Theory Tree2

(*<*)
theory Tree2 imports Tree begin
(*>*)

text‹\noindent In Exercise~\ref{ex:Tree} we defined a function
\<^term>‹flatten› from trees to lists. The straightforward version of
\<^term>‹flatten› is based on ‹@› and is thus, like \<^term>‹rev›,
quadratic. A linear time version of \<^term>‹flatten› again reqires an extra
argument, the accumulator. Define›
(*<*)primrec(*>*)flatten2 :: "'a tree ⇒ 'a list ⇒ 'a list"(*<*)where
"flatten2 Tip xs = xs" |
"flatten2 (Node l x r) xs = flatten2 l (x#(flatten2 r xs))"
(*>*)

text‹\noindent and prove›
(*<*)
lemma [simp]: "∀xs. flatten2 t xs = flatten t @ xs"
apply(induct_tac t)
by(auto)
(*>*)
lemma "flatten2 t [] = flatten t"
(*<*)
by(simp)

end
(*>*)