diff -r 0c86acc069ad -r 5deda0549f97 doc-src/TutorialI/document/Pairs.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/document/Pairs.tex	Thu Jul 26 17:16:02 2012 +0200
@@ -0,0 +1,394 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Pairs}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupsection{Pairs and Tuples%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec:products}
+Ordered pairs were already introduced in \S\ref{sec:pairs}, but only with a minimal
+repertoire of operations: pairing and the two projections \isa{fst} and
+\isa{snd}. In any non-trivial application of pairs you will find that this
+quickly leads to unreadable nests of projections. This
+section introduces syntactic sugar to overcome this
+problem: pattern matching with tuples.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Pattern Matching with Tuples%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Tuples may be used as patterns in $\lambda$-abstractions,
+for example \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{2C}{\isacharcomma}}z{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}x{\isaliteral{2B}{\isacharplus}}y{\isaliteral{2B}{\isacharplus}}z} and \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}z{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}x{\isaliteral{2B}{\isacharplus}}y{\isaliteral{2B}{\isacharplus}}z}. In fact,
+tuple patterns can be used in most variable binding constructs,
+and they can be nested. Here are
+some typical examples:
+\begin{quote}
+\isa{let\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ z\ in\ {\isaliteral{28}{\isacharparenleft}}y{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}}\\
+\isa{case\ xs\ of\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isadigit{0}}\ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{23}{\isacharhash}}\ zs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ x\ {\isaliteral{2B}{\isacharplus}}\ y}\\
+\isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E3E}{\isasymin}}A{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}y}\\
+\isa{{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{2C}{\isacharcomma}}z{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}z{\isaliteral{7D}{\isacharbraceright}}}\\
+\isa{{\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}\isaliteral{5C3C5E627375623E}{}\isactrlbsub {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E3E}{\isasymin}}A\isaliteral{5C3C5E657375623E}{}\isactrlesub \ {\isaliteral{7B}{\isacharbraceleft}}x\ {\isaliteral{2B}{\isacharplus}}\ y{\isaliteral{7D}{\isacharbraceright}}}
+\end{quote}
+The intuitive meanings of these expressions should be obvious.
+Unfortunately, we need to know in more detail what the notation really stands
+for once we have to reason about it.  Abstraction
+over pairs and tuples is merely a convenient shorthand for a more complex
+internal representation.  Thus the internal and external form of a term may
+differ, which can affect proofs. If you want to avoid this complication,
+stick to \isa{fst} and \isa{snd} and write \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}p{\isaliteral{2E}{\isachardot}}\ fst\ p\ {\isaliteral{2B}{\isacharplus}}\ snd\ p}
+instead of \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{2B}{\isacharplus}}y}.  These terms are distinct even though they
+denote the same function.
+
+Internally, \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ t} becomes \isa{split\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x\ y{\isaliteral{2E}{\isachardot}}\ t{\isaliteral{29}{\isacharparenright}}}, where
+\cdx{split} is the uncurrying function of type \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}c} defined as
+\begin{center}
+\isa{prod{\isaliteral{5F}{\isacharunderscore}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}c\ p{\isaliteral{2E}{\isachardot}}\ c\ {\isaliteral{28}{\isacharparenleft}}fst\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}snd\ p{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}
+\hfill(\isa{split{\isaliteral{5F}{\isacharunderscore}}def})
+\end{center}
+Pattern matching in
+other variable binding constructs is translated similarly. Thus we need to
+understand how to reason about such constructs.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Theorem Proving%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The most obvious approach is the brute force expansion of \isa{prod{\isaliteral{5F}{\isacharunderscore}}case}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}x{\isaliteral{29}{\isacharparenright}}\ p\ {\isaliteral{3D}{\isacharequal}}\ fst\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+This works well if rewriting with \isa{split{\isaliteral{5F}{\isacharunderscore}}def} finishes the
+proof, as it does above.  But if it does not, you end up with exactly what
+we are trying to avoid: nests of \isa{fst} and \isa{snd}. Thus this
+approach is neither elegant nor very practical in large examples, although it
+can be effective in small ones.
+
+If we consider why this lemma presents a problem, 
+we realize that we need to replace variable~\isa{p} by some pair \isa{{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}}.  Then both sides of the
+equation would simplify to \isa{a} by the simplification rules
+\isa{{\isaliteral{28}{\isacharparenleft}}case\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ of\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ xa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ f\ x\ xa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a\ b} and \isa{fst\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a}.  
+To reason about tuple patterns requires some way of
+converting a variable of product type into a pair.
+In case of a subterm of the form \isa{case\ p\ of\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ xa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ f\ x\ xa} this is easy: the split
+rule \isa{split{\isaliteral{5F}{\isacharunderscore}}split} replaces \isa{p} by a pair:%
+\index{*split (method)}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}y{\isaliteral{29}{\isacharparenright}}\ p\ {\isaliteral{3D}{\isacharequal}}\ snd\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}split\ split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ p\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ snd\ p%
+\end{isabelle}
+This subgoal is easily proved by simplification. Thus we could have combined
+simplification and splitting in one command that proves the goal outright:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ split{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Let us look at a second example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}let\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ p\ in\ fst\ p\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ Let{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ case\ p\ of\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ fst\ p\ {\isaliteral{3D}{\isacharequal}}\ x%
+\end{isabelle}
+A paired \isa{let} reduces to a paired $\lambda$-abstraction, which
+can be split as above. The same is true for paired set comprehension:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}p\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}y{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ fst\ p\ {\isaliteral{3D}{\isacharequal}}\ snd\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+\ simp%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}case\ p\ of\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ xa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ x\ {\isaliteral{3D}{\isacharequal}}\ xa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ fst\ p\ {\isaliteral{3D}{\isacharequal}}\ snd\ p%
+\end{isabelle}
+Again, simplification produces a term suitable for \isa{split{\isaliteral{5F}{\isacharunderscore}}split}
+as above. If you are worried about the strange form of the premise:
+\isa{split\ {\isaliteral{28}{\isacharparenleft}}op\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{29}{\isacharparenright}}} is short for \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y}.
+The same proof procedure works for%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}p\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}y{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ fst\ p\ {\isaliteral{3D}{\isacharequal}}\ snd\ p{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+\noindent
+except that we now have to use \isa{split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{5F}{\isacharunderscore}}asm}, because
+\isa{prod{\isaliteral{5F}{\isacharunderscore}}case} occurs in the assumptions.
+
+However, splitting \isa{prod{\isaliteral{5F}{\isacharunderscore}}case} is not always a solution, as no \isa{prod{\isaliteral{5F}{\isacharunderscore}}case}
+may be present in the goal. Consider the following function:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{primrec}\isamarkupfalse%
+\ swap\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}swap\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}y{\isaliteral{2C}{\isacharcomma}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\noindent
+Note that the above \isacommand{primrec} definition is admissible
+because \isa{{\isaliteral{5C3C74696D65733E}{\isasymtimes}}} is a datatype. When we now try to prove%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}swap{\isaliteral{28}{\isacharparenleft}}swap\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+\noindent
+simplification will do nothing, because the defining equation for
+\isa{swap} expects a pair. Again, we need to turn \isa{p}
+into a pair first, but this time there is no \isa{prod{\isaliteral{5F}{\isacharunderscore}}case} in sight.
+The only thing we can do is to split the term by hand:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}case{\isaliteral{5F}{\isacharunderscore}}tac\ p{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\noindent
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ b{\isaliteral{2E}{\isachardot}}\ p\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ swap\ {\isaliteral{28}{\isacharparenleft}}swap\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ p%
+\end{isabelle}
+Again, \methdx{case_tac} is applicable because \isa{{\isaliteral{5C3C74696D65733E}{\isasymtimes}}} is a datatype.
+The subgoal is easily proved by \isa{simp}.
+
+Splitting by \isa{case{\isaliteral{5F}{\isacharunderscore}}tac} also solves the previous examples and may thus
+appear preferable to the more arcane methods introduced first. However, see
+the warning about \isa{case{\isaliteral{5F}{\isacharunderscore}}tac} in \S\ref{sec:struct-ind-case}.
+
+Alternatively, you can split \emph{all} \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}}-quantified variables
+in a goal with the rewrite rule \isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all}:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}p\ q{\isaliteral{2E}{\isachardot}}\ swap{\isaliteral{28}{\isacharparenleft}}swap\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ q\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ q{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\noindent
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ b\ aa\ ba{\isaliteral{2E}{\isachardot}}\ swap\ {\isaliteral{28}{\isacharparenleft}}swap\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}aa{\isaliteral{2C}{\isacharcomma}}\ ba{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}aa{\isaliteral{2C}{\isacharcomma}}\ ba{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+Note that we have intentionally included only \isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all}
+in the first simplification step, and then we simplify again. 
+This time the reason was not merely
+pedagogical:
+\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all} may interfere with other functions
+of the simplifier.
+The following command could fail (here it does not)
+where two separate \isa{simp} applications succeed.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+Finally, the simplifier automatically splits all \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}} and
+\isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}}-quantified variables:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}p{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}q{\isaliteral{2E}{\isachardot}}\ swap\ p\ {\isaliteral{3D}{\isacharequal}}\ swap\ q{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ simp%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\noindent
+To turn off this automatic splitting, disable the
+responsible simplification rules:
+\begin{center}
+\isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}a\ b{\isaliteral{2E}{\isachardot}}\ P\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}
+\hfill
+(\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}All})\\
+\isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}a\ b{\isaliteral{2E}{\isachardot}}\ P\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}
+\hfill
+(\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}Ex})
+\end{center}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End: