MAYBE

Problem:
 app(app(.(),1()),x) -> x
 app(app(.(),x),1()) -> x
 app(app(.(),app(i(),x)),x) -> 1()
 app(app(.(),x),app(i(),x)) -> 1()
 app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
 app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
 app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
 app(i(),1()) -> 1()
 app(i(),app(i(),x)) -> x
 app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
 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(.(),app(app(.(),x),y)),z) -> app#(.(),y)
   app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
   app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
   app#(i(),app(app(.(),x),y)) -> app#(i(),x)
   app#(i(),app(app(.(),x),y)) -> app#(i(),y)
   app#(i(),app(app(.(),x),y)) -> app#(.(),app(i(),y))
   app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x))
   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(app(.(),1()),x) -> x
   app(app(.(),x),1()) -> x
   app(app(.(),app(i(),x)),x) -> 1()
   app(app(.(),x),app(i(),x)) -> 1()
   app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
   app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
   app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
   app(i(),1()) -> 1()
   app(i(),app(i(),x)) -> x
   app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
   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)
  EDG Processor:
   DPs:
    app#(app(.(),app(app(.(),x),y)),z) -> app#(.(),y)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
    app#(i(),app(app(.(),x),y)) -> app#(i(),x)
    app#(i(),app(app(.(),x),y)) -> app#(i(),y)
    app#(i(),app(app(.(),x),y)) -> app#(.(),app(i(),y))
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x))
    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(app(.(),1()),x) -> x
    app(app(.(),x),1()) -> x
    app(app(.(),app(i(),x)),x) -> 1()
    app(app(.(),x),app(i(),x)) -> 1()
    app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
    app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
    app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
    app(i(),1()) -> 1()
    app(i(),app(i(),x)) -> x
    app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
    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)
   graph:
    app#(i(),app(app(.(),x),y)) -> app#(i(),y) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),x)
    app#(i(),app(app(.(),x),y)) -> app#(i(),y) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),y)
    app#(i(),app(app(.(),x),y)) -> app#(i(),y) ->
    app#(i(),app(app(.(),x),y)) -> app#(.(),app(i(),y))
    app#(i(),app(app(.(),x),y)) -> app#(i(),y) ->
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x))
    app#(i(),app(app(.(),x),y)) -> app#(i(),x) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),x)
    app#(i(),app(app(.(),x),y)) -> app#(i(),x) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),y)
    app#(i(),app(app(.(),x),y)) -> app#(i(),x) ->
    app#(i(),app(app(.(),x),y)) -> app#(.(),app(i(),y))
    app#(i(),app(app(.(),x),y)) -> app#(i(),x) ->
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x))
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x)) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(.(),y)
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x)) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x)) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
    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(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(filter(),f),xs)
    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#(cons(),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#(app(cons(),x),app(app(filter(),f),xs))
    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(),false()),f),x),xs) -> app#(filter(),f)
    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(),false()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(.(),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(.(),app(i(),y))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(filter(),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(filter(),f),app(app(cons(),x),xs)) -> 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#(f,x)
    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#(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#(f,x) ->
    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#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    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#(app(map(),f),xs) ->
    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(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(.(),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(i(),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(.(),app(i(),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(i(),app(app(.(),x),y)) -> app#(app(.(),app(i(),y)),app(i(),x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    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#(f,x)
    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#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),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(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    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(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(.(),y)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z)) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(.(),y)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z)) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z)) ->
    app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
   SCC Processor:
    #sccs: 3
    #rules: 10
    #arcs: 81/484
    DPs:
     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(),false()),f),x),xs) -> app#(app(filter(),f),xs)
     app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
     app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    TRS:
     app(app(.(),1()),x) -> x
     app(app(.(),x),1()) -> x
     app(app(.(),app(i(),x)),x) -> 1()
     app(app(.(),x),app(i(),x)) -> 1()
     app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
     app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
     app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
     app(i(),1()) -> 1()
     app(i(),app(i(),x)) -> x
     app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
     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
    
    DPs:
     app#(i(),app(app(.(),x),y)) -> app#(i(),y)
     app#(i(),app(app(.(),x),y)) -> app#(i(),x)
    TRS:
     app(app(.(),1()),x) -> x
     app(app(.(),x),1()) -> x
     app(app(.(),app(i(),x)),x) -> 1()
     app(app(.(),x),app(i(),x)) -> 1()
     app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
     app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
     app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
     app(i(),1()) -> 1()
     app(i(),app(i(),x)) -> x
     app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
     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
    
    DPs:
     app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),x),app(app(.(),y),z))
     app#(app(.(),app(app(.(),x),y)),z) -> app#(app(.(),y),z)
    TRS:
     app(app(.(),1()),x) -> x
     app(app(.(),x),1()) -> x
     app(app(.(),app(i(),x)),x) -> 1()
     app(app(.(),x),app(i(),x)) -> 1()
     app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
     app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
     app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
     app(i(),1()) -> 1()
     app(i(),app(i(),x)) -> x
     app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
     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