YES(O(1),O(n^1))

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

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

We add following weak dependency pairs:

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

and mark the set of starting terms.

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

Strict DPs:
  { g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
  , g^#(0(), f(x, x)) -> c_2()
  , g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
  , g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) }
Strict Trs:
  { g(x, s(y)) -> g(f(x, y), 0())
  , g(0(), f(x, x)) -> x
  , g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0()))
  , g(s(x), y) -> g(f(x, y), 0()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

No rule is usable, rules are removed from the input problem.

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

Strict DPs:
  { g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
  , g^#(0(), f(x, x)) -> c_2()
  , g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
  , g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_1) = {1}, Uargs(c_3) = {1, 2}, Uargs(c_4) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

            [0] = [1]                  
                                       
    [f](x1, x2) = [1] x1 + [1] x2 + [0]
                                       
        [s](x1) = [1] x1 + [0]         
                                       
  [g^#](x1, x2) = [1] x1 + [1] x2 + [0]
                                       
      [c_1](x1) = [1] x1 + [0]         
                                       
          [c_2] = [0]                  
                                       
  [c_3](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
      [c_4](x1) = [1] x1 + [0]         

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

Strict DPs:
  { g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
  , g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
  , g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) }
Weak DPs: { g^#(0(), f(x, x)) -> c_2() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ g^#(0(), f(x, x)) -> c_2() }

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

Strict DPs:
  { g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
  , g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
  , g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

Consider the dependency graph

  1: g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
     -->_1 g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0())) :2
  
  2: g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
     -->_2 g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) :3
     -->_1 g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) :3
     -->_2 g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0())) :2
     -->_1 g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0())) :2
  
  3: g^#(s(x), y) -> c_4(g^#(f(x, y), 0()))
     -->_1 g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0())) :2
  

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).

  { g^#(x, s(y)) -> c_1(g^#(f(x, y), 0())) }


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

Strict DPs:
  { g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
  , g^#(s(x), y) -> c_4(g^#(f(x, y), 0())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following argument positions are usable:
  Uargs(c_3) = {1, 2}, Uargs(c_4) = {1}

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

            [0] = [0]                  
                                       
    [f](x1, x2) = [1] x1 + [1] x2 + [1]
                                       
        [s](x1) = [1] x1 + [2]         
                                       
  [g^#](x1, x2) = [2] x1 + [2] x2 + [0]
                                       
  [c_3](x1, x2) = [1] x1 + [1] x2 + [1]
                                       
      [c_4](x1) = [1] x1 + [1]         

This order satisfies following ordering constraints:

  [g^#(f(x, y), 0())] = [2] x + [2] y + [2]            
                      > [2] x + [2] y + [1]            
                      = [c_3(g^#(x, 0()), g^#(y, 0()))]
                                                       
       [g^#(s(x), y)] = [2] x + [2] y + [4]            
                      > [2] x + [2] y + [3]            
                      = [c_4(g^#(f(x, y), 0()))]       
                                                       

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