YES(?,ELEMENTARY)

We are left with following problem, upon which TcT provides the
certificate YES(?,ELEMENTARY).

Strict Trs:
  { f(0()) -> s(0())
  , f(s(0())) -> s(s(0()))
  , f(s(0())) -> *(s(s(0())), f(0()))
  , f(+(x, y)) -> *(f(x), f(y))
  , f(+(x, s(0()))) -> +(s(s(0())), f(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,ELEMENTARY)

The input was oriented with the instance of 'Lightweight Multiset
Path Order' as induced by the safe mapping

 safe(f) = {}, safe(0) = {}, safe(s) = {1}, safe(*) = {1, 2},
 safe(+) = {1, 2}

and precedence

 empty .

Following symbols are considered recursive:

 {f}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

                f(0();) > s(; 0())                    
                                                      
           f(s(; 0());) > s(; s(; 0()))               
                                                      
           f(s(; 0());) > *(; s(; s(; 0())),  f(0();))
                                                      
         f(+(; x,  y);) > *(; f(x;),  f(y;))          
                                                      
  f(+(; x,  s(; 0()));) > +(; s(; s(; 0())),  f(x;))  
                                                      

Hurray, we answered YES(?,ELEMENTARY)