MAYBE

Problem:
 app(p(),0()) -> 0()
 app(p(),app(s(),x)) -> x
 app(app(le(),0()),y) -> true()
 app(app(le(),app(s(),x)),0()) -> false()
 app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
 app(app(minus(),x),y) -> app(app(app(if(),app(app(le(),x),y)),x),y)
 app(app(app(if(),true()),x),y) -> 0()
 app(app(app(if(),false()),x),y) -> app(s(),app(app(minus(),app(p(),x)),y))
 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:
 DP Processor:
  DPs:
   app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
   app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
   app#(app(minus(),x),y) -> app#(le(),x)
   app#(app(minus(),x),y) -> app#(app(le(),x),y)
   app#(app(minus(),x),y) -> app#(if(),app(app(le(),x),y))
   app#(app(minus(),x),y) -> app#(app(if(),app(app(le(),x),y)),x)
   app#(app(minus(),x),y) -> app#(app(app(if(),app(app(le(),x),y)),x),y)
   app#(app(app(if(),false()),x),y) -> app#(p(),x)
   app#(app(app(if(),false()),x),y) -> app#(minus(),app(p(),x))
   app#(app(app(if(),false()),x),y) -> app#(app(minus(),app(p(),x)),y)
   app#(app(app(if(),false()),x),y) -> app#(s(),app(app(minus(),app(p(),x)),y))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
   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#(filter(),f)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
   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#(filter(),f)
   app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
  TRS:
   app(p(),0()) -> 0()
   app(p(),app(s(),x)) -> x
   app(app(le(),0()),y) -> true()
   app(app(le(),app(s(),x)),0()) -> false()
   app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
   app(app(minus(),x),y) -> app(app(app(if(),app(app(le(),x),y)),x),y)
   app(app(app(if(),true()),x),y) -> 0()
   app(app(app(if(),false()),x),y) -> app(s(),app(app(minus(),app(p(),x)),y))
   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)
  SCC Processor:
   #sccs: 1
   #rules: 26
   #arcs: 676/676
   DPs:
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
    app#(app(minus(),x),y) -> app#(le(),x)
    app#(app(minus(),x),y) -> app#(app(le(),x),y)
    app#(app(minus(),x),y) -> app#(if(),app(app(le(),x),y))
    app#(app(minus(),x),y) -> app#(app(if(),app(app(le(),x),y)),x)
    app#(app(minus(),x),y) -> app#(app(app(if(),app(app(le(),x),y)),x),y)
    app#(app(app(if(),false()),x),y) -> app#(p(),x)
    app#(app(app(if(),false()),x),y) -> app#(minus(),app(p(),x))
    app#(app(app(if(),false()),x),y) -> app#(app(minus(),app(p(),x)),y)
    app#(app(app(if(),false()),x),y) -> app#(s(),app(app(minus(),app(p(),x)),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    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#(filter(),f)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    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#(filter(),f)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
   TRS:
    app(p(),0()) -> 0()
    app(p(),app(s(),x)) -> x
    app(app(le(),0()),y) -> true()
    app(app(le(),app(s(),x)),0()) -> false()
    app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
    app(app(minus(),x),y) -> app(app(app(if(),app(app(le(),x),y)),x),y)
    app(app(app(if(),true()),x),y) -> 0()
    app(app(app(if(),false()),x),y) -> app(s(),app(app(minus(),app(p(),x)),y))
    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)
   Open