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)))
  Usable Rule Processor:
   DPs:
    sum#(s(x)) -> sum#(x)
    sum1#(s(x)) -> sum1#(x)
   TRS:
    
   CDG Processor:
    DPs:
     sum#(s(x)) -> sum#(x)
     sum1#(s(x)) -> sum1#(x)
    TRS:
     
    graph:
     
    Qed