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

We add following weak dependency pairs:

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

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

TcT has computed following constructor-restricted matrix
interpretation.

    [g](x1) = [1] x1 + [1]
                          
  [f^#](x1) = [1] x1 + [2]
                          
  [c_1](x1) = [1] x1 + [1]
                          
      [c_2] = [2]         

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: { f^#(g(x)) -> c_1(f^#(x)) }
Weak DPs: { f^#(g(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.

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

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

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

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

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

    [g](x1) = [1] x1 + [3]
                          
  [f^#](x1) = [1] x1 + [3]
                          
  [c_1](x1) = [1] x1 + [2]

This order satisfies following ordering constraints:

  [f^#(g(x))] = [1] x + [6]  
              > [1] x + [5]  
              = [c_1(f^#(x))]
                             

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