theory FP0 = PreList:
datatype 'a list = Nil ("[]")
| Cons 'a "'a list" (infixr "#" 65)
consts app :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
rev :: "'a list \<Rightarrow> 'a list"
primrec
"[] @ ys = ys"
"(x # xs) @ ys = x # (xs @ ys)"
primrec
"rev [] = []"
"rev (x # xs) = (rev xs) @ (x # [])"
theorem rev_rev [simp]: "rev(rev xs) = xs"
(*<*)oops(*>*)
text{*
\begin{exercise}
Define a datatype of binary trees and a function @{term mirror}
that mirrors a binary tree by swapping subtrees recursively. Prove
@{prop"mirror(mirror t) = t"}.
Define a function @{term flatten} that flattens a tree into a list
by traversing it in infix order. Prove
@{prop"flatten(mirror t) = rev(flatten t)"}.
\end{exercise}
*}
end