MAYBE

Problem:
 app'(app'(minus(),x),0()) -> x
 app'(app'(minus(),app'(s(),x)),app'(s(),y)) -> app'(app'(minus(),x),y)
 app'(app'(minus(),app'(app'(minus(),x),y)),z) -> app'(app'(minus(),x),app'(app'(plus(),y),z))
 app'(app'(quot(),0()),app'(s(),y)) -> 0()
 app'(app'(quot(),app'(s(),x)),app'(s(),y)) ->
 app'(s(),app'(app'(quot(),app'(app'(minus(),x),y)),app'(s(),y)))
 app'(app'(plus(),0()),y) -> y
 app'(app'(plus(),app'(s(),x)),y) -> app'(s(),app'(app'(plus(),x),y))
 app'(app'(app(),nil()),k) -> k
 app'(app'(app(),l),nil()) -> l
 app'(app'(app(),app'(app'(cons(),x),l)),k) -> app'(app'(cons(),x),app'(app'(app(),l),k))
 app'(sum(),app'(app'(cons(),x),nil())) -> app'(app'(cons(),x),nil())
 app'(sum(),app'(app'(cons(),x),app'(app'(cons(),y),l))) ->
 app'(sum(),app'(app'(cons(),app'(app'(plus(),x),y)),l))
 app'(sum(),app'(app'(app(),l),app'(app'(cons(),x),app'(app'(cons(),y),k)))) ->
 app'(sum(),app'(app'(app(),l),app'(sum(),app'(app'(cons(),x),app'(app'(cons(),y),k)))))
 app'(app'(map(),f),nil()) -> nil()
 app'(app'(map(),f),app'(app'(cons(),x),xs)) -> app'(app'(cons(),app'(f,x)),app'(app'(map(),f),xs))
 app'(app'(filter(),f),nil()) -> nil()
 app'(app'(filter(),f),app'(app'(cons(),x),xs)) ->
 app'(app'(app'(app'(filter2(),app'(f,x)),f),x),xs)
 app'(app'(app'(app'(filter2(),true()),f),x),xs) -> app'(app'(cons(),x),app'(app'(filter(),f),xs))
 app'(app'(app'(app'(filter2(),false()),f),x),xs) -> app'(app'(filter(),f),xs)

Proof:
 Open