YES(?,O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

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}

and precedence

 sum > + .

Following symbols are considered recursive:

 {sum, +}

The recursion depth is 2.

For your convenience, here are the satisfied ordering constraints:

     sum(0();) > 0()               
                                   
  sum(s(; x);) > +(s(; x); sum(x;))
                                   
     +(0(); x) > x                 
                                   
  +(s(; y); x) > s(; +(y; x))      
                                   

Hurray, we answered YES(?,O(n^2))