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 theory simp2 imports Main begin

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 section{*Simplification*}

 text{*\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.

 *}

 subsection{*Advanced Features*}

 subsubsection{*Congruence Rules*}

 text{*\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, @{prop"xs = [] --> xs@xs = xs"} simplifies to @{term

 True} because we may use @{prop"xs = []"} when simplifying @{prop"xs@xs =

 xs"}. The generation of contextual information during simplification is

 controlled by so-called \bfindex{congruence rules}. This is the one for

 @{text"\<longrightarrow>"}:

 @{thm[display]imp_cong[no_vars]}

 It should be read as follows:

 In order to simplify @{prop"P-->Q"} to @{prop"P'-->Q'"},

 simplify @{prop P} to @{prop P'}

 and assume @{prop"P'"} when simplifying @{prop Q} to @{prop"Q'"}.

 Here are some more examples.  The congruence rules for bounded

 quantifiers supply contextual information about the bound variable:

 @{thm[display,eta_contract=false,margin=60]ball_cong[no_vars]}

 One congruence rule for conditional expressions supplies contextual

 information for simplifying the @{text then} and @{text else} cases:

 @{thm[display]if_cong[no_vars]}

 An alternative congruence rule for conditional expressions

 actually \emph{prevents} simplification of some arguments:

 @{thm[display]if_weak_cong[no_vars]}

 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 @{text "if"}. Analogous rules control the evaluation of

 @{text 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} @{text"[cong]"}

 \end{quote}

 or locally in a @{text"simp"} call by adding the modifier

 \begin{quote}

 @{text"cong:"} \textit{list of theorem names}

 \end{quote}

 The effect is reversed by @{text"cong del"} instead of @{text cong}.

 \begin{warn}

 The congruence rule @{thm[source]conj_cong}

 @{thm[display]conj_cong[no_vars]}

 \par\noindent

 is occasionally useful but is not a default rule; you have to declare it explicitly.

 \end{warn}

 *}

 subsubsection{*Permutative Rewrite Rules*}

 text{*

 \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: @{prop"x+y = y+x"}.  Other examples

 include @{prop"(x-y)-z = (x-z)-y"} in arithmetic and @{prop"insert x (insert

 y A) = insert y (insert x A)"} 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

 @{term"b+a"} to @{term"a+b"}, but then stops because @{term"a+b"} is strictly

 smaller than @{term"b+a"}.  Permutative rewrite rules can be turned into

 simplification rules in the usual manner via the @{text 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 @{text"+"} and @{text"*"} on numbers is rarely

 necessary because the built-in arithmetic prover often succeeds without

 such tricks.

 *}

 subsection{*How the Simplifier Works*}

 text{*\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. 

 *}

 subsubsection{*Higher-Order Patterns*}

 text{*\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 @{thm add_assoc}, are acceptable: if an unknown is

 of base type, it cannot have any arguments. Additionally, the rule

 @{text"(\<forall>x. ?P x \<and> ?Q x) = ((\<forall>x. ?P x) \<and> (\<forall>x. ?Q x))"} is also acceptable, in

 both directions: all arguments of the unknowns @{text"?P"} and

 @{text"?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, @{text"(?f ?x \<in> range ?f) = True"} rewrites

 @{term"g a \<in> range g"} to @{const True}, but will fail to match

 @{text"g(h b) \<in> range(\<lambda>x. g(h x))"}.  However, you can

 eliminate the offending subterms --- those that are not patterns ---

 by adding new variables and conditions.

 In our example, we eliminate @{text"?f ?x"} and obtain

  @{text"?y =

 ?f ?x \<Longrightarrow> (?y \<in> range ?f) = 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.

 *}

 subsubsection{*The Preprocessor*}

 text{*\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, @{prop"f x =

 g x & h x = k x"} becomes the two separate

 simplification rules @{prop"f x = g x"} and @{prop"h x = 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\\

 @{text "if"}\ P\ @{text then}\ Q\ @{text 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}@{prop"(p \<longrightarrow> t=u \<and> ~r) \<and> s"}\end{center}

 is converted into the three rules

 \begin{center}

 @{prop"p \<Longrightarrow> t = u"},\quad  @{prop"p \<Longrightarrow> r = False"},\quad  @{prop"s = True"}.

 \end{center}

 \index{simplification rule|)}

 \index{simplification|)}

 *}

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 end

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