YES
Input TRS:
      1: app(app(map(),f),nil()) -> nil()
      2: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
      3: app(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
      4: app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
      5: app(app(plus(),0()),x) -> 0()
      6: app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
Number of strict rules: 6
Direct POLO(bPol) ... failed.
Uncurrying app
      1: app^2_map(f,nil()) -> nil()
      2: app^2_map(f,app^2_cons(x,xs)) -> app^2_cons(app(f,x),app^2_map(f,xs))
      3: app^1_sum(app^2_cons(x,xs)) -> app^2_plus(x,app^1_sum(xs))
      4: app^1_size(app^2_node(x,xs)) -> app^1_s(app^1_sum(app^2_map(size(),xs)))
      5: app^2_plus(0(),x) -> 0()
      6: app^2_plus(app^1_s(x),y) -> app^1_s(app^2_plus(x,y))
      7: app(plus(),_1) ->= app^1_plus(_1)
      8: app(app^1_plus(_1),_2) ->= app^2_plus(_1,_2)
      9: app(cons(),_1) ->= app^1_cons(_1)
      10: app(app^1_cons(_1),_2) ->= app^2_cons(_1,_2)
      11: app(s(),_1) ->= app^1_s(_1)
      12: app(map(),_1) ->= app^1_map(_1)
      13: app(app^1_map(_1),_2) ->= app^2_map(_1,_2)
      14: app(node(),_1) ->= app^1_node(_1)
      15: app(app^1_node(_1),_2) ->= app^2_node(_1,_2)
      16: app(sum(),_1) ->= app^1_sum(_1)
      17: app(size(),_1) ->= app^1_size(_1)
Number of strict rules: 6
Direct POLO(bPol) ... failed.
Dependency Pairs:
     #1: #app^2_map(f,app^2_cons(x,xs)) -> #app(f,x)
     #2: #app^2_map(f,app^2_cons(x,xs)) -> #app^2_map(f,xs)
     #3: #app^2_plus(app^1_s(x),y) -> #app^2_plus(x,y)
     #4: #app(app^1_map(_1),_2) ->? #app^2_map(_1,_2)
     #5: #app(size(),_1) ->? #app^1_size(_1)
     #6: #app(sum(),_1) ->? #app^1_sum(_1)
     #7: #app^1_sum(app^2_cons(x,xs)) -> #app^2_plus(x,app^1_sum(xs))
     #8: #app^1_sum(app^2_cons(x,xs)) -> #app^1_sum(xs)
     #9: #app(app^1_plus(_1),_2) ->? #app^2_plus(_1,_2)
     #10: #app^1_size(app^2_node(x,xs)) -> #app^1_sum(app^2_map(size(),xs))
     #11: #app^1_size(app^2_node(x,xs)) -> #app^2_map(size(),xs)
Number of SCCs: 3, DPs: 7
  SCC { #8 }
POLO(Sum)... succeeded.
      s 	w: 0
      app^1_cons	w: 0
      app^1_plus	w: 0
      #app^2_map	w: 0
      app^1_sum	w: 0
      #app^1_size	w: 0
      #app^2_plus	w: 0
      sum	w: 0
      app^2_plus	w: 0
      app^2_node	w: 0
      0 	w: 0
      nil	w: 0
      #app	w: 0
      map	w: 0
      plus	w: 0
      cons	w: 0
      app^1_size	w: 0
      #app^1_sum	w: x1
      app^1_map	w: 0
      app^2_cons	w: x2 + 1
      node	w: 0
      app^1_s	w: 0
      app^2_map	w: 0
      size	w: 0
      app	w: 0
      app^1_node	w: 0
    USABLE RULES: { }
    Removed DPs: #8
Number of SCCs: 2, DPs: 6
  SCC { #3 }
POLO(Sum)... succeeded.
      s 	w: 0
      app^1_cons	w: 0
      app^1_plus	w: 0
      #app^2_map	w: 0
      app^1_sum	w: 0
      #app^1_size	w: 0
      #app^2_plus	w: x1
      sum	w: 0
      app^2_plus	w: 0
      app^2_node	w: 0
      0 	w: 0
      nil	w: 0
      #app	w: 0
      map	w: 0
      plus	w: 0
      cons	w: 0
      app^1_size	w: 0
      #app^1_sum	w: 0
      app^1_map	w: 0
      app^2_cons	w: 1
      node	w: 0
      app^1_s	w: x1 + 1
      app^2_map	w: 0
      size	w: 0
      app	w: 0
      app^1_node	w: 0
    USABLE RULES: { }
    Removed DPs: #3
Number of SCCs: 1, DPs: 5
  SCC { #1 #2 #4 #5 #11 }
POLO(Sum)... succeeded.
      s 	w: 0
      app^1_cons	w: 0
      app^1_plus	w: 0
      #app^2_map	w: x1
      app^1_sum	w: 0
      #app^1_size	w: 7758
      #app^2_plus	w: 0
      sum	w: 0
      app^2_plus	w: 0
      app^2_node	w: 1
      0 	w: 0
      nil	w: 0
      #app	w: x1
      map	w: 0
      plus	w: 0
      cons	w: 0
      app^1_size	w: 0
      #app^1_sum	w: 0
      app^1_map	w: x1 + 1
      app^2_cons	w: 1
      node	w: 0
      app^1_s	w: 1
      app^2_map	w: 0
      size	w: 7758
      app	w: 0
      app^1_node	w: 0
    USABLE RULES: { }
    Removed DPs: #4
Number of SCCs: 1, DPs: 4
  SCC { #1 #2 #5 #11 }
POLO(Sum)... succeeded.
      s 	w: 0
      app^1_cons	w: 0
      app^1_plus	w: 0
      #app^2_map	w: x2
      app^1_sum	w: 0
      #app^1_size	w: x1
      #app^2_plus	w: 0
      sum	w: 0
      app^2_plus	w: 0
      app^2_node	w: x2 + 1
      0 	w: 0
      nil	w: 0
      #app	w: x2 + 1
      map	w: 0
      plus	w: 0
      cons	w: 0
      app^1_size	w: 0
      #app^1_sum	w: 0
      app^1_map	w: 1
      app^2_cons	w: x1 + x2 + 2
      node	w: 0
      app^1_s	w: 1
      app^2_map	w: 0
      size	w: 7758
      app	w: 0
      app^1_node	w: 0
    USABLE RULES: { }
    Removed DPs: #1 #2 #5 #11
Number of SCCs: 0, DPs: 0