diff -r 0c86acc069ad -r 5deda0549f97 doc-src/TutorialI/document/simp2.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/doc-src/TutorialI/document/simp2.tex Thu Jul 26 17:16:02 2012 +0200 @@ -0,0 +1,249 @@ +% +\begin{isabellebody}% +\def\isabellecontext{simp{\isadigit{2}}}% +% +\isadelimtheory +% +\endisadelimtheory +% +\isatagtheory +% +\endisatagtheory +{\isafoldtheory}% +% +\isadelimtheory +% +\endisadelimtheory +% +\isamarkupsection{Simplification% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\label{sec:simplification-II}\index{simplification|(} +This section describes features not covered until now. It also +outlines the simplification process itself, which can be helpful +when the simplifier does not do what you expect of it.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsection{Advanced Features% +} +\isamarkuptrue% +% +\isamarkupsubsubsection{Congruence Rules% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\label{sec:simp-cong} +While simplifying the conclusion $Q$ +of $P \Imp Q$, it is legal to use the assumption $P$. +For $\Imp$ this policy is hardwired, but +contextual information can also be made available for other +operators. For example, \isa{xs\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ xs\ {\isaliteral{40}{\isacharat}}\ xs\ {\isaliteral{3D}{\isacharequal}}\ xs} simplifies to \isa{True} because we may use \isa{xs\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} when simplifying \isa{xs\ {\isaliteral{40}{\isacharat}}\ xs\ {\isaliteral{3D}{\isacharequal}}\ xs}. The generation of contextual information during simplification is +controlled by so-called \bfindex{congruence rules}. This is the one for +\isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}}: +\begin{isabelle}% +\ \ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}P\ {\isaliteral{3D}{\isacharequal}}\ P{\isaliteral{27}{\isacharprime}}{\isaliteral{3B}{\isacharsemicolon}}\ P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q\ {\isaliteral{3D}{\isacharequal}}\ Q{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}% +\end{isabelle} +It should be read as follows: +In order to simplify \isa{P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q} to \isa{P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{27}{\isacharprime}}}, +simplify \isa{P} to \isa{P{\isaliteral{27}{\isacharprime}}} +and assume \isa{P{\isaliteral{27}{\isacharprime}}} when simplifying \isa{Q} to \isa{Q{\isaliteral{27}{\isacharprime}}}. + +Here are some more examples. The congruence rules for bounded +quantifiers supply contextual information about the bound variable: +\begin{isabelle}% +\ \ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}A\ {\isaliteral{3D}{\isacharequal}}\ B{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ x\ {\isaliteral{3D}{\isacharequal}}\ Q\ x{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\isaindent{\ \ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{5C3C696E3E}{\isasymin}}A{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{5C3C696E3E}{\isasymin}}B{\isaliteral{2E}{\isachardot}}\ Q\ x{\isaliteral{29}{\isacharparenright}}% +\end{isabelle} +One congruence rule for conditional expressions supplies contextual +information for simplifying the \isa{then} and \isa{else} cases: +\begin{isabelle}% +\ \ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}b\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{3B}{\isacharsemicolon}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{3D}{\isacharequal}}\ u{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ v{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline +\isaindent{\ \ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}if\ b\ then\ x\ else\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ c\ then\ u\ else\ v{\isaliteral{29}{\isacharparenright}}% +\end{isabelle} +An alternative congruence rule for conditional expressions +actually \emph{prevents} simplification of some arguments: +\begin{isabelle}% +\ \ \ \ \ b\ {\isaliteral{3D}{\isacharequal}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}if\ b\ then\ x\ else\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ c\ then\ x\ else\ y{\isaliteral{29}{\isacharparenright}}% +\end{isabelle} +Only the first argument is simplified; the others remain unchanged. +This makes simplification much faster and is faithful to the evaluation +strategy in programming languages, which is why this is the default +congruence rule for \isa{if}. Analogous rules control the evaluation of +\isa{case} expressions. + +You can declare your own congruence rules with the attribute \attrdx{cong}, +either globally, in the usual manner, +\begin{quote} +\isacommand{declare} \textit{theorem-name} \isa{{\isaliteral{5B}{\isacharbrackleft}}cong{\isaliteral{5D}{\isacharbrackright}}} +\end{quote} +or locally in a \isa{simp} call by adding the modifier +\begin{quote} +\isa{cong{\isaliteral{3A}{\isacharcolon}}} \textit{list of theorem names} +\end{quote} +The effect is reversed by \isa{cong\ del} instead of \isa{cong}. + +\begin{warn} +The congruence rule \isa{conj{\isaliteral{5F}{\isacharunderscore}}cong} +\begin{isabelle}% +\ \ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}P\ {\isaliteral{3D}{\isacharequal}}\ P{\isaliteral{27}{\isacharprime}}{\isaliteral{3B}{\isacharsemicolon}}\ P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q\ {\isaliteral{3D}{\isacharequal}}\ Q{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C616E643E}{\isasymand}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ Q{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}% +\end{isabelle} +\par\noindent +is occasionally useful but is not a default rule; you have to declare it explicitly. +\end{warn}% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{Permutative Rewrite Rules% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\index{rewrite rules!permutative|bold}% +An equation is a \textbf{permutative rewrite rule} if the left-hand +side and right-hand side are the same up to renaming of variables. The most +common permutative rule is commutativity: \isa{x\ {\isaliteral{2B}{\isacharplus}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{2B}{\isacharplus}}\ x}. Other examples +include \isa{x\ {\isaliteral{2D}{\isacharminus}}\ y\ {\isaliteral{2D}{\isacharminus}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{2D}{\isacharminus}}\ z\ {\isaliteral{2D}{\isacharminus}}\ y} in arithmetic and \isa{insert\ x\ {\isaliteral{28}{\isacharparenleft}}insert\ y\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ insert\ y\ {\isaliteral{28}{\isacharparenleft}}insert\ x\ A{\isaliteral{29}{\isacharparenright}}} for sets. Such rules are problematic because +once they apply, they can be used forever. The simplifier is aware of this +danger and treats permutative rules by means of a special strategy, called +\bfindex{ordered rewriting}: a permutative rewrite +rule is only applied if the term becomes smaller with respect to a fixed +lexicographic ordering on terms. For example, commutativity rewrites +\isa{b\ {\isaliteral{2B}{\isacharplus}}\ a} to \isa{a\ {\isaliteral{2B}{\isacharplus}}\ b}, but then stops because \isa{a\ {\isaliteral{2B}{\isacharplus}}\ b} is strictly +smaller than \isa{b\ {\isaliteral{2B}{\isacharplus}}\ a}. Permutative rewrite rules can be turned into +simplification rules in the usual manner via the \isa{simp} attribute; the +simplifier recognizes their special status automatically. + +Permutative rewrite rules are most effective in the case of +associative-commutative functions. (Associativity by itself is not +permutative.) When dealing with an AC-function~$f$, keep the +following points in mind: +\begin{itemize}\index{associative-commutative function} + +\item The associative law must always be oriented from left to right, + namely $f(f(x,y),z) = f(x,f(y,z))$. The opposite orientation, if + used with commutativity, can lead to nontermination. + +\item To complete your set of rewrite rules, you must add not just + associativity~(A) and commutativity~(C) but also a derived rule, {\bf + left-com\-mut\-ativ\-ity} (LC): $f(x,f(y,z)) = f(y,f(x,z))$. +\end{itemize} +Ordered rewriting with the combination of A, C, and LC sorts a term +lexicographically: +\[\def\maps#1{~\stackrel{#1}{\leadsto}~} + f(f(b,c),a) \maps{A} f(b,f(c,a)) \maps{C} f(b,f(a,c)) \maps{LC} f(a,f(b,c)) \] + +Note that ordered rewriting for \isa{{\isaliteral{2B}{\isacharplus}}} and \isa{{\isaliteral{2A}{\isacharasterisk}}} on numbers is rarely +necessary because the built-in arithmetic prover often succeeds without +such tricks.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsection{How the Simplifier Works% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\label{sec:SimpHow} +Roughly speaking, the simplifier proceeds bottom-up: subterms are simplified +first. A conditional equation is only applied if its condition can be +proved, again by simplification. Below we explain some special features of +the rewriting process.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{Higher-Order Patterns% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\index{simplification rule|(} +So far we have pretended the simplifier can deal with arbitrary +rewrite rules. This is not quite true. For reasons of feasibility, +the simplifier expects the +left-hand side of each rule to be a so-called \emph{higher-order +pattern}~\cite{nipkow-patterns}\indexbold{patterns!higher-order}. +This restricts where +unknowns may occur. Higher-order patterns are terms in $\beta$-normal +form. (This means there are no subterms of the form $(\lambda x. M)(N)$.) +Each occurrence of an unknown is of the form +$\Var{f}~x@1~\dots~x@n$, where the $x@i$ are distinct bound +variables. Thus all ordinary rewrite rules, where all unknowns are +of base type, for example \isa{{\isaliteral{3F}{\isacharquery}}a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{3F}{\isacharquery}}b\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{3F}{\isacharquery}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}b\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{3F}{\isacharquery}}c{\isaliteral{29}{\isacharparenright}}}, are acceptable: if an unknown is +of base type, it cannot have any arguments. Additionally, the rule +\isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ x\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3F}{\isacharquery}}Q\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}Q\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}} is also acceptable, in +both directions: all arguments of the unknowns \isa{{\isaliteral{3F}{\isacharquery}}P} and +\isa{{\isaliteral{3F}{\isacharquery}}Q} are distinct bound variables. + +If the left-hand side is not a higher-order pattern, all is not lost. +The simplifier will still try to apply the rule provided it +matches directly: without much $\lambda$-calculus hocus +pocus. For example, \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ range\ {\isaliteral{3F}{\isacharquery}}f{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True} rewrites +\isa{g\ a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ range\ g} to \isa{True}, but will fail to match +\isa{g{\isaliteral{28}{\isacharparenleft}}h\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ range{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ g{\isaliteral{28}{\isacharparenleft}}h\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}. However, you can +eliminate the offending subterms --- those that are not patterns --- +by adding new variables and conditions. +In our example, we eliminate \isa{{\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}x} and obtain + \isa{{\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{5C3C696E3E}{\isasymin}}\ range\ {\isaliteral{3F}{\isacharquery}}f{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True}, which is fine +as a conditional rewrite rule since conditions can be arbitrary +terms. However, this trick is not a panacea because the newly +introduced conditions may be hard to solve. + +There is no restriction on the form of the right-hand +sides. They may not contain extraneous term or type variables, though.% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isamarkupsubsubsection{The Preprocessor% +} +\isamarkuptrue% +% +\begin{isamarkuptext}% +\label{sec:simp-preprocessor} +When a theorem is declared a simplification rule, it need not be a +conditional equation already. The simplifier will turn it into a set of +conditional equations automatically. For example, \isa{f\ x\ {\isaliteral{3D}{\isacharequal}}\ g\ x\ {\isaliteral{5C3C616E643E}{\isasymand}}\ h\ x\ {\isaliteral{3D}{\isacharequal}}\ k\ x} becomes the two separate +simplification rules \isa{f\ x\ {\isaliteral{3D}{\isacharequal}}\ g\ x} and \isa{h\ x\ {\isaliteral{3D}{\isacharequal}}\ k\ x}. In +general, the input theorem is converted as follows: +\begin{eqnarray} +\neg P &\mapsto& P = \hbox{\isa{False}} \nonumber\\ +P \longrightarrow Q &\mapsto& P \Longrightarrow Q \nonumber\\ +P \land Q &\mapsto& P,\ Q \nonumber\\ +\forall x.~P~x &\mapsto& P~\Var{x}\nonumber\\ +\forall x \in A.\ P~x &\mapsto& \Var{x} \in A \Longrightarrow P~\Var{x} \nonumber\\ +\isa{if}\ P\ \isa{then}\ Q\ \isa{else}\ R &\mapsto& + P \Longrightarrow Q,\ \neg P \Longrightarrow R \nonumber +\end{eqnarray} +Once this conversion process is finished, all remaining non-equations +$P$ are turned into trivial equations $P =\isa{True}$. +For example, the formula +\begin{center}\isa{{\isaliteral{28}{\isacharparenleft}}p\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ t\ {\isaliteral{3D}{\isacharequal}}\ u\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ r{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ s}\end{center} +is converted into the three rules +\begin{center} +\isa{p\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ t\ {\isaliteral{3D}{\isacharequal}}\ u},\quad \isa{p\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ r\ {\isaliteral{3D}{\isacharequal}}\ False},\quad \isa{s\ {\isaliteral{3D}{\isacharequal}}\ True}. +\end{center} +\index{simplification rule|)} +\index{simplification|)}% +\end{isamarkuptext}% +\isamarkuptrue% +% +\isadelimtheory +% +\endisadelimtheory +% +\isatagtheory +% +\endisatagtheory +{\isafoldtheory}% +% +\isadelimtheory +% +\endisadelimtheory +\end{isabellebody}% +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "root" +%%% End: