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(*<*)


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theory Fundata = Main:


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(*>*)

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datatype ('a,'i)bigtree = Tip  Branch 'a "'i \<Rightarrow> ('a,'i)bigtree"

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text{*\noindent


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Parameter @{typ"'a"} is the type of values stored in


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the @{term"Branch"}es of the tree, whereas @{typ"'i"} is the index


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type over which the tree branches. If @{typ"'i"} is instantiated to


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@{typ"bool"}, the result is a binary tree; if it is instantiated to


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@{typ"nat"}, we have an infinitely branching tree because each node

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has as many subtrees as there are natural numbers. How can we possibly

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write down such a tree? Using functional notation! For example, the term


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@{term[display]"Branch 0 (%i. Branch i (%n. Tip))"}


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of type @{typ"(nat,nat)bigtree"} is the tree whose

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root is labeled with 0 and whose $i$th subtree is labeled with $i$ and

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has merely @{term"Tip"}s as further subtrees.

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Function @{term"map_bt"} applies a function to all labels in a @{text"bigtree"}:

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*}


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consts map_bt :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a,'i)bigtree \<Rightarrow> ('b,'i)bigtree"

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primrec


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"map_bt f Tip = Tip"

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"map_bt f (Branch a F) = Branch (f a) (\<lambda>i. map_bt f (F i))"

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text{*\noindent This is a valid \isacommand{primrec} definition because the

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recursive calls of @{term"map_bt"} involve only subtrees obtained from


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@{term"F"}, i.e.\ the lefthand side. Thus termination is assured. The

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seasoned functional programmer might have written @{term"map_bt f o F"}


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instead of @{term"%i. map_bt f (F i)"}, but the former is not accepted by

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Isabelle because the termination proof is not as obvious since

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@{term"map_bt"} is only partially applied.

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The following lemma has a canonical proof *}


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lemma "map_bt (g o f) T = map_bt g (map_bt f T)";

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apply(induct_tac T, simp_all)


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done

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text{*\noindent

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%apply(induct_tac T);


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%pr(latex xsymbols symbols)

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but it is worth taking a look at the proof state after the induction step


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to understand what the presence of the function type entails:

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\begin{isabelle}

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\ \isadigit{1}{\isachardot}\ map{\isacharunderscore}bt\ {\isacharparenleft}g\ {\isasymcirc}\ f{\isacharparenright}\ Tip\ {\isacharequal}\ map{\isacharunderscore}bt\ g\ {\isacharparenleft}map{\isacharunderscore}bt\ f\ Tip{\isacharparenright}\isanewline


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\ \isadigit{2}{\isachardot}\ {\isasymAnd}a\ F{\isachardot}\isanewline


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\ \ \ \ \ \ \ {\isasymforall}x{\isachardot}\ map{\isacharunderscore}bt\ {\isacharparenleft}g\ {\isasymcirc}\ f{\isacharparenright}\ {\isacharparenleft}F\ x{\isacharparenright}\ {\isacharequal}\ map{\isacharunderscore}bt\ g\ {\isacharparenleft}map{\isacharunderscore}bt\ f\ {\isacharparenleft}F\ x{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline


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\ \ \ \ \ \ \ map{\isacharunderscore}bt\ {\isacharparenleft}g\ {\isasymcirc}\ f{\isacharparenright}\ {\isacharparenleft}Branch\ a\ F{\isacharparenright}\ {\isacharequal}\ map{\isacharunderscore}bt\ g\ {\isacharparenleft}map{\isacharunderscore}bt\ f\ {\isacharparenleft}Branch\ a\ F{\isacharparenright}{\isacharparenright}

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\end{isabelle}

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*}


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(*<*)


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end


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(*>*)
