author | wenzelm |
Wed, 04 Aug 1999 18:20:24 +0200 | |
changeset 7175 | 8263d0b50e12 |
parent 6618 | 13293a7d4a57 |
child 7491 | 95a4af0e10a7 |
permissions | -rw-r--r-- |
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%% $Id$ |
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\chapter{Tactics} \label{tactics} |
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\index{tactics|(} Tactics have type \mltydx{tactic}. This is just an |
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abbreviation for functions from theorems to theorem sequences, where |
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the theorems represent states of a backward proof. Tactics seldom |
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need to be coded from scratch, as functions; instead they are |
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expressed using basic tactics and tacticals. |
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This chapter only presents the primitive tactics. Substantial proofs |
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require the power of automatic tools like simplification |
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(Chapter~\ref{chap:simplification}) and classical tableau 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, proves its first premise by |
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assumption, and finally \emph{deletes} that assumption from any new |
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subgoals. (To rotate a rule's premises, |
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see \texttt{rotate_prems} in~\S\ref{MiscellaneousForwardRules}.) |
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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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\emph{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 \emph{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} \label{sec:rewrite_goals} |
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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 also 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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|
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284 |
\item[\ttindexbold{rewrite_tac} {\it defs}] |
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285 |
unfolds the {\it defs} throughout the proof state, including the main goal |
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286 |
--- not normally desirable! |
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287 |
|
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288 |
\item[\ttindexbold{fold_goals_tac} {\it defs}] |
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289 |
folds the {\it defs} throughout the subgoals of the proof state, while |
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290 |
leaving the main goal unchanged. |
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|
291 |
|
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292 |
\item[\ttindexbold{fold_tac} {\it defs}] |
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293 |
folds the {\it defs} throughout the proof state. |
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294 |
\end{ttdescription} |
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295 |
|
4317 | 296 |
\begin{warn} |
297 |
These tactics only cope with definitions expressed as meta-level |
|
298 |
equalities ($\equiv$). More general equivalences are handled by the |
|
299 |
simplifier, provided that it is set up appropriately for your logic |
|
300 |
(see Chapter~\ref{chap:simplification}). |
|
301 |
\end{warn} |
|
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|
302 |
|
104 | 303 |
\subsection{Theorems useful with tactics} |
323 | 304 |
\index{theorems!of pure theory} |
104 | 305 |
\begin{ttbox} |
306 |
asm_rl: thm |
|
307 |
cut_rl: thm |
|
308 |
\end{ttbox} |
|
323 | 309 |
\begin{ttdescription} |
310 |
\item[\tdx{asm_rl}] |
|
104 | 311 |
is $\psi\Imp\psi$. Under elim-resolution it does proof by assumption, and |
312 |
\hbox{\tt eresolve_tac (asm_rl::{\it thms}) {\it i}} is equivalent to |
|
313 |
\begin{ttbox} |
|
314 |
assume_tac {\it i} ORELSE eresolve_tac {\it thms} {\it i} |
|
315 |
\end{ttbox} |
|
316 |
||
323 | 317 |
\item[\tdx{cut_rl}] |
104 | 318 |
is $\List{\psi\Imp\theta,\psi}\Imp\theta$. It is useful for inserting |
323 | 319 |
assumptions; it underlies {\tt forward_tac}, {\tt cut_facts_tac} |
320 |
and {\tt subgoal_tac}. |
|
321 |
\end{ttdescription} |
|
104 | 322 |
|
323 |
||
324 |
\section{Obscure tactics} |
|
1212 | 325 |
|
323 | 326 |
\subsection{Renaming parameters in a goal} \index{parameters!renaming} |
104 | 327 |
\begin{ttbox} |
328 |
rename_tac : string -> int -> tactic |
|
329 |
rename_last_tac : string -> string list -> int -> tactic |
|
330 |
Logic.set_rename_prefix : string -> unit |
|
331 |
Logic.auto_rename : bool ref \hfill{\bf initially false} |
|
332 |
\end{ttbox} |
|
333 |
When creating a parameter, Isabelle chooses its name by matching variable |
|
334 |
names via the object-rule. Given the rule $(\forall I)$ formalized as |
|
335 |
$\left(\Forall x. P(x)\right) \Imp \forall x.P(x)$, Isabelle will note that |
|
336 |
the $\Forall$-bound variable in the premise has the same name as the |
|
337 |
$\forall$-bound variable in the conclusion. |
|
338 |
||
339 |
Sometimes there is insufficient information and Isabelle chooses an |
|
340 |
arbitrary name. The renaming tactics let you override Isabelle's choice. |
|
341 |
Because renaming parameters has no logical effect on the proof state, the |
|
323 | 342 |
{\tt by} command prints the message {\tt Warning:\ same as previous |
104 | 343 |
level}. |
344 |
||
345 |
Alternatively, you can suppress the naming mechanism described above and |
|
346 |
have Isabelle generate uniform names for parameters. These names have the |
|
347 |
form $p${\tt a}, $p${\tt b}, $p${\tt c},~\ldots, where $p$ is any desired |
|
348 |
prefix. They are ugly but predictable. |
|
349 |
||
323 | 350 |
\begin{ttdescription} |
104 | 351 |
\item[\ttindexbold{rename_tac} {\it str} {\it i}] |
352 |
interprets the string {\it str} as a series of blank-separated variable |
|
353 |
names, and uses them to rename the parameters of subgoal~$i$. The names |
|
354 |
must be distinct. If there are fewer names than parameters, then the |
|
355 |
tactic renames the innermost parameters and may modify the remaining ones |
|
356 |
to ensure that all the parameters are distinct. |
|
357 |
||
358 |
\item[\ttindexbold{rename_last_tac} {\it prefix} {\it suffixes} {\it i}] |
|
359 |
generates a list of names by attaching each of the {\it suffixes\/} to the |
|
360 |
{\it prefix}. It is intended for coding structural induction tactics, |
|
361 |
where several of the new parameters should have related names. |
|
362 |
||
363 |
\item[\ttindexbold{Logic.set_rename_prefix} {\it prefix};] |
|
364 |
sets the prefix for uniform renaming to~{\it prefix}. The default prefix |
|
365 |
is {\tt"k"}. |
|
366 |
||
4317 | 367 |
\item[set \ttindexbold{Logic.auto_rename};] |
104 | 368 |
makes Isabelle generate uniform names for parameters. |
323 | 369 |
\end{ttdescription} |
104 | 370 |
|
371 |
||
2612 | 372 |
\subsection{Manipulating assumptions} |
373 |
\index{assumptions!rotating} |
|
374 |
\begin{ttbox} |
|
375 |
thin_tac : string -> int -> tactic |
|
376 |
rotate_tac : int -> int -> tactic |
|
377 |
\end{ttbox} |
|
378 |
\begin{ttdescription} |
|
379 |
\item[\ttindexbold{thin_tac} {\it formula} $i$] |
|
380 |
\index{assumptions!deleting} |
|
381 |
deletes the specified assumption from subgoal $i$. Often the assumption |
|
382 |
can be abbreviated, replacing subformul{\ae} by unknowns; the first matching |
|
383 |
assumption will be deleted. Removing useless assumptions from a subgoal |
|
384 |
increases its readability and can make search tactics run faster. |
|
385 |
||
386 |
\item[\ttindexbold{rotate_tac} $n$ $i$] |
|
387 |
\index{assumptions!rotating} |
|
388 |
rotates the assumptions of subgoal $i$ by $n$ positions: from right to left |
|
389 |
if $n$ is positive, and from left to right if $n$ is negative. This is |
|
390 |
sometimes necessary in connection with \ttindex{asm_full_simp_tac}, which |
|
391 |
processes assumptions from left to right. |
|
392 |
\end{ttdescription} |
|
393 |
||
394 |
||
395 |
\subsection{Tidying the proof state} |
|
3400 | 396 |
\index{duplicate subgoals!removing} |
2612 | 397 |
\index{parameters!removing unused} |
398 |
\index{flex-flex constraints} |
|
399 |
\begin{ttbox} |
|
3400 | 400 |
distinct_subgoals_tac : tactic |
401 |
prune_params_tac : tactic |
|
402 |
flexflex_tac : tactic |
|
2612 | 403 |
\end{ttbox} |
404 |
\begin{ttdescription} |
|
3400 | 405 |
\item[\ttindexbold{distinct_subgoals_tac}] |
406 |
removes duplicate subgoals from a proof state. (These arise especially |
|
407 |
in \ZF{}, where the subgoals are essentially type constraints.) |
|
408 |
||
2612 | 409 |
\item[\ttindexbold{prune_params_tac}] |
410 |
removes unused parameters from all subgoals of the proof state. It works |
|
411 |
by rewriting with the theorem $(\Forall x. V)\equiv V$. This tactic can |
|
412 |
make the proof state more readable. It is used with |
|
413 |
\ttindex{rule_by_tactic} to simplify the resulting theorem. |
|
414 |
||
415 |
\item[\ttindexbold{flexflex_tac}] |
|
416 |
removes all flex-flex pairs from the proof state by applying the trivial |
|
417 |
unifier. This drastic step loses information, and should only be done as |
|
418 |
the last step of a proof. |
|
419 |
||
420 |
Flex-flex constraints arise from difficult cases of higher-order |
|
421 |
unification. To prevent this, use \ttindex{res_inst_tac} to instantiate |
|
422 |
some variables in a rule~(\S\ref{res_inst_tac}). Normally flex-flex |
|
423 |
constraints can be ignored; they often disappear as unknowns get |
|
424 |
instantiated. |
|
425 |
\end{ttdescription} |
|
426 |
||
427 |
||
104 | 428 |
\subsection{Composition: resolution without lifting} |
323 | 429 |
\index{tactics!for composition} |
104 | 430 |
\begin{ttbox} |
431 |
compose_tac: (bool * thm * int) -> int -> tactic |
|
432 |
\end{ttbox} |
|
332 | 433 |
{\bf Composing} two rules means resolving them without prior lifting or |
104 | 434 |
renaming of unknowns. This low-level operation, which underlies the |
435 |
resolution tactics, may occasionally be useful for special effects. |
|
436 |
A typical application is \ttindex{res_inst_tac}, which lifts and instantiates a |
|
437 |
rule, then passes the result to {\tt compose_tac}. |
|
323 | 438 |
\begin{ttdescription} |
104 | 439 |
\item[\ttindexbold{compose_tac} ($flag$, $rule$, $m$) $i$] |
440 |
refines subgoal~$i$ using $rule$, without lifting. The $rule$ is taken to |
|
441 |
have the form $\List{\psi@1; \ldots; \psi@m} \Imp \psi$, where $\psi$ need |
|
323 | 442 |
not be atomic; thus $m$ determines the number of new subgoals. If |
104 | 443 |
$flag$ is {\tt true} then it performs elim-resolution --- it solves the |
444 |
first premise of~$rule$ by assumption and deletes that assumption. |
|
323 | 445 |
\end{ttdescription} |
104 | 446 |
|
447 |
||
4276 | 448 |
\section{*Managing lots of rules} |
104 | 449 |
These operations are not intended for interactive use. They are concerned |
450 |
with the processing of large numbers of rules in automatic proof |
|
451 |
strategies. Higher-order resolution involving a long list of rules is |
|
452 |
slow. Filtering techniques can shorten the list of rules given to |
|
2039
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|
453 |
resolution, and can also detect whether a subgoal is too flexible, |
104 | 454 |
with too many rules applicable. |
455 |
||
456 |
\subsection{Combined resolution and elim-resolution} \label{biresolve_tac} |
|
457 |
\index{tactics!resolution} |
|
458 |
\begin{ttbox} |
|
459 |
biresolve_tac : (bool*thm)list -> int -> tactic |
|
460 |
bimatch_tac : (bool*thm)list -> int -> tactic |
|
461 |
subgoals_of_brl : bool*thm -> int |
|
462 |
lessb : (bool*thm) * (bool*thm) -> bool |
|
463 |
\end{ttbox} |
|
464 |
{\bf Bi-resolution} takes a list of $\it (flag,rule)$ pairs. For each |
|
465 |
pair, it applies resolution if the flag is~{\tt false} and |
|
466 |
elim-resolution if the flag is~{\tt true}. A single tactic call handles a |
|
467 |
mixture of introduction and elimination rules. |
|
468 |
||
323 | 469 |
\begin{ttdescription} |
104 | 470 |
\item[\ttindexbold{biresolve_tac} {\it brls} {\it i}] |
471 |
refines the proof state by resolution or elim-resolution on each rule, as |
|
472 |
indicated by its flag. It affects subgoal~$i$ of the proof state. |
|
473 |
||
474 |
\item[\ttindexbold{bimatch_tac}] |
|
475 |
is like {\tt biresolve_tac}, but performs matching: unknowns in the |
|
476 |
proof state are never updated (see~\S\ref{match_tac}). |
|
477 |
||
478 |
\item[\ttindexbold{subgoals_of_brl}({\it flag},{\it rule})] |
|
4597
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|
479 |
returns the number of new subgoals that bi-res\-o\-lu\-tion would yield for the |
104 | 480 |
pair (if applied to a suitable subgoal). This is $n$ if the flag is |
481 |
{\tt false} and $n-1$ if the flag is {\tt true}, where $n$ is the number |
|
482 |
of premises of the rule. Elim-resolution yields one fewer subgoal than |
|
483 |
ordinary resolution because it solves the major premise by assumption. |
|
484 |
||
485 |
\item[\ttindexbold{lessb} ({\it brl1},{\it brl2})] |
|
486 |
returns the result of |
|
487 |
\begin{ttbox} |
|
332 | 488 |
subgoals_of_brl{\it brl1} < subgoals_of_brl{\it brl2} |
104 | 489 |
\end{ttbox} |
323 | 490 |
\end{ttdescription} |
104 | 491 |
Note that \hbox{\tt sort lessb {\it brls}} sorts a list of $\it |
492 |
(flag,rule)$ pairs by the number of new subgoals they will yield. Thus, |
|
493 |
those that yield the fewest subgoals should be tried first. |
|
494 |
||
495 |
||
323 | 496 |
\subsection{Discrimination nets for fast resolution}\label{filt_resolve_tac} |
104 | 497 |
\index{discrimination nets|bold} |
498 |
\index{tactics!resolution} |
|
499 |
\begin{ttbox} |
|
500 |
net_resolve_tac : thm list -> int -> tactic |
|
501 |
net_match_tac : thm list -> int -> tactic |
|
502 |
net_biresolve_tac: (bool*thm) list -> int -> tactic |
|
503 |
net_bimatch_tac : (bool*thm) list -> int -> tactic |
|
504 |
filt_resolve_tac : thm list -> int -> int -> tactic |
|
505 |
could_unify : term*term->bool |
|
506 |
filter_thms : (term*term->bool) -> int*term*thm list -> thm list |
|
507 |
\end{ttbox} |
|
323 | 508 |
The module {\tt Net} implements a discrimination net data structure for |
104 | 509 |
fast selection of rules \cite[Chapter 14]{charniak80}. A term is |
510 |
classified by the symbol list obtained by flattening it in preorder. |
|
511 |
The flattening takes account of function applications, constants, and free |
|
512 |
and bound variables; it identifies all unknowns and also regards |
|
323 | 513 |
\index{lambda abs@$\lambda$-abstractions} |
104 | 514 |
$\lambda$-abstractions as unknowns, since they could $\eta$-contract to |
515 |
anything. |
|
516 |
||
517 |
A discrimination net serves as a polymorphic dictionary indexed by terms. |
|
518 |
The module provides various functions for inserting and removing items from |
|
519 |
nets. It provides functions for returning all items whose term could match |
|
520 |
or unify with a target term. The matching and unification tests are |
|
521 |
overly lax (due to the identifications mentioned above) but they serve as |
|
522 |
useful filters. |
|
523 |
||
524 |
A net can store introduction rules indexed by their conclusion, and |
|
525 |
elimination rules indexed by their major premise. Isabelle provides |
|
323 | 526 |
several functions for `compiling' long lists of rules into fast |
104 | 527 |
resolution tactics. When supplied with a list of theorems, these functions |
528 |
build a discrimination net; the net is used when the tactic is applied to a |
|
332 | 529 |
goal. To avoid repeatedly constructing the nets, use currying: bind the |
104 | 530 |
resulting tactics to \ML{} identifiers. |
531 |
||
323 | 532 |
\begin{ttdescription} |
104 | 533 |
\item[\ttindexbold{net_resolve_tac} {\it thms}] |
534 |
builds a discrimination net to obtain the effect of a similar call to {\tt |
|
535 |
resolve_tac}. |
|
536 |
||
537 |
\item[\ttindexbold{net_match_tac} {\it thms}] |
|
538 |
builds a discrimination net to obtain the effect of a similar call to {\tt |
|
539 |
match_tac}. |
|
540 |
||
541 |
\item[\ttindexbold{net_biresolve_tac} {\it brls}] |
|
542 |
builds a discrimination net to obtain the effect of a similar call to {\tt |
|
543 |
biresolve_tac}. |
|
544 |
||
545 |
\item[\ttindexbold{net_bimatch_tac} {\it brls}] |
|
546 |
builds a discrimination net to obtain the effect of a similar call to {\tt |
|
547 |
bimatch_tac}. |
|
548 |
||
549 |
\item[\ttindexbold{filt_resolve_tac} {\it thms} {\it maxr} {\it i}] |
|
550 |
uses discrimination nets to extract the {\it thms} that are applicable to |
|
551 |
subgoal~$i$. If more than {\it maxr\/} theorems are applicable then the |
|
552 |
tactic fails. Otherwise it calls {\tt resolve_tac}. |
|
553 |
||
554 |
This tactic helps avoid runaway instantiation of unknowns, for example in |
|
555 |
type inference. |
|
556 |
||
557 |
\item[\ttindexbold{could_unify} ({\it t},{\it u})] |
|
323 | 558 |
returns {\tt false} if~$t$ and~$u$ are `obviously' non-unifiable, and |
104 | 559 |
otherwise returns~{\tt true}. It assumes all variables are distinct, |
560 |
reporting that {\tt ?a=?a} may unify with {\tt 0=1}. |
|
561 |
||
562 |
\item[\ttindexbold{filter_thms} $could\; (limit,prem,thms)$] |
|
563 |
returns the list of potentially resolvable rules (in {\it thms\/}) for the |
|
564 |
subgoal {\it prem}, using the predicate {\it could\/} to compare the |
|
565 |
conclusion of the subgoal with the conclusion of each rule. The resulting list |
|
566 |
is no longer than {\it limit}. |
|
323 | 567 |
\end{ttdescription} |
104 | 568 |
|
569 |
||
4317 | 570 |
\section{Programming tools for proof strategies} |
104 | 571 |
Do not consider using the primitives discussed in this section unless you |
323 | 572 |
really need to code tactics from scratch. |
104 | 573 |
|
6618 | 574 |
\subsection{Operations on tactics} |
3561 | 575 |
\index{tactics!primitives for coding} A tactic maps theorems to sequences of |
576 |
theorems. The type constructor for sequences (lazy lists) is called |
|
4276 | 577 |
\mltydx{Seq.seq}. To simplify the types of tactics and tacticals, |
3561 | 578 |
Isabelle defines a type abbreviation: |
104 | 579 |
\begin{ttbox} |
4276 | 580 |
type tactic = thm -> thm Seq.seq |
104 | 581 |
\end{ttbox} |
3108 | 582 |
The following operations provide means for coding tactics in a clean style. |
104 | 583 |
\begin{ttbox} |
584 |
PRIMITIVE : (thm -> thm) -> tactic |
|
585 |
SUBGOAL : ((term*int) -> tactic) -> int -> tactic |
|
586 |
\end{ttbox} |
|
323 | 587 |
\begin{ttdescription} |
3561 | 588 |
\item[\ttindexbold{PRIMITIVE} $f$] packages the meta-rule~$f$ as a tactic that |
589 |
applies $f$ to the proof state and returns the result as a one-element |
|
590 |
sequence. If $f$ raises an exception, then the tactic's result is the empty |
|
591 |
sequence. |
|
104 | 592 |
|
593 |
\item[\ttindexbold{SUBGOAL} $f$ $i$] |
|
594 |
extracts subgoal~$i$ from the proof state as a term~$t$, and computes a |
|
595 |
tactic by calling~$f(t,i)$. It applies the resulting tactic to the same |
|
323 | 596 |
state. The tactic body is expressed using tactics and tacticals, but may |
597 |
peek at a particular subgoal: |
|
104 | 598 |
\begin{ttbox} |
323 | 599 |
SUBGOAL (fn (t,i) => {\it tactic-valued expression}) |
104 | 600 |
\end{ttbox} |
323 | 601 |
\end{ttdescription} |
104 | 602 |
|
603 |
||
604 |
\subsection{Tracing} |
|
323 | 605 |
\index{tactics!tracing} |
104 | 606 |
\index{tracing!of tactics} |
607 |
\begin{ttbox} |
|
608 |
pause_tac: tactic |
|
609 |
print_tac: tactic |
|
610 |
\end{ttbox} |
|
332 | 611 |
These tactics print tracing information when they are applied to a proof |
612 |
state. Their output may be difficult to interpret. Note that certain of |
|
613 |
the searching tacticals, such as {\tt REPEAT}, have built-in tracing |
|
614 |
options. |
|
323 | 615 |
\begin{ttdescription} |
104 | 616 |
\item[\ttindexbold{pause_tac}] |
332 | 617 |
prints {\footnotesize\tt** Press RETURN to continue:} and then reads a line |
618 |
from the terminal. If this line is blank then it returns the proof state |
|
619 |
unchanged; otherwise it fails (which may terminate a repetition). |
|
104 | 620 |
|
621 |
\item[\ttindexbold{print_tac}] |
|
622 |
returns the proof state unchanged, with the side effect of printing it at |
|
623 |
the terminal. |
|
323 | 624 |
\end{ttdescription} |
104 | 625 |
|
626 |
||
4276 | 627 |
\section{*Sequences} |
104 | 628 |
\index{sequences (lazy lists)|bold} |
4276 | 629 |
The module {\tt Seq} declares a type of lazy lists. It uses |
323 | 630 |
Isabelle's type \mltydx{option} to represent the possible presence |
104 | 631 |
(\ttindexbold{Some}) or absence (\ttindexbold{None}) of |
632 |
a value: |
|
633 |
\begin{ttbox} |
|
634 |
datatype 'a option = None | Some of 'a; |
|
635 |
\end{ttbox} |
|
4276 | 636 |
The {\tt Seq} structure is supposed to be accessed via fully qualified |
4317 | 637 |
names and should not be \texttt{open}. |
104 | 638 |
|
323 | 639 |
\subsection{Basic operations on sequences} |
104 | 640 |
\begin{ttbox} |
4276 | 641 |
Seq.empty : 'a seq |
642 |
Seq.make : (unit -> ('a * 'a seq) option) -> 'a seq |
|
643 |
Seq.single : 'a -> 'a seq |
|
644 |
Seq.pull : 'a seq -> ('a * 'a seq) option |
|
104 | 645 |
\end{ttbox} |
323 | 646 |
\begin{ttdescription} |
4276 | 647 |
\item[Seq.empty] is the empty sequence. |
104 | 648 |
|
4276 | 649 |
\item[\tt Seq.make (fn () => Some ($x$, $xq$))] constructs the |
650 |
sequence with head~$x$ and tail~$xq$, neither of which is evaluated. |
|
104 | 651 |
|
4276 | 652 |
\item[Seq.single $x$] |
104 | 653 |
constructs the sequence containing the single element~$x$. |
654 |
||
4276 | 655 |
\item[Seq.pull $xq$] returns {\tt None} if the sequence is empty and |
656 |
{\tt Some ($x$, $xq'$)} if the sequence has head~$x$ and tail~$xq'$. |
|
657 |
Warning: calling \hbox{Seq.pull $xq$} again will {\it recompute\/} |
|
658 |
the value of~$x$; it is not stored! |
|
323 | 659 |
\end{ttdescription} |
104 | 660 |
|
661 |
||
323 | 662 |
\subsection{Converting between sequences and lists} |
104 | 663 |
\begin{ttbox} |
4276 | 664 |
Seq.chop : int * 'a seq -> 'a list * 'a seq |
665 |
Seq.list_of : 'a seq -> 'a list |
|
666 |
Seq.of_list : 'a list -> 'a seq |
|
104 | 667 |
\end{ttbox} |
323 | 668 |
\begin{ttdescription} |
4276 | 669 |
\item[Seq.chop ($n$, $xq$)] returns the first~$n$ elements of~$xq$ as a |
670 |
list, paired with the remaining elements of~$xq$. If $xq$ has fewer |
|
671 |
than~$n$ elements, then so will the list. |
|
672 |
||
673 |
\item[Seq.list_of $xq$] returns the elements of~$xq$, which must be |
|
674 |
finite, as a list. |
|
675 |
||
676 |
\item[Seq.of_list $xs$] creates a sequence containing the elements |
|
677 |
of~$xs$. |
|
323 | 678 |
\end{ttdescription} |
104 | 679 |
|
680 |
||
323 | 681 |
\subsection{Combining sequences} |
104 | 682 |
\begin{ttbox} |
4276 | 683 |
Seq.append : 'a seq * 'a seq -> 'a seq |
684 |
Seq.interleave : 'a seq * 'a seq -> 'a seq |
|
685 |
Seq.flat : 'a seq seq -> 'a seq |
|
686 |
Seq.map : ('a -> 'b) -> 'a seq -> 'b seq |
|
687 |
Seq.filter : ('a -> bool) -> 'a seq -> 'a seq |
|
104 | 688 |
\end{ttbox} |
323 | 689 |
\begin{ttdescription} |
4276 | 690 |
\item[Seq.append ($xq$, $yq$)] concatenates $xq$ to $yq$. |
691 |
||
692 |
\item[Seq.interleave ($xq$, $yq$)] joins $xq$ with $yq$ by |
|
693 |
interleaving their elements. The result contains all the elements |
|
694 |
of the sequences, even if both are infinite. |
|
695 |
||
696 |
\item[Seq.flat $xqq$] concatenates a sequence of sequences. |
|
697 |
||
698 |
\item[Seq.map $f$ $xq$] applies $f$ to every element |
|
699 |
of~$xq=x@1,x@2,\ldots$, yielding the sequence $f(x@1),f(x@2),\ldots$. |
|
700 |
||
701 |
\item[Seq.filter $p$ $xq$] returns the sequence consisting of all |
|
702 |
elements~$x$ of~$xq$ such that $p(x)$ is {\tt true}. |
|
323 | 703 |
\end{ttdescription} |
104 | 704 |
|
705 |
\index{tactics|)} |
|
5371 | 706 |
|
707 |
||
708 |
%%% Local Variables: |
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709 |
%%% mode: latex |
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%%% TeX-master: "ref" |
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%%% End: |