We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , sum1(0()) -> 0()
  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping

 safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
 safe(sum1) = {}

and precedence

 empty .

Following symbols are considered recursive:

 {sum, sum1}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

      sum(0();) > 0()                            
                                                 
   sum(s(; x);) > +(; sum(x;),  s(; x))          
                                                 
     sum1(0();) > 0()                            
                                                 
  sum1(s(; x);) > s(; +(; sum1(x;),  +(; x,  x)))
                                                 

Hurray, we answered YES(?,O(n^1))