YES

Problem:
 sum(0()) -> 0()
 sum(s(x)) -> +(sum(x),s(x))
 sum1(0()) -> 0()
 sum1(s(x)) -> s(+(sum1(x),+(x,x)))

Proof:
 DP Processor:
  DPs:
   sum#(s(x)) -> sum#(x)
   sum1#(s(x)) -> sum1#(x)
  TRS:
   sum(0()) -> 0()
   sum(s(x)) -> +(sum(x),s(x))
   sum1(0()) -> 0()
   sum1(s(x)) -> s(+(sum1(x),+(x,x)))
  TDG Processor:
   DPs:
    sum#(s(x)) -> sum#(x)
    sum1#(s(x)) -> sum1#(x)
   TRS:
    sum(0()) -> 0()
    sum(s(x)) -> +(sum(x),s(x))
    sum1(0()) -> 0()
    sum1(s(x)) -> s(+(sum1(x),+(x,x)))
   graph:
    sum1#(s(x)) -> sum1#(x) -> sum1#(s(x)) -> sum1#(x)
    sum#(s(x)) -> sum#(x) -> sum#(s(x)) -> sum#(x)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 2/4
    DPs:
     sum#(s(x)) -> sum#(x)
    TRS:
     sum(0()) -> 0()
     sum(s(x)) -> +(sum(x),s(x))
     sum1(0()) -> 0()
     sum1(s(x)) -> s(+(sum1(x),+(x,x)))
    Subterm Criterion Processor:
     simple projection:
      pi(sum#) = 0
     problem:
      DPs:
       
      TRS:
       sum(0()) -> 0()
       sum(s(x)) -> +(sum(x),s(x))
       sum1(0()) -> 0()
       sum1(s(x)) -> s(+(sum1(x),+(x,x)))
     Qed
    
    DPs:
     sum1#(s(x)) -> sum1#(x)
    TRS:
     sum(0()) -> 0()
     sum(s(x)) -> +(sum(x),s(x))
     sum1(0()) -> 0()
     sum1(s(x)) -> s(+(sum1(x),+(x,x)))
    Subterm Criterion Processor:
     simple projection:
      pi(sum1#) = 0
     problem:
      DPs:
       
      TRS:
       sum(0()) -> 0()
       sum(s(x)) -> +(sum(x),s(x))
       sum1(0()) -> 0()
       sum1(s(x)) -> s(+(sum1(x),+(x,x)))
     Qed