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(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(maxlist(),x),app(app(cons(),y),ys)) ->
 app(app(if(),app(app(le(),x),y)),app(app(maxlist(),y),ys))
 app(app(maxlist(),x),nil()) -> x
 app(height(),app(app(node(),x),xs)) ->
 app(s(),app(app(maxlist(),0()),app(app(map(),height()),xs)))

Proof:
 Uncurry Processor:
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  le2(0(),y) -> true()
  le2(s1(x),0()) -> false()
  le2(s1(x),s1(y)) -> le2(x,y)
  maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
  maxlist2(x,nil()) -> x
  height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
  app(map1(x5),x6) -> map2(x5,x6)
  app(map(),x6) -> map1(x6)
  app(cons1(x5),x6) -> cons2(x5,x6)
  app(cons(),x6) -> cons1(x6)
  app(le1(x5),x6) -> le2(x5,x6)
  app(le(),x6) -> le1(x6)
  app(s(),x6) -> s1(x6)
  app(maxlist1(x5),x6) -> maxlist2(x5,x6)
  app(maxlist(),x6) -> maxlist1(x6)
  app(if1(x5),x6) -> if2(x5,x6)
  app(if(),x6) -> if1(x6)
  app(height(),x6) -> height1(x6)
  app(node1(x5),x6) -> node2(x5,x6)
  app(node(),x6) -> node1(x6)
  DP Processor:
   DPs:
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
    maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys)
    maxlist{2,#}(x,cons2(y,ys)) -> le{2,#}(x,y)
    height{1,#}(node2(x,xs)) -> map{2,#}(height(),xs)
    height{1,#}(node2(x,xs)) -> maxlist{2,#}(0(),map2(height(),xs))
    app#(map1(x5),x6) -> map{2,#}(x5,x6)
    app#(le1(x5),x6) -> le{2,#}(x5,x6)
    app#(maxlist1(x5),x6) -> maxlist{2,#}(x5,x6)
    app#(height(),x6) -> height{1,#}(x6)
   TRS:
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    le2(0(),y) -> true()
    le2(s1(x),0()) -> false()
    le2(s1(x),s1(y)) -> le2(x,y)
    maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
    maxlist2(x,nil()) -> x
    height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
    app(map1(x5),x6) -> map2(x5,x6)
    app(map(),x6) -> map1(x6)
    app(cons1(x5),x6) -> cons2(x5,x6)
    app(cons(),x6) -> cons1(x6)
    app(le1(x5),x6) -> le2(x5,x6)
    app(le(),x6) -> le1(x6)
    app(s(),x6) -> s1(x6)
    app(maxlist1(x5),x6) -> maxlist2(x5,x6)
    app(maxlist(),x6) -> maxlist1(x6)
    app(if1(x5),x6) -> if2(x5,x6)
    app(if(),x6) -> if1(x6)
    app(height(),x6) -> height1(x6)
    app(node1(x5),x6) -> node2(x5,x6)
    app(node(),x6) -> node1(x6)
   TDG Processor:
    DPs:
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys)
     maxlist{2,#}(x,cons2(y,ys)) -> le{2,#}(x,y)
     height{1,#}(node2(x,xs)) -> map{2,#}(height(),xs)
     height{1,#}(node2(x,xs)) -> maxlist{2,#}(0(),map2(height(),xs))
     app#(map1(x5),x6) -> map{2,#}(x5,x6)
     app#(le1(x5),x6) -> le{2,#}(x5,x6)
     app#(maxlist1(x5),x6) -> maxlist{2,#}(x5,x6)
     app#(height(),x6) -> height{1,#}(x6)
    TRS:
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     le2(0(),y) -> true()
     le2(s1(x),0()) -> false()
     le2(s1(x),s1(y)) -> le2(x,y)
     maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
     maxlist2(x,nil()) -> x
     height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
     app(map1(x5),x6) -> map2(x5,x6)
     app(map(),x6) -> map1(x6)
     app(cons1(x5),x6) -> cons2(x5,x6)
     app(cons(),x6) -> cons1(x6)
     app(le1(x5),x6) -> le2(x5,x6)
     app(le(),x6) -> le1(x6)
     app(s(),x6) -> s1(x6)
     app(maxlist1(x5),x6) -> maxlist2(x5,x6)
     app(maxlist(),x6) -> maxlist1(x6)
     app(if1(x5),x6) -> if2(x5,x6)
     app(if(),x6) -> if1(x6)
     app(height(),x6) -> height1(x6)
     app(node1(x5),x6) -> node2(x5,x6)
     app(node(),x6) -> node1(x6)
    graph:
     height{1,#}(node2(x,xs)) -> maxlist{2,#}(0(),map2(height(),xs)) ->
     maxlist{2,#}(x,cons2(y,ys)) -> le{2,#}(x,y)
     height{1,#}(node2(x,xs)) -> maxlist{2,#}(0(),map2(height(),xs)) ->
     maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys)
     height{1,#}(node2(x,xs)) -> map{2,#}(height(),xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     height{1,#}(node2(x,xs)) -> map{2,#}(height(),xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys) ->
     maxlist{2,#}(x,cons2(y,ys)) -> le{2,#}(x,y)
     maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys) ->
     maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys)
     maxlist{2,#}(x,cons2(y,ys)) -> le{2,#}(x,y) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     app#(maxlist1(x5),x6) -> maxlist{2,#}(x5,x6) ->
     maxlist{2,#}(x,cons2(y,ys)) -> le{2,#}(x,y)
     app#(maxlist1(x5),x6) -> maxlist{2,#}(x5,x6) ->
     maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys)
     app#(le1(x5),x6) -> le{2,#}(x5,x6) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     app#(map1(x5),x6) -> map{2,#}(x5,x6) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x5),x6) -> map{2,#}(x5,x6) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(height(),x6) -> height{1,#}(x6) ->
     height{1,#}(node2(x,xs)) -> maxlist{2,#}(0(),map2(height(),xs))
     app#(height(),x6) -> height{1,#}(x6) ->
     height{1,#}(node2(x,xs)) -> map{2,#}(height(),xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(height(),x6) -> height{1,#}(x6)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(maxlist1(x5),x6) -> maxlist{2,#}(x5,x6)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(le1(x5),x6) -> le{2,#}(x5,x6)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x5),x6) -> map{2,#}(x5,x6)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) -> map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    SCC Processor:
     #sccs: 3
     #rules: 7
     #arcs: 21/121
     DPs:
      height{1,#}(node2(x,xs)) -> map{2,#}(height(),xs)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x5),x6) -> map{2,#}(x5,x6)
      app#(height(),x6) -> height{1,#}(x6)
     TRS:
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
      maxlist2(x,nil()) -> x
      height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
      app(map1(x5),x6) -> map2(x5,x6)
      app(map(),x6) -> map1(x6)
      app(cons1(x5),x6) -> cons2(x5,x6)
      app(cons(),x6) -> cons1(x6)
      app(le1(x5),x6) -> le2(x5,x6)
      app(le(),x6) -> le1(x6)
      app(s(),x6) -> s1(x6)
      app(maxlist1(x5),x6) -> maxlist2(x5,x6)
      app(maxlist(),x6) -> maxlist1(x6)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if(),x6) -> if1(x6)
      app(height(),x6) -> height1(x6)
      app(node1(x5),x6) -> node2(x5,x6)
      app(node(),x6) -> node1(x6)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
       pi(height{1,#}) = 0
      problem:
       DPs:
        app#(map1(x5),x6) -> map{2,#}(x5,x6)
        app#(height(),x6) -> height{1,#}(x6)
       TRS:
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
        maxlist2(x,nil()) -> x
        height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
        app(map1(x5),x6) -> map2(x5,x6)
        app(map(),x6) -> map1(x6)
        app(cons1(x5),x6) -> cons2(x5,x6)
        app(cons(),x6) -> cons1(x6)
        app(le1(x5),x6) -> le2(x5,x6)
        app(le(),x6) -> le1(x6)
        app(s(),x6) -> s1(x6)
        app(maxlist1(x5),x6) -> maxlist2(x5,x6)
        app(maxlist(),x6) -> maxlist1(x6)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if(),x6) -> if1(x6)
        app(height(),x6) -> height1(x6)
        app(node1(x5),x6) -> node2(x5,x6)
        app(node(),x6) -> node1(x6)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 9/4
       
     
     DPs:
      maxlist{2,#}(x,cons2(y,ys)) -> maxlist{2,#}(y,ys)
     TRS:
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
      maxlist2(x,nil()) -> x
      height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
      app(map1(x5),x6) -> map2(x5,x6)
      app(map(),x6) -> map1(x6)
      app(cons1(x5),x6) -> cons2(x5,x6)
      app(cons(),x6) -> cons1(x6)
      app(le1(x5),x6) -> le2(x5,x6)
      app(le(),x6) -> le1(x6)
      app(s(),x6) -> s1(x6)
      app(maxlist1(x5),x6) -> maxlist2(x5,x6)
      app(maxlist(),x6) -> maxlist1(x6)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if(),x6) -> if1(x6)
      app(height(),x6) -> height1(x6)
      app(node1(x5),x6) -> node2(x5,x6)
      app(node(),x6) -> node1(x6)
     Subterm Criterion Processor:
      simple projection:
       pi(maxlist{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
        maxlist2(x,nil()) -> x
        height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
        app(map1(x5),x6) -> map2(x5,x6)
        app(map(),x6) -> map1(x6)
        app(cons1(x5),x6) -> cons2(x5,x6)
        app(cons(),x6) -> cons1(x6)
        app(le1(x5),x6) -> le2(x5,x6)
        app(le(),x6) -> le1(x6)
        app(s(),x6) -> s1(x6)
        app(maxlist1(x5),x6) -> maxlist2(x5,x6)
        app(maxlist(),x6) -> maxlist1(x6)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if(),x6) -> if1(x6)
        app(height(),x6) -> height1(x6)
        app(node1(x5),x6) -> node2(x5,x6)
        app(node(),x6) -> node1(x6)
      Qed
     
     DPs:
      le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     TRS:
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
      maxlist2(x,nil()) -> x
      height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
      app(map1(x5),x6) -> map2(x5,x6)
      app(map(),x6) -> map1(x6)
      app(cons1(x5),x6) -> cons2(x5,x6)
      app(cons(),x6) -> cons1(x6)
      app(le1(x5),x6) -> le2(x5,x6)
      app(le(),x6) -> le1(x6)
      app(s(),x6) -> s1(x6)
      app(maxlist1(x5),x6) -> maxlist2(x5,x6)
      app(maxlist(),x6) -> maxlist1(x6)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if(),x6) -> if1(x6)
      app(height(),x6) -> height1(x6)
      app(node1(x5),x6) -> node2(x5,x6)
      app(node(),x6) -> node1(x6)
     Subterm Criterion Processor:
      simple projection:
       pi(le{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        maxlist2(x,cons2(y,ys)) -> if2(le2(x,y),maxlist2(y,ys))
        maxlist2(x,nil()) -> x
        height1(node2(x,xs)) -> s1(maxlist2(0(),map2(height(),xs)))
        app(map1(x5),x6) -> map2(x5,x6)
        app(map(),x6) -> map1(x6)
        app(cons1(x5),x6) -> cons2(x5,x6)
        app(cons(),x6) -> cons1(x6)
        app(le1(x5),x6) -> le2(x5,x6)
        app(le(),x6) -> le1(x6)
        app(s(),x6) -> s1(x6)
        app(maxlist1(x5),x6) -> maxlist2(x5,x6)
        app(maxlist(),x6) -> maxlist1(x6)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if(),x6) -> if1(x6)
        app(height(),x6) -> height1(x6)
        app(node1(x5),x6) -> node2(x5,x6)
        app(node(),x6) -> node1(x6)
      Qed