author | wenzelm |
Fri, 10 Nov 2000 19:02:37 +0100 | |
changeset 10432 | 3dfbc913d184 |
parent 10318 | e47c221beded |
child 10548 | e8c774c12105 |
permissions | -rw-r--r-- |
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\chapter{Generic Tools and Packages}\label{ch:gen-tools} |
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\section{Axiomatic Type Classes}\label{sec:axclass} |
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%FIXME |
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% - qualified names |
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% - class intro rules; |
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% - class axioms; |
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\indexisarcmd{axclass}\indexisarcmd{instance}\indexisarmeth{intro-classes} |
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\begin{matharray}{rcl} |
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\isarcmd{axclass} & : & \isartrans{theory}{theory} \\ |
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\isarcmd{instance} & : & \isartrans{theory}{proof(prove)} \\ |
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intro_classes & : & \isarmeth \\ |
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\end{matharray} |
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Axiomatic type classes are provided by Isabelle/Pure as a \emph{definitional} |
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interface to type classes (cf.~\S\ref{sec:classes}). Thus any object logic |
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may make use of this light-weight mechanism of abstract theories |
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\cite{Wenzel:1997:TPHOL}. There is also a tutorial on using axiomatic type |
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classes in isabelle \cite{isabelle-axclass} that is part of the standard |
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Isabelle documentation. |
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\begin{rail} |
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'axclass' classdecl (axmdecl prop comment? +) |
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; |
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'instance' (nameref '<' nameref | nameref '::' simplearity) comment? |
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; |
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\end{rail} |
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\begin{descr} |
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\item [$\AXCLASS~c < \vec c~axms$] defines an axiomatic type class as the |
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intersection of existing classes, with additional axioms holding. Class |
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axioms may not contain more than one type variable. The class axioms (with |
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implicit sort constraints added) are bound to the given names. Furthermore |
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a class introduction rule is generated, which is employed by method |
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$intro_classes$ to support instantiation proofs of this class. |
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\item [$\INSTANCE~c@1 < c@2$ and $\INSTANCE~t :: (\vec s)c$] setup a goal |
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stating a class relation or type arity. The proof would usually proceed by |
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$intro_classes$, and then establish the characteristic theorems of the type |
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classes involved. After finishing the proof, the theory will be augmented |
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by a type signature declaration corresponding to the resulting theorem. |
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\item [$intro_classes$] repeatedly expands all class introduction rules of |
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this theory. |
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\end{descr} |
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|
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\section{Calculational proof}\label{sec:calculation} |
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\indexisarcmd{also}\indexisarcmd{finally} |
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\indexisarcmd{moreover}\indexisarcmd{ultimately} |
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\indexisarcmd{print-trans-rules}\indexisaratt{trans} |
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\begin{matharray}{rcl} |
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\isarcmd{also} & : & \isartrans{proof(state)}{proof(state)} \\ |
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\isarcmd{finally} & : & \isartrans{proof(state)}{proof(chain)} \\ |
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\isarcmd{moreover} & : & \isartrans{proof(state)}{proof(state)} \\ |
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\isarcmd{ultimately} & : & \isartrans{proof(state)}{proof(chain)} \\ |
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\isarcmd{print_trans_rules}^* & : & \isarkeep{theory~|~proof} \\ |
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trans & : & \isaratt \\ |
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\end{matharray} |
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Calculational proof is forward reasoning with implicit application of |
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transitivity rules (such those of $=$, $\le$, $<$). Isabelle/Isar maintains |
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an auxiliary register $calculation$\indexisarthm{calculation} for accumulating |
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results obtained by transitivity composed with the current result. Command |
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$\ALSO$ updates $calculation$ involving $this$, while $\FINALLY$ exhibits the |
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final $calculation$ by forward chaining towards the next goal statement. Both |
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commands require valid current facts, i.e.\ may occur only after commands that |
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produce theorems such as $\ASSUMENAME$, $\NOTENAME$, or some finished proof of |
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$\HAVENAME$, $\SHOWNAME$ etc. The $\MOREOVER$ and $\ULTIMATELY$ commands are |
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similar to $\ALSO$ and $\FINALLY$, but only collect further results in |
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$calculation$ without applying any rules yet. |
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Also note that the automatic term abbreviation ``$\dots$'' has its canonical |
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application with calculational proofs. It refers to the argument\footnote{The |
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argument of a curried infix expression is its right-hand side.} of the |
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preceding statement. |
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Isabelle/Isar calculations are implicitly subject to block structure in the |
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sense that new threads of calculational reasoning are commenced for any new |
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block (as opened by a local goal, for example). This means that, apart from |
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being able to nest calculations, there is no separate \emph{begin-calculation} |
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command required. |
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\medskip |
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The Isar calculation proof commands may be defined as |
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follows:\footnote{Internal bookkeeping such as proper handling of |
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block-structure has been suppressed.} |
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\begin{matharray}{rcl} |
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\ALSO@0 & \equiv & \NOTE{calculation}{this} \\ |
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\ALSO@{n+1} & \equiv & \NOTE{calculation}{trans~[OF~calculation~this]} \\[0.5ex] |
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\FINALLY & \equiv & \ALSO~\FROM{calculation} \\ |
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\MOREOVER & \equiv & \NOTE{calculation}{calculation~this} \\ |
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\ULTIMATELY & \equiv & \MOREOVER~\FROM{calculation} \\ |
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\end{matharray} |
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\begin{rail} |
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('also' | 'finally') transrules? comment? |
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; |
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('moreover' | 'ultimately') comment? |
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; |
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'trans' (() | 'add' | 'del') |
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; |
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transrules: '(' thmrefs ')' interest? |
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; |
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\end{rail} |
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\begin{descr} |
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\item [$\ALSO~(\vec a)$] maintains the auxiliary $calculation$ register as |
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follows. The first occurrence of $\ALSO$ in some calculational thread |
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initializes $calculation$ by $this$. Any subsequent $\ALSO$ on the same |
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level of block-structure updates $calculation$ by some transitivity rule |
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applied to $calculation$ and $this$ (in that order). Transitivity rules are |
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picked from the current context plus those given as explicit arguments (the |
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latter have precedence). |
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\item [$\FINALLY~(\vec a)$] maintaining $calculation$ in the same way as |
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$\ALSO$, and concludes the current calculational thread. The final result |
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is exhibited as fact for forward chaining towards the next goal. Basically, |
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$\FINALLY$ just abbreviates $\ALSO~\FROM{calculation}$. Note that |
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``$\FINALLY~\SHOW{}{\Var{thesis}}~\DOT$'' and |
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``$\FINALLY~\HAVE{}{\phi}~\DOT$'' are typical idioms for concluding |
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calculational proofs. |
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\item [$\MOREOVER$ and $\ULTIMATELY$] are analogous to $\ALSO$ and $\FINALLY$, |
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but collect results only, without applying rules. |
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\item [$\isarkeyword{print_trans_rules}$] prints the list of transitivity |
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rules declared in the current context. |
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\item [$trans$] declares theorems as transitivity rules. |
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\end{descr} |
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\section{Named local contexts (cases)}\label{sec:cases} |
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\indexisarcmd{case}\indexisarcmd{print-cases} |
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\indexisaratt{case-names}\indexisaratt{params} |
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\begin{matharray}{rcl} |
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\isarcmd{case} & : & \isartrans{proof(state)}{proof(state)} \\ |
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\isarcmd{print_cases}^* & : & \isarkeep{proof} \\ |
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case_names & : & \isaratt \\ |
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params & : & \isaratt \\ |
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\end{matharray} |
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Basically, Isar proof contexts are built up explicitly using commands like |
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$\FIXNAME$, $\ASSUMENAME$ etc.\ (see \S\ref{sec:proof-context}). In typical |
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verification tasks this can become hard to manage, though. In particular, a |
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large number of local contexts may emerge from case analysis or induction over |
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inductive sets and types. |
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\medskip |
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The $\CASENAME$ command provides a shorthand to refer to certain parts of |
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logical context symbolically. Proof methods may provide an environment of |
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named ``cases'' of the form $c\colon \vec x, \vec \phi$. Then the effect of |
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$\CASE{c}$ is exactly the same as $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$. |
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It is important to note that $\CASENAME$ does \emph{not} provide any means to |
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peek at the current goal state, which is treated as strictly non-observable in |
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Isar! Instead, the cases considered here usually emerge in a canonical way |
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from certain pieces of specification that appear in the theory somewhere else |
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(e.g.\ in an inductive definition, or recursive function). See also |
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\S\ref{sec:induct-method} for more details of how this works in HOL. |
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\medskip |
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Named cases may be exhibited in the current proof context only if both the |
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proof method and the rules involved support this. Case names and parameters |
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of basic rules may be declared by hand as well, by using appropriate |
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attributes. Thus variant versions of rules that have been derived manually |
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may be used in advanced case analysis later. |
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\railalias{casenames}{case\_names} |
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\railterm{casenames} |
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\begin{rail} |
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'case' nameref attributes? |
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; |
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casenames (name + ) |
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; |
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'params' ((name * ) + 'and') |
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; |
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\end{rail} |
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%FIXME bug in rail |
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\begin{descr} |
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\item [$\CASE{c}$] invokes a named local context $c\colon \vec x, \vec \phi$, |
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as provided by an appropriate proof method (such as $cases$ and $induct$ in |
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Isabelle/HOL, see \S\ref{sec:induct-method}). The command $\CASE{c}$ |
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abbreviates $\FIX{\vec x}~\ASSUME{c}{\vec\phi}$. |
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\item [$\isarkeyword{print_cases}$] prints all local contexts of the current |
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state, using Isar proof language notation. This is a diagnostic command; |
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$undo$ does not apply. |
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\item [$case_names~\vec c$] declares names for the local contexts of premises |
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of some theorem; $\vec c$ refers to the \emph{suffix} of the list premises. |
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\item [$params~\vec p@1 \dots \vec p@n$] renames the innermost parameters of |
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premises $1, \dots, n$ of some theorem. An empty list of names may be given |
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to skip positions, leaving the present parameters unchanged. |
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Note that the default usage of case rules does \emph{not} directly expose |
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parameters to the proof context (see also \S\ref{sec:induct-method-proper}). |
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\end{descr} |
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\section{Generalized existence}\label{sec:obtain} |
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\indexisarcmd{obtain} |
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\begin{matharray}{rcl} |
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\isarcmd{obtain} & : & \isartrans{proof(state)}{proof(prove)} \\ |
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\end{matharray} |
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Generalized existence means that additional elements with certain properties |
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may introduced in the current context. Technically, the $\OBTAINNAME$ |
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language element is like a declaration of $\FIXNAME$ and $\ASSUMENAME$ (see |
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also see \S\ref{sec:proof-context}), together with a soundness proof of its |
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additional claim. According to the nature of existential reasoning, |
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assumptions get eliminated from any result exported from the context later, |
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provided that the corresponding parameters do \emph{not} occur in the |
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conclusion. |
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\begin{rail} |
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'obtain' (vars + 'and') comment? \\ 'where' (assm comment? + 'and') |
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; |
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\end{rail} |
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$\OBTAINNAME$ is defined as a derived Isar command as follows, where $\vec b$ |
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shall refer to (optional) facts indicated for forward chaining. |
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\begin{matharray}{l} |
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\langle facts~\vec b\rangle \\ |
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\OBTAIN{\vec x}{a}{\vec \phi}~~\langle proof\rangle \equiv {} \\[1ex] |
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\quad \BG \\ |
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\qquad \FIX{thesis} \\ |
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\qquad \ASSUME{that~[simp, intro]}{\All{\vec x} \vec\phi \Imp thesis} \\ |
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\qquad \FROM{\vec b}~\HAVE{}{thesis}~~\langle proof\rangle \\ |
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\quad \EN \\ |
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\quad \FIX{\vec x}~\ASSUMENAME^\ast~a\colon~\vec\phi \\ |
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\end{matharray} |
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Typically, the soundness proof is relatively straight-forward, often just by |
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canonical automated tools such as $\BY{simp}$ (see \S\ref{sec:simp}) or |
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$\BY{blast}$ (see \S\ref{sec:classical-auto}). Accordingly, the ``$that$'' |
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reduction above is declared as simplification and introduction rule. |
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\medskip |
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In a sense, $\OBTAINNAME$ represents at the level of Isar proofs what would be |
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meta-logical existential quantifiers and conjunctions. This concept has a |
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broad range of useful applications, ranging from plain elimination (or even |
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introduction) of object-level existentials and conjunctions, to elimination |
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over results of symbolic evaluation of recursive definitions, for example. |
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Also note that $\OBTAINNAME$ without parameters acts much like $\HAVENAME$, |
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where the result is treated as an assumption. |
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\section{Miscellaneous methods and attributes}\label{sec:misc-methods} |
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\indexisarmeth{unfold}\indexisarmeth{fold}\indexisarmeth{insert} |
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\indexisarmeth{erule}\indexisarmeth{drule}\indexisarmeth{frule} |
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\indexisarmeth{fail}\indexisarmeth{succeed} |
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\begin{matharray}{rcl} |
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unfold & : & \isarmeth \\ |
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fold & : & \isarmeth \\[0.5ex] |
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insert^* & : & \isarmeth \\[0.5ex] |
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erule^* & : & \isarmeth \\ |
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drule^* & : & \isarmeth \\ |
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frule^* & : & \isarmeth \\[0.5ex] |
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succeed & : & \isarmeth \\ |
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fail & : & \isarmeth \\ |
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\end{matharray} |
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\begin{rail} |
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('fold' | 'unfold' | 'insert' | 'erule' | 'drule' | 'frule') thmrefs |
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; |
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\end{rail} |
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\begin{descr} |
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\item [$unfold~\vec a$ and $fold~\vec a$] expand and fold back again the given |
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meta-level definitions throughout all goals; any facts provided are inserted |
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into the goal and subject to rewriting as well. |
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\item [$erule~\vec a$, $drule~\vec a$, and $frule~\vec a$] are similar to the |
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basic $rule$ method (see \S\ref{sec:pure-meth-att}), but apply rules by |
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elim-resolution, destruct-resolution, and forward-resolution, respectively |
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\cite{isabelle-ref}. These are improper method, mainly for experimentation |
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and emulating tactic scripts. |
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|
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Different modes of basic rule application are usually expressed in Isar at |
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the proof language level, rather than via implicit proof state |
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manipulations. For example, a proper single-step elimination would be done |
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using the basic $rule$ method, with forward chaining of current facts. |
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\item [$insert~\vec a$] inserts theorems as facts into all goals of the proof |
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state. Note that current facts indicated for forward chaining are ignored. |
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\item [$succeed$] yields a single (unchanged) result; it is the identity of |
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the ``\texttt{,}'' method combinator (cf.\ \S\ref{sec:syn-meth}). |
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\item [$fail$] yields an empty result sequence; it is the identity of the |
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``\texttt{|}'' method combinator (cf.\ \S\ref{sec:syn-meth}). |
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\end{descr} |
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|
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\indexisaratt{tagged}\indexisaratt{untagged} |
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\indexisaratt{THEN}\indexisaratt{COMP} |
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\indexisaratt{where}\indexisaratt{unfolded}\indexisaratt{folded} |
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\indexisaratt{standard}\indexisaratt{elim-format} |
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\indexisaratt{no-vars}\indexisaratt{exported} |
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\begin{matharray}{rcl} |
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tagged & : & \isaratt \\ |
311 |
untagged & : & \isaratt \\[0.5ex] |
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THEN & : & \isaratt \\ |
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COMP & : & \isaratt \\[0.5ex] |
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where & : & \isaratt \\[0.5ex] |
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unfolded & : & \isaratt \\ |
316 |
folded & : & \isaratt \\[0.5ex] |
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standard & : & \isaratt \\ |
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elim_format & : & \isaratt \\ |
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no_vars^* & : & \isaratt \\ |
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exported^* & : & \isaratt \\ |
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\end{matharray} |
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||
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\begin{rail} |
|
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'tagged' (nameref+) |
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; |
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'untagged' name |
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; |
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('THEN' | 'COMP') ('[' nat ']')? thmref |
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; |
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'where' (name '=' term * 'and') |
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; |
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('unfolded' | 'folded') thmrefs |
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; |
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\end{rail} |
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\begin{descr} |
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\item [$tagged~name~args$ and $untagged~name$] add and remove $tags$ of some |
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theorem. Tags may be any list of strings that serve as comment for some |
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tools (e.g.\ $\LEMMANAME$ causes the tag ``$lemma$'' to be added to the |
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result). The first string is considered the tag name, the rest its |
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arguments. Note that untag removes any tags of the same name. |
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\item [$THEN~n~a$ and $COMP~n~a$] compose rules. $THEN$ resolves with the |
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$n$-th premise of $a$; the $COMP$ version skips the automatic lifting |
|
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process that is normally intended (cf.\ \texttt{RS} and \texttt{COMP} in |
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\cite[\S5]{isabelle-ref}). |
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\item [$where~\vec x = \vec t$] perform named instantiation of schematic |
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variables occurring in a theorem. Unlike instantiation tactics such as |
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$rule_tac$ (see \S\ref{sec:tactic-commands}), actual schematic variables |
|
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have to be specified (e.g.\ $\Var{x@3}$). |
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\item [$unfolded~\vec a$ and $folded~\vec a$] expand and fold back again the |
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given meta-level definitions throughout a rule. |
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\item [$standard$] puts a theorem into the standard form of object-rules, just |
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as the ML function \texttt{standard} (see \cite[\S5]{isabelle-ref}). |
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|
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\item [$elim_format$] turns a destruction rule into elimination rule format; |
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|
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see also the ML function \texttt{make\_elim} (see \cite{isabelle-ref}). |
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\item [$no_vars$] replaces schematic variables by free ones; this is mainly |
357 |
for tuning output of pretty printed theorems. |
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\item [$exported$] lifts a local result out of the current proof context, |
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generalizing all fixed variables and discharging all assumptions. Note that |
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proper incremental export is already done as part of the basic Isar |
361 |
machinery. This attribute is mainly for experimentation. |
|
8517 | 362 |
\end{descr} |
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||
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\section{Tactic emulations}\label{sec:tactics} |
366 |
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The following improper proof methods emulate traditional tactics. These admit |
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direct access to the goal state, which is normally considered harmful! In |
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particular, this may involve both numbered goal addressing (default 1), and |
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dynamic instantiation within the scope of some subgoal. |
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372 |
\begin{warn} |
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373 |
Dynamic instantiations are read and type-checked according to a subgoal of |
|
374 |
the current dynamic goal state, rather than the static proof context! In |
|
375 |
particular, locally fixed variables and term abbreviations may not be |
|
376 |
included in the term specifications. Thus schematic variables are left to |
|
377 |
be solved by unification with certain parts of the subgoal involved. |
|
378 |
\end{warn} |
|
379 |
||
380 |
Note that the tactic emulation proof methods in Isabelle/Isar are consistently |
|
381 |
named $foo_tac$. |
|
382 |
||
383 |
\indexisarmeth{rule-tac}\indexisarmeth{erule-tac} |
|
384 |
\indexisarmeth{drule-tac}\indexisarmeth{frule-tac} |
|
385 |
\indexisarmeth{cut-tac}\indexisarmeth{thin-tac} |
|
9642 | 386 |
\indexisarmeth{subgoal-tac}\indexisarmeth{rename-tac} |
9614 | 387 |
\indexisarmeth{rotate-tac}\indexisarmeth{tactic} |
9606 | 388 |
\begin{matharray}{rcl} |
389 |
rule_tac^* & : & \isarmeth \\ |
|
390 |
erule_tac^* & : & \isarmeth \\ |
|
391 |
drule_tac^* & : & \isarmeth \\ |
|
392 |
frule_tac^* & : & \isarmeth \\ |
|
393 |
cut_tac^* & : & \isarmeth \\ |
|
394 |
thin_tac^* & : & \isarmeth \\ |
|
395 |
subgoal_tac^* & : & \isarmeth \\ |
|
9614 | 396 |
rename_tac^* & : & \isarmeth \\ |
397 |
rotate_tac^* & : & \isarmeth \\ |
|
9606 | 398 |
tactic^* & : & \isarmeth \\ |
399 |
\end{matharray} |
|
400 |
||
401 |
\railalias{ruletac}{rule\_tac} |
|
402 |
\railterm{ruletac} |
|
403 |
||
404 |
\railalias{eruletac}{erule\_tac} |
|
405 |
\railterm{eruletac} |
|
406 |
||
407 |
\railalias{druletac}{drule\_tac} |
|
408 |
\railterm{druletac} |
|
409 |
||
410 |
\railalias{fruletac}{frule\_tac} |
|
411 |
\railterm{fruletac} |
|
412 |
||
413 |
\railalias{cuttac}{cut\_tac} |
|
414 |
\railterm{cuttac} |
|
415 |
||
416 |
\railalias{thintac}{thin\_tac} |
|
417 |
\railterm{thintac} |
|
418 |
||
419 |
\railalias{subgoaltac}{subgoal\_tac} |
|
420 |
\railterm{subgoaltac} |
|
421 |
||
9614 | 422 |
\railalias{renametac}{rename\_tac} |
423 |
\railterm{renametac} |
|
424 |
||
425 |
\railalias{rotatetac}{rotate\_tac} |
|
426 |
\railterm{rotatetac} |
|
427 |
||
9606 | 428 |
\begin{rail} |
429 |
( ruletac | eruletac | druletac | fruletac | cuttac | thintac ) goalspec? |
|
430 |
( insts thmref | thmrefs ) |
|
431 |
; |
|
432 |
subgoaltac goalspec? (prop +) |
|
433 |
; |
|
9614 | 434 |
renametac goalspec? (name +) |
435 |
; |
|
436 |
rotatetac goalspec? int? |
|
437 |
; |
|
9606 | 438 |
'tactic' text |
439 |
; |
|
440 |
||
441 |
insts: ((name '=' term) + 'and') 'in' |
|
442 |
; |
|
443 |
\end{rail} |
|
444 |
||
445 |
\begin{descr} |
|
446 |
\item [$rule_tac$ etc.] do resolution of rules with explicit instantiation. |
|
447 |
This works the same way as the ML tactics \texttt{res_inst_tac} etc. (see |
|
448 |
\cite[\S3]{isabelle-ref}). |
|
9614 | 449 |
|
9606 | 450 |
Note that multiple rules may be only given there is no instantiation. Then |
451 |
$rule_tac$ is the same as \texttt{resolve_tac} in ML (see |
|
452 |
\cite[\S3]{isabelle-ref}). |
|
453 |
\item [$cut_tac$] inserts facts into the proof state as assumption of a |
|
454 |
subgoal, see also \texttt{cut_facts_tac} in \cite[\S3]{isabelle-ref}. Note |
|
455 |
that the scope of schmatic variables is spread over the main goal statement. |
|
456 |
Instantiations may be given as well, see also ML tactic |
|
457 |
\texttt{cut_inst_tac} in \cite[\S3]{isabelle-ref}. |
|
458 |
\item [$thin_tac~\phi$] deletes the specified assumption from a subgoal; note |
|
459 |
that $\phi$ may contain schematic variables. See also \texttt{thin_tac} in |
|
460 |
\cite[\S3]{isabelle-ref}. |
|
461 |
\item [$subgoal_tac~\phi$] adds $\phi$ as an assumption to a subgoal. See |
|
462 |
also \texttt{subgoal_tac} and \texttt{subgoals_tac} in |
|
463 |
\cite[\S3]{isabelle-ref}. |
|
9614 | 464 |
\item [$rename_tac~\vec x$] renames parameters of a goal according to the list |
465 |
$\vec x$, which refers to the \emph{suffix} of variables. |
|
466 |
\item [$rotate_tac~n$] rotates the assumptions of a goal by $n$ positions: |
|
467 |
from right to left if $n$ is positive, and from left to right if $n$ is |
|
468 |
negative; the default value is $1$. See also \texttt{rotate_tac} in |
|
469 |
\cite[\S3]{isabelle-ref}. |
|
9606 | 470 |
\item [$tactic~text$] produces a proof method from any ML text of type |
471 |
\texttt{tactic}. Apart from the usual ML environment and the current |
|
472 |
implicit theory context, the ML code may refer to the following locally |
|
473 |
bound values: |
|
474 |
||
475 |
%%FIXME ttbox produces too much trailing space (why?) |
|
476 |
{\footnotesize\begin{verbatim} |
|
477 |
val ctxt : Proof.context |
|
478 |
val facts : thm list |
|
479 |
val thm : string -> thm |
|
480 |
val thms : string -> thm list |
|
481 |
\end{verbatim}} |
|
482 |
Here \texttt{ctxt} refers to the current proof context, \texttt{facts} |
|
483 |
indicates any current facts for forward-chaining, and |
|
484 |
\texttt{thm}~/~\texttt{thms} retrieve named facts (including global |
|
485 |
theorems) from the context. |
|
486 |
\end{descr} |
|
487 |
||
488 |
||
9614 | 489 |
\section{The Simplifier}\label{sec:simplifier} |
7135 | 490 |
|
7321 | 491 |
\subsection{Simplification methods}\label{sec:simp} |
7315 | 492 |
|
8483 | 493 |
\indexisarmeth{simp}\indexisarmeth{simp-all} |
7315 | 494 |
\begin{matharray}{rcl} |
495 |
simp & : & \isarmeth \\ |
|
8483 | 496 |
simp_all & : & \isarmeth \\ |
7315 | 497 |
\end{matharray} |
498 |
||
8483 | 499 |
\railalias{simpall}{simp\_all} |
500 |
\railterm{simpall} |
|
501 |
||
8704 | 502 |
\railalias{noasm}{no\_asm} |
503 |
\railterm{noasm} |
|
504 |
||
505 |
\railalias{noasmsimp}{no\_asm\_simp} |
|
506 |
\railterm{noasmsimp} |
|
507 |
||
508 |
\railalias{noasmuse}{no\_asm\_use} |
|
509 |
\railterm{noasmuse} |
|
510 |
||
7315 | 511 |
\begin{rail} |
8706 | 512 |
('simp' | simpall) ('!' ?) opt? (simpmod * ) |
7315 | 513 |
; |
514 |
||
8811 | 515 |
opt: '(' (noasm | noasmsimp | noasmuse) ')' |
8704 | 516 |
; |
9711 | 517 |
simpmod: ('add' | 'del' | 'only' | 'cong' (() | 'add' | 'del') | |
9847 | 518 |
'split' (() | 'add' | 'del')) ':' thmrefs |
7315 | 519 |
; |
520 |
\end{rail} |
|
521 |
||
7321 | 522 |
\begin{descr} |
8547 | 523 |
\item [$simp$] invokes Isabelle's simplifier, after declaring additional rules |
8594 | 524 |
according to the arguments given. Note that the \railtterm{only} modifier |
8547 | 525 |
first removes all other rewrite rules, congruences, and looper tactics |
8594 | 526 |
(including splits), and then behaves like \railtterm{add}. |
9711 | 527 |
|
528 |
\medskip The \railtterm{cong} modifiers add or delete Simplifier congruence |
|
529 |
rules (see also \cite{isabelle-ref}), the default is to add. |
|
530 |
||
531 |
\medskip The \railtterm{split} modifiers add or delete rules for the |
|
532 |
Splitter (see also \cite{isabelle-ref}), the default is to add. This works |
|
533 |
only if the Simplifier method has been properly setup to include the |
|
534 |
Splitter (all major object logics such HOL, HOLCF, FOL, ZF do this already). |
|
8483 | 535 |
\item [$simp_all$] is similar to $simp$, but acts on all goals. |
7321 | 536 |
\end{descr} |
537 |
||
8704 | 538 |
By default, the Simplifier methods are based on \texttt{asm_full_simp_tac} |
8706 | 539 |
internally \cite[\S10]{isabelle-ref}, which means that assumptions are both |
540 |
simplified as well as used in simplifying the conclusion. In structured |
|
541 |
proofs this is usually quite well behaved in practice: just the local premises |
|
542 |
of the actual goal are involved, additional facts may inserted via explicit |
|
543 |
forward-chaining (using $\THEN$, $\FROMNAME$ etc.). The full context of |
|
544 |
assumptions is only included if the ``$!$'' (bang) argument is given, which |
|
545 |
should be used with some care, though. |
|
7321 | 546 |
|
8704 | 547 |
Additional Simplifier options may be specified to tune the behavior even |
9614 | 548 |
further: $(no_asm)$ means assumptions are ignored completely (cf.\ |
8811 | 549 |
\texttt{simp_tac}), $(no_asm_simp)$ means assumptions are used in the |
9614 | 550 |
simplification of the conclusion but are not themselves simplified (cf.\ |
8811 | 551 |
\texttt{asm_simp_tac}), and $(no_asm_use)$ means assumptions are simplified |
552 |
but are not used in the simplification of each other or the conclusion (cf. |
|
8704 | 553 |
\texttt{full_simp_tac}). |
554 |
||
555 |
\medskip |
|
556 |
||
557 |
The Splitter package is usually configured to work as part of the Simplifier. |
|
9711 | 558 |
The effect of repeatedly applying \texttt{split_tac} can be simulated by |
559 |
$(simp~only\colon~split\colon~\vec a)$. There is also a separate $split$ |
|
560 |
method available for single-step case splitting, see \S\ref{sec:basic-eq}. |
|
8483 | 561 |
|
562 |
||
563 |
\subsection{Declaring rules} |
|
564 |
||
8667 | 565 |
\indexisarcmd{print-simpset} |
8638 | 566 |
\indexisaratt{simp}\indexisaratt{split}\indexisaratt{cong} |
7321 | 567 |
\begin{matharray}{rcl} |
10154 | 568 |
print_simpset^* & : & \isarkeep{theory~|~proof} \\ |
7321 | 569 |
simp & : & \isaratt \\ |
9711 | 570 |
cong & : & \isaratt \\ |
8483 | 571 |
split & : & \isaratt \\ |
7321 | 572 |
\end{matharray} |
573 |
||
574 |
\begin{rail} |
|
9711 | 575 |
('simp' | 'cong' | 'split') (() | 'add' | 'del') |
7321 | 576 |
; |
577 |
\end{rail} |
|
578 |
||
579 |
\begin{descr} |
|
8667 | 580 |
\item [$print_simpset$] prints the collection of rules declared to the |
581 |
Simplifier, which is also known as ``simpset'' internally |
|
582 |
\cite{isabelle-ref}. This is a diagnostic command; $undo$ does not apply. |
|
8547 | 583 |
\item [$simp$] declares simplification rules. |
8638 | 584 |
\item [$cong$] declares congruence rules. |
9711 | 585 |
\item [$split$] declares case split rules. |
7321 | 586 |
\end{descr} |
7319 | 587 |
|
7315 | 588 |
|
589 |
\subsection{Forward simplification} |
|
590 |
||
9905 | 591 |
\indexisaratt{simplified} |
7315 | 592 |
\begin{matharray}{rcl} |
9905 | 593 |
simplified & : & \isaratt \\ |
7315 | 594 |
\end{matharray} |
595 |
||
9905 | 596 |
\begin{rail} |
597 |
'simplified' opt? |
|
598 |
; |
|
599 |
||
600 |
opt: '(' (noasm | noasmsimp | noasmuse) ')' |
|
601 |
; |
|
602 |
\end{rail} |
|
7905 | 603 |
|
9905 | 604 |
\begin{descr} |
605 |
\item [$simplified$] causes a theorem to be simplified according to the |
|
606 |
current Simplifier context (there are no separate arguments for declaring |
|
607 |
additional rules). By default the result is fully simplified, including |
|
608 |
assumptions and conclusion. The options $no_asm$ etc.\ restrict the |
|
609 |
Simplifier in the same way as the for the $simp$ method (see |
|
610 |
\S\ref{sec:simp}). |
|
611 |
||
612 |
The $simplified$ operation should be used only very rarely, usually for |
|
613 |
experimentation only. |
|
614 |
\end{descr} |
|
7315 | 615 |
|
616 |
||
9711 | 617 |
\section{Basic equational reasoning}\label{sec:basic-eq} |
9614 | 618 |
|
9703 | 619 |
\indexisarmeth{subst}\indexisarmeth{hypsubst}\indexisarmeth{split}\indexisaratt{symmetric} |
9614 | 620 |
\begin{matharray}{rcl} |
621 |
subst & : & \isarmeth \\ |
|
622 |
hypsubst^* & : & \isarmeth \\ |
|
9703 | 623 |
split & : & \isarmeth \\ |
9614 | 624 |
symmetric & : & \isaratt \\ |
625 |
\end{matharray} |
|
626 |
||
627 |
\begin{rail} |
|
628 |
'subst' thmref |
|
629 |
; |
|
9799 | 630 |
'split' ('(' 'asm' ')')? thmrefs |
9703 | 631 |
; |
9614 | 632 |
\end{rail} |
633 |
||
634 |
These methods and attributes provide basic facilities for equational reasoning |
|
635 |
that are intended for specialized applications only. Normally, single step |
|
636 |
reasoning would be performed by calculation (see \S\ref{sec:calculation}), |
|
637 |
while the Simplifier is the canonical tool for automated normalization (see |
|
638 |
\S\ref{sec:simplifier}). |
|
639 |
||
640 |
\begin{descr} |
|
641 |
\item [$subst~thm$] performs a single substitution step using rule $thm$, |
|
642 |
which may be either a meta or object equality. |
|
643 |
\item [$hypsubst$] performs substitution using some assumption. |
|
9703 | 644 |
\item [$split~thms$] performs single-step case splitting using rules $thms$. |
9799 | 645 |
By default, splitting is performed in the conclusion of a goal; the $asm$ |
646 |
option indicates to operate on assumptions instead. |
|
647 |
||
9703 | 648 |
Note that the $simp$ method already involves repeated application of split |
649 |
rules as declared in the current context (see \S\ref{sec:simp}). |
|
9614 | 650 |
\item [$symmetric$] applies the symmetry rule of meta or object equality. |
651 |
\end{descr} |
|
652 |
||
653 |
||
9847 | 654 |
\section{The Classical Reasoner}\label{sec:classical} |
7135 | 655 |
|
7335 | 656 |
\subsection{Basic methods}\label{sec:classical-basic} |
7321 | 657 |
|
7974 | 658 |
\indexisarmeth{rule}\indexisarmeth{intro} |
659 |
\indexisarmeth{elim}\indexisarmeth{default}\indexisarmeth{contradiction} |
|
7321 | 660 |
\begin{matharray}{rcl} |
661 |
rule & : & \isarmeth \\ |
|
662 |
intro & : & \isarmeth \\ |
|
663 |
elim & : & \isarmeth \\ |
|
664 |
contradiction & : & \isarmeth \\ |
|
665 |
\end{matharray} |
|
666 |
||
667 |
\begin{rail} |
|
8547 | 668 |
('rule' | 'intro' | 'elim') thmrefs? |
7321 | 669 |
; |
670 |
\end{rail} |
|
671 |
||
672 |
\begin{descr} |
|
7466 | 673 |
\item [$rule$] as offered by the classical reasoner is a refinement over the |
8517 | 674 |
primitive one (see \S\ref{sec:pure-meth-att}). In case that no rules are |
7466 | 675 |
provided as arguments, it automatically determines elimination and |
7321 | 676 |
introduction rules from the context (see also \S\ref{sec:classical-mod}). |
8517 | 677 |
This is made the default method for basic proof steps, such as $\PROOFNAME$ |
678 |
and ``$\DDOT$'' (two dots), see also \S\ref{sec:proof-steps} and |
|
679 |
\S\ref{sec:pure-meth-att}. |
|
9614 | 680 |
|
7466 | 681 |
\item [$intro$ and $elim$] repeatedly refine some goal by intro- or |
7905 | 682 |
elim-resolution, after having inserted any facts. Omitting the arguments |
8547 | 683 |
refers to any suitable rules declared in the context, otherwise only the |
684 |
explicitly given ones may be applied. The latter form admits better control |
|
685 |
of what actually happens, thus it is very appropriate as an initial method |
|
686 |
for $\PROOFNAME$ that splits up certain connectives of the goal, before |
|
687 |
entering the actual sub-proof. |
|
9614 | 688 |
|
7466 | 689 |
\item [$contradiction$] solves some goal by contradiction, deriving any result |
690 |
from both $\neg A$ and $A$. Facts, which are guaranteed to participate, may |
|
691 |
appear in either order. |
|
7321 | 692 |
\end{descr} |
693 |
||
694 |
||
7981 | 695 |
\subsection{Automated methods}\label{sec:classical-auto} |
7315 | 696 |
|
9799 | 697 |
\indexisarmeth{blast}\indexisarmeth{fast}\indexisarmeth{slow} |
698 |
\indexisarmeth{best}\indexisarmeth{safe}\indexisarmeth{clarify} |
|
7321 | 699 |
\begin{matharray}{rcl} |
9780 | 700 |
blast & : & \isarmeth \\ |
701 |
fast & : & \isarmeth \\ |
|
9799 | 702 |
slow & : & \isarmeth \\ |
9780 | 703 |
best & : & \isarmeth \\ |
704 |
safe & : & \isarmeth \\ |
|
705 |
clarify & : & \isarmeth \\ |
|
7321 | 706 |
\end{matharray} |
707 |
||
708 |
\begin{rail} |
|
7905 | 709 |
'blast' ('!' ?) nat? (clamod * ) |
7321 | 710 |
; |
9799 | 711 |
('fast' | 'slow' | 'best' | 'safe' | 'clarify') ('!' ?) (clamod * ) |
7321 | 712 |
; |
713 |
||
9408 | 714 |
clamod: (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' thmrefs |
7321 | 715 |
; |
716 |
\end{rail} |
|
717 |
||
718 |
\begin{descr} |
|
719 |
\item [$blast$] refers to the classical tableau prover (see \texttt{blast_tac} |
|
7335 | 720 |
in \cite[\S11]{isabelle-ref}). The optional argument specifies a |
9606 | 721 |
user-supplied search bound (default 20). Note that $blast$ is the only |
722 |
classical proof procedure in Isabelle that can handle actual object-logic |
|
723 |
rules as local assumptions ($fast$ etc.\ would just ignore non-atomic |
|
724 |
facts). |
|
9799 | 725 |
\item [$fast$, $slow$, $best$, $safe$, and $clarify$] refer to the generic |
726 |
classical reasoner. See \texttt{fast_tac}, \texttt{slow_tac}, |
|
727 |
\texttt{best_tac}, \texttt{safe_tac}, and \texttt{clarify_tac} in |
|
728 |
\cite[\S11]{isabelle-ref} for more information. |
|
7321 | 729 |
\end{descr} |
730 |
||
731 |
Any of above methods support additional modifiers of the context of classical |
|
8517 | 732 |
rules. Their semantics is analogous to the attributes given in |
8547 | 733 |
\S\ref{sec:classical-mod}. Facts provided by forward chaining are |
734 |
inserted\footnote{These methods usually cannot make proper use of actual rules |
|
735 |
inserted that way, though.} into the goal before doing the search. The |
|
736 |
``!''~argument causes the full context of assumptions to be included as well. |
|
737 |
This is slightly less hazardous than for the Simplifier (see |
|
738 |
\S\ref{sec:simp}). |
|
7321 | 739 |
|
7315 | 740 |
|
9847 | 741 |
\subsection{Combined automated methods}\label{sec:clasimp} |
7315 | 742 |
|
9799 | 743 |
\indexisarmeth{auto}\indexisarmeth{force}\indexisarmeth{clarsimp} |
744 |
\indexisarmeth{fastsimp}\indexisarmeth{slowsimp}\indexisarmeth{bestsimp} |
|
7321 | 745 |
\begin{matharray}{rcl} |
9606 | 746 |
auto & : & \isarmeth \\ |
7321 | 747 |
force & : & \isarmeth \\ |
9438 | 748 |
clarsimp & : & \isarmeth \\ |
9606 | 749 |
fastsimp & : & \isarmeth \\ |
9799 | 750 |
slowsimp & : & \isarmeth \\ |
751 |
bestsimp & : & \isarmeth \\ |
|
7321 | 752 |
\end{matharray} |
753 |
||
754 |
\begin{rail} |
|
9780 | 755 |
'auto' '!'? (nat nat)? (clasimpmod * ) |
756 |
; |
|
9799 | 757 |
('force' | 'clarsimp' | 'fastsimp' | 'slowsimp' | 'bestsimp') '!'? (clasimpmod * ) |
7321 | 758 |
; |
7315 | 759 |
|
9711 | 760 |
clasimpmod: ('simp' (() | 'add' | 'del' | 'only') | |
10031 | 761 |
('cong' | 'split') (() | 'add' | 'del') | |
762 |
'iff' (((() | 'add') '?'?) | 'del') | |
|
9408 | 763 |
(('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' thmrefs |
7321 | 764 |
\end{rail} |
7315 | 765 |
|
7321 | 766 |
\begin{descr} |
9799 | 767 |
\item [$auto$, $force$, $clarsimp$, $fastsimp$, $slowsimp$, and $bestsimp$] |
768 |
provide access to Isabelle's combined simplification and classical reasoning |
|
769 |
tactics. These correspond to \texttt{auto_tac}, \texttt{force_tac}, |
|
770 |
\texttt{clarsimp_tac}, and Classical Reasoner tactics with the Simplifier |
|
771 |
added as wrapper, see \cite[\S11]{isabelle-ref} for more information. The |
|
772 |
modifier arguments correspond to those given in \S\ref{sec:simp} and |
|
9606 | 773 |
\S\ref{sec:classical-auto}. Just note that the ones related to the |
774 |
Simplifier are prefixed by \railtterm{simp} here. |
|
9614 | 775 |
|
7987 | 776 |
Facts provided by forward chaining are inserted into the goal before doing |
777 |
the search. The ``!''~argument causes the full context of assumptions to be |
|
778 |
included as well. |
|
7321 | 779 |
\end{descr} |
780 |
||
7987 | 781 |
|
8483 | 782 |
\subsection{Declaring rules}\label{sec:classical-mod} |
7135 | 783 |
|
8667 | 784 |
\indexisarcmd{print-claset} |
7391 | 785 |
\indexisaratt{intro}\indexisaratt{elim}\indexisaratt{dest} |
9936 | 786 |
\indexisaratt{iff}\indexisaratt{rule} |
7321 | 787 |
\begin{matharray}{rcl} |
10154 | 788 |
print_claset^* & : & \isarkeep{theory~|~proof} \\ |
7321 | 789 |
intro & : & \isaratt \\ |
790 |
elim & : & \isaratt \\ |
|
791 |
dest & : & \isaratt \\ |
|
9936 | 792 |
rule & : & \isaratt \\ |
7391 | 793 |
iff & : & \isaratt \\ |
7321 | 794 |
\end{matharray} |
7135 | 795 |
|
7321 | 796 |
\begin{rail} |
9408 | 797 |
('intro' | 'elim' | 'dest') ('!' | () | '?') |
7321 | 798 |
; |
9936 | 799 |
'rule' 'del' |
800 |
; |
|
10031 | 801 |
'iff' (((() | 'add') '?'?) | 'del') |
9936 | 802 |
; |
7321 | 803 |
\end{rail} |
7135 | 804 |
|
7321 | 805 |
\begin{descr} |
8667 | 806 |
\item [$print_claset$] prints the collection of rules declared to the |
807 |
Classical Reasoner, which is also known as ``simpset'' internally |
|
808 |
\cite{isabelle-ref}. This is a diagnostic command; $undo$ does not apply. |
|
8517 | 809 |
\item [$intro$, $elim$, and $dest$] declare introduction, elimination, and |
810 |
destruct rules, respectively. By default, rules are considered as |
|
9408 | 811 |
\emph{unsafe} (i.e.\ not applied blindly without backtracking), while a |
812 |
single ``!'' classifies as \emph{safe}, and ``?'' as \emph{extra} (i.e.\ not |
|
813 |
applied in the search-oriented automated methods, but only in single-step |
|
814 |
methods such as $rule$). |
|
9936 | 815 |
\item [$rule~del$] deletes introduction, elimination, or destruct rules from |
816 |
the context. |
|
10031 | 817 |
\item [$iff$] declares equivalence rules to the context. The default behavior |
818 |
is to declare a rewrite rule to the Simplifier, and the two corresponding |
|
819 |
implications to the Classical Reasoner (as ``safe'' rules that are used |
|
820 |
aggressively, which would normally be indicated by ``!''). |
|
821 |
||
822 |
The ``?'' version of $iff$ declares ``extra'' Classical Reasoner rules only, |
|
823 |
and omits the Simplifier declaration. Thus the declaration does not have |
|
824 |
any effect on automated proof tools, but only on simple methods such as |
|
825 |
$rule$ (see \S\ref{sec:misc-methods}). |
|
7321 | 826 |
\end{descr} |
7135 | 827 |
|
8203
2fcc6017cb72
intro/elim/dest attributes: changed ! / !! flags to ? / ??;
wenzelm
parents:
8195
diff
changeset
|
828 |
|
9614 | 829 |
%%% Local Variables: |
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%%% mode: latex |
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%%% TeX-master: "isar-ref" |
|
9614 | 832 |
%%% End: |