--- a/doc-src/TutorialI/document/Fundata.tex Tue Aug 28 13:15:15 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,115 +0,0 @@
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-\begin{isabellebody}%
-\def\isabellecontext{Fundata}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isacommand{datatype}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}i{\isaliteral{29}{\isacharparenright}}bigtree\ {\isaliteral{3D}{\isacharequal}}\ Tip\ {\isaliteral{7C}{\isacharbar}}\ Br\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}i{\isaliteral{29}{\isacharparenright}}bigtree{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent
-Parameter \isa{{\isaliteral{27}{\isacharprime}}a} is the type of values stored in
-the \isa{Br}anches of the tree, whereas \isa{{\isaliteral{27}{\isacharprime}}i} is the index
-type over which the tree branches. If \isa{{\isaliteral{27}{\isacharprime}}i} is instantiated to
-\isa{bool}, the result is a binary tree; if it is instantiated to
-\isa{nat}, we have an infinitely branching tree because each node
-has as many subtrees as there are natural numbers. How can we possibly
-write down such a tree? Using functional notation! For example, the term
-\begin{isabelle}%
-\ \ \ \ \ Br\ {\isadigit{0}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}i{\isaliteral{2E}{\isachardot}}\ Br\ i\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}n{\isaliteral{2E}{\isachardot}}\ Tip{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-of type \isa{{\isaliteral{28}{\isacharparenleft}}nat{\isaliteral{2C}{\isacharcomma}}\ nat{\isaliteral{29}{\isacharparenright}}\ bigtree} is the tree whose
-root is labeled with 0 and whose $i$th subtree is labeled with $i$ and
-has merely \isa{Tip}s as further subtrees.
-
-Function \isa{map{\isaliteral{5F}{\isacharunderscore}}bt} applies a function to all labels in a \isa{bigtree}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\ map{\isaliteral{5F}{\isacharunderscore}}bt\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}i{\isaliteral{29}{\isacharparenright}}bigtree\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}b{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}i{\isaliteral{29}{\isacharparenright}}bigtree{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-{\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ Tip\ \ \ \ \ \ {\isaliteral{3D}{\isacharequal}}\ Tip{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-{\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ {\isaliteral{28}{\isacharparenleft}}Br\ a\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Br\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}i{\isaliteral{2E}{\isachardot}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ {\isaliteral{28}{\isacharparenleft}}F\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent This is a valid \isacommand{primrec} definition because the
-recursive calls of \isa{map{\isaliteral{5F}{\isacharunderscore}}bt} involve only subtrees of
-\isa{F}, which is itself a subterm of the left-hand side. Thus termination
-is assured. The seasoned functional programmer might try expressing
-\isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}i{\isaliteral{2E}{\isachardot}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ {\isaliteral{28}{\isacharparenleft}}F\ i{\isaliteral{29}{\isacharparenright}}} as \isa{map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ F}, which Isabelle
-however will reject. Applying \isa{map{\isaliteral{5F}{\isacharunderscore}}bt} to only one of its arguments
-makes the termination proof less obvious.
-
-The following lemma has a simple proof by induction:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}bt\ {\isaliteral{28}{\isacharparenleft}}g\ o\ f{\isaliteral{29}{\isacharparenright}}\ T\ {\isaliteral{3D}{\isacharequal}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ g\ {\isaliteral{28}{\isacharparenleft}}map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ T{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ T{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent
-Because of the function type, the proof state after induction looks unusual.
-Notice the quantified induction hypothesis:
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ {\isaliteral{28}{\isacharparenleft}}g\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isaliteral{29}{\isacharparenright}}\ Tip\ {\isaliteral{3D}{\isacharequal}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ g\ {\isaliteral{28}{\isacharparenleft}}map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ Tip{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ F{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ {\isaliteral{28}{\isacharparenleft}}g\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}F\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ g\ {\isaliteral{28}{\isacharparenleft}}map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ {\isaliteral{28}{\isacharparenleft}}F\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ F{\isaliteral{2E}{\isachardot}}\ }map{\isaliteral{5F}{\isacharunderscore}}bt\ {\isaliteral{28}{\isacharparenleft}}g\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Br\ a\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ map{\isaliteral{5F}{\isacharunderscore}}bt\ g\ {\isaliteral{28}{\isacharparenleft}}map{\isaliteral{5F}{\isacharunderscore}}bt\ f\ {\isaliteral{28}{\isacharparenleft}}Br\ a\ F{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
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-\endisadelimproof
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-\isadelimtheory
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-\endisadelimtheory
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-\isatagtheory
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-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
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-\endisadelimtheory
-\end{isabellebody}%
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