(*<*)

 theory simp = Main:

 (*>*)

 subsubsection{*Simplification rules*}

 text{*\indexbold{simplification rule}

 To facilitate simplification, theorems can be declared to be simplification

 rules (with the help of the attribute @{text"[simp]"}\index{*simp

   (attribute)}), in which case proofs by simplification make use of these

 rules automatically. In addition the constructs \isacommand{datatype} and

 \isacommand{primrec} (and a few others) invisibly declare useful

 simplification rules. Explicit definitions are \emph{not} declared

 simplification rules automatically!

 Not merely equations but pretty much any theorem can become a simplification

 rule. The simplifier will try to make sense of it.  For example, a theorem

 @{prop"~P"} is automatically turned into @{prop"P = False"}. The details

 are explained in \S\ref{sec:SimpHow}.

 The simplification attribute of theorems can be turned on and off as follows:

 \begin{quote}

 \isacommand{declare} \textit{theorem-name}@{text"[simp]"}\\

 \isacommand{declare} \textit{theorem-name}@{text"[simp del]"}

 \end{quote}

 As a rule of thumb, equations that really simplify (like @{prop"rev(rev xs) =

  xs"} and @{prop"xs @ [] = xs"}) should be made simplification

 rules.  Those of a more specific nature (e.g.\ distributivity laws, which

 alter the structure of terms considerably) should only be used selectively,

 i.e.\ they should not be default simplification rules.  Conversely, it may

 also happen that a simplification rule needs to be disabled in certain

 proofs.  Frequent changes in the simplification status of a theorem may

 indicate a badly designed theory.

 \begin{warn}

   Simplification may not terminate, for example if both $f(x) = g(x)$ and

   $g(x) = f(x)$ are simplification rules. It is the user's responsibility not

   to include simplification rules that can lead to nontermination, either on

   their own or in combination with other simplification rules.

 \end{warn}

 *}

 subsubsection{*The simplification method*}

 text{*\index{*simp (method)|bold}

 The general format of the simplification method is

 \begin{quote}

 @{text simp} \textit{list of modifiers}

 \end{quote}

 where the list of \emph{modifiers} helps to fine tune the behaviour and may

 be empty. Most if not all of the proofs seen so far could have been performed

 with @{text simp} instead of \isa{auto}, except that @{text simp} attacks

 only the first subgoal and may thus need to be repeated---use

 \isaindex{simp_all} to simplify all subgoals.

 Note that @{text simp} fails if nothing changes.

 *}

 subsubsection{*Adding and deleting simplification rules*}

 text{*

 If a certain theorem is merely needed in a few proofs by simplification,

 we do not need to make it a global simplification rule. Instead we can modify

 the set of simplification rules used in a simplification step by adding rules

 to it and/or deleting rules from it. The two modifiers for this are

 \begin{quote}

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

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

 \end{quote}

 In case you want to use only a specific list of theorems and ignore all

 others:

 \begin{quote}

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

 \end{quote}

 *}

 subsubsection{*Assumptions*}

 text{*\index{simplification!with/of assumptions}

 By default, assumptions are part of the simplification process: they are used

 as simplification rules and are simplified themselves. For example:

 *}

 lemma "\<lbrakk> xs @ zs = ys @ xs; [] @ xs = [] @ [] \<rbrakk> \<Longrightarrow> ys = zs";

 apply simp;

 done

 text{*\noindent

 The second assumption simplifies to @{term"xs = []"}, which in turn

 simplifies the first assumption to @{term"zs = ys"}, thus reducing the

 conclusion to @{term"ys = ys"} and hence to @{term"True"}.

 In some cases this may be too much of a good thing and may lead to

 nontermination:

 *}

 lemma "\<forall>x. f x = g (f (g x)) \<Longrightarrow> f [] = f [] @ []";

 txt{*\noindent

 cannot be solved by an unmodified application of @{text"simp"} because the

 simplification rule @{term"f x = g (f (g x))"} extracted from the assumption

 does not terminate. Isabelle notices certain simple forms of

 nontermination but not this one. The problem can be circumvented by

 explicitly telling the simplifier to ignore the assumptions:

 *}

 apply(simp (no_asm));

 done

 text{*\noindent

 There are three options that influence the treatment of assumptions:

 \begin{description}

 \item[@{text"(no_asm)"}]\indexbold{*no_asm}

  means that assumptions are completely ignored.

 \item[@{text"(no_asm_simp)"}]\indexbold{*no_asm_simp}

  means that the assumptions are not simplified but

   are used in the simplification of the conclusion.

 \item[@{text"(no_asm_use)"}]\indexbold{*no_asm_use}

  means that the assumptions are simplified but are not

   used in the simplification of each other or the conclusion.

 \end{description}

 Neither @{text"(no_asm_simp)"} nor @{text"(no_asm_use)"} allow to simplify

 the above problematic subgoal.

 Note that only one of the above options is allowed, and it must precede all

 other arguments.

 *}

 subsubsection{*Rewriting with definitions*}

 text{*\index{simplification!with definitions}

 Constant definitions (\S\ref{sec:ConstDefinitions}) can

 be used as simplification rules, but by default they are not.  Hence the

 simplifier does not expand them automatically, just as it should be:

 definitions are introduced for the purpose of abbreviating complex

 concepts. Of course we need to expand the definitions initially to derive

 enough lemmas that characterize the concept sufficiently for us to forget the

 original definition. For example, given

 *}

 constdefs exor :: "bool \<Rightarrow> bool \<Rightarrow> bool"

          "exor A B \<equiv> (A \<and> \<not>B) \<or> (\<not>A \<and> B)";

 text{*\noindent

 we may want to prove

 *}

 lemma "exor A (\<not>A)";

 txt{*\noindent

 Typically, the opening move consists in \emph{unfolding} the definition(s), which we need to

 get started, but nothing else:\indexbold{*unfold}\indexbold{definition!unfolding}

 *}

 apply(simp only:exor_def);

 txt{*\noindent

 In this particular case, the resulting goal

 \begin{isabelle}

 ~1.~A~{\isasymand}~{\isasymnot}~{\isasymnot}~A~{\isasymor}~{\isasymnot}~A~{\isasymand}~{\isasymnot}~A%

 \end{isabelle}

 can be proved by simplification. Thus we could have proved the lemma outright by

 *}(*<*)oops;lemma "exor A (\<not>A)";(*>*)

 apply(simp add: exor_def)

 (*<*)done(*>*)

 text{*\noindent

 Of course we can also unfold definitions in the middle of a proof.

 You should normally not turn a definition permanently into a simplification

 rule because this defeats the whole purpose of an abbreviation.

 \begin{warn}

   If you have defined $f\,x\,y~\isasymequiv~t$ then you can only expand

   occurrences of $f$ with at least two arguments. Thus it is safer to define

   $f$~\isasymequiv~\isasymlambda$x\,y.\;t$.

 \end{warn}

 *}

 subsubsection{*Simplifying let-expressions*}

 text{*\index{simplification!of let-expressions}

 Proving a goal containing \isaindex{let}-expressions almost invariably

 requires the @{text"let"}-con\-structs to be expanded at some point. Since

 @{text"let"}-@{text"in"} is just syntactic sugar for a predefined constant

 (called @{term"Let"}), expanding @{text"let"}-constructs means rewriting with

 @{thm[source]Let_def}:

 *}

 lemma "(let xs = [] in xs@ys@xs) = ys";

 apply(simp add: Let_def);

 done

 text{*

 If, in a particular context, there is no danger of a combinatorial explosion

 of nested @{text"let"}s one could even simlify with @{thm[source]Let_def} by

 default:

 *}

 declare Let_def [simp]

 subsubsection{*Conditional equations*}

 text{*

 So far all examples of rewrite rules were equations. The simplifier also

 accepts \emph{conditional} equations, for example

 *}

 lemma hd_Cons_tl[simp]: "xs \<noteq> []  \<Longrightarrow>  hd xs # tl xs = xs";

 apply(case_tac xs, simp, simp);

 done

 text{*\noindent

 Note the use of ``\ttindexboldpos{,}{$Isar}'' to string together a

 sequence of methods. Assuming that the simplification rule

 @{term"(rev xs = []) = (xs = [])"}

 is present as well,

 *}

 lemma "xs \<noteq> [] \<Longrightarrow> hd(rev xs) # tl(rev xs) = rev xs";

 (*<*)

 by(simp);

 (*>*)

 text{*\noindent

 is proved by plain simplification:

 the conditional equation @{thm[source]hd_Cons_tl} above

 can simplify @{term"hd(rev xs) # tl(rev xs)"} to @{term"rev xs"}

 because the corresponding precondition @{term"rev xs ~= []"}

 simplifies to @{term"xs ~= []"}, which is exactly the local

 assumption of the subgoal.

 *}

 subsubsection{*Automatic case splits*}

 text{*\indexbold{case splits}\index{*split|(}

 Goals containing @{text"if"}-expressions are usually proved by case

 distinction on the condition of the @{text"if"}. For example the goal

 *}

 lemma "\<forall>xs. if xs = [] then rev xs = [] else rev xs \<noteq> []";

 txt{*\noindent

 can be split into

 \begin{isabelle}

 ~1.~{\isasymforall}xs.~(xs~=~[]~{\isasymlongrightarrow}~rev~xs~=~[])~{\isasymand}~(xs~{\isasymnoteq}~[]~{\isasymlongrightarrow}~rev~xs~{\isasymnoteq}~[])

 \end{isabelle}

 by a degenerate form of simplification

 *}

 apply(simp only: split: split_if);

 (*<*)oops;(*>*)

 text{*\noindent

 where no simplification rules are included (@{text"only:"} is followed by the

 empty list of theorems) but the rule \isaindexbold{split_if} for

 splitting @{text"if"}s is added (via the modifier @{text"split:"}). Because

 case-splitting on @{text"if"}s is almost always the right proof strategy, the

 simplifier performs it automatically. Try \isacommand{apply}@{text"(simp)"}

 on the initial goal above.

 This splitting idea generalizes from @{text"if"} to \isaindex{case}:

 *}

 lemma "(case xs of [] \<Rightarrow> zs | y#ys \<Rightarrow> y#(ys@zs)) = xs@zs";

 txt{*\noindent

 becomes

 \begin{isabelle}\makeatother

 ~1.~(xs~=~[]~{\isasymlongrightarrow}~zs~=~xs~@~zs)~{\isasymand}\isanewline

 ~~~~({\isasymforall}a~list.~xs~=~a~\#~list~{\isasymlongrightarrow}~a~\#~list~@~zs~=~xs~@~zs)

 \end{isabelle}

 by typing

 *}

 apply(simp only: split: list.split);

 (*<*)oops;(*>*)

 text{*\noindent

 In contrast to @{text"if"}-expressions, the simplifier does not split

 @{text"case"}-expressions by default because this can lead to nontermination

 in case of recursive datatypes. Again, if the @{text"only:"} modifier is

 dropped, the above goal is solved,

 *}

 (*<*)

 lemma "(case xs of [] \<Rightarrow> zs | y#ys \<Rightarrow> y#(ys@zs)) = xs@zs";

 (*>*)

 apply(simp split: list.split);

 (*<*)done(*>*)

 text{*\noindent%

 which \isacommand{apply}@{text"(simp)"} alone will not do.

 In general, every datatype $t$ comes with a theorem

 $t$@{text".split"} which can be declared to be a \bfindex{split rule} either

 locally as above, or by giving it the @{text"split"} attribute globally:

 *}

 declare list.split [split]

 text{*\noindent

 The @{text"split"} attribute can be removed with the @{text"del"} modifier,

 either locally

 *}

 (*<*)

 lemma "dummy=dummy";

 (*>*)

 apply(simp split del: split_if);

 (*<*)

 oops;

 (*>*)

 text{*\noindent

 or globally:

 *}

 declare list.split [split del]

 text{*

 The above split rules intentionally only affect the conclusion of a

 subgoal.  If you want to split an @{text"if"} or @{text"case"}-expression in

 the assumptions, you have to apply @{thm[source]split_if_asm} or

 $t$@{text".split_asm"}:

 *}

 lemma "if xs = [] then ys ~= [] else ys = [] ==> xs @ ys ~= []"

 apply(simp only: split: split_if_asm);

 (*<*)

 by(simp_all)

 (*>*)

 text{*\noindent

 In contrast to splitting the conclusion, this actually creates two

 separate subgoals (which are solved by @{text"simp_all"}):

 \begin{isabelle}

 \ \isadigit{1}{\isachardot}\ {\isasymlbrakk}\mbox{xs}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ \mbox{ys}\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharat}\ \mbox{ys}\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}\isanewline

 \ \isadigit{2}{\isachardot}\ {\isasymlbrakk}\mbox{xs}\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ \mbox{ys}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \mbox{xs}\ {\isacharat}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymnoteq}\ {\isacharbrackleft}{\isacharbrackright}

 \end{isabelle}

 If you need to split both in the assumptions and the conclusion,

 use $t$@{text".splits"} which subsumes $t$@{text".split"} and

 $t$@{text".split_asm"}. Analogously, there is @{thm[source]if_splits}.

 \begin{warn}

   The simplifier merely simplifies the condition of an \isa{if} but not the

   \isa{then} or \isa{else} parts. The latter are simplified only after the

   condition reduces to \isa{True} or \isa{False}, or after splitting. The

   same is true for \isaindex{case}-expressions: only the selector is

   simplified at first, until either the expression reduces to one of the

   cases or it is split.

 \end{warn}

 \index{*split|)}

 *}

 subsubsection{*Arithmetic*}

 text{*\index{arithmetic}

 The simplifier routinely solves a small class of linear arithmetic formulae

 (over type \isa{nat} and other numeric types): it only takes into account

 assumptions and conclusions that are (possibly negated) (in)equalities

 (@{text"="}, \isasymle, @{text"<"}) and it only knows about addition. Thus

 *}

 lemma "\<lbrakk> \<not> m < n; m < n+1 \<rbrakk> \<Longrightarrow> m = n"

 (*<*)by(auto)(*>*)

 text{*\noindent

 is proved by simplification, whereas the only slightly more complex

 *}

 lemma "\<not> m < n \<and> m < n+1 \<Longrightarrow> m = n";

 (*<*)by(arith)(*>*)

 text{*\noindent

 is not proved by simplification and requires @{text arith}.

 *}

 subsubsection{*Tracing*}

 text{*\indexbold{tracing the simplifier}

 Using the simplifier effectively may take a bit of experimentation.  Set the

 \isaindexbold{trace_simp} \rmindex{flag} to get a better idea of what is going

 on:

 *}

 ML "set trace_simp";

 lemma "rev [a] = []";

 apply(simp);

 (*<*)oops(*>*)

 text{*\noindent

 produces the trace

 \begin{ttbox}\makeatother

 Applying instance of rewrite rule:

 rev (?x1 \# ?xs1) == rev ?xs1 @ [?x1]

 Rewriting:

 rev [x] == rev [] @ [x]

 Applying instance of rewrite rule:

 rev [] == []

 Rewriting:

 rev [] == []

 Applying instance of rewrite rule:

 [] @ ?y == ?y

 Rewriting:

 [] @ [x] == [x]

 Applying instance of rewrite rule:

 ?x3 \# ?t3 = ?t3 == False

 Rewriting:

 [x] = [] == False

 \end{ttbox}

 In more complicated cases, the trace can be quite lenghty, especially since

 invocations of the simplifier are often nested (e.g.\ when solving conditions

 of rewrite rules). Thus it is advisable to reset it:

 *}

 ML "reset trace_simp";

 (*<*)

 end

 (*>*)