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\begin{isabellebody}%
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\def\isabellecontext{Pairs}%
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%
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\isadelimtheory
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\endisadelimtheory
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%
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\isatagtheory
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupsection{Pairs and Tuples%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\label{sec:products}
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Ordered pairs were already introduced in \S\ref{sec:pairs}, but only with a minimal
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repertoire of operations: pairing and the two projections \isa{fst} and
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\isa{snd}. In any non-trivial application of pairs you will find that this
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quickly leads to unreadable nests of projections. This
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section introduces syntactic sugar to overcome this
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problem: pattern matching with tuples.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Pattern Matching with Tuples%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Tuples may be used as patterns in $\lambda$-abstractions,
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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,
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tuple patterns can be used in most variable binding constructs,
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and they can be nested. Here are
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some typical examples:
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\begin{quote}
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\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}}}\\
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\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}\\
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\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}\\
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\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}}}\\
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\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}}}
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\end{quote}
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The intuitive meanings of these expressions should be obvious.
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Unfortunately, we need to know in more detail what the notation really stands
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for once we have to reason about it. Abstraction
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over pairs and tuples is merely a convenient shorthand for a more complex
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internal representation. Thus the internal and external form of a term may
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differ, which can affect proofs. If you want to avoid this complication,
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stick to \isa{fst} and \isa{snd} and write \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}p{\isaliteral{2E}{\isachardot}}\ fst\ p\ {\isaliteral{2B}{\isacharplus}}\ snd\ p}
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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
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denote the same function.
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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
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\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
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\begin{center}
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\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}}}
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\hfill(\isa{split{\isaliteral{5F}{\isacharunderscore}}def})
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\end{center}
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Pattern matching in
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other variable binding constructs is translated similarly. Thus we need to
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understand how to reason about such constructs.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Theorem Proving%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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The most obvious approach is the brute force expansion of \isa{prod{\isaliteral{5F}{\isacharunderscore}}case}:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ {\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
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{by}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\begin{isamarkuptext}%
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\noindent
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This works well if rewriting with \isa{split{\isaliteral{5F}{\isacharunderscore}}def} finishes the
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proof, as it does above. But if it does not, you end up with exactly what
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we are trying to avoid: nests of \isa{fst} and \isa{snd}. Thus this
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approach is neither elegant nor very practical in large examples, although it
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can be effective in small ones.
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If we consider why this lemma presents a problem,
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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
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equation would simplify to \isa{a} by the simplification rules
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\isa{prod{\isaliteral{5F}{\isacharunderscore}}case\ f\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\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}.
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To reason about tuple patterns requires some way of
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converting a variable of product type into a pair.
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In case of a subterm of the form \isa{prod{\isaliteral{5F}{\isacharunderscore}}case\ f\ p} this is easy: the split
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rule \isa{split{\isaliteral{5F}{\isacharunderscore}}split} replaces \isa{p} by a pair:%
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\index{*split (method)}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ {\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
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}split\ split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
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\begin{isamarkuptxt}%
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\begin{isabelle}%
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\ {\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%
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\end{isabelle}
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This subgoal is easily proved by simplification. Thus we could have combined
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simplification and splitting in one command that proves the goal outright:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{by}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}simp\ split{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\begin{isamarkuptext}%
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Let us look at a second example:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ {\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
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{apply}\isamarkupfalse%
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{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ Let{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
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\begin{isamarkuptxt}%
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\begin{isabelle}%
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\ {\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%
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\end{isabelle}
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A paired \isa{let} reduces to a paired $\lambda$-abstraction, which
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can be split as above. The same is true for paired set comprehension:%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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\isacommand{lemma}\isamarkupfalse%
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\ {\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
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{apply}\isamarkupfalse%
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\ simp%
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\begin{isamarkuptxt}%
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\begin{isabelle}%
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ prod{\isaliteral{5F}{\isacharunderscore}}case\ op\ {\isaliteral{3D}{\isacharequal}}\ p\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ fst\ p\ {\isaliteral{3D}{\isacharequal}}\ snd\ p%
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\end{isabelle}
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Again, simplification produces a term suitable for \isa{split{\isaliteral{5F}{\isacharunderscore}}split}
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as above. If you are worried about the strange form of the premise:
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\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}.
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The same proof procedure works for%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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\isacommand{lemma}\isamarkupfalse%
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\ {\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}}%
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\isadelimproof
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%
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\endisadelimproof
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%
|
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\isatagproof
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%
|
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\begin{isamarkuptxt}%
|
|
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\noindent
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except that we now have to use \isa{split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{5F}{\isacharunderscore}}asm}, because
|
|
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\isa{prod{\isaliteral{5F}{\isacharunderscore}}case} occurs in the assumptions.
|
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|
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|
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However, splitting \isa{prod{\isaliteral{5F}{\isacharunderscore}}case} is not always a solution, as no \isa{prod{\isaliteral{5F}{\isacharunderscore}}case}
|
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|
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may be present in the goal. Consider the following function:%
|
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\end{isamarkuptxt}%
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\isamarkuptrue%
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%
|
|
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\endisatagproof
|
|
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{\isafoldproof}%
|
|
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%
|
|
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\isadelimproof
|
|
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%
|
|
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\endisadelimproof
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|
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\isacommand{primrec}\isamarkupfalse%
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\ 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}}%
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\begin{isamarkuptext}%
|
|
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\noindent
|
|
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Note that the above \isacommand{primrec} definition is admissible
|
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|
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because \isa{{\isaliteral{5C3C74696D65733E}{\isasymtimes}}} is a datatype. When we now try to prove%
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|
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\end{isamarkuptext}%
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|
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\isamarkuptrue%
|
|
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\isacommand{lemma}\isamarkupfalse%
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|
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\ {\isaliteral{22}{\isachardoublequoteopen}}swap{\isaliteral{28}{\isacharparenleft}}swap\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}%
|
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|
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\isadelimproof
|
|
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%
|
|
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\endisadelimproof
|
|
249 |
%
|
|
250 |
\isatagproof
|
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|
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%
|
|
252 |
\begin{isamarkuptxt}%
|
|
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\noindent
|
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|
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simplification will do nothing, because the defining equation for
|
|
255 |
\isa{swap} expects a pair. Again, we need to turn \isa{p}
|
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|
256 |
into a pair first, but this time there is no \isa{prod{\isaliteral{5F}{\isacharunderscore}}case} in sight.
|
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|
257 |
The only thing we can do is to split the term by hand:%
|
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|
258 |
\end{isamarkuptxt}%
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|
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\isamarkuptrue%
|
|
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\isacommand{apply}\isamarkupfalse%
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|
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{\isaliteral{28}{\isacharparenleft}}case{\isaliteral{5F}{\isacharunderscore}}tac\ p{\isaliteral{29}{\isacharparenright}}%
|
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|
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\begin{isamarkuptxt}%
|
|
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\noindent
|
|
264 |
\begin{isabelle}%
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|
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\ {\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%
|
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|
266 |
\end{isabelle}
|
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|
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Again, \methdx{case_tac} is applicable because \isa{{\isaliteral{5C3C74696D65733E}{\isasymtimes}}} is a datatype.
|
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|
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The subgoal is easily proved by \isa{simp}.
|
|
269 |
|
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|
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Splitting by \isa{case{\isaliteral{5F}{\isacharunderscore}}tac} also solves the previous examples and may thus
|
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|
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appear preferable to the more arcane methods introduced first. However, see
|
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|
272 |
the warning about \isa{case{\isaliteral{5F}{\isacharunderscore}}tac} in \S\ref{sec:struct-ind-case}.
|
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|
273 |
|
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|
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Alternatively, you can split \emph{all} \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}}-quantified variables
|
|
275 |
in a goal with the rewrite rule \isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all}:%
|
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|
276 |
\end{isamarkuptxt}%
|
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|
277 |
\isamarkuptrue%
|
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|
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%
|
|
279 |
\endisatagproof
|
|
280 |
{\isafoldproof}%
|
|
281 |
%
|
|
282 |
\isadelimproof
|
|
283 |
%
|
|
284 |
\endisadelimproof
|
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|
285 |
\isacommand{lemma}\isamarkupfalse%
|
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|
286 |
\ {\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
|
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|
287 |
%
|
|
288 |
\isadelimproof
|
|
289 |
%
|
|
290 |
\endisadelimproof
|
|
291 |
%
|
|
292 |
\isatagproof
|
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|
293 |
\isacommand{apply}\isamarkupfalse%
|
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|
294 |
{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
|
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|
295 |
\begin{isamarkuptxt}%
|
|
296 |
\noindent
|
|
297 |
\begin{isabelle}%
|
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|
298 |
\ {\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}}%
|
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|
299 |
\end{isabelle}%
|
|
300 |
\end{isamarkuptxt}%
|
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|
301 |
\isamarkuptrue%
|
|
302 |
\isacommand{apply}\isamarkupfalse%
|
|
303 |
\ simp\isanewline
|
|
304 |
\isacommand{done}\isamarkupfalse%
|
|
305 |
%
|
17056
|
306 |
\endisatagproof
|
|
307 |
{\isafoldproof}%
|
|
308 |
%
|
|
309 |
\isadelimproof
|
|
310 |
%
|
|
311 |
\endisadelimproof
|
11866
|
312 |
%
|
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|
313 |
\begin{isamarkuptext}%
|
|
314 |
\noindent
|
40406
|
315 |
Note that we have intentionally included only \isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all}
|
11494
|
316 |
in the first simplification step, and then we simplify again.
|
|
317 |
This time the reason was not merely
|
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|
318 |
pedagogical:
|
40406
|
319 |
\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all} may interfere with other functions
|
11494
|
320 |
of the simplifier.
|
|
321 |
The following command could fail (here it does not)
|
|
322 |
where two separate \isa{simp} applications succeed.%
|
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|
323 |
\end{isamarkuptext}%
|
17175
|
324 |
\isamarkuptrue%
|
17056
|
325 |
%
|
|
326 |
\isadelimproof
|
|
327 |
%
|
|
328 |
\endisadelimproof
|
|
329 |
%
|
|
330 |
\isatagproof
|
17175
|
331 |
\isacommand{apply}\isamarkupfalse%
|
40406
|
332 |
{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
|
17056
|
333 |
\endisatagproof
|
|
334 |
{\isafoldproof}%
|
|
335 |
%
|
|
336 |
\isadelimproof
|
|
337 |
%
|
|
338 |
\endisadelimproof
|
11866
|
339 |
%
|
10560
|
340 |
\begin{isamarkuptext}%
|
|
341 |
\noindent
|
40406
|
342 |
Finally, the simplifier automatically splits all \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}} and
|
|
343 |
\isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}}-quantified variables:%
|
10560
|
344 |
\end{isamarkuptext}%
|
17175
|
345 |
\isamarkuptrue%
|
|
346 |
\isacommand{lemma}\isamarkupfalse%
|
40406
|
347 |
\ {\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
|
17056
|
348 |
%
|
|
349 |
\isadelimproof
|
|
350 |
%
|
|
351 |
\endisadelimproof
|
|
352 |
%
|
|
353 |
\isatagproof
|
17175
|
354 |
\isacommand{by}\isamarkupfalse%
|
|
355 |
\ simp%
|
17056
|
356 |
\endisatagproof
|
|
357 |
{\isafoldproof}%
|
|
358 |
%
|
|
359 |
\isadelimproof
|
|
360 |
%
|
|
361 |
\endisadelimproof
|
11866
|
362 |
%
|
10560
|
363 |
\begin{isamarkuptext}%
|
|
364 |
\noindent
|
27027
|
365 |
To turn off this automatic splitting, disable the
|
10560
|
366 |
responsible simplification rules:
|
|
367 |
\begin{center}
|
40406
|
368 |
\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}}}
|
10560
|
369 |
\hfill
|
40406
|
370 |
(\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}All})\\
|
|
371 |
\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}}}
|
10560
|
372 |
\hfill
|
40406
|
373 |
(\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}Ex})
|
10560
|
374 |
\end{center}%
|
|
375 |
\end{isamarkuptext}%
|
17175
|
376 |
\isamarkuptrue%
|
17056
|
377 |
%
|
|
378 |
\isadelimtheory
|
|
379 |
%
|
|
380 |
\endisadelimtheory
|
|
381 |
%
|
|
382 |
\isatagtheory
|
|
383 |
%
|
|
384 |
\endisatagtheory
|
|
385 |
{\isafoldtheory}%
|
|
386 |
%
|
|
387 |
\isadelimtheory
|
|
388 |
%
|
|
389 |
\endisadelimtheory
|
10560
|
390 |
\end{isabellebody}%
|
|
391 |
%%% Local Variables:
|
|
392 |
%%% mode: latex
|
|
393 |
%%% TeX-master: "root"
|
|
394 |
%%% End:
|