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

Strict Trs:
  { add0(S(x), x2) -> +(S(0()), add0(x2, x))
  , add0(0(), x2) -> x2 }
Weak Trs:
  { +(x, S(0())) -> S(x)
  , +(S(0()), y) -> S(y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping

 safe(+) = {1, 2}, safe(S) = {1}, safe(0) = {}, safe(add0) = {}

and precedence

 add0 > + .

Following symbols are considered recursive:

 {add0}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

   +(; x,  S(; 0())) > S(; x)                       
                                                    
   +(; S(; 0()),  y) > S(; y)                       
                                                    
  add0(S(; x),  x2;) > +(; S(; 0()),  add0(x2,  x;))
                                                    
     add0(0(),  x2;) > x2                           
                                                    

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