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

We add the following weak dependency pairs:

Strict DPs: { f^#(g(x), y, y) -> c_1(f^#(x, 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^#(g(x), y, y) -> c_1(f^#(x, x, y)) }
Strict Trs: { f(g(x), y, y) -> g(f(x, x, y)) }
Obligation:
  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), y, y) -> c_1(f^#(x, x, y)) }
Obligation:
  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 the following constructor-restricted matrix
interpretation.

            [g](x1) = [1 0] x1 + [2]           
                      [0 1]      [1]           
                                               
  [f^#](x1, x2, x3) = [1 0] x1 + [2 2] x3 + [2]
                      [2 2]      [2 2]      [1]
                                               
          [c_1](x1) = [1 0] x1 + [1]           
                      [0 1]      [1]           

The following symbols are considered usable

  {f^#}

The order satisfies the following ordering constraints:

  [f^#(g(x), y, y)] = [1 0] x + [2 2] y + [4]
                      [2 2]     [2 2]     [7]
                    > [1 0] x + [2 2] y + [3]
                      [2 2]     [2 2]     [2]
                    = [c_1(f^#(x, x, y))]    
                                             

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)).

Weak DPs: { f^#(g(x), y, y) -> c_1(f^#(x, x, y)) }
Obligation:
  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), y, y) -> c_1(f^#(x, x, y)) }

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

Rules: Empty
Obligation:
  runtime complexity
Answer:
  YES(?,O(n^1))

We employ 'linear path analysis' using the following approximated
dependency graph:
empty


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