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author | wenzelm |

Sat, 18 Mar 2000 19:11:34 +0100 | |

changeset 8517 | 062e6cd78534 |

parent 8516 | f5f6a97ee43f |

child 8518 | daaedc7b56a9 |

obtain;
moved pure methods / atts to pure.tex;
tuned;

--- a/doc-src/IsarRef/generic.tex Sat Mar 18 19:10:02 2000 +0100 +++ b/doc-src/IsarRef/generic.tex Sat Mar 18 19:11:34 2000 +0100 @@ -1,146 +1,47 @@ \chapter{Generic Tools and Packages}\label{ch:gen-tools} -\section{Basic proof methods}\label{sec:pure-meth} +\section{Axiomatic Type Classes}\label{sec:axclass} -\indexisarmeth{fail}\indexisarmeth{succeed}\indexisarmeth{$-$} -\indexisarmeth{assumption}\indexisarmeth{this} -\indexisarmeth{fold}\indexisarmeth{unfold} -\indexisarmeth{rule}\indexisarmeth{erule} +\indexisarcmd{axclass}\indexisarcmd{instance}\indexisarmeth{intro-classes} \begin{matharray}{rcl} - - & : & \isarmeth \\ - assumption & : & \isarmeth \\ - this & : & \isarmeth \\ - rule & : & \isarmeth \\ - erule^* & : & \isarmeth \\[0.5ex] - fold & : & \isarmeth \\ - unfold & : & \isarmeth \\[0.5ex] - succeed & : & \isarmeth \\ - fail & : & \isarmeth \\ + \isarcmd{axclass} & : & \isartrans{theory}{theory} \\ + \isarcmd{instance} & : & \isartrans{theory}{proof(prove)} \\ + intro_classes & : & \isarmeth \\ \end{matharray} +Axiomatic type classes are provided by Isabelle/Pure as a \emph{definitional} +interface to type classes (cf.~\S\ref{sec:classes}). Thus any object logic +may make use of this light-weight mechanism of abstract theories. See +\cite{Wenzel:1997:TPHOL} for more information. There is also a tutorial on +\emph{Using Axiomatic Type Classes in Isabelle} that is part of the standard +Isabelle documentation. +%FIXME cite + \begin{rail} - ('fold' | 'unfold' | 'rule' | 'erule') thmrefs + 'axclass' classdecl (axmdecl prop comment? +) + ; + 'instance' (nameref '<' nameref | nameref '::' simplearity) comment? ; \end{rail} \begin{descr} -\item [``$-$''] does nothing but insert the forward chaining facts as premises - into the goal. Note that command $\PROOFNAME$ without any method actually - performs a single reduction step using the $rule$ method (see below); thus a - plain \emph{do-nothing} proof step would be $\PROOF{-}$ rather than - $\PROOFNAME$ alone. -\item [$assumption$] solves some goal by assumption. Any facts given are - guaranteed to participate in the refinement. -\item [$this$] applies the current facts directly as rules. Note that - ``$\DOT$'' (dot) abbreviates $\BY{this}$. -\item [$rule~thms$] applies some rule given as argument in backward manner; - facts are used to reduce the rule before applying it to the goal. Thus - $rule$ without facts is plain \emph{introduction}, while with facts it - becomes a (generalized) \emph{elimination}. +\item [$\isarkeyword{axclass}~c < \vec c~axms$] defines an axiomatic type + class as the intersection of existing classes, with additional axioms + holding. Class axioms may not contain more than one type variable. The + class axioms (with implicit sort constraints added) are bound to the given + names. Furthermore a class introduction rule is generated, which is + employed by method $intro_classes$ to support instantiation proofs of this + class. - Note that the classical reasoner introduces another version of $rule$ that - is able to pick appropriate rules automatically, whenever $thms$ are omitted - (see \S\ref{sec:classical-basic}); that method is the default for - $\PROOFNAME$ steps. Note that ``$\DDOT$'' (double-dot) abbreviates - $\BY{default}$. -\item [$erule~thms$] is similar to $rule$, but applies rules by - elim-resolution. This is an improper method, mainly for experimentation and - porting of old scripts. Actual elimination proofs are usually done with - $rule$ (single step, involving facts) or $elim$ (repeated steps, see - \S\ref{sec:classical-basic}). -\item [$unfold~thms$ and $fold~thms$] expand and fold back again the given - meta-level definitions throughout all goals; any facts provided are inserted - into the goal and subject to rewriting as well. -\item [$succeed$] yields a single (unchanged) result; it is the identity of - the ``\texttt{,}'' method combinator (cf.\ \S\ref{sec:syn-meth}). -\item [$fail$] yields an empty result sequence; it is the identity of the - ``\texttt{|}'' method combinator (cf.\ \S\ref{sec:syn-meth}). -\end{descr} - - -\section{Miscellaneous attributes} - -\indexisaratt{tag}\indexisaratt{untag}\indexisaratt{COMP}\indexisaratt{RS} -\indexisaratt{OF}\indexisaratt{where}\indexisaratt{of}\indexisaratt{standard} -\indexisaratt{elimify}\indexisaratt{transfer}\indexisaratt{export} -\indexisaratt{unfold}\indexisaratt{fold} -\begin{matharray}{rcl} - tag & : & \isaratt \\ - untag & : & \isaratt \\[0.5ex] - OF & : & \isaratt \\ - RS & : & \isaratt \\ - COMP & : & \isaratt \\[0.5ex] - of & : & \isaratt \\ - where & : & \isaratt \\[0.5ex] - unfold & : & \isaratt \\ - fold & : & \isaratt \\[0.5ex] - standard & : & \isaratt \\ - elimify & : & \isaratt \\ - export^* & : & \isaratt \\ - transfer & : & \isaratt \\[0.5ex] -\end{matharray} - -\begin{rail} - 'tag' (nameref+) - ; - 'untag' name - ; - 'OF' thmrefs - ; - ('RS' | 'COMP') nat? thmref - ; - 'of' (inst * ) ('concl' ':' (inst * ))? - ; - 'where' (name '=' term * 'and') - ; - ('unfold' | 'fold') thmrefs - ; - - inst: underscore | term - ; -\end{rail} - -\begin{descr} -\item [$tag~name~args$ and $untag~name$] add and remove $tags$ of some - theorem. Tags may be any list of strings that serve as comment for some - tools (e.g.\ $\LEMMANAME$ causes the tag ``$lemma$'' to be added to the - result). The first string is considered the tag name, the rest its - arguments. Note that untag removes any tags of the same name. -\item [$OF~thms$, $RS~n~thm$, and $COMP~n~thm$] compose rules. $OF$ applies - $thms$ in parallel (cf.\ \texttt{MRS} in \cite[\S5]{isabelle-ref}, but note - the reversed order). Note that premises may be skipped by including - ``$\_$'' (underscore) as argument. - - $RS$ resolves with the $n$-th premise of $thm$; $COMP$ is a version of $RS$ - that skips the automatic lifting process that is normally intended (cf.\ - \texttt{RS} and \texttt{COMP} in \cite[\S5]{isabelle-ref}). - -\item [$of~\vec t$ and $where~\vec x = \vec t$] perform positional and named - instantiation, respectively. The terms given in $of$ are substituted for - any schematic variables occurring in a theorem from left to right; - ``\texttt{_}'' (underscore) indicates to skip a position. Arguments - following a ``$concl\colon$'' specification refer to positions of the - conclusion of a rule. - -\item [$unfold~thms$ and $fold~thms$] expand and fold back again the given - meta-level definitions throughout a rule. - -\item [$standard$] puts a theorem into the standard form of object-rules, just - as the ML function \texttt{standard} (see \cite[\S5]{isabelle-ref}). - -\item [$elimify$] turns an destruction rule into an elimination, just as the - ML function \texttt{make\_elim} (see \cite{isabelle-ref}). - -\item [$export$] lifts a local result out of the current proof context, - generalizing all fixed variables and discharging all assumptions. Note that - (partial) export is usually done automatically behind the scenes. This - attribute is mainly for experimentation. - -\item [$transfer$] promotes a theorem to the current theory context, which has - to enclose the former one. Normally, this is done automatically when rules - are joined by inference. - +\item [$\isarkeyword{instance}~c@1 < c@2$ and $\isarkeyword{instance}~t :: + (\vec s)c$] setup up a goal stating the class relation or type arity. The + proof would usually proceed by $intro_classes$, and then establish the + characteristic theorems of the type classes involved. After finishing the + proof, the theory will be augmented by a type signature declaration + corresponding to the resulting theorem. +\item [$intro_classes$] repeatedly expands all class introduction rules of + this theory. \end{descr} @@ -205,12 +106,6 @@ context, by adding or deleting theorems (the default is to add). \end{descr} -%FIXME -%See theory \texttt{HOL/Isar_examples/Group} for a simple application of -%calculations for basic equational reasoning. -%\texttt{HOL/Isar_examples/KnasterTarski} involves a few more advanced -%calculational steps in combination with natural deduction. - \section{Named local contexts (cases)}\label{sec:cases} @@ -218,7 +113,7 @@ \indexisaratt{case-names}\indexisaratt{params} \begin{matharray}{rcl} \isarcmd{case} & : & \isartrans{proof(state)}{proof(state)} \\ - \isarcmd{print_cases} & : & \isarkeep{proof} \\[0.5ex] + \isarcmd{print_cases}^* & : & \isarkeep{proof} \\ case_names & : & \isaratt \\ params & : & \isaratt \\ \end{matharray} @@ -279,52 +174,172 @@ \end{descr} -\section{Axiomatic Type Classes}\label{sec:axclass} +\section{Generalized existence} -\indexisarcmd{axclass}\indexisarcmd{instance}\indexisarmeth{intro-classes} +\indexisarcmd{obtain} \begin{matharray}{rcl} - \isarcmd{axclass} & : & \isartrans{theory}{theory} \\ - \isarcmd{instance} & : & \isartrans{theory}{proof(prove)} \\ - intro_classes & : & \isarmeth \\ + \isarcmd{obtain} & : & \isartrans{proof(prove)}{proof(state)} \\ +\end{matharray} + +Generalized existence reasoning means that additional elements with certain +properties are introduced, together with a soundness proof of that context +change (the rest of the main goal is left unchanged). + +Syntactically, the $\OBTAINNAME$ language element is like a proof method to +the present goal, followed by a proof of its additional claim, followed by the +actual context commands (cf.\ $\FIXNAME$ and $\ASSUMENAME$, +\S\ref{sec:proof-context}). + +\begin{rail} + 'obtain' (vars + 'and') comment? \\ 'where' (assm comment? + 'and') + ; +\end{rail} + +$\OBTAINNAME$ is defined as a derived Isar command as follows, where the +preceding goal shall be $\psi$, with (optional) facts $\vec b$ indicated for +forward chaining. +\begin{matharray}{l} + \OBTAIN{\vec x}{a}{\vec \phi}~~\langle proof\rangle \equiv {} \\[0.5ex] + \quad \PROOF{succeed} \\ + \qquad \DEF{}{thesis \equiv \psi} \\ + \qquad \PRESUME{that}{\All{\vec x} \vec\phi \Imp thesis} \\ + \qquad \FROM{\vec b}~\SHOW{}{thesis}~~\langle proof\rangle \\ + \quad \NEXT \\ + \qquad \FIX{\vec x}~\ASSUME{a}{\vec\phi} \\ \end{matharray} -Axiomatic type classes are provided by Isabelle/Pure as a \emph{definitional} -interface to type classes (cf.~\S\ref{sec:classes}). Thus any object logic -may make use of this light-weight mechanism of abstract theories. See -\cite{Wenzel:1997:TPHOL} for more information. There is also a tutorial on -\emph{Using Axiomatic Type Classes in Isabelle} that is part of the standard -Isabelle documentation. -%FIXME cite +Typically, the soundness proof is relatively straight-forward, often just by +canonical automated tools such as $\BY{simp}$ (see \S\ref{sec:simp}) or +$\BY{blast}$ (see \S\ref{sec:classical-auto}). Note that the ``$that$'' +presumption above is usually declared as simplification and (unsafe) +introduction rule, somewhat depending on the object-logic's policy, +though.\footnote{Major object-logics such as HOL and HOLCF do this already.} + +The original goal statement is wrapped into a local definition in order to +avoid any automated tools descending into it. Usually, any statement would +admit the intended reduction; only in very rare cases $thesis_def$ has to be +expanded to complete the soundness proof. + +\medskip + +In a sense, $\OBTAINNAME$ represents at the level of Isar proofs what would be +meta-logical existential quantifiers and conjunctions. This concept has a +broad range of useful applications, ranging from plain elimination (or even +introduction) of object-level existentials and conjunctions, to elimination +over results of symbolic evaluation of recursive definitions, for example. + + +\section{Miscellaneous methods and attributes} + +\indexisarmeth{unfold}\indexisarmeth{fold} +\indexisarmeth{erule}\indexisarmeth{drule}\indexisarmeth{frule} +\indexisarmeth{fail}\indexisarmeth{succeed} +\begin{matharray}{rcl} + unfold & : & \isarmeth \\ + fold & : & \isarmeth \\[0.5ex] + erule^* & : & \isarmeth \\ + drule^* & : & \isarmeth \\ + frule^* & : & \isarmeth \\[0.5ex] + succeed & : & \isarmeth \\ + fail & : & \isarmeth \\ +\end{matharray} \begin{rail} - 'axclass' classdecl (axmdecl prop comment? +) - ; - 'instance' (nameref '<' nameref | nameref '::' simplearity) comment? + ('fold' | 'unfold' | 'erule' | 'drule' | 'frule') thmrefs ; \end{rail} \begin{descr} -\item [$\isarkeyword{axclass}~c < \vec c~axms$] defines an axiomatic type - class as the intersection of existing classes, with additional axioms - holding. Class axioms may not contain more than one type variable. The - class axioms (with implicit sort constraints added) are bound to the given - names. Furthermore a class introduction rule is generated, which is - employed by method $intro_classes$ to support instantiation proofs of this - class. +\item [$unfold~thms$ and $fold~thms$] expand and fold back again the given + meta-level definitions throughout all goals; any facts provided are inserted + into the goal and subject to rewriting as well. +\item [$erule~thms$, $drule~thms$, and $frule~thms$] are similar to the basic + $rule$ method (see \S\ref{sec:pure-meth-att}), but apply rules by + elim-resolution, destruct-resolution, and forward-resolution, respectively + \cite{isabelle-ref}. These are improper method, mainly for experimentation + and emulating tactic scripts. -\item [$\isarkeyword{instance}~c@1 < c@2$ and $\isarkeyword{instance}~t :: - (\vec s)c$] setup up a goal stating the class relation or type arity. The - proof would usually proceed by $intro_classes$, and then establish the - characteristic theorems of the type classes involved. After finishing the - proof, the theory will be augmented by a type signature declaration - corresponding to the resulting theorem. -\item [$intro_classes$] repeatedly expands all class introduction rules of - this theory. + Different modes of basic rule application are usually expressed in Isar at + the proof language level, rather than via implicit proof state + modifications. For example, a proper single-step elimination would be done + using the basic $rule$ method, with forward chaining of current facts. +\item [$succeed$] yields a single (unchanged) result; it is the identity of + the ``\texttt{,}'' method combinator (cf.\ \S\ref{sec:syn-meth}). +\item [$fail$] yields an empty result sequence; it is the identity of the + ``\texttt{|}'' method combinator (cf.\ \S\ref{sec:syn-meth}). \end{descr} -%FIXME -%See theory \texttt{HOL/Isar_examples/Group} for a simple example of using -%axiomatic type classes, including instantiation proofs. + +\indexisaratt{standard} +\indexisaratt{elimify} + +\indexisaratt{RS}\indexisaratt{COMP} +\indexisaratt{where} +\indexisaratt{tag}\indexisaratt{untag} +\indexisaratt{transfer} +\indexisaratt{export} +\indexisaratt{unfold}\indexisaratt{fold} +\begin{matharray}{rcl} + tag & : & \isaratt \\ + untag & : & \isaratt \\[0.5ex] + RS & : & \isaratt \\ + COMP & : & \isaratt \\[0.5ex] + where & : & \isaratt \\[0.5ex] + unfold & : & \isaratt \\ + fold & : & \isaratt \\[0.5ex] + standard & : & \isaratt \\ + elimify & : & \isaratt \\ + export^* & : & \isaratt \\ + transfer & : & \isaratt \\[0.5ex] +\end{matharray} + +\begin{rail} + 'tag' (nameref+) + ; + 'untag' name + ; + ('RS' | 'COMP') nat? thmref + ; + 'where' (name '=' term * 'and') + ; + ('unfold' | 'fold') thmrefs + ; +\end{rail} + +\begin{descr} +\item [$tag~name~args$ and $untag~name$] add and remove $tags$ of some + theorem. Tags may be any list of strings that serve as comment for some + tools (e.g.\ $\LEMMANAME$ causes the tag ``$lemma$'' to be added to the + result). The first string is considered the tag name, the rest its + arguments. Note that untag removes any tags of the same name. +\item [$RS~n~thm$ and $COMP~n~thm$] compose rules. $RS$ resolves with the + $n$-th premise of $thm$; $COMP$ is a version of $RS$ that skips the + automatic lifting process that is normally intended (cf.\ \texttt{RS} and + \texttt{COMP} in \cite[\S5]{isabelle-ref}). +\item [$where~\vec x = \vec t$] perform named instantiation of schematic + variables occurring in a theorem. Unlike instantiation tactics (such as + \texttt{res_inst_tac}, see \cite{isabelle-ref}), actual schematic variables + have to be specified (e.g.\ $\Var{x@3}$). + +\item [$unfold~thms$ and $fold~thms$] expand and fold back again the given + meta-level definitions throughout a rule. + +\item [$standard$] puts a theorem into the standard form of object-rules, just + as the ML function \texttt{standard} (see \cite[\S5]{isabelle-ref}). + +\item [$elimify$] turns an destruction rule into an elimination, just as the + ML function \texttt{make\_elim} (see \cite{isabelle-ref}). + +\item [$export$] lifts a local result out of the current proof context, + generalizing all fixed variables and discharging all assumptions. Note that + (partial) export is usually done automatically behind the scenes. This + attribute is mainly for experimentation. + +\item [$transfer$] promotes a theorem to the current theory context, which has + to enclose the former one. Normally, this is done automatically when rules + are joined by inference. + +\end{descr} \section{The Simplifier} @@ -374,7 +389,7 @@ should be used with some care, though. Note that there is no separate $split$ method. The effect of -\texttt{split_tac} can be simulated via $(simp~only\colon~split\colon~thms)$. +\texttt{split_tac} can be simulated by $(simp~only\colon~split\colon~thms)$. \subsection{Declaring rules} @@ -436,11 +451,12 @@ \begin{descr} \item [$rule$] as offered by the classical reasoner is a refinement over the - primitive one (see \S\ref{sec:pure-meth}). In case that no rules are + primitive one (see \S\ref{sec:pure-meth-att}). In case that no rules are provided as arguments, it automatically determines elimination and introduction rules from the context (see also \S\ref{sec:classical-mod}). - In that form it is the default method for basic proof steps, such as - $\PROOFNAME$ and ``$\DDOT$'' (two dots). + This is made the default method for basic proof steps, such as $\PROOFNAME$ + and ``$\DDOT$'' (two dots), see also \S\ref{sec:proof-steps} and + \S\ref{sec:pure-meth-att}. \item [$intro$ and $elim$] repeatedly refine some goal by intro- or elim-resolution, after having inserted any facts. Omitting the arguments @@ -490,7 +506,7 @@ \end{descr} Any of above methods support additional modifiers of the context of classical -rules. There semantics is analogous to the attributes given in +rules. Their semantics is analogous to the attributes given in \S\ref{sec:classical-mod}. Facts provided by forward chaining are inserted into the goal before doing the search. The ``!''~argument causes the full context of assumptions to be included as well.\footnote{This is slightly less @@ -546,11 +562,11 @@ \end{rail} \begin{descr} -\item [$intro$, $elim$, $dest$] add introduction, elimination, destruct rules, - respectively. By default, rules are considered as \emph{safe}, while a - single ``?'' classifies as \emph{unsafe}, and ``??'' as \emph{extra} (i.e.\ - not applied in the search-oriented automated methods, but only in - single-step methods such as $rule$). +\item [$intro$, $elim$, and $dest$] declare introduction, elimination, and + destruct rules, respectively. By default, rules are considered as + \emph{safe}, while a single ``?'' classifies as \emph{unsafe}, and ``??'' as + \emph{extra} (i.e.\ not applied in the search-oriented automated methods, + but only in single-step methods such as $rule$). \item [$iff$] declares equations both as rewrite rules for the simplifier and classical reasoning rules.