doc-src/TutorialI/ToyList2/ToyList.thy

abstracted lemmas

theory ToyList
imports Datatype
begin

datatype 'a list = Nil                          ("[]")
                 | Cons 'a "'a list"            (infixr "#" 65)

(* This is the append function: *)
primrec app :: "'a list => 'a list => 'a list"  (infixr "@" 65)
where
"[] @ ys       = ys" |
"(x # xs) @ ys = x # (xs @ ys)"

primrec rev :: "'a list => 'a list" where
"rev []        = []" |
"rev (x # xs)  = (rev xs) @ (x # [])"
lemma app_Nil2 [simp]: "xs @ [] = xs"
apply(induct_tac xs)
apply(auto)
done

lemma app_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
apply(induct_tac xs)
apply(auto)
done

lemma rev_app [simp]: "rev(xs @ ys) = (rev ys) @ (rev xs)"
apply(induct_tac xs)
apply(auto)
done

theorem rev_rev [simp]: "rev(rev xs) = xs"
apply(induct_tac xs)
apply(auto)
done

end