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

Strict Trs:
  { add0(x', Cons(x, xs)) -> add0(Cons(Cons(Nil(), Nil()), x'), xs)
  , add0(x, Nil()) -> x
  , notEmpty(Cons(x, xs)) -> True()
  , notEmpty(Nil()) -> False()
  , goal(x, y) -> add0(x, 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(add0) = {1}, safe(Cons) = {1, 2}, safe(Nil) = {},
 safe(notEmpty) = {}, safe(True) = {}, safe(False) = {},
 safe(goal) = {}

and precedence

 goal > add0 .

Following symbols are considered recursive:

 {add0}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

   add0(Cons(; x,  xs); x') > add0(xs; Cons(; Cons(; Nil(),  Nil()),  x'))
                                                                          
             add0(Nil(); x) > x                                           
                                                                          
  notEmpty(Cons(; x,  xs);) > True()                                      
                                                                          
           notEmpty(Nil();) > False()                                     
                                                                          
               goal(x,  y;) > add0(y; x)                                  
                                                                          

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