YES(?,O(n^1))

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

Strict Trs:
  { dbl(S(0()), S(0())) -> S(S(S(S(0()))))
  , dbl(0(), y) -> y
  , unsafe(S(x)) -> dbl(unsafe(x), 0())
  , unsafe(0()) -> 0() }
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(dbl) = {1, 2}, safe(S) = {1}, safe(0) = {}, safe(unsafe) = {}

and precedence

 unsafe > dbl .

Following symbols are considered recursive:

 {unsafe}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

  dbl(; S(; 0()),  S(; 0())) > S(; S(; S(; S(; 0()))))
                                                      
              dbl(; 0(),  y) > y                      
                                                      
             unsafe(S(; x);) > dbl(; unsafe(x;),  0())
                                                      
                unsafe(0();) > 0()                    
                                                      

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