YES

Problem:
 app(app(twice(),f),x) -> app(f,app(f,x))
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),h),t)) -> app(app(cons(),app(f,h)),app(app(map(),f),t))
 app(app(fmap(),nil()),x) -> nil()
 app(app(fmap(),app(app(cons(),f),t_f())),x) -> app(app(cons(),app(f,x)),app(app(fmap(),t_f()),x))

Proof:
 Uncurry Processor:
  twice2(f,x) -> app(f,app(f,x))
  map2(f,nil()) -> nil()
  map2(f,cons2(h,t)) -> cons2(app(f,h),map2(f,t))
  fmap2(nil(),x) -> nil()
  fmap2(cons2(f,t_f()),x) -> cons2(app(f,x),fmap2(t_f(),x))
  app(twice1(x4),x5) -> twice2(x4,x5)
  app(twice(),x5) -> twice1(x5)
  app(map1(x4),x5) -> map2(x4,x5)
  app(map(),x5) -> map1(x5)
  app(cons1(x4),x5) -> cons2(x4,x5)
  app(cons(),x5) -> cons1(x5)
  app(fmap1(x4),x5) -> fmap2(x4,x5)
  app(fmap(),x5) -> fmap1(x5)
  DP Processor:
   DPs:
    twice{2,#}(f,x) -> app#(f,x)
    twice{2,#}(f,x) -> app#(f,app(f,x))
    map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t)
    map{2,#}(f,cons2(h,t)) -> app#(f,h)
    fmap{2,#}(cons2(f,t_f()),x) -> fmap{2,#}(t_f(),x)
    fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x)
    app#(twice1(x4),x5) -> twice{2,#}(x4,x5)
    app#(map1(x4),x5) -> map{2,#}(x4,x5)
    app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5)
   TRS:
    twice2(f,x) -> app(f,app(f,x))
    map2(f,nil()) -> nil()
    map2(f,cons2(h,t)) -> cons2(app(f,h),map2(f,t))
    fmap2(nil(),x) -> nil()
    fmap2(cons2(f,t_f()),x) -> cons2(app(f,x),fmap2(t_f(),x))
    app(twice1(x4),x5) -> twice2(x4,x5)
    app(twice(),x5) -> twice1(x5)
    app(map1(x4),x5) -> map2(x4,x5)
    app(map(),x5) -> map1(x5)
    app(cons1(x4),x5) -> cons2(x4,x5)
    app(cons(),x5) -> cons1(x5)
    app(fmap1(x4),x5) -> fmap2(x4,x5)
    app(fmap(),x5) -> fmap1(x5)
   TDG Processor:
    DPs:
     twice{2,#}(f,x) -> app#(f,x)
     twice{2,#}(f,x) -> app#(f,app(f,x))
     map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t)
     map{2,#}(f,cons2(h,t)) -> app#(f,h)
     fmap{2,#}(cons2(f,t_f()),x) -> fmap{2,#}(t_f(),x)
     fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x)
     app#(twice1(x4),x5) -> twice{2,#}(x4,x5)
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5)
    TRS:
     twice2(f,x) -> app(f,app(f,x))
     map2(f,nil()) -> nil()
     map2(f,cons2(h,t)) -> cons2(app(f,h),map2(f,t))
     fmap2(nil(),x) -> nil()
     fmap2(cons2(f,t_f()),x) -> cons2(app(f,x),fmap2(t_f(),x))
     app(twice1(x4),x5) -> twice2(x4,x5)
     app(twice(),x5) -> twice1(x5)
     app(map1(x4),x5) -> map2(x4,x5)
     app(map(),x5) -> map1(x5)
     app(cons1(x4),x5) -> cons2(x4,x5)
     app(cons(),x5) -> cons1(x5)
     app(fmap1(x4),x5) -> fmap2(x4,x5)
     app(fmap(),x5) -> fmap1(x5)
    graph:
     fmap{2,#}(cons2(f,t_f()),x) -> fmap{2,#}(t_f(),x) ->
     fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x)
     fmap{2,#}(cons2(f,t_f()),x) -> fmap{2,#}(t_f(),x) ->
     fmap{2,#}(cons2(f,t_f()),x) -> fmap{2,#}(t_f(),x)
     fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x) ->
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5)
     fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x) ->
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x) ->
     app#(twice1(x4),x5) -> twice{2,#}(x4,x5)
     map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t) ->
     map{2,#}(f,cons2(h,t)) -> app#(f,h)
     map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t) ->
     map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t)
     map{2,#}(f,cons2(h,t)) -> app#(f,h) ->
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5)
     map{2,#}(f,cons2(h,t)) -> app#(f,h) ->
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     map{2,#}(f,cons2(h,t)) -> app#(f,h) ->
     app#(twice1(x4),x5) -> twice{2,#}(x4,x5)
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5) ->
     fmap{2,#}(cons2(f,t_f()),x) -> app#(f,x)
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5) ->
     fmap{2,#}(cons2(f,t_f()),x) -> fmap{2,#}(t_f(),x)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(h,t)) -> app#(f,h)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t)
     app#(twice1(x4),x5) -> twice{2,#}(x4,x5) ->
     twice{2,#}(f,x) -> app#(f,app(f,x))
     app#(twice1(x4),x5) -> twice{2,#}(x4,x5) ->
     twice{2,#}(f,x) -> app#(f,x)
     twice{2,#}(f,x) -> app#(f,app(f,x)) ->
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5)
     twice{2,#}(f,x) -> app#(f,app(f,x)) ->
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     twice{2,#}(f,x) -> app#(f,app(f,x)) ->
     app#(twice1(x4),x5) -> twice{2,#}(x4,x5)
     twice{2,#}(f,x) -> app#(f,x) ->
     app#(fmap1(x4),x5) -> fmap{2,#}(x4,x5)
     twice{2,#}(f,x) -> app#(f,x) ->
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     twice{2,#}(f,x) -> app#(f,x) -> app#(twice1(x4),x5) -> twice{2,#}(x4,x5)
    Subterm Criterion Processor:
     simple projection:
      pi(twice{2,#}) = 0
      pi(app#) = 0
      pi(map{2,#}) = 0
      pi(fmap{2,#}) = 0
     problem:
      DPs:
       twice{2,#}(f,x) -> app#(f,x)
       twice{2,#}(f,x) -> app#(f,app(f,x))
       map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t)
       map{2,#}(f,cons2(h,t)) -> app#(f,h)
      TRS:
       twice2(f,x) -> app(f,app(f,x))
       map2(f,nil()) -> nil()
       map2(f,cons2(h,t)) -> cons2(app(f,h),map2(f,t))
       fmap2(nil(),x) -> nil()
       fmap2(cons2(f,t_f()),x) -> cons2(app(f,x),fmap2(t_f(),x))
       app(twice1(x4),x5) -> twice2(x4,x5)
       app(twice(),x5) -> twice1(x5)
       app(map1(x4),x5) -> map2(x4,x5)
       app(map(),x5) -> map1(x5)
       app(cons1(x4),x5) -> cons2(x4,x5)
       app(cons(),x5) -> cons1(x5)
       app(fmap1(x4),x5) -> fmap2(x4,x5)
       app(fmap(),x5) -> fmap1(x5)
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 22/16
      DPs:
       map{2,#}(f,cons2(h,t)) -> map{2,#}(f,t)
      TRS:
       twice2(f,x) -> app(f,app(f,x))
       map2(f,nil()) -> nil()
       map2(f,cons2(h,t)) -> cons2(app(f,h),map2(f,t))
       fmap2(nil(),x) -> nil()
       fmap2(cons2(f,t_f()),x) -> cons2(app(f,x),fmap2(t_f(),x))
       app(twice1(x4),x5) -> twice2(x4,x5)
       app(twice(),x5) -> twice1(x5)
       app(map1(x4),x5) -> map2(x4,x5)
       app(map(),x5) -> map1(x5)
       app(cons1(x4),x5) -> cons2(x4,x5)
       app(cons(),x5) -> cons1(x5)
       app(fmap1(x4),x5) -> fmap2(x4,x5)
       app(fmap(),x5) -> fmap1(x5)
      Subterm Criterion Processor:
       simple projection:
        pi(map{2,#}) = 1
       problem:
        DPs:
         
        TRS:
         twice2(f,x) -> app(f,app(f,x))
         map2(f,nil()) -> nil()
         map2(f,cons2(h,t)) -> cons2(app(f,h),map2(f,t))
         fmap2(nil(),x) -> nil()
         fmap2(cons2(f,t_f()),x) -> cons2(app(f,x),fmap2(t_f(),x))
         app(twice1(x4),x5) -> twice2(x4,x5)
         app(twice(),x5) -> twice1(x5)
         app(map1(x4),x5) -> map2(x4,x5)
         app(map(),x5) -> map1(x5)
         app(cons1(x4),x5) -> cons2(x4,x5)
         app(cons(),x5) -> cons1(x5)
         app(fmap1(x4),x5) -> fmap2(x4,x5)
         app(fmap(),x5) -> fmap1(x5)
       Qed