YES(?,O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict Trs:
  { f(0()) -> 1()
  , f(s(x)) -> g(x, s(x))
  , g(0(), y) -> y
  , g(s(x), y) -> g(x, s(+(y, x)))
  , g(s(x), y) -> g(x, +(y, s(x)))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping

 safe(f) = {}, safe(0) = {}, safe(1) = {}, safe(s) = {1},
 safe(g) = {2}, safe(+) = {1}

and precedence

 f > +, g > +, f ~ g .

Following symbols are considered recursive:

 {f, g, +}

The recursion depth is 2.

For your convenience, here are the satisfied ordering constraints:

       f(0();) > 1()               
                                   
    f(s(; x);) > g(x; s(; x))      
                                   
     g(0(); y) > y                 
                                   
  g(s(; x); y) > g(x; s(; +(x; y)))
                                   
  g(s(; x); y) > g(x; +(s(; x); y))
                                   
     +(0(); x) > x                 
                                   
  +(s(; y); x) > s(; +(y; x))      
                                   

Hurray, we answered YES(?,O(n^2))