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))

We add following dependency tuples:

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(x)) -> c_2(g^#(x, s(x)))
  , g^#(0(), y) -> c_3()
  , g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
  , g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
  , +^#(x, 0()) -> c_6()
  , +^#(x, s(y)) -> c_7(+^#(x, y)) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(x)) -> c_2(g^#(x, s(x)))
  , g^#(0(), y) -> c_3()
  , g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
  , g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
  , +^#(x, 0()) -> c_6()
  , +^#(x, s(y)) -> c_7(+^#(x, y)) }
Weak 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))

We estimate the number of application of {1,3,6} by applications of
Pre({1,3,6}) = {2,4,5,7}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(0()) -> c_1()
    , 2: f^#(s(x)) -> c_2(g^#(x, s(x)))
    , 3: g^#(0(), y) -> c_3()
    , 4: g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
    , 5: g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
    , 6: +^#(x, 0()) -> c_6()
    , 7: +^#(x, s(y)) -> c_7(+^#(x, y)) }

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

Strict DPs:
  { f^#(s(x)) -> c_2(g^#(x, s(x)))
  , g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
  , g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
  , +^#(x, s(y)) -> c_7(+^#(x, y)) }
Weak DPs:
  { f^#(0()) -> c_1()
  , g^#(0(), y) -> c_3()
  , +^#(x, 0()) -> c_6() }
Weak 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 following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ f^#(0()) -> c_1()
, g^#(0(), y) -> c_3()
, +^#(x, 0()) -> c_6() }

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

Strict DPs:
  { f^#(s(x)) -> c_2(g^#(x, s(x)))
  , g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
  , g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
  , +^#(x, s(y)) -> c_7(+^#(x, y)) }
Weak 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))

Consider the dependency graph

  1: f^#(s(x)) -> c_2(g^#(x, s(x)))
     -->_1 g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x))) :3
     -->_1 g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x)) :2
  
  2: g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
     -->_2 +^#(x, s(y)) -> c_7(+^#(x, y)) :4
     -->_1 g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x))) :3
     -->_1 g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x)) :2
  
  3: g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
     -->_2 +^#(x, s(y)) -> c_7(+^#(x, y)) :4
     -->_1 g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x))) :3
     -->_1 g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x)) :2
  
  4: +^#(x, s(y)) -> c_7(+^#(x, y))
     -->_1 +^#(x, s(y)) -> c_7(+^#(x, y)) :4
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { f^#(s(x)) -> c_2(g^#(x, s(x))) }


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

Strict DPs:
  { g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
  , g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
  , +^#(x, s(y)) -> c_7(+^#(x, y)) }
Weak 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))

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { +(x, 0()) -> x
    , +(x, s(y)) -> s(+(x, y)) }

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

Strict DPs:
  { g^#(s(x), y) -> c_4(g^#(x, s(+(y, x))), +^#(y, x))
  , g^#(s(x), y) -> c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))
  , +^#(x, s(y)) -> c_7(+^#(x, y)) }
Weak Trs:
  { +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

The following argument positions are usable:
  Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}, Uargs(c_7) = {1}

TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).

            [0] = [0]                      
                  [0]                      
                                           
        [s](x1) = [1 1] x1 + [0]           
                  [0 1]      [1]           
                                           
    [+](x1, x2) = [0]                      
                  [0]                      
                                           
  [g^#](x1, x2) = [1 3] x1 + [0]           
                  [0 0]      [0]           
                                           
  [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                  [0 0]      [0 0]      [0]
                                           
  [+^#](x1, x2) = [0 1] x2 + [0]           
                  [0 0]      [0]           
                                           
  [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                  [0 0]      [0 0]      [0]
                                           
      [c_7](x1) = [1 0] x1 + [0]           
                  [0 0]      [0]           

This order satisfies following ordering constraints:

  [g^#(s(x), y)] = [1 4] x + [3]                          
                   [0 0]     [0]                          
                 > [1 4] x + [1]                          
                   [0 0]     [0]                          
                 = [c_4(g^#(x, s(+(y, x))), +^#(y, x))]   
                                                          
  [g^#(s(x), y)] = [1 4] x + [3]                          
                   [0 0]     [0]                          
                 > [1 4] x + [2]                          
                   [0 0]     [0]                          
                 = [c_5(g^#(x, +(y, s(x))), +^#(y, s(x)))]
                                                          
  [+^#(x, s(y))] = [0 1] y + [1]                          
                   [0 0]     [0]                          
                 > [0 1] y + [0]                          
                   [0 0]     [0]                          
                 = [c_7(+^#(x, y))]                       
                                                          

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