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(treemap(),f),app(app(node(),x),xs)) ->
 app(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))

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(treemap(),f),app(app(node(),x),xs)) -> app#(map(),app(treemap(),f))
   app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs)
   app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x)
   app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(node(),app(f,x))
   app#(app(treemap(),f),app(app(node(),x),xs)) ->
   app#(app(node(),app(f,x)),app(app(map(),app(treemap(),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(treemap(),f),app(app(node(),x),xs)) ->
   app(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))
  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(treemap(),f),app(app(node(),x),xs)) -> app#(map(),app(treemap(),f))
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(node(),app(f,x))
    app#(app(treemap(),f),app(app(node(),x),xs)) ->
    app#(app(node(),app(f,x)),app(app(map(),app(treemap(),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(treemap(),f),app(app(node(),x),xs)) ->
    app(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))
   graph:
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(treemap(),f),app(app(node(),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(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(map(),app(treemap(),f))
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x)
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(node(),app(f,x))
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) ->
    app#(app(node(),app(f,x)),app(app(map(),app(treemap(),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(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(treemap(),f),app(app(node(),x),xs)) -> app#(map(),app(treemap(),f))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(node(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(treemap(),f),app(app(node(),x),xs)) ->
    app#(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 26/81
    DPs:
     app#(app(treemap(),f),app(app(node(),x),xs)) -> app#(app(map(),app(treemap(),f)),xs)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
     app#(app(treemap(),f),app(app(node(),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(treemap(),f),app(app(node(),x),xs)) ->
     app(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))
    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(treemap(),f),app(app(node(),x),xs)) ->
       app(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))
     Qed