(*<*)

theory simp = Main:

(*>*)

subsection{*Simplification Rules*}

text{*\indexbold{simplification rule}

To facilitate simplification, theorems can be declared to be simplification

rules (by 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 run forever, 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}

*}

subsection{*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} fine tunes 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.

*}

subsection{*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}

*}

subsection{*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 modifiers 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}

Both @{text"(no_asm_simp)"} and @{text"(no_asm_use)"} run forever on

the problematic subgoal above.

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

other arguments.

\begin{warn}

Assumptions are simplified in a left-to-right fashion. If an

assumption can help in simplifying one to the left of it, this may get

overlooked. In such cases you have to rotate the assumptions explicitly:

\isacommand{apply}@{text"(rotate_tac"}~$n$@{text")"}\indexbold{*rotate_tac}

causes a cyclic shift by $n$ positions from right to left, if $n$ is

positive, and from left to right, if $n$ is negative.

Beware that such rotations make proofs quite brittle.

\end{warn}

*}

subsection{*Rewriting with Definitions*}

text{*\label{sec:Simp-with-Defs}\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 xor :: "bool \<Rightarrow> bool \<Rightarrow> bool"

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

text{*\noindent

we may want to prove

*}

lemma "xor 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:xor_def);

txt{*\noindent

In this particular case, the resulting goal

@{subgoals[display,indent=0]}

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

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

apply(simp add: xor_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.

\begin{warn}

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

  occurrences of $f$ with at least two arguments. This may be helpful for unfolding

  $f$ selectively, but it may also get in the way. Defining

  $f$~\isasymequiv~\isasymlambda$x\,y.\;t$ allows to unfold all occurrences of $f$.

\end{warn}

*}

subsection{*Simplifying {\tt\slshape 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"}\ldots\isa{=}\ldots@{text"in"}{\ldots} 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]

subsection{*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,

the lemma below is proved by plain simplification:

*}

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

(*<*)

by(simp);

(*>*)

text{*\noindent

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.

*}

subsection{*Automatic Case Splits*}

text{*\label{sec:AutoCaseSplits}\indexbold{case splits}\index{*split (method, attr.)|(}

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 by a special method @{text split}:

*}

apply(split split_if)

txt{*\noindent

@{subgoals[display,indent=0]}

where \isaindexbold{split_if} is a theorem that expresses splitting of

@{text"if"}s. 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}:

*}(*<*)by simp(*>*)

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

apply(split list.split);

txt{*

@{subgoals[display,indent=0]}

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. Therefore the simplifier has a modifier

@{text split} for adding further splitting rules explicitly. This means the

above lemma can be proved in one step by

*}

(*<*)oops;

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

(*>*)

apply(simp split: list.split);

(*<*)done(*>*)

text{*\noindent

whereas \isacommand{apply}@{text"(simp)"} alone will not succeed.

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{*

In polished proofs the @{text split} method is rarely used on its own

but always as part of the simplifier. However, if a goal contains

multiple splittable constructs, the @{text split} method can be

helpful in selectively exploring the effects of splitting.

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 \<noteq> [] else ys = [] \<Longrightarrow> xs @ ys \<noteq> []"

apply(split split_if_asm)

txt{*\noindent

In contrast to splitting the conclusion, this actually creates two

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

@{subgoals[display,indent=0]}

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 (method, attr.)|)}

*}

(*<*)

by(simp_all)

(*>*)

subsection{*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 relations

($=$, $\le$, $<$, possibly negated) 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}.

*}

subsection{*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 [a] == rev [] @ [a]

Applying instance of rewrite rule:

rev [] == []

Rewriting:

rev [] == []

Applying instance of rewrite rule:

[] @ ?y == ?y

Rewriting:

[] @ [a] == [a]

Applying instance of rewrite rule:

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

Rewriting:

[a] = [] == 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

(*>*)