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

We add following weak dependency pairs:

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

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^#(x, a()) -> c_1()
  , f^#(x, g(y)) -> c_2(f^#(g(x), y)) }
Strict Trs:
  { f(x, a()) -> x
  , f(x, g(y)) -> f(g(x), y) }
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^#(x, a()) -> c_1()
  , f^#(x, g(y)) -> c_2(f^#(g(x), y)) }
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_2) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

            [a] = [2]         
                              
        [g](x1) = [1] x1 + [0]
                              
  [f^#](x1, x2) = [2] x1 + [1]
                              
          [c_1] = [0]         
                              
      [c_2](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: { f^#(x, g(y)) -> c_2(f^#(g(x), y)) }
Weak DPs: { f^#(x, a()) -> c_1() }
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^#(x, a()) -> c_1() }

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

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

The following argument positions are usable:
  Uargs(c_2) = {1}

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

        [g](x1) = [1] x1 + [2]
                              
  [f^#](x1, x2) = [2] x2 + [0]
                              
      [c_2](x1) = [1] x1 + [3]

This order satisfies following ordering constraints:

  [f^#(x, g(y))] = [2] y + [4]        
                 > [2] y + [3]        
                 = [c_2(f^#(g(x), y))]
                                      

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