YES(?,ELEMENTARY)

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

Strict Trs:
  { sqr(0()) -> 0()
  , sqr(s(x)) -> s(+(sqr(x), double(x)))
  , sqr(s(x)) -> +(sqr(x), s(double(x)))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(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(sqr) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1},
 safe(double) = {}

and precedence

 sqr > +, sqr > double, + ~ double .

Following symbols are considered recursive:

 {sqr, +, double}

The recursion depth is 2.

For your convenience, here are the satisfied ordering constraints:

        sqr(0();) > 0()                        
                                               
     sqr(s(; x);) > s(; +(double(x;); sqr(x;)))
                                               
     sqr(s(; x);) > +(s(; double(x;)); sqr(x;))
                                               
        +(0(); x) > x                          
                                               
     +(s(; y); x) > s(; +(y; x))               
                                               
     double(0();) > 0()                        
                                               
  double(s(; x);) > s(; s(; double(x;)))       
                                               

Hurray, we answered YES(?,ELEMENTARY)