YES

Problem:
 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(app(comp(),f),g),x) -> app(f,app(g,x))
 app(twice(),f) -> app(app(comp(),f),f)

Proof:
 DP Processor:
  DPs:
   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(app(comp(),f),g),x) -> app#(g,x)
   app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
   app#(twice(),f) -> app#(comp(),f)
   app#(twice(),f) -> app#(app(comp(),f),f)
  TRS:
   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(app(comp(),f),g),x) -> app(f,app(g,x))
   app(twice(),f) -> app(app(comp(),f),f)
  EDG Processor:
   DPs:
    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(app(comp(),f),g),x) -> app#(g,x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(twice(),f) -> app#(comp(),f)
    app#(twice(),f) -> app#(app(comp(),f),f)
   TRS:
    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(app(comp(),f),g),x) -> app(f,app(g,x))
    app(twice(),f) -> app(app(comp(),f),f)
   graph:
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(app(comp(),f),g),x) -> app#(g,x)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(twice(),f) -> app#(comp(),f)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(twice(),f) -> app#(app(comp(),f),f)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(app(comp(),f),g),x) -> app#(g,x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(twice(),f) -> app#(comp(),f)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(twice(),f) -> app#(app(comp(),f),f)
    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(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(app(comp(),f),g),x) -> app#(g,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(twice(),f) -> app#(comp(),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(twice(),f) -> app#(app(comp(),f),f)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 28/64
    DPs:
     app#(app(app(comp(),f),g),x) -> app#(g,x)
     app#(app(app(comp(),f),g),x) -> app#(f,app(g,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)
    TRS:
     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(app(comp(),f),g),x) -> app(f,app(g,x))
     app(twice(),f) -> app(app(comp(),f),f)
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     problem:
      DPs:
       app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
      TRS:
       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(app(comp(),f),g),x) -> app(f,app(g,x))
       app(twice(),f) -> app(app(comp(),f),f)
     Subterm Criterion Processor:
      simple projection:
       pi(app#) = 1
      problem:
       DPs:
        
       TRS:
        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(app(comp(),f),g),x) -> app(f,app(g,x))
        app(twice(),f) -> app(app(comp(),f),f)
      Qed