1.1 --- a/doc-src/TutorialI/Types/document/Pairs.tex Thu Jul 26 16:08:16 2012 +0200
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,394 +0,0 @@
1.4 -%
1.5 -\begin{isabellebody}%
1.6 -\def\isabellecontext{Pairs}%
1.7 -%
1.8 -\isadelimtheory
1.9 -%
1.10 -\endisadelimtheory
1.11 -%
1.12 -\isatagtheory
1.13 -%
1.14 -\endisatagtheory
1.15 -{\isafoldtheory}%
1.16 -%
1.17 -\isadelimtheory
1.18 -%
1.19 -\endisadelimtheory
1.20 -%
1.21 -\isamarkupsection{Pairs and Tuples%
1.22 -}
1.23 -\isamarkuptrue%
1.24 -%
1.25 -\begin{isamarkuptext}%
1.26 -\label{sec:products}
1.27 -Ordered pairs were already introduced in \S\ref{sec:pairs}, but only with a minimal
1.28 -repertoire of operations: pairing and the two projections \isa{fst} and
1.29 -\isa{snd}. In any non-trivial application of pairs you will find that this
1.30 -quickly leads to unreadable nests of projections. This
1.31 -section introduces syntactic sugar to overcome this
1.32 -problem: pattern matching with tuples.%
1.33 -\end{isamarkuptext}%
1.34 -\isamarkuptrue%
1.35 -%
1.36 -\isamarkupsubsection{Pattern Matching with Tuples%
1.37 -}
1.38 -\isamarkuptrue%
1.39 -%
1.40 -\begin{isamarkuptext}%
1.41 -Tuples may be used as patterns in $\lambda$-abstractions,
1.42 -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,
1.43 -tuple patterns can be used in most variable binding constructs,
1.44 -and they can be nested. Here are
1.45 -some typical examples:
1.46 -\begin{quote}
1.47 -\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}}}\\
1.48 -\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}\\
1.49 -\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}\\
1.50 -\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}}}\\
1.51 -\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}}}
1.52 -\end{quote}
1.53 -The intuitive meanings of these expressions should be obvious.
1.54 -Unfortunately, we need to know in more detail what the notation really stands
1.55 -for once we have to reason about it. Abstraction
1.56 -over pairs and tuples is merely a convenient shorthand for a more complex
1.57 -internal representation. Thus the internal and external form of a term may
1.58 -differ, which can affect proofs. If you want to avoid this complication,
1.59 -stick to \isa{fst} and \isa{snd} and write \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}p{\isaliteral{2E}{\isachardot}}\ fst\ p\ {\isaliteral{2B}{\isacharplus}}\ snd\ p}
1.60 -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
1.61 -denote the same function.
1.62 -
1.63 -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
1.64 -\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
1.65 -\begin{center}
1.66 -\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}}}
1.67 -\hfill(\isa{split{\isaliteral{5F}{\isacharunderscore}}def})
1.68 -\end{center}
1.69 -Pattern matching in
1.70 -other variable binding constructs is translated similarly. Thus we need to
1.71 -understand how to reason about such constructs.%
1.72 -\end{isamarkuptext}%
1.73 -\isamarkuptrue%
1.74 -%
1.75 -\isamarkupsubsection{Theorem Proving%
1.76 -}
1.77 -\isamarkuptrue%
1.78 -%
1.79 -\begin{isamarkuptext}%
1.80 -The most obvious approach is the brute force expansion of \isa{prod{\isaliteral{5F}{\isacharunderscore}}case}:%
1.81 -\end{isamarkuptext}%
1.82 -\isamarkuptrue%
1.83 -\isacommand{lemma}\isamarkupfalse%
1.84 -\ {\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
1.85 -%
1.86 -\isadelimproof
1.87 -%
1.88 -\endisadelimproof
1.89 -%
1.90 -\isatagproof
1.91 -\isacommand{by}\isamarkupfalse%
1.92 -{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
1.93 -\endisatagproof
1.94 -{\isafoldproof}%
1.95 -%
1.96 -\isadelimproof
1.97 -%
1.98 -\endisadelimproof
1.99 -%
1.100 -\begin{isamarkuptext}%
1.101 -\noindent
1.102 -This works well if rewriting with \isa{split{\isaliteral{5F}{\isacharunderscore}}def} finishes the
1.103 -proof, as it does above. But if it does not, you end up with exactly what
1.104 -we are trying to avoid: nests of \isa{fst} and \isa{snd}. Thus this
1.105 -approach is neither elegant nor very practical in large examples, although it
1.106 -can be effective in small ones.
1.107 -
1.108 -If we consider why this lemma presents a problem,
1.109 -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
1.110 -equation would simplify to \isa{a} by the simplification rules
1.111 -\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}.
1.112 -To reason about tuple patterns requires some way of
1.113 -converting a variable of product type into a pair.
1.114 -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
1.115 -rule \isa{split{\isaliteral{5F}{\isacharunderscore}}split} replaces \isa{p} by a pair:%
1.116 -\index{*split (method)}%
1.117 -\end{isamarkuptext}%
1.118 -\isamarkuptrue%
1.119 -\isacommand{lemma}\isamarkupfalse%
1.120 -\ {\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
1.121 -%
1.122 -\isadelimproof
1.123 -%
1.124 -\endisadelimproof
1.125 -%
1.126 -\isatagproof
1.127 -\isacommand{apply}\isamarkupfalse%
1.128 -{\isaliteral{28}{\isacharparenleft}}split\ split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
1.129 -\begin{isamarkuptxt}%
1.130 -\begin{isabelle}%
1.131 -\ {\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%
1.132 -\end{isabelle}
1.133 -This subgoal is easily proved by simplification. Thus we could have combined
1.134 -simplification and splitting in one command that proves the goal outright:%
1.135 -\end{isamarkuptxt}%
1.136 -\isamarkuptrue%
1.137 -%
1.138 -\endisatagproof
1.139 -{\isafoldproof}%
1.140 -%
1.141 -\isadelimproof
1.142 -%
1.143 -\endisadelimproof
1.144 -%
1.145 -\isadelimproof
1.146 -%
1.147 -\endisadelimproof
1.148 -%
1.149 -\isatagproof
1.150 -\isacommand{by}\isamarkupfalse%
1.151 -{\isaliteral{28}{\isacharparenleft}}simp\ split{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
1.152 -\endisatagproof
1.153 -{\isafoldproof}%
1.154 -%
1.155 -\isadelimproof
1.156 -%
1.157 -\endisadelimproof
1.158 -%
1.159 -\begin{isamarkuptext}%
1.160 -Let us look at a second example:%
1.161 -\end{isamarkuptext}%
1.162 -\isamarkuptrue%
1.163 -\isacommand{lemma}\isamarkupfalse%
1.164 -\ {\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
1.165 -%
1.166 -\isadelimproof
1.167 -%
1.168 -\endisadelimproof
1.169 -%
1.170 -\isatagproof
1.171 -\isacommand{apply}\isamarkupfalse%
1.172 -{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ Let{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
1.173 -\begin{isamarkuptxt}%
1.174 -\begin{isabelle}%
1.175 -\ {\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%
1.176 -\end{isabelle}
1.177 -A paired \isa{let} reduces to a paired $\lambda$-abstraction, which
1.178 -can be split as above. The same is true for paired set comprehension:%
1.179 -\end{isamarkuptxt}%
1.180 -\isamarkuptrue%
1.181 -%
1.182 -\endisatagproof
1.183 -{\isafoldproof}%
1.184 -%
1.185 -\isadelimproof
1.186 -%
1.187 -\endisadelimproof
1.188 -\isacommand{lemma}\isamarkupfalse%
1.189 -\ {\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
1.190 -%
1.191 -\isadelimproof
1.192 -%
1.193 -\endisadelimproof
1.194 -%
1.195 -\isatagproof
1.196 -\isacommand{apply}\isamarkupfalse%
1.197 -\ simp%
1.198 -\begin{isamarkuptxt}%
1.199 -\begin{isabelle}%
1.200 -\ {\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%
1.201 -\end{isabelle}
1.202 -Again, simplification produces a term suitable for \isa{split{\isaliteral{5F}{\isacharunderscore}}split}
1.203 -as above. If you are worried about the strange form of the premise:
1.204 -\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}.
1.205 -The same proof procedure works for%
1.206 -\end{isamarkuptxt}%
1.207 -\isamarkuptrue%
1.208 -%
1.209 -\endisatagproof
1.210 -{\isafoldproof}%
1.211 -%
1.212 -\isadelimproof
1.213 -%
1.214 -\endisadelimproof
1.215 -\isacommand{lemma}\isamarkupfalse%
1.216 -\ {\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}}%
1.217 -\isadelimproof
1.218 -%
1.219 -\endisadelimproof
1.220 -%
1.221 -\isatagproof
1.222 -%
1.223 -\begin{isamarkuptxt}%
1.224 -\noindent
1.225 -except that we now have to use \isa{split{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{5F}{\isacharunderscore}}asm}, because
1.226 -\isa{prod{\isaliteral{5F}{\isacharunderscore}}case} occurs in the assumptions.
1.227 -
1.228 -However, splitting \isa{prod{\isaliteral{5F}{\isacharunderscore}}case} is not always a solution, as no \isa{prod{\isaliteral{5F}{\isacharunderscore}}case}
1.229 -may be present in the goal. Consider the following function:%
1.230 -\end{isamarkuptxt}%
1.231 -\isamarkuptrue%
1.232 -%
1.233 -\endisatagproof
1.234 -{\isafoldproof}%
1.235 -%
1.236 -\isadelimproof
1.237 -%
1.238 -\endisadelimproof
1.239 -\isacommand{primrec}\isamarkupfalse%
1.240 -\ 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}}%
1.241 -\begin{isamarkuptext}%
1.242 -\noindent
1.243 -Note that the above \isacommand{primrec} definition is admissible
1.244 -because \isa{{\isaliteral{5C3C74696D65733E}{\isasymtimes}}} is a datatype. When we now try to prove%
1.245 -\end{isamarkuptext}%
1.246 -\isamarkuptrue%
1.247 -\isacommand{lemma}\isamarkupfalse%
1.248 -\ {\isaliteral{22}{\isachardoublequoteopen}}swap{\isaliteral{28}{\isacharparenleft}}swap\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}%
1.249 -\isadelimproof
1.250 -%
1.251 -\endisadelimproof
1.252 -%
1.253 -\isatagproof
1.254 -%
1.255 -\begin{isamarkuptxt}%
1.256 -\noindent
1.257 -simplification will do nothing, because the defining equation for
1.258 -\isa{swap} expects a pair. Again, we need to turn \isa{p}
1.259 -into a pair first, but this time there is no \isa{prod{\isaliteral{5F}{\isacharunderscore}}case} in sight.
1.260 -The only thing we can do is to split the term by hand:%
1.261 -\end{isamarkuptxt}%
1.262 -\isamarkuptrue%
1.263 -\isacommand{apply}\isamarkupfalse%
1.264 -{\isaliteral{28}{\isacharparenleft}}case{\isaliteral{5F}{\isacharunderscore}}tac\ p{\isaliteral{29}{\isacharparenright}}%
1.265 -\begin{isamarkuptxt}%
1.266 -\noindent
1.267 -\begin{isabelle}%
1.268 -\ {\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%
1.269 -\end{isabelle}
1.270 -Again, \methdx{case_tac} is applicable because \isa{{\isaliteral{5C3C74696D65733E}{\isasymtimes}}} is a datatype.
1.271 -The subgoal is easily proved by \isa{simp}.
1.272 -
1.273 -Splitting by \isa{case{\isaliteral{5F}{\isacharunderscore}}tac} also solves the previous examples and may thus
1.274 -appear preferable to the more arcane methods introduced first. However, see
1.275 -the warning about \isa{case{\isaliteral{5F}{\isacharunderscore}}tac} in \S\ref{sec:struct-ind-case}.
1.276 -
1.277 -Alternatively, you can split \emph{all} \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}}-quantified variables
1.278 -in a goal with the rewrite rule \isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all}:%
1.279 -\end{isamarkuptxt}%
1.280 -\isamarkuptrue%
1.281 -%
1.282 -\endisatagproof
1.283 -{\isafoldproof}%
1.284 -%
1.285 -\isadelimproof
1.286 -%
1.287 -\endisadelimproof
1.288 -\isacommand{lemma}\isamarkupfalse%
1.289 -\ {\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
1.290 -%
1.291 -\isadelimproof
1.292 -%
1.293 -\endisadelimproof
1.294 -%
1.295 -\isatagproof
1.296 -\isacommand{apply}\isamarkupfalse%
1.297 -{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
1.298 -\begin{isamarkuptxt}%
1.299 -\noindent
1.300 -\begin{isabelle}%
1.301 -\ {\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}}%
1.302 -\end{isabelle}%
1.303 -\end{isamarkuptxt}%
1.304 -\isamarkuptrue%
1.305 -\isacommand{apply}\isamarkupfalse%
1.306 -\ simp\isanewline
1.307 -\isacommand{done}\isamarkupfalse%
1.308 -%
1.309 -\endisatagproof
1.310 -{\isafoldproof}%
1.311 -%
1.312 -\isadelimproof
1.313 -%
1.314 -\endisadelimproof
1.315 -%
1.316 -\begin{isamarkuptext}%
1.317 -\noindent
1.318 -Note that we have intentionally included only \isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all}
1.319 -in the first simplification step, and then we simplify again.
1.320 -This time the reason was not merely
1.321 -pedagogical:
1.322 -\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all} may interfere with other functions
1.323 -of the simplifier.
1.324 -The following command could fail (here it does not)
1.325 -where two separate \isa{simp} applications succeed.%
1.326 -\end{isamarkuptext}%
1.327 -\isamarkuptrue%
1.328 -%
1.329 -\isadelimproof
1.330 -%
1.331 -\endisadelimproof
1.332 -%
1.333 -\isatagproof
1.334 -\isacommand{apply}\isamarkupfalse%
1.335 -{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
1.336 -\endisatagproof
1.337 -{\isafoldproof}%
1.338 -%
1.339 -\isadelimproof
1.340 -%
1.341 -\endisadelimproof
1.342 -%
1.343 -\begin{isamarkuptext}%
1.344 -\noindent
1.345 -Finally, the simplifier automatically splits all \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}} and
1.346 -\isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}}-quantified variables:%
1.347 -\end{isamarkuptext}%
1.348 -\isamarkuptrue%
1.349 -\isacommand{lemma}\isamarkupfalse%
1.350 -\ {\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
1.351 -%
1.352 -\isadelimproof
1.353 -%
1.354 -\endisadelimproof
1.355 -%
1.356 -\isatagproof
1.357 -\isacommand{by}\isamarkupfalse%
1.358 -\ simp%
1.359 -\endisatagproof
1.360 -{\isafoldproof}%
1.361 -%
1.362 -\isadelimproof
1.363 -%
1.364 -\endisadelimproof
1.365 -%
1.366 -\begin{isamarkuptext}%
1.367 -\noindent
1.368 -To turn off this automatic splitting, disable the
1.369 -responsible simplification rules:
1.370 -\begin{center}
1.371 -\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}}}
1.372 -\hfill
1.373 -(\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}All})\\
1.374 -\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}}}
1.375 -\hfill
1.376 -(\isa{split{\isaliteral{5F}{\isacharunderscore}}paired{\isaliteral{5F}{\isacharunderscore}}Ex})
1.377 -\end{center}%
1.378 -\end{isamarkuptext}%
1.379 -\isamarkuptrue%
1.380 -%
1.381 -\isadelimtheory
1.382 -%
1.383 -\endisadelimtheory
1.384 -%
1.385 -\isatagtheory
1.386 -%
1.387 -\endisatagtheory
1.388 -{\isafoldtheory}%
1.389 -%
1.390 -\isadelimtheory
1.391 -%
1.392 -\endisadelimtheory
1.393 -\end{isabellebody}%
1.394 -%%% Local Variables:
1.395 -%%% mode: latex
1.396 -%%% TeX-master: "root"
1.397 -%%% End: