(*<*)
theory Tree imports Main begin
(*>*)
text\<open>\noindent
Define the datatype of \rmindex{binary trees}:
\<close>
datatype 'a tree = Tip | Node "'a tree" 'a "'a tree"(*<*)
primrec mirror :: "'a tree \<Rightarrow> 'a tree" where
"mirror Tip = Tip" |
"mirror (Node l x r) = Node (mirror r) x (mirror l)"(*>*)
text\<open>\noindent
Define a function \<^term>\<open>mirror\<close> that mirrors a binary tree
by swapping subtrees recursively. Prove
\<close>
lemma mirror_mirror: "mirror(mirror t) = t"
(*<*)
apply(induct_tac t)
by(auto)
primrec flatten :: "'a tree => 'a list" where
"flatten Tip = []" |
"flatten (Node l x r) = flatten l @ [x] @ flatten r"
(*>*)
text\<open>\noindent
Define a function \<^term>\<open>flatten\<close> that flattens a tree into a list
by traversing it in infix order. Prove
\<close>
lemma "flatten(mirror t) = rev(flatten t)"
(*<*)
apply(induct_tac t)
by(auto)
end
(*>*)