YES(?,O(n^1))

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

Strict Trs:
  { div2(S(S(x))) -> +(S(0()), div2(x))
  , div2(S(0())) -> 0()
  , div2(0()) -> 0() }
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(div2) = {}, safe(+) = {1, 2}, safe(S) = {1}, safe(0) = {}

and precedence

 div2 > + .

Following symbols are considered recursive:

 {div2}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

  div2(S(; S(; x));) > +(; S(; 0()),  div2(x;))
                                               
     div2(S(; 0());) > 0()                     
                                               
          div2(0();) > 0()                     
                                               
   +(; x,  S(; 0())) > S(; x)                  
                                               
   +(; S(; 0()),  y) > S(; y)                  
                                               

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