YES(?,ELEMENTARY)

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

Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, 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(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
 safe(sqr) = {1}, safe(*) = {1, 2}

and precedence

 sum > sqr .

Following symbols are considered recursive:

 {sum, sqr}

The recursion depth is 2.

For your convenience, here are the satisfied ordering constraints:

     sum(0();) > 0()                                
                                                    
  sum(s(; x);) > +(; sqr(; s(; x)),  sum(x;))       
                                                    
  sum(s(; x);) > +(; *(; s(; x),  s(; x)),  sum(x;))
                                                    
      sqr(; x) > *(; x,  x)                         
                                                    

Hurray, we answered YES(?,ELEMENTARY)