YES(?,O(n^1))

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

Strict Trs:
  { addlist(Cons(x, xs'), Cons(S(0()), xs)) ->
    Cons(S(x), addlist(xs', xs))
  , addlist(Cons(S(0()), xs'), Cons(x, xs)) ->
    Cons(S(x), addlist(xs', xs))
  , addlist(Nil(), ys) -> Nil()
  , notEmpty(Cons(x, xs)) -> True()
  , notEmpty(Nil()) -> False()
  , goal(xs, ys) -> addlist(xs, ys) }
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(addlist) = {1}, safe(Cons) = {1, 2}, safe(S) = {1},
 safe(0) = {}, safe(Nil) = {}, safe(notEmpty) = {}, safe(True) = {},
 safe(False) = {}, safe(goal) = {1}

and precedence

 notEmpty > addlist, goal > addlist, notEmpty ~ goal .

Following symbols are considered recursive:

 {addlist}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

  addlist(Cons(; S(; 0()),  xs); Cons(; x,  xs')) > Cons(; S(; x),  addlist(xs; xs'))
                                                                                     
  addlist(Cons(; x,  xs); Cons(; S(; 0()),  xs')) > Cons(; S(; x),  addlist(xs; xs'))
                                                                                     
                               addlist(ys; Nil()) > Nil()                            
                                                                                     
                        notEmpty(Cons(; x,  xs);) > True()                           
                                                                                     
                                 notEmpty(Nil();) > False()                          
                                                                                     
                                     goal(ys; xs) > addlist(ys; xs)                  
                                                                                     

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