nipkow@9932
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(*<*)
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haftmann@16417
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theory simp imports Main begin
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nipkow@9932
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(*>*)
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wenzelm@9922
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nipkow@11214
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subsection{*Simplification Rules*}
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wenzelm@9922
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paulson@11458
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text{*\index{simplification rules}
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paulson@11458
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To facilitate simplification,
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paulson@11458
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the attribute @{text"[simp]"}\index{*simp (attribute)}
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paulson@11458
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declares theorems to be simplification rules, which the simplifier
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will use automatically. In addition, \isacommand{datatype} and
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paulson@11458
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\isacommand{primrec} declarations (and a few others)
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paulson@11458
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implicitly declare some simplification rules.
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paulson@11458
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Explicit definitions are \emph{not} declared as
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nipkow@9932
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simplification rules automatically!
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nipkow@9932
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paulson@11458
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Nearly any theorem can become a simplification
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paulson@11458
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rule. The simplifier will try to transform it into an equation.
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paulson@11458
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For example, the theorem
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paulson@11458
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@{prop"~P"} is turned into @{prop"P = False"}. The details
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nipkow@9932
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are explained in \S\ref{sec:SimpHow}.
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nipkow@9932
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paulson@11458
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The simplification attribute of theorems can be turned on and off:%
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nipkow@12489
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\index{*simp del (attribute)}
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\begin{quote}
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nipkow@9932
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\isacommand{declare} \textit{theorem-name}@{text"[simp]"}\\
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\isacommand{declare} \textit{theorem-name}@{text"[simp del]"}
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\end{quote}
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paulson@11309
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Only equations that really simplify, like \isa{rev\
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paulson@11309
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{\isacharparenleft}rev\ xs{\isacharparenright}\ {\isacharequal}\ xs} and
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paulson@11309
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\isa{xs\ {\isacharat}\ {\isacharbrackleft}{\isacharbrackright}\
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paulson@11309
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{\isacharequal}\ xs}, should be declared as default simplification rules.
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paulson@11309
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More specific ones should only be used selectively and should
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paulson@11309
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not be made default. Distributivity laws, for example, alter
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paulson@11309
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the structure of terms and can produce an exponential blow-up instead of
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paulson@11309
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simplification. A default simplification rule may
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paulson@11309
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need to be disabled in certain proofs. Frequent changes in the simplification
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paulson@11309
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status of a theorem may indicate an unwise use of defaults.
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\begin{warn}
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paulson@11458
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Simplification can run forever, for example if both $f(x) = g(x)$ and
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nipkow@9932
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$g(x) = f(x)$ are simplification rules. It is the user's responsibility not
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nipkow@9932
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to include simplification rules that can lead to nontermination, either on
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nipkow@9932
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their own or in combination with other simplification rules.
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\end{warn}
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nipkow@12332
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\begin{warn}
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It is inadvisable to toggle the simplification attribute of a
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nipkow@12332
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theorem from a parent theory $A$ in a child theory $B$ for good.
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nipkow@12332
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The reason is that if some theory $C$ is based both on $B$ and (via a
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nipkow@13814
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different path) on $A$, it is not defined what the simplification attribute
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nipkow@12332
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of that theorem will be in $C$: it could be either.
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\end{warn}
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paulson@11458
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*}
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nipkow@9932
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paulson@11458
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subsection{*The {\tt\slshape simp} Method*}
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text{*\index{*simp (method)|bold}
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The general format of the simplification method is
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\begin{quote}
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@{text simp} \textit{list of modifiers}
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\end{quote}
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paulson@10795
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where the list of \emph{modifiers} fine tunes the behaviour and may
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paulson@11309
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be empty. Specific modifiers are discussed below. Most if not all of the
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paulson@11309
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proofs seen so far could have been performed
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nipkow@9932
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with @{text simp} instead of \isa{auto}, except that @{text simp} attacks
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nipkow@10971
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only the first subgoal and may thus need to be repeated --- use
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paulson@11428
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\methdx{simp_all} to simplify all subgoals.
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paulson@11458
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If nothing changes, @{text simp} fails.
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*}
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nipkow@11214
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subsection{*Adding and Deleting Simplification Rules*}
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text{*
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\index{simplification rules!adding and deleting}%
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nipkow@9932
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If a certain theorem is merely needed in a few proofs by simplification,
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we do not need to make it a global simplification rule. Instead we can modify
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the set of simplification rules used in a simplification step by adding rules
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to it and/or deleting rules from it. The two modifiers for this are
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\begin{quote}
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paulson@11458
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@{text"add:"} \textit{list of theorem names}\index{*add (modifier)}\\
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@{text"del:"} \textit{list of theorem names}\index{*del (modifier)}
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\end{quote}
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paulson@11458
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Or you can use a specific list of theorems and omit all others:
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nipkow@9932
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\begin{quote}
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paulson@11458
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@{text"only:"} \textit{list of theorem names}\index{*only (modifier)}
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\end{quote}
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paulson@11309
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In this example, we invoke the simplifier, adding two distributive
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paulson@11309
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laws:
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paulson@11309
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\begin{quote}
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paulson@11309
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\isacommand{apply}@{text"(simp add: mod_mult_distrib add_mult_distrib)"}
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\end{quote}
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*}
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nipkow@11214
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subsection{*Assumptions*}
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text{*\index{simplification!with/of assumptions}
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By default, assumptions are part of the simplification process: they are used
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as simplification rules and are simplified themselves. For example:
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*}
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wenzelm@12631
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lemma "\<lbrakk> xs @ zs = ys @ xs; [] @ xs = [] @ [] \<rbrakk> \<Longrightarrow> ys = zs"
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wenzelm@12631
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apply simp
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done
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text{*\noindent
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The second assumption simplifies to @{term"xs = []"}, which in turn
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nipkow@9932
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simplifies the first assumption to @{term"zs = ys"}, thus reducing the
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conclusion to @{term"ys = ys"} and hence to @{term"True"}.
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nipkow@9932
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paulson@11458
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In some cases, using the assumptions can lead to nontermination:
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nipkow@9932
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*}
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wenzelm@12631
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lemma "\<forall>x. f x = g (f (g x)) \<Longrightarrow> f [] = f [] @ []"
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nipkow@9932
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nipkow@9932
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txt{*\noindent
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paulson@11458
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An unmodified application of @{text"simp"} loops. The culprit is the
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paulson@11458
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simplification rule @{term"f x = g (f (g x))"}, which is extracted from
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paulson@11458
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the assumption. (Isabelle notices certain simple forms of
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paulson@11458
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nontermination but not this one.) The problem can be circumvented by
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paulson@11458
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telling the simplifier to ignore the assumptions:
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nipkow@9932
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*}
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nipkow@9932
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wenzelm@12631
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apply(simp (no_asm))
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done
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text{*\noindent
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paulson@11458
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Three modifiers influence the treatment of assumptions:
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nipkow@9932
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\begin{description}
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paulson@11458
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\item[@{text"(no_asm)"}]\index{*no_asm (modifier)}
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means that assumptions are completely ignored.
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paulson@11458
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\item[@{text"(no_asm_simp)"}]\index{*no_asm_simp (modifier)}
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nipkow@9932
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means that the assumptions are not simplified but
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nipkow@9932
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are used in the simplification of the conclusion.
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paulson@11458
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\item[@{text"(no_asm_use)"}]\index{*no_asm_use (modifier)}
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means that the assumptions are simplified but are not
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used in the simplification of each other or the conclusion.
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\end{description}
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paulson@11458
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Only one of the modifiers is allowed, and it must precede all
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paulson@11309
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other modifiers.
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nipkow@13623
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%\begin{warn}
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nipkow@13623
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%Assumptions are simplified in a left-to-right fashion. If an
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nipkow@13623
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%assumption can help in simplifying one to the left of it, this may get
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nipkow@13623
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%overlooked. In such cases you have to rotate the assumptions explicitly:
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nipkow@13623
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%\isacommand{apply}@ {text"("}\methdx{rotate_tac}~$n$@ {text")"}
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nipkow@13623
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%causes a cyclic shift by $n$ positions from right to left, if $n$ is
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nipkow@13623
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%positive, and from left to right, if $n$ is negative.
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nipkow@13623
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%Beware that such rotations make proofs quite brittle.
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nipkow@13623
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%\end{warn}
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*}
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nipkow@9932
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nipkow@11214
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subsection{*Rewriting with Definitions*}
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nipkow@11215
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text{*\label{sec:Simp-with-Defs}\index{simplification!with definitions}
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paulson@11458
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Constant definitions (\S\ref{sec:ConstDefinitions}) can be used as
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paulson@11458
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simplification rules, but by default they are not: the simplifier does not
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paulson@11458
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expand them automatically. Definitions are intended for introducing abstract
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wenzelm@12582
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concepts and not merely as abbreviations. Of course, we need to expand
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paulson@11458
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the definition initially, but once we have proved enough abstract properties
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paulson@11458
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of the new constant, we can forget its original definition. This style makes
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paulson@11458
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proofs more robust: if the definition has to be changed,
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paulson@11458
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only the proofs of the abstract properties will be affected.
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paulson@11458
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For example, given *}
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nipkow@27027
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definition xor :: "bool \<Rightarrow> bool \<Rightarrow> bool" where
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nipkow@27027
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"xor A B \<equiv> (A \<and> \<not>B) \<or> (\<not>A \<and> B)"
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nipkow@9932
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text{*\noindent
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nipkow@9932
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we may want to prove
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*}
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wenzelm@12631
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lemma "xor A (\<not>A)"
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nipkow@9932
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txt{*\noindent
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paulson@11428
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Typically, we begin by unfolding some definitions:
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paulson@11428
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\indexbold{definitions!unfolding}
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*}
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wenzelm@12631
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apply(simp only: xor_def)
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txt{*\noindent
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In this particular case, the resulting goal
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nipkow@10362
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@{subgoals[display,indent=0]}
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can be proved by simplification. Thus we could have proved the lemma outright by
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wenzelm@12631
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*}(*<*)oops lemma "xor A (\<not>A)"(*>*)
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apply(simp add: xor_def)
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(*<*)done(*>*)
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text{*\noindent
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Of course we can also unfold definitions in the middle of a proof.
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\begin{warn}
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nipkow@10971
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If you have defined $f\,x\,y~\isasymequiv~t$ then you can only unfold
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nipkow@10971
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occurrences of $f$ with at least two arguments. This may be helpful for unfolding
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nipkow@10971
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$f$ selectively, but it may also get in the way. Defining
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nipkow@10971
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$f$~\isasymequiv~\isasymlambda$x\,y.\;t$ allows to unfold all occurrences of $f$.
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nipkow@9932
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\end{warn}
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nipkow@12473
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nipkow@12473
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There is also the special method \isa{unfold}\index{*unfold (method)|bold}
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nipkow@12473
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which merely unfolds
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nipkow@12473
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one or several definitions, as in \isacommand{apply}\isa{(unfold xor_def)}.
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nipkow@12473
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This is can be useful in situations where \isa{simp} does too much.
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Warning: \isa{unfold} acts on all subgoals!
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*}
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nipkow@11214
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subsection{*Simplifying {\tt\slshape let}-Expressions*}
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paulson@11458
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text{*\index{simplification!of \isa{let}-expressions}\index{*let expressions}%
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paulson@11458
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Proving a goal containing \isa{let}-expressions almost invariably requires the
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paulson@11458
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@{text"let"}-con\-structs to be expanded at some point. Since
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paulson@11458
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@{text"let"}\ldots\isa{=}\ldots@{text"in"}{\ldots} is just syntactic sugar for
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paulson@11458
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the predefined constant @{term"Let"}, expanding @{text"let"}-constructs
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paulson@11458
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means rewriting with \tdx{Let_def}: *}
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nipkow@9932
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wenzelm@12631
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lemma "(let xs = [] in xs@ys@xs) = ys"
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wenzelm@12631
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apply(simp add: Let_def)
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done
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text{*
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If, in a particular context, there is no danger of a combinatorial explosion
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paulson@11458
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of nested @{text"let"}s, you could even simplify with @{thm[source]Let_def} by
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default:
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*}
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declare Let_def [simp]
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paulson@11458
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subsection{*Conditional Simplification Rules*}
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text{*
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paulson@11458
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\index{conditional simplification rules}%
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So far all examples of rewrite rules were equations. The simplifier also
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accepts \emph{conditional} equations, for example
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*}
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wenzelm@12631
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lemma hd_Cons_tl[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs # tl xs = xs"
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wenzelm@12631
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apply(case_tac xs, simp, simp)
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done
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text{*\noindent
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Note the use of ``\ttindexboldpos{,}{$Isar}'' to string together a
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sequence of methods. Assuming that the simplification rule
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@{term"(rev xs = []) = (xs = [])"}
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is present as well,
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paulson@10795
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the lemma below is proved by plain simplification:
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*}
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nipkow@9932
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wenzelm@12631
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lemma "xs \<noteq> [] \<Longrightarrow> hd(rev xs) # tl(rev xs) = rev xs"
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(*<*)
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wenzelm@12631
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by(simp)
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(*>*)
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text{*\noindent
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paulson@10795
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The conditional equation @{thm[source]hd_Cons_tl} above
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can simplify @{term"hd(rev xs) # tl(rev xs)"} to @{term"rev xs"}
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because the corresponding precondition @{term"rev xs ~= []"}
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simplifies to @{term"xs ~= []"}, which is exactly the local
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assumption of the subgoal.
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*}
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nipkow@9932
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nipkow@9932
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nipkow@11214
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subsection{*Automatic Case Splits*}
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paulson@11458
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text{*\label{sec:AutoCaseSplits}\indexbold{case splits}%
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paulson@11458
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Goals containing @{text"if"}-expressions\index{*if expressions!splitting of}
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are usually proved by case
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distinction on the boolean condition. Here is an example:
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*}
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nipkow@9932
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wenzelm@12631
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lemma "\<forall>xs. if xs = [] then rev xs = [] else rev xs \<noteq> []"
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txt{*\noindent
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paulson@11458
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The goal can be split by a special method, \methdx{split}:
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*}
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nipkow@9932
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nipkow@10654
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apply(split split_if)
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nipkow@10362
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txt{*\noindent
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@{subgoals[display,indent=0]}
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paulson@11428
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where \tdx{split_if} is a theorem that expresses splitting of
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@{text"if"}s. Because
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paulson@11458
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splitting the @{text"if"}s is usually the right proof strategy, the
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paulson@11458
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simplifier does it automatically. Try \isacommand{apply}@{text"(simp)"}
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on the initial goal above.
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paulson@11428
|
281 |
This splitting idea generalizes from @{text"if"} to \sdx{case}.
|
paulson@11458
|
282 |
Let us simplify a case analysis over lists:\index{*list.split (theorem)}
|
nipkow@10654
|
283 |
*}(*<*)by simp(*>*)
|
wenzelm@12631
|
284 |
lemma "(case xs of [] \<Rightarrow> zs | y#ys \<Rightarrow> y#(ys@zs)) = xs@zs"
|
wenzelm@12631
|
285 |
apply(split list.split)
|
paulson@11309
|
286 |
|
nipkow@10362
|
287 |
txt{*
|
nipkow@10362
|
288 |
@{subgoals[display,indent=0]}
|
paulson@11458
|
289 |
The simplifier does not split
|
paulson@11458
|
290 |
@{text"case"}-expressions, as it does @{text"if"}-expressions,
|
paulson@11458
|
291 |
because with recursive datatypes it could lead to nontermination.
|
paulson@11458
|
292 |
Instead, the simplifier has a modifier
|
paulson@11494
|
293 |
@{text split}\index{*split (modifier)}
|
paulson@11458
|
294 |
for adding splitting rules explicitly. The
|
paulson@11458
|
295 |
lemma above can be proved in one step by
|
nipkow@9932
|
296 |
*}
|
wenzelm@12631
|
297 |
(*<*)oops
|
wenzelm@12631
|
298 |
lemma "(case xs of [] \<Rightarrow> zs | y#ys \<Rightarrow> y#(ys@zs)) = xs@zs"
|
nipkow@9932
|
299 |
(*>*)
|
wenzelm@12631
|
300 |
apply(simp split: list.split)
|
nipkow@10171
|
301 |
(*<*)done(*>*)
|
nipkow@10654
|
302 |
text{*\noindent
|
nipkow@10654
|
303 |
whereas \isacommand{apply}@{text"(simp)"} alone will not succeed.
|
nipkow@9932
|
304 |
|
paulson@11458
|
305 |
Every datatype $t$ comes with a theorem
|
nipkow@9932
|
306 |
$t$@{text".split"} which can be declared to be a \bfindex{split rule} either
|
paulson@11458
|
307 |
locally as above, or by giving it the \attrdx{split} attribute globally:
|
nipkow@9932
|
308 |
*}
|
nipkow@9932
|
309 |
|
nipkow@9932
|
310 |
declare list.split [split]
|
nipkow@9932
|
311 |
|
nipkow@9932
|
312 |
text{*\noindent
|
nipkow@9932
|
313 |
The @{text"split"} attribute can be removed with the @{text"del"} modifier,
|
nipkow@9932
|
314 |
either locally
|
nipkow@9932
|
315 |
*}
|
nipkow@9932
|
316 |
(*<*)
|
wenzelm@12631
|
317 |
lemma "dummy=dummy"
|
nipkow@9932
|
318 |
(*>*)
|
wenzelm@12631
|
319 |
apply(simp split del: split_if)
|
nipkow@9932
|
320 |
(*<*)
|
wenzelm@12631
|
321 |
oops
|
nipkow@9932
|
322 |
(*>*)
|
nipkow@9932
|
323 |
text{*\noindent
|
nipkow@9932
|
324 |
or globally:
|
nipkow@9932
|
325 |
*}
|
nipkow@9932
|
326 |
declare list.split [split del]
|
nipkow@9932
|
327 |
|
nipkow@9932
|
328 |
text{*
|
paulson@11458
|
329 |
Polished proofs typically perform splitting within @{text simp} rather than
|
paulson@11458
|
330 |
invoking the @{text split} method. However, if a goal contains
|
wenzelm@19792
|
331 |
several @{text "if"} and @{text case} expressions,
|
paulson@11458
|
332 |
the @{text split} method can be
|
nipkow@10654
|
333 |
helpful in selectively exploring the effects of splitting.
|
nipkow@10654
|
334 |
|
paulson@11458
|
335 |
The split rules shown above are intended to affect only the subgoal's
|
paulson@11458
|
336 |
conclusion. If you want to split an @{text"if"} or @{text"case"}-expression
|
paulson@11458
|
337 |
in the assumptions, you have to apply \tdx{split_if_asm} or
|
paulson@11458
|
338 |
$t$@{text".split_asm"}: *}
|
nipkow@9932
|
339 |
|
nipkow@10654
|
340 |
lemma "if xs = [] then ys \<noteq> [] else ys = [] \<Longrightarrow> xs @ ys \<noteq> []"
|
nipkow@10654
|
341 |
apply(split split_if_asm)
|
nipkow@9932
|
342 |
|
nipkow@10362
|
343 |
txt{*\noindent
|
paulson@11458
|
344 |
Unlike splitting the conclusion, this step creates two
|
paulson@11458
|
345 |
separate subgoals, which here can be solved by @{text"simp_all"}:
|
nipkow@10362
|
346 |
@{subgoals[display,indent=0]}
|
nipkow@9932
|
347 |
If you need to split both in the assumptions and the conclusion,
|
nipkow@9932
|
348 |
use $t$@{text".splits"} which subsumes $t$@{text".split"} and
|
nipkow@9932
|
349 |
$t$@{text".split_asm"}. Analogously, there is @{thm[source]if_splits}.
|
nipkow@9932
|
350 |
|
nipkow@9932
|
351 |
\begin{warn}
|
paulson@11458
|
352 |
The simplifier merely simplifies the condition of an
|
paulson@11458
|
353 |
\isa{if}\index{*if expressions!simplification of} but not the
|
nipkow@9932
|
354 |
\isa{then} or \isa{else} parts. The latter are simplified only after the
|
nipkow@9932
|
355 |
condition reduces to \isa{True} or \isa{False}, or after splitting. The
|
paulson@11428
|
356 |
same is true for \sdx{case}-expressions: only the selector is
|
nipkow@9932
|
357 |
simplified at first, until either the expression reduces to one of the
|
nipkow@9932
|
358 |
cases or it is split.
|
paulson@11458
|
359 |
\end{warn}
|
nipkow@9932
|
360 |
*}
|
nipkow@10362
|
361 |
(*<*)
|
nipkow@10362
|
362 |
by(simp_all)
|
nipkow@10362
|
363 |
(*>*)
|
nipkow@9932
|
364 |
|
nipkow@11214
|
365 |
subsection{*Tracing*}
|
nipkow@9932
|
366 |
text{*\indexbold{tracing the simplifier}
|
nipkow@9932
|
367 |
Using the simplifier effectively may take a bit of experimentation. Set the
|
nipkow@16523
|
368 |
Proof General flag \pgmenu{Isabelle} $>$ \pgmenu{Settings} $>$ \pgmenu{Trace Simplifier} to get a better idea of what is going on:
|
nipkow@9932
|
369 |
*}
|
nipkow@9932
|
370 |
|
wenzelm@12631
|
371 |
lemma "rev [a] = []"
|
wenzelm@12631
|
372 |
apply(simp)
|
nipkow@9932
|
373 |
(*<*)oops(*>*)
|
nipkow@9932
|
374 |
|
nipkow@9932
|
375 |
text{*\noindent
|
nipkow@16523
|
376 |
produces the following trace in Proof General's \pgmenu{Trace} buffer:
|
nipkow@9932
|
377 |
|
nipkow@9932
|
378 |
\begin{ttbox}\makeatother
|
nipkow@16518
|
379 |
[1]Applying instance of rewrite rule "List.rev.simps_2":
|
nipkow@16359
|
380 |
rev (?x1 # ?xs1) \(\equiv\) rev ?xs1 @ [?x1]
|
nipkow@16359
|
381 |
|
nipkow@16518
|
382 |
[1]Rewriting:
|
nipkow@16359
|
383 |
rev [a] \(\equiv\) rev [] @ [a]
|
nipkow@16359
|
384 |
|
nipkow@16518
|
385 |
[1]Applying instance of rewrite rule "List.rev.simps_1":
|
nipkow@16359
|
386 |
rev [] \(\equiv\) []
|
nipkow@16359
|
387 |
|
nipkow@16518
|
388 |
[1]Rewriting:
|
nipkow@16359
|
389 |
rev [] \(\equiv\) []
|
nipkow@16359
|
390 |
|
nipkow@16518
|
391 |
[1]Applying instance of rewrite rule "List.op @.append_Nil":
|
nipkow@16359
|
392 |
[] @ ?y \(\equiv\) ?y
|
nipkow@16359
|
393 |
|
nipkow@16518
|
394 |
[1]Rewriting:
|
nipkow@16359
|
395 |
[] @ [a] \(\equiv\) [a]
|
nipkow@16359
|
396 |
|
nipkow@16518
|
397 |
[1]Applying instance of rewrite rule
|
nipkow@16359
|
398 |
?x2 # ?t1 = ?t1 \(\equiv\) False
|
nipkow@16359
|
399 |
|
nipkow@16518
|
400 |
[1]Rewriting:
|
nipkow@16359
|
401 |
[a] = [] \(\equiv\) False
|
nipkow@9932
|
402 |
\end{ttbox}
|
nipkow@16359
|
403 |
The trace lists each rule being applied, both in its general form and
|
nipkow@16359
|
404 |
the instance being used. The \texttt{[}$i$\texttt{]} in front (where
|
nipkow@16518
|
405 |
above $i$ is always \texttt{1}) indicates that we are inside the $i$th
|
nipkow@16518
|
406 |
invocation of the simplifier. Each attempt to apply a
|
nipkow@16359
|
407 |
conditional rule shows the rule followed by the trace of the
|
nipkow@16359
|
408 |
(recursive!) simplification of the conditions, the latter prefixed by
|
nipkow@16359
|
409 |
\texttt{[}$i+1$\texttt{]} instead of \texttt{[}$i$\texttt{]}.
|
nipkow@16359
|
410 |
Another source of recursive invocations of the simplifier are
|
nipkow@35992
|
411 |
proofs of arithmetic formulae. By default, recursive invocations are not shown,
|
nipkow@35992
|
412 |
you must increase the trace depth via \pgmenu{Isabelle} $>$ \pgmenu{Settings} $>$ \pgmenu{Trace Simplifier Depth}.
|
nipkow@9932
|
413 |
|
nipkow@16359
|
414 |
Many other hints about the simplifier's actions may appear.
|
paulson@11309
|
415 |
|
nipkow@16359
|
416 |
In more complicated cases, the trace can be very lengthy. Thus it is
|
nipkow@16523
|
417 |
advisable to reset the \pgmenu{Trace Simplifier} flag after having
|
nipkow@35992
|
418 |
obtained the desired trace.
|
nipkow@36069
|
419 |
Since this is easily forgotten (and may have the unpleasant effect of
|
nipkow@36069
|
420 |
swamping the interface with trace information), here is how you can switch
|
nipkow@36069
|
421 |
the trace on locally in a proof: *}
|
nipkow@36069
|
422 |
|
nipkow@36069
|
423 |
(*<*)lemma "x=x"
|
nipkow@36069
|
424 |
(*>*)
|
nipkow@36069
|
425 |
using [[trace_simp=true]]
|
nipkow@36069
|
426 |
apply simp
|
nipkow@36069
|
427 |
(*<*)oops(*>*)
|
nipkow@36069
|
428 |
|
nipkow@36069
|
429 |
text{* \noindent
|
nipkow@36069
|
430 |
Within the current proof, all simplifications in subsequent proof steps
|
nipkow@36069
|
431 |
will be traced, but the text reminds you to remove the \isa{using} clause
|
nipkow@36069
|
432 |
after it has done its job. *}
|
nipkow@9932
|
433 |
|
nipkow@16544
|
434 |
subsection{*Finding Theorems\label{sec:find}*}
|
nipkow@16518
|
435 |
|
nipkow@16523
|
436 |
text{*\indexbold{finding theorems}\indexbold{searching theorems}
|
paulson@16560
|
437 |
Isabelle's large database of proved theorems
|
paulson@16560
|
438 |
offers a powerful search engine. Its chief limitation is
|
nipkow@16518
|
439 |
its restriction to the theories currently loaded.
|
nipkow@16518
|
440 |
|
nipkow@16518
|
441 |
\begin{pgnote}
|
nipkow@16523
|
442 |
The search engine is started by clicking on Proof General's \pgmenu{Find} icon.
|
nipkow@16518
|
443 |
You specify your search textually in the input buffer at the bottom
|
nipkow@16518
|
444 |
of the window.
|
nipkow@16518
|
445 |
\end{pgnote}
|
nipkow@16518
|
446 |
|
paulson@16560
|
447 |
The simplest form of search finds theorems containing specified
|
paulson@16560
|
448 |
patterns. A pattern can be any term (even
|
paulson@16560
|
449 |
a single identifier). It may contain ``\texttt{\_}'', a wildcard standing
|
paulson@16560
|
450 |
for any term. Here are some
|
nipkow@16518
|
451 |
examples:
|
nipkow@16518
|
452 |
\begin{ttbox}
|
nipkow@16518
|
453 |
length
|
nipkow@16518
|
454 |
"_ # _ = _ # _"
|
nipkow@16518
|
455 |
"_ + _"
|
nipkow@16518
|
456 |
"_ * (_ - (_::nat))"
|
nipkow@16518
|
457 |
\end{ttbox}
|
paulson@16560
|
458 |
Specifying types, as shown in the last example,
|
paulson@16560
|
459 |
constrains searches involving overloaded operators.
|
nipkow@16518
|
460 |
|
nipkow@16518
|
461 |
\begin{warn}
|
nipkow@16518
|
462 |
Always use ``\texttt{\_}'' rather than variable names: searching for
|
nipkow@16518
|
463 |
\texttt{"x + y"} will usually not find any matching theorems
|
paulson@16560
|
464 |
because they would need to contain \texttt{x} and~\texttt{y} literally.
|
paulson@16560
|
465 |
When searching for infix operators, do not just type in the symbol,
|
paulson@16560
|
466 |
such as~\texttt{+}, but a proper term such as \texttt{"_ + _"}.
|
paulson@16560
|
467 |
This remark applies to more complicated syntaxes, too.
|
nipkow@16518
|
468 |
\end{warn}
|
nipkow@16518
|
469 |
|
paulson@16560
|
470 |
If you are looking for rewrite rules (possibly conditional) that could
|
paulson@16560
|
471 |
simplify some term, prefix the pattern with \texttt{simp:}.
|
nipkow@16518
|
472 |
\begin{ttbox}
|
nipkow@16518
|
473 |
simp: "_ * (_ + _)"
|
nipkow@16518
|
474 |
\end{ttbox}
|
paulson@16560
|
475 |
This finds \emph{all} equations---not just those with a \isa{simp} attribute---whose conclusion has the form
|
nipkow@16518
|
476 |
@{text[display]"_ * (_ + _) = \<dots>"}
|
paulson@16560
|
477 |
It only finds equations that can simplify the given pattern
|
paulson@16560
|
478 |
at the root, not somewhere inside: for example, equations of the form
|
paulson@16560
|
479 |
@{text"_ + _ = \<dots>"} do not match.
|
nipkow@16518
|
480 |
|
paulson@16560
|
481 |
You may also search for theorems by name---you merely
|
paulson@16560
|
482 |
need to specify a substring. For example, you could search for all
|
nipkow@16518
|
483 |
commutativity theorems like this:
|
nipkow@16518
|
484 |
\begin{ttbox}
|
nipkow@16518
|
485 |
name: comm
|
nipkow@16518
|
486 |
\end{ttbox}
|
nipkow@16518
|
487 |
This retrieves all theorems whose name contains \texttt{comm}.
|
nipkow@16518
|
488 |
|
paulson@16560
|
489 |
Search criteria can also be negated by prefixing them with ``\texttt{-}''.
|
paulson@16560
|
490 |
For example,
|
nipkow@16518
|
491 |
\begin{ttbox}
|
nipkow@16523
|
492 |
-name: List
|
nipkow@16518
|
493 |
\end{ttbox}
|
paulson@16560
|
494 |
finds theorems whose name does not contain \texttt{List}. You can use this
|
paulson@16560
|
495 |
to exclude particular theories from the search: the long name of
|
nipkow@16518
|
496 |
a theorem contains the name of the theory it comes from.
|
nipkow@16518
|
497 |
|
paulson@16560
|
498 |
Finallly, different search criteria can be combined arbitrarily.
|
webertj@20143
|
499 |
The effect is conjuctive: Find returns the theorems that satisfy all of
|
paulson@16560
|
500 |
the criteria. For example,
|
nipkow@16518
|
501 |
\begin{ttbox}
|
nipkow@16518
|
502 |
"_ + _" -"_ - _" -simp: "_ * (_ + _)" name: assoc
|
nipkow@16518
|
503 |
\end{ttbox}
|
paulson@16560
|
504 |
looks for theorems containing plus but not minus, and which do not simplify
|
paulson@16560
|
505 |
\mbox{@{text"_ * (_ + _)"}} at the root, and whose name contains \texttt{assoc}.
|
nipkow@16518
|
506 |
|
nipkow@16544
|
507 |
Further search criteria are explained in \S\ref{sec:find2}.
|
nipkow@16523
|
508 |
|
nipkow@16523
|
509 |
\begin{pgnote}
|
nipkow@16523
|
510 |
Proof General keeps a history of all your search expressions.
|
nipkow@16523
|
511 |
If you click on \pgmenu{Find}, you can use the arrow keys to scroll
|
nipkow@16523
|
512 |
through previous searches and just modify them. This saves you having
|
nipkow@16523
|
513 |
to type in lengthy expressions again and again.
|
nipkow@16523
|
514 |
\end{pgnote}
|
nipkow@16518
|
515 |
*}
|
nipkow@9932
|
516 |
(*<*)
|
wenzelm@9922
|
517 |
end
|
nipkow@9932
|
518 |
(*>*)
|