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