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lcp
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   427
strategies.  Higher-order resolution involving a long list of rules is
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diff changeset
   428
slow.  Filtering techniques can shorten the list of rules given to
2039
79c86b966257 Documented defer_tac and moved back the obsolete tactics like fold_tac
paulson
parents: 1212
diff changeset
   429
resolution, and can also detect whether a subgoal is too flexible,
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with too many rules applicable.
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lcp
parents:
diff changeset
   431
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\subsection{Combined resolution and elim-resolution} \label{biresolve_tac}
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   433
\index{tactics!resolution}
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   434
\begin{ttbox} 
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   435
biresolve_tac   : (bool*thm)list -> int -> tactic
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lcp
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   436
bimatch_tac     : (bool*thm)list -> int -> tactic
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lcp
parents:
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   437
subgoals_of_brl : bool*thm -> int
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   438
lessb           : (bool*thm) * (bool*thm) -> bool
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   439
\end{ttbox}
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   440
{\bf Bi-resolution} takes a list of $\it (flag,rule)$ pairs.  For each
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lcp
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diff changeset
   441
pair, it applies resolution if the flag is~{\tt false} and
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diff changeset
   442
elim-resolution if the flag is~{\tt true}.  A single tactic call handles a
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diff changeset
   443
mixture of introduction and elimination rules.
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parents:
diff changeset
   444
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   445
\begin{ttdescription}
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   446
\item[\ttindexbold{biresolve_tac} {\it brls} {\it i}] 
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   447
refines the proof state by resolution or elim-resolution on each rule, as
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diff changeset
   448
indicated by its flag.  It affects subgoal~$i$ of the proof state.
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lcp
parents:
diff changeset
   449
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   450
\item[\ttindexbold{bimatch_tac}] 
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   451
is like {\tt biresolve_tac}, but performs matching: unknowns in the
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   452
proof state are never updated (see~\S\ref{match_tac}).
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   453
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lcp
parents:
diff changeset
   454
\item[\ttindexbold{subgoals_of_brl}({\it flag},{\it rule})] 
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diff changeset
   455
returns the number of new subgoals that bi-resolution would yield for the
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diff changeset
   456
pair (if applied to a suitable subgoal).  This is $n$ if the flag is
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lcp
parents:
diff changeset
   457
{\tt false} and $n-1$ if the flag is {\tt true}, where $n$ is the number
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lcp
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diff changeset
   458
of premises of the rule.  Elim-resolution yields one fewer subgoal than
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lcp
parents:
diff changeset
   459
ordinary resolution because it solves the major premise by assumption.
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diff changeset
   460
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lcp
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diff changeset
   461
\item[\ttindexbold{lessb} ({\it brl1},{\it brl2})] 
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   462
returns the result of 
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   463
\begin{ttbox}
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subgoals_of_brl{\it brl1} < subgoals_of_brl{\it brl2}
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   465
\end{ttbox}
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diff changeset
   466
\end{ttdescription}
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   467
Note that \hbox{\tt sort lessb {\it brls}} sorts a list of $\it
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diff changeset
   468
(flag,rule)$ pairs by the number of new subgoals they will yield.  Thus,
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lcp
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diff changeset
   469
those that yield the fewest subgoals should be tried first.
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lcp
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diff changeset
   470
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diff changeset
   471
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   472
\subsection{Discrimination nets for fast resolution}\label{filt_resolve_tac}
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\index{discrimination nets|bold}
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   474
\index{tactics!resolution}
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   475
\begin{ttbox} 
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   476
net_resolve_tac  : thm list -> int -> tactic
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lcp
parents:
diff changeset
   477
net_match_tac    : thm list -> int -> tactic
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lcp
parents:
diff changeset
   478
net_biresolve_tac: (bool*thm) list -> int -> tactic
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lcp
parents:
diff changeset
   479
net_bimatch_tac  : (bool*thm) list -> int -> tactic
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lcp
parents:
diff changeset
   480
filt_resolve_tac : thm list -> int -> int -> tactic
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lcp
parents:
diff changeset
   481
could_unify      : term*term->bool
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lcp
parents:
diff changeset
   482
filter_thms      : (term*term->bool) -> int*term*thm list -> thm list
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lcp
parents:
diff changeset
   483
\end{ttbox}
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diff changeset
   484
The module {\tt Net} implements a discrimination net data structure for
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diff changeset
   485
fast selection of rules \cite[Chapter 14]{charniak80}.  A term is
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lcp
parents:
diff changeset
   486
classified by the symbol list obtained by flattening it in preorder.
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lcp
parents:
diff changeset
   487
The flattening takes account of function applications, constants, and free
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lcp
parents:
diff changeset
   488
and bound variables; it identifies all unknowns and also regards
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diff changeset
   489
\index{lambda abs@$\lambda$-abstractions}
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   490
$\lambda$-abstractions as unknowns, since they could $\eta$-contract to
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lcp
parents:
diff changeset
   491
anything.  
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lcp
parents:
diff changeset
   492
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lcp
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   493
A discrimination net serves as a polymorphic dictionary indexed by terms.
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lcp
parents:
diff changeset
   494
The module provides various functions for inserting and removing items from
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   495
nets.  It provides functions for returning all items whose term could match
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lcp
parents:
diff changeset
   496
or unify with a target term.  The matching and unification tests are
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lcp
parents:
diff changeset
   497
overly lax (due to the identifications mentioned above) but they serve as
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lcp
parents:
diff changeset
   498
useful filters.
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lcp
parents:
diff changeset
   499
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lcp
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diff changeset
   500
A net can store introduction rules indexed by their conclusion, and
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lcp
parents:
diff changeset
   501
elimination rules indexed by their major premise.  Isabelle provides
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diff changeset
   502
several functions for `compiling' long lists of rules into fast
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parents:
diff changeset
   503
resolution tactics.  When supplied with a list of theorems, these functions
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lcp
parents:
diff changeset
   504
build a discrimination net; the net is used when the tactic is applied to a
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diff changeset
   505
goal.  To avoid repeatedly constructing the nets, use currying: bind the
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parents:
diff changeset
   506
resulting tactics to \ML{} identifiers.
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lcp
parents:
diff changeset
   507
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diff changeset
   508
\begin{ttdescription}
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diff changeset
   509
\item[\ttindexbold{net_resolve_tac} {\it thms}] 
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lcp
parents:
diff changeset
   510
builds a discrimination net to obtain the effect of a similar call to {\tt
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lcp
parents:
diff changeset
   511
resolve_tac}.
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lcp
parents:
diff changeset
   512
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lcp
parents:
diff changeset
   513
\item[\ttindexbold{net_match_tac} {\it thms}] 
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lcp
parents:
diff changeset
   514
builds a discrimination net to obtain the effect of a similar call to {\tt
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lcp
parents:
diff changeset
   515
match_tac}.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   516
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   517
\item[\ttindexbold{net_biresolve_tac} {\it brls}] 
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lcp
parents:
diff changeset
   518
builds a discrimination net to obtain the effect of a similar call to {\tt
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lcp
parents:
diff changeset
   519
biresolve_tac}.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   520
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   521
\item[\ttindexbold{net_bimatch_tac} {\it brls}] 
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lcp
parents:
diff changeset
   522
builds a discrimination net to obtain the effect of a similar call to {\tt
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   523
bimatch_tac}.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   524
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lcp
parents:
diff changeset
   525
\item[\ttindexbold{filt_resolve_tac} {\it thms} {\it maxr} {\it i}] 
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lcp
parents:
diff changeset
   526
uses discrimination nets to extract the {\it thms} that are applicable to
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lcp
parents:
diff changeset
   527
subgoal~$i$.  If more than {\it maxr\/} theorems are applicable then the
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lcp
parents:
diff changeset
   528
tactic fails.  Otherwise it calls {\tt resolve_tac}.  
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   529
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   530
This tactic helps avoid runaway instantiation of unknowns, for example in
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   531
type inference.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   532
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   533
\item[\ttindexbold{could_unify} ({\it t},{\it u})] 
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   534
returns {\tt false} if~$t$ and~$u$ are `obviously' non-unifiable, and
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parents:
diff changeset
   535
otherwise returns~{\tt true}.  It assumes all variables are distinct,
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   536
reporting that {\tt ?a=?a} may unify with {\tt 0=1}.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   537
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   538
\item[\ttindexbold{filter_thms} $could\; (limit,prem,thms)$] 
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   539
returns the list of potentially resolvable rules (in {\it thms\/}) for the
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   540
subgoal {\it prem}, using the predicate {\it could\/} to compare the
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   541
conclusion of the subgoal with the conclusion of each rule.  The resulting list
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   542
is no longer than {\it limit}.
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diff changeset
   543
\end{ttdescription}
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\section{Programming tools for proof strategies}
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Do not consider using the primitives discussed in this section unless you
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really need to code tactics from scratch.
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\subsection{Operations on type {\tt tactic}}
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\index{tactics!primitives for coding}
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A tactic maps theorems to theorem sequences (lazy lists).  The type
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constructor for sequences is called \mltydx{Sequence.seq}.  To simplify the
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types of tactics and tacticals, Isabelle defines a type of tactics:
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   555
\begin{ttbox} 
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datatype tactic = Tactic of thm -> thm Sequence.seq
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\end{ttbox} 
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{\tt Tactic} and {\tt tapply} convert between tactics and functions.  The
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other operations provide means for coding tactics in a clean style.
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\begin{ttbox} 
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tapply    : tactic * thm -> thm Sequence.seq
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Tactic    :     (thm -> thm Sequence.seq) -> tactic
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PRIMITIVE :                  (thm -> thm) -> tactic  
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parents:
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STATE     :               (thm -> tactic) -> tactic
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lcp
parents:
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SUBGOAL   : ((term*int) -> tactic) -> int -> tactic
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   566
\end{ttbox} 
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\begin{ttdescription}
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   568
\item[\ttindexbold{tapply}({\it tac}, {\it thm})]  
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returns the result of applying the tactic, as a function, to {\it thm}.
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   570
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parents:
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\item[\ttindexbold{Tactic} {\it f}]  
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packages {\it f} as a tactic.
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parents:
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   573
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\item[\ttindexbold{PRIMITIVE} $f$] 
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applies $f$ to the proof state and returns the result as a
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parents:
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   576
one-element sequence.  This packages the meta-rule~$f$ as a tactic.
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lcp
parents:
diff changeset
   577
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lcp
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   578
\item[\ttindexbold{STATE} $f$] 
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   579
applies $f$ to the proof state and then applies the resulting tactic to the
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   580
same state.  It supports the following style, where the tactic body is
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   581
expressed using tactics and tacticals, but may peek at the proof state:
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   582
\begin{ttbox} 
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   583
STATE (fn state => {\it tactic-valued expression})
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   584
\end{ttbox} 
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parents:
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   585
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parents:
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   586
\item[\ttindexbold{SUBGOAL} $f$ $i$] 
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extracts subgoal~$i$ from the proof state as a term~$t$, and computes a
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lcp
parents:
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   588
tactic by calling~$f(t,i)$.  It applies the resulting tactic to the same
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parents: 286
diff changeset
   589
state.  The tactic body is expressed using tactics and tacticals, but may
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diff changeset
   590
peek at a particular subgoal:
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parents:
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   591
\begin{ttbox} 
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diff changeset
   592
SUBGOAL (fn (t,i) => {\it tactic-valued expression})
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parents:
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   593
\end{ttbox} 
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diff changeset
   594
\end{ttdescription}
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parents:
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   595
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parents:
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   596
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parents:
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   597
\subsection{Tracing}
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diff changeset
   598
\index{tactics!tracing}
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   599
\index{tracing!of tactics}
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   600
\begin{ttbox} 
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   601
pause_tac: tactic
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parents:
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   602
print_tac: tactic
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parents:
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   603
\end{ttbox}
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   604
These tactics print tracing information when they are applied to a proof
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parents: 323
diff changeset
   605
state.  Their output may be difficult to interpret.  Note that certain of
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parents: 323
diff changeset
   606
the searching tacticals, such as {\tt REPEAT}, have built-in tracing
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parents: 323
diff changeset
   607
options.
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   608
\begin{ttdescription}
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   609
\item[\ttindexbold{pause_tac}] 
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parents: 323
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   610
prints {\footnotesize\tt** Press RETURN to continue:} and then reads a line
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parents: 323
diff changeset
   611
from the terminal.  If this line is blank then it returns the proof state
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parents: 323
diff changeset
   612
unchanged; otherwise it fails (which may terminate a repetition).
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parents:
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   613
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parents:
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   614
\item[\ttindexbold{print_tac}] 
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parents:
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   615
returns the proof state unchanged, with the side effect of printing it at
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parents:
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   616
the terminal.
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diff changeset
   617
\end{ttdescription}
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   618
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   619
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   620
\section{Sequences}
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   621
\index{sequences (lazy lists)|bold}
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diff changeset
   622
The module {\tt Sequence} declares a type of lazy lists.  It uses
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   623
Isabelle's type \mltydx{option} to represent the possible presence
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   624
(\ttindexbold{Some}) or absence (\ttindexbold{None}) of
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   625
a value:
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parents:
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   626
\begin{ttbox}
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parents:
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   627
datatype 'a option = None  |  Some of 'a;
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parents:
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   628
\end{ttbox}
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   629
For clarity, the module name {\tt Sequence} is omitted from the signature
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parents: 104
diff changeset
   630
specifications below; for instance, {\tt null} appears instead of {\tt
e7efbf03562b first draft of Springer book
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   631
  Sequence.null}.
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   632
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   633
\subsection{Basic operations on sequences}
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   634
\begin{ttbox} 
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   635
null   : 'a seq
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parents: 104
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   636
seqof  : (unit -> ('a * 'a seq) option) -> 'a seq
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lcp
parents: 104
diff changeset
   637
single : 'a -> 'a seq
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lcp
parents: 104
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   638
pull   : 'a seq -> ('a * 'a seq) option
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   639
\end{ttbox}
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diff changeset
   640
\begin{ttdescription}
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   641
\item[Sequence.null] 
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is the empty sequence.
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   643
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lcp
parents:
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   644
\item[\tt Sequence.seqof (fn()=> Some($x$,$s$))] 
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lcp
parents:
diff changeset
   645
constructs the sequence with head~$x$ and tail~$s$, neither of which is
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lcp
parents:
diff changeset
   646
evaluated.
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parents:
diff changeset
   647
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   648
\item[Sequence.single $x$] 
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constructs the sequence containing the single element~$x$.
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parents:
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   650
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   651
\item[Sequence.pull $s$] 
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returns {\tt None} if the sequence is empty and {\tt Some($x$,$s'$)} if the
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lcp
parents:
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   653
sequence has head~$x$ and tail~$s'$.  Warning: calling \hbox{Sequence.pull
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   654
$s$} again will {\it recompute\/} the value of~$x$; it is not stored!
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diff changeset
   655
\end{ttdescription}
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   656
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diff changeset
   657
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   658
\subsection{Converting between sequences and lists}
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   659
\begin{ttbox} 
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   660
chop      : int * 'a seq -> 'a list * 'a seq
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   661
list_of_s : 'a seq -> 'a list
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diff changeset
   662
s_of_list : 'a list -> 'a seq
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parents:
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   663
\end{ttbox}
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diff changeset
   664
\begin{ttdescription}
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   665
\item[Sequence.chop($n$,$s$)] 
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   666
returns the first~$n$ elements of~$s$ as a list, paired with the remaining
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lcp
parents:
diff changeset
   667
elements of~$s$.  If $s$ has fewer than~$n$ elements, then so will the
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lcp
parents:
diff changeset
   668
list.
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lcp
parents:
diff changeset
   669
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parents: 286
diff changeset
   670
\item[Sequence.list_of_s $s$] 
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   671
returns the elements of~$s$, which must be finite, as a list.
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lcp
parents:
diff changeset
   672
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parents: 286
diff changeset
   673
\item[Sequence.s_of_list $l$] 
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parents:
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   674
creates a sequence containing the elements of~$l$.
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diff changeset
   675
\end{ttdescription}
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parents:
diff changeset
   676
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lcp
parents:
diff changeset
   677
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   678
\subsection{Combining sequences}
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   679
\begin{ttbox} 
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   680
append     : 'a seq * 'a seq -> 'a seq
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parents: 104
diff changeset
   681
interleave : 'a seq * 'a seq -> 'a seq
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lcp
parents: 104
diff changeset
   682
flats      : 'a seq seq -> 'a seq
e7efbf03562b first draft of Springer book
lcp
parents: 104
diff changeset
   683
maps       : ('a -> 'b) -> 'a seq -> 'b seq
e7efbf03562b first draft of Springer book
lcp
parents: 104
diff changeset
   684
filters    : ('a -> bool) -> 'a seq -> 'a seq
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parents:
diff changeset
   685
\end{ttbox} 
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diff changeset
   686
\begin{ttdescription}
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parents: 323
diff changeset
   687
\item[Sequence.append($s@1$,$s@2$)] 
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parents:
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   688
concatenates $s@1$ to $s@2$.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   689
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parents: 323
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   690
\item[Sequence.interleave($s@1$,$s@2$)] 
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parents:
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   691
joins $s@1$ with $s@2$ by interleaving their elements.  The result contains
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   692
all the elements of the sequences, even if both are infinite.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   693
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parents: 286
diff changeset
   694
\item[Sequence.flats $ss$] 
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parents:
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   695
concatenates a sequence of sequences.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   696
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parents: 286
diff changeset
   697
\item[Sequence.maps $f$ $s$] 
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parents:
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   698
applies $f$ to every element of~$s=x@1,x@2,\ldots$, yielding the sequence
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lcp
parents:
diff changeset
   699
$f(x@1),f(x@2),\ldots$.
d8205bb279a7 Initial revision
lcp
parents:
diff changeset
   700
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parents: 286
diff changeset
   701
\item[Sequence.filters $p$ $s$] 
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parents:
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   702
returns the sequence consisting of all elements~$x$ of~$s$ such that $p(x)$
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lcp
parents:
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   703
is {\tt true}.
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diff changeset
   704
\end{ttdescription}
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lcp
parents:
diff changeset
   705
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lcp
parents:
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   706
\index{tactics|)}