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

We add following weak dependency pairs:

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

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

TcT has computed following constructor-restricted matrix
interpretation.

    [s](x1) = [1] x1 + [0]
                          
        [0] = [0]         
                          
  [g^#](x1) = [0]         
                          
  [c_1](x1) = [1] x1 + [1]
                          
  [f^#](x1) = [1]         
                          
      [c_2] = [1]         
                          
  [c_3](x1) = [1] x1 + [1]
                          
      [c_4] = [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^#(s(x)) -> c_1(f^#(x))
  , g^#(0()) -> c_2()
  , f^#(s(x)) -> c_3(g^#(x)) }
Weak DPs: { f^#(0()) -> c_4() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

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

  DPs:
    { 1: g^#(s(x)) -> c_1(f^#(x))
    , 2: g^#(0()) -> c_2()
    , 3: f^#(s(x)) -> c_3(g^#(x))
    , 4: f^#(0()) -> c_4() }

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

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

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

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

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

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

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

This order satisfies following ordering constraints:

  [g^#(s(x))] = [2] x + [4]  
              > [2] x + [1]  
              = [c_1(f^#(x))]
                             
  [f^#(s(x))] = [2] x + [4]  
              > [2] x + [1]  
              = [c_3(g^#(x))]
                             

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