YES(?,POLY)

'Pop* with parameter subtitution (timeout of 60.0 seconds)'
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Answer:           YES(?,POLY)
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  sum(0()) -> 0()
     , sum(s(x)) -> +(sum(x), s(x))
     , sum1(0()) -> 0()
     , sum1(s(x)) -> s(+(sum1(x), +(x, x)))}

Proof Output:    
  The input was oriented with the instance of POP* as induced by the precedence
  
   sum ~ sum1
  
  and safe mapping
  
   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sum1) = {} .
  
  For your convenience, here is the input in predicative notation:
  
   Rules:
    {  sum(0();) -> 0()
     , sum(s(; x);) -> +(; sum(x;), s(; x))
     , sum1(0();) -> 0()
     , sum1(s(; x);) -> s(; +(; sum1(x;), +(; x, x)))}