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\begin{isabellebody}%
\def\isabellecontext{simp}%
%
\isamarkupsection{Simplification%
}
%
\begin{isamarkuptext}%
\label{sec:simplification-II}\index{simplification|(}
This section discusses some additional nifty features not covered so far and
gives a short introduction to the simplification process itself. The latter
is helpful to understand why a particular rule does or does not apply in some
situation.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Advanced Features%
}
%
\isamarkupsubsubsection{Congruence Rules%
}
%
\begin{isamarkuptext}%
\label{sec:simp-cong}
It is hardwired into the simplifier that while simplifying the conclusion $Q$
of $P \Imp Q$ it is legal to make uses of the assumption $P$. This
kind of contextual information can also be made available for other
operators. For example, \isa{xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongrightarrow}\ xs\ {\isacharat}\ xs\ {\isacharequal}\ xs} simplifies to \isa{True} because we may use \isa{xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}} when simplifying \isa{xs\ {\isacharat}\ xs\ {\isacharequal}\ xs}. The generation of contextual information during simplification is
controlled by so-called \bfindex{congruence rules}. This is the one for
\isa{{\isasymlongrightarrow}}:
\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}P\ {\isacharequal}\ P{\isacharprime}{\isacharsemicolon}\ P{\isacharprime}\ {\isasymLongrightarrow}\ Q\ {\isacharequal}\ Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymlongrightarrow}\ Q{\isacharprime}{\isacharparenright}%
\end{isabelle}
It should be read as follows:
In order to simplify \isa{P\ {\isasymlongrightarrow}\ Q} to \isa{P{\isacharprime}\ {\isasymlongrightarrow}\ Q{\isacharprime}},
simplify \isa{P} to \isa{P{\isacharprime}}
and assume \isa{P{\isacharprime}} when simplifying \isa{Q} to \isa{Q{\isacharprime}}.
Here are some more examples. The congruence rules for bounded
quantifiers supply contextual information about the bound variable:
\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}A\ {\isacharequal}\ B{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ B\ {\isasymLongrightarrow}\ P\ x\ {\isacharequal}\ Q\ x{\isasymrbrakk}\isanewline
\isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ {\isacharparenleft}{\isasymforall}x{\isasymin}A{\isachardot}\ P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymforall}x{\isasymin}B{\isachardot}\ Q\ x{\isacharparenright}%
\end{isabelle}
The congruence rule for conditional expressions supplies contextual
information for simplifying the \isa{then} and \isa{else} cases:
\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}b\ {\isacharequal}\ c{\isacharsemicolon}\ c\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ u{\isacharsemicolon}\ {\isasymnot}\ c\ {\isasymLongrightarrow}\ y\ {\isacharequal}\ v{\isasymrbrakk}\isanewline
\isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ {\isacharparenleft}if\ b\ then\ x\ else\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ c\ then\ u\ else\ v{\isacharparenright}%
\end{isabelle}
A congruence rule can also \emph{prevent} simplification of some arguments.
Here is an alternative congruence rule for conditional expressions:
\begin{isabelle}%
\ \ \ \ \ b\ {\isacharequal}\ c\ {\isasymLongrightarrow}\ {\isacharparenleft}if\ b\ then\ x\ else\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ c\ then\ x\ else\ y{\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{{\isacharbrackleft}cong{\isacharbrackright}}
\end{quote}
or locally in a \isa{simp} call by adding the modifier
\begin{quote}
\isa{cong{\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{\isacharunderscore}cong}
\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}P\ {\isacharequal}\ P{\isacharprime}{\isacharsemicolon}\ P{\isacharprime}\ {\isasymLongrightarrow}\ Q\ {\isacharequal}\ Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymand}\ Q{\isacharprime}{\isacharparenright}%
\end{isabelle}
\par\noindent
is occasionally useful but is not a default rule; you have to declare it explicitly.
\end{warn}%
\end{isamarkuptext}%
%
\isamarkupsubsubsection{Permutative Rewrite Rules%
}
%
\begin{isamarkuptext}%
\index{rewrite rule!permutative|bold}
\index{rewriting!ordered|bold}
\index{ordered rewriting|bold}
\index{simplification!ordered|bold}
An equation is a \bfindex{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\ {\isacharplus}\ y\ {\isacharequal}\ y\ {\isacharplus}\ x}. Other examples
include \isa{x\ {\isacharminus}\ y\ {\isacharminus}\ z\ {\isacharequal}\ x\ {\isacharminus}\ z\ {\isacharminus}\ y} in arithmetic and \isa{insert\ x\ {\isacharparenleft}insert\ y\ A{\isacharparenright}\ {\isacharequal}\ insert\ y\ {\isacharparenleft}insert\ x\ A{\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\ {\isacharplus}\ a} to \isa{a\ {\isacharplus}\ b}, but then stops because \isa{a\ {\isacharplus}\ b} is strictly
smaller than \isa{b\ {\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{{\isacharplus}} and \isa{{\isacharasterisk}} on numbers is rarely
necessary because the built-in arithmetic prover often succeeds without
such tricks.%
\end{isamarkuptext}%
%
\isamarkupsubsection{How it Works%
}
%
\begin{isamarkuptext}%
\label{sec:SimpHow}
Roughly speaking, the simplifier proceeds bottom-up (subterms are simplified
first) and a conditional equation is only applied if its condition could be
proved (again by simplification). Below we explain some special features of the rewriting process.%
\end{isamarkuptext}%
%
\isamarkupsubsubsection{Higher-Order Patterns%
}
%
\begin{isamarkuptext}%
\index{simplification rule|(}
So far we have pretended the simplifier can deal with arbitrary
rewrite rules. This is not quite true. Due to efficiency (and
potentially also computability) reasons, the simplifier expects the
left-hand side of each rule to be a so-called \emph{higher-order
pattern}~\cite{nipkow-patterns}\indexbold{higher-order
pattern}\indexbold{pattern, higher-order}. This restricts where
unknowns may occur. Higher-order patterns are terms in $\beta$-normal
form (this will always be the case unless you have done something
strange) where 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{{\isacharquery}m\ {\isacharplus}\ {\isacharquery}n\ {\isacharplus}\ {\isacharquery}k\ {\isacharequal}\ {\isacharquery}m\ {\isacharplus}\ {\isacharparenleft}{\isacharquery}n\ {\isacharplus}\ {\isacharquery}k{\isacharparenright}}, are OK: if an unknown is
of base type, it cannot have any arguments. Additionally, the rule
\isa{{\isacharparenleft}{\isasymforall}x{\isachardot}\ {\isacharquery}P\ x\ {\isasymand}\ {\isacharquery}Q\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}{\isasymforall}x{\isachardot}\ {\isacharquery}P\ x{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}x{\isachardot}\ {\isacharquery}Q\ x{\isacharparenright}{\isacharparenright}} is also OK, in
both directions: all arguments of the unknowns \isa{{\isacharquery}P} and
\isa{{\isacharquery}Q} are distinct bound variables.
If the left-hand side is not a higher-order pattern, not all is lost
and the simplifier will still try to apply the rule, but only if it
matches \emph{directly}, i.e.\ without much $\lambda$-calculus hocus
pocus. For example, \isa{{\isacharquery}f\ {\isacharquery}x\ {\isasymin}\ range\ {\isacharquery}f\ {\isacharequal}\ True} rewrites
\isa{g\ a\ {\isasymin}\ range\ g} to \isa{True}, but will fail to match
\isa{g{\isacharparenleft}h\ b{\isacharparenright}\ {\isasymin}\ range{\isacharparenleft}{\isasymlambda}x{\isachardot}\ g{\isacharparenleft}h\ x{\isacharparenright}{\isacharparenright}}. However, you can
replace the offending subterms (in our case \isa{{\isacharquery}f\ {\isacharquery}x}, which
is not a pattern) by adding new variables and conditions: \isa{{\isacharquery}y\ {\isacharequal}\ {\isacharquery}f\ {\isacharquery}x\ {\isasymLongrightarrow}\ {\isacharquery}y\ {\isasymin}\ range\ {\isacharquery}f\ {\isacharequal}\ True} 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}%
%
\isamarkupsubsubsection{The Preprocessor%
}
%
\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\ {\isacharequal}\ g\ x\ {\isasymand}\ h\ x\ {\isacharequal}\ k\ x} becomes the two separate
simplification rules \isa{f\ x\ {\isacharequal}\ g\ x} and \isa{h\ x\ {\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{{\isacharparenleft}p\ {\isasymlongrightarrow}\ t\ {\isacharequal}\ u\ {\isasymand}\ {\isasymnot}\ r{\isacharparenright}\ {\isasymand}\ s}\end{center}
is converted into the three rules
\begin{center}
\isa{p\ {\isasymLongrightarrow}\ t\ {\isacharequal}\ u},\quad \isa{p\ {\isasymLongrightarrow}\ r\ {\isacharequal}\ False},\quad \isa{s\ {\isacharequal}\ True}.
\end{center}
\index{simplification rule|)}
\index{simplification|)}%
\end{isamarkuptext}%
\end{isabellebody}%
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