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doc-src/IsarRef/generic.tex

author | wenzelm |

Mon, 27 Mar 2000 18:10:11 +0200 | |

changeset 8594 | d2e2a3df6871 |

parent 8547 | 93b8685d004b |

child 8619 | 63a0e1502e41 |

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

rail token vs. terminal;

\chapter{Generic Tools and Packages}\label{ch:gen-tools} \section{Axiomatic Type Classes}\label{sec:axclass} \indexisarcmd{axclass}\indexisarcmd{instance}\indexisarmeth{intro-classes} \begin{matharray}{rcl} \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 \cite{Wenzel:1997:TPHOL}. 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} 'axclass' classdecl (axmdecl prop comment? +) ; 'instance' (nameref '<' nameref | nameref '::' simplearity) comment? ; \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 [$\isarkeyword{instance}~c@1 < c@2$ and $\isarkeyword{instance}~t :: (\vec s)c$] setup a goal stating a 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} \section{Calculational proof}\label{sec:calculation} \indexisarcmd{also}\indexisarcmd{finally}\indexisaratt{trans} \begin{matharray}{rcl} \isarcmd{also} & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{finally} & : & \isartrans{proof(state)}{proof(chain)} \\ trans & : & \isaratt \\ \end{matharray} Calculational proof is forward reasoning with implicit application of transitivity rules (such those of $=$, $\le$, $<$). Isabelle/Isar maintains an auxiliary register $calculation$\indexisarthm{calculation} for accumulating results obtained by transitivity composed with the current result. Command $\ALSO$ updates $calculation$ involving $this$, while $\FINALLY$ exhibits the final $calculation$ by forward chaining towards the next goal statement. Both commands require valid current facts, i.e.\ may occur only after commands that produce theorems such as $\ASSUMENAME$, $\NOTENAME$, or some finished proof of $\HAVENAME$, $\SHOWNAME$ etc. Also note that the automatic term abbreviation ``$\dots$'' has its canonical application with calculational proofs. It automatically refers to the argument\footnote{The argument of a curried infix expression is its right-hand side.} of the preceding statement. Isabelle/Isar calculations are implicitly subject to block structure in the sense that new threads of calculational reasoning are commenced for any new block (as opened by a local goal, for example). This means that, apart from being able to nest calculations, there is no separate \emph{begin-calculation} command required. \begin{rail} ('also' | 'finally') transrules? comment? ; 'trans' (() | 'add' | 'del') ; transrules: '(' thmrefs ')' interest? ; \end{rail} \begin{descr} \item [$\ALSO~(\vec a)$] maintains the auxiliary $calculation$ register as follows. The first occurrence of $\ALSO$ in some calculational thread initializes $calculation$ by $this$. Any subsequent $\ALSO$ on the same level of block-structure updates $calculation$ by some transitivity rule applied to $calculation$ and $this$ (in that order). Transitivity rules are picked from the current context plus those given as explicit arguments (the latter have precedence). \item [$\FINALLY~(\vec a)$] maintaining $calculation$ in the same way as $\ALSO$, and concludes the current calculational thread. The final result is exhibited as fact for forward chaining towards the next goal. Basically, $\FINALLY$ just abbreviates $\ALSO~\FROM{calculation}$. Note that ``$\FINALLY~\SHOW{}{\Var{thesis}}~\DOT$'' and ``$\FINALLY~\HAVE{}{\phi}~\DOT$'' are typical idioms for concluding calculational proofs. \item [$trans$] declares theorems as transitivity rules. \end{descr} \section{Named local contexts (cases)}\label{sec:cases} \indexisarcmd{case}\indexisarcmd{print-cases} \indexisaratt{case-names}\indexisaratt{params} \begin{matharray}{rcl} \isarcmd{case} & : & \isartrans{proof(state)}{proof(state)} \\ \isarcmd{print_cases}^* & : & \isarkeep{proof} \\ case_names & : & \isaratt \\ params & : & \isaratt \\ \end{matharray} Basically, Isar proof contexts are built up explicitly using commands like $\FIXNAME$, $\ASSUMENAME$ etc.\ (see \S\ref{sec:proof-context}). In typical verification tasks this can become hard to manage, though. In particular, a large number of local contexts may emerge from case analysis or induction over inductive sets and types. \medskip The $\CASENAME$ command provides a shorthand to refer to certain parts of logical context symbolically. Proof methods may provide an environment of named ``cases'' of the form $c\colon \vec x, \vec \phi$. Then the effect of $\CASE{c}$ is exactly the same as $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$. It is important to note that $\CASENAME$ does \emph{not} provide any means to peek at the current goal state, which is treated as strictly non-observable in Isar! Instead, the cases considered here usually emerge in a canonical way from certain pieces of specification that appear in the theory somewhere else (e.g.\ in an inductive definition, or recursive function). See also \S\ref{sec:induct-method} for more details of how this works in HOL. \medskip Named cases may be exhibited in the current proof context only if both the proof method and the rules involved support this. Case names and parameters of basic rules may be declared by hand as well, by using appropriate attributes. Thus variant versions of rules that have been derived manually may be used in advanced case analysis later. \railalias{casenames}{case\_names} \railterm{casenames} \begin{rail} 'case' nameref attributes? ; casenames (name + ) ; 'params' ((name * ) + 'and') ; \end{rail} %FIXME bug in rail \begin{descr} \item [$\CASE{c}$] invokes a named local context $c\colon \vec x, \vec \phi$, as provided by an appropriate proof method (such as $cases$ and $induct$ in Isabelle/HOL, see \S\ref{sec:induct-method}). The command $\CASE{c}$ abbreviates $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$. \item [$\isarkeyword{print_cases}$] prints all local contexts of the current state, using Isar proof language notation. This is a diagnostic command; $undo$ does not apply. \item [$case_names~\vec c$] declares names for the local contexts of premises of some theorem; $\vec c$ refers to the \emph{suffix} of the list premises. \item [$params~\vec p@1 \dots \vec p@n$] renames the innermost parameters of premises $1, \dots, n$ of some theorem. An empty list of names may be given to skip positions, leaving the present parameters unchanged. \end{descr} \section{Generalized existence} \indexisarcmd{obtain} \begin{matharray}{rcl} \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 an initial proof method to the present goal, followed by a proof of its additional claim, followed by the actual context commands (using the syntax of $\FIXNAME$ and $\ASSUMENAME$, see \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; here 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} 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, depending on the object-logic's policy, though.\footnote{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 anyway; 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} ('fold' | 'unfold' | 'erule' | 'drule' | 'frule') thmrefs ; \end{rail} \begin{descr} \item [$unfold~\vec a$ and $fold~\vec a$] 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 [$erule~\vec a$, $drule~\vec a$, and $frule~\vec a$] 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. Different modes of basic rule application are usually expressed in Isar at the proof language level, rather than via implicit proof state manipulations. 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} \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 [$RS~n~a$ and $COMP~n~a$] compose rules. $RS$ resolves with the $n$-th premise of $a$; $COMP$ 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 [$unfold~\vec a$ and $fold~\vec a$] 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 proper incremental export is already done as part of the basic Isar machinery. This attribute is mainly for experimentation. \item [$transfer$] promotes a theorem to the current theory context, which has to enclose the former one. This is done automatically whenever rules are joined by inference. \end{descr} \section{The Simplifier} \subsection{Simplification methods}\label{sec:simp} \indexisarmeth{simp}\indexisarmeth{simp-all} \begin{matharray}{rcl} simp & : & \isarmeth \\ simp_all & : & \isarmeth \\ \end{matharray} \railalias{simpall}{simp\_all} \railterm{simpall} \begin{rail} ('simp' | simpall) ('!' ?) (simpmod * ) ; simpmod: ('add' | 'del' | 'only' | 'split' (() | 'add' | 'del') | 'other') ':' thmrefs ; \end{rail} \begin{descr} \item [$simp$] invokes Isabelle's simplifier, after declaring additional rules according to the arguments given. Note that the \railtterm{only} modifier first removes all other rewrite rules, congruences, and looper tactics (including splits), and then behaves like \railtterm{add}. The \railtterm{split} modifiers add or delete rules for the Splitter (see also \cite{isabelle-ref}), the default is to add. This works only if the Simplifier method has been properly setup to include the Splitter (all major object logics such HOL, HOLCF, FOL, ZF do this already). The \railtterm{other} modifier ignores its arguments. Nevertheless, additional kinds of rules may be declared by including appropriate attributes in the specification. \item [$simp_all$] is similar to $simp$, but acts on all goals. \end{descr} Internally, the $simp$ method is based on \texttt{asm_full_simp_tac} \cite[\S10]{isabelle-ref}, but is much better behaved in practice. Just the local premises of the actual goal are involved by default. Additional facts may be inserted via forward-chaining (using $\THEN$, $\FROMNAME$ etc.). The full context of assumptions is only included in the $simp!$ version, which should be used with some care, though. Note that there is no separate $split$ method. The effect of \texttt{split_tac} can be simulated by $(simp~only\colon~split\colon~\vec a)$. \subsection{Declaring rules} \indexisaratt{simp}\indexisaratt{split} \begin{matharray}{rcl} simp & : & \isaratt \\ split & : & \isaratt \\ \end{matharray} \begin{rail} ('simp' | 'split') (() | 'add' | 'del') ; \end{rail} \begin{descr} \item [$simp$] declares simplification rules. \item [$split$] declares split rules. \end{descr} \subsection{Forward simplification} \indexisaratt{simplify}\indexisaratt{asm-simplify} \indexisaratt{full-simplify}\indexisaratt{asm-full-simplify} \begin{matharray}{rcl} simplify & : & \isaratt \\ asm_simplify & : & \isaratt \\ full_simplify & : & \isaratt \\ asm_full_simplify & : & \isaratt \\ \end{matharray} These attributes provide forward rules for simplification, which should be used only very rarely. There are no separate options for declaring simplification rules locally. See the ML functions of the same name in \cite[\S10]{isabelle-ref} for more information. \section{The Classical Reasoner} \subsection{Basic methods}\label{sec:classical-basic} \indexisarmeth{rule}\indexisarmeth{intro} \indexisarmeth{elim}\indexisarmeth{default}\indexisarmeth{contradiction} \begin{matharray}{rcl} rule & : & \isarmeth \\ intro & : & \isarmeth \\ elim & : & \isarmeth \\ contradiction & : & \isarmeth \\ \end{matharray} \begin{rail} ('rule' | 'intro' | 'elim') thmrefs? ; \end{rail} \begin{descr} \item [$rule$] as offered by the classical reasoner is a refinement over the 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}). 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 refers to any suitable rules declared in the context, otherwise only the explicitly given ones may be applied. The latter form admits better control of what actually happens, thus it is very appropriate as an initial method for $\PROOFNAME$ that splits up certain connectives of the goal, before entering the actual sub-proof. \item [$contradiction$] solves some goal by contradiction, deriving any result from both $\neg A$ and $A$. Facts, which are guaranteed to participate, may appear in either order. \end{descr} \subsection{Automated methods}\label{sec:classical-auto} \indexisarmeth{blast} \indexisarmeth{fast}\indexisarmeth{best}\indexisarmeth{slow}\indexisarmeth{slow-best} \begin{matharray}{rcl} blast & : & \isarmeth \\ fast & : & \isarmeth \\ best & : & \isarmeth \\ slow & : & \isarmeth \\ slow_best & : & \isarmeth \\ \end{matharray} \railalias{slowbest}{slow\_best} \railterm{slowbest} \begin{rail} 'blast' ('!' ?) nat? (clamod * ) ; ('fast' | 'best' | 'slow' | slowbest) ('!' ?) (clamod * ) ; clamod: (('intro' | 'elim' | 'dest') (() | '?' | '??') | 'del') ':' thmrefs ; \end{rail} \begin{descr} \item [$blast$] refers to the classical tableau prover (see \texttt{blast_tac} in \cite[\S11]{isabelle-ref}). The optional argument specifies a user-supplied search bound (default 20). \item [$fast$, $best$, $slow$, $slow_best$] refer to the generic classical reasoner (see \cite[\S11]{isabelle-ref}, tactic \texttt{fast_tac} etc). \end{descr} Any of above methods support additional modifiers of the context of classical rules. Their semantics is analogous to the attributes given in \S\ref{sec:classical-mod}. Facts provided by forward chaining are inserted\footnote{These methods usually cannot make proper use of actual rules inserted that way, though.} into the goal before doing the search. The ``!''~argument causes the full context of assumptions to be included as well. This is slightly less hazardous than for the Simplifier (see \S\ref{sec:simp}). \subsection{Combined automated methods} \indexisarmeth{auto}\indexisarmeth{force} \begin{matharray}{rcl} force & : & \isarmeth \\ auto & : & \isarmeth \\ \end{matharray} \begin{rail} ('force' | 'auto') ('!' ?) (clasimpmod * ) ; clasimpmod: ('simp' (() | 'add' | 'del' | 'only') | 'other' | ('split' (() | 'add' | 'del')) | (('intro' | 'elim' | 'dest') (() | '?' | '??') | 'del')) ':' thmrefs \end{rail} \begin{descr} \item [$force$ and $auto$] provide access to Isabelle's combined simplification and classical reasoning tactics. See \texttt{force_tac} and \texttt{auto_tac} in \cite[\S11]{isabelle-ref} for more information. The modifier arguments correspond to those given in \S\ref{sec:simp} and \S\ref{sec:classical-auto}. Just note that the ones related to the Simplifier are prefixed by \railtterm{simp} here. 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. \end{descr} \subsection{Declaring rules}\label{sec:classical-mod} \indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest} \indexisaratt{iff}\indexisaratt{delrule} \begin{matharray}{rcl} intro & : & \isaratt \\ elim & : & \isaratt \\ dest & : & \isaratt \\ iff & : & \isaratt \\ delrule & : & \isaratt \\ \end{matharray} \begin{rail} ('intro' | 'elim' | 'dest') (() | '?' | '??') ; \end{rail} \begin{descr} \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 rules for the Simplifier and Classical Reasoner. \item [$delrule$] deletes introduction or elimination rules from the context. Note that destruction rules would have to be turned into elimination rules first, e.g.\ by using the $elimify$ attribute. \end{descr} %%% Local Variables: %%% mode: latex %%% TeX-master: "isar-ref" %%% End: