YES(O(1),O(1))

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

Strict Trs:
  { f(a(), n__b(), X) -> f(X, X, X)
  , c() -> a()
  , c() -> b()
  , b() -> n__b()
  , activate(X) -> X
  , activate(n__b()) -> b() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#()) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#()) }
Strict Trs:
  { f(a(), n__b(), X) -> f(X, X, X)
  , c() -> a()
  , c() -> b()
  , b() -> n__b()
  , activate(X) -> X
  , activate(n__b()) -> b() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(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(1)).

Strict DPs:
  { f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

The following argument positions are usable:
  Uargs(c_3) = {1}, Uargs(c_6) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

                [a] = [1]         
                                  
             [n__b] = [2]         
                                  
  [f^#](x1, x2, x3) = [2] x3 + [2]
                                  
          [c_1](x1) = [1] x1 + [2]
                                  
              [c^#] = [1]         
                                  
              [c_2] = [0]         
                                  
          [c_3](x1) = [1] x1 + [1]
                                  
              [b^#] = [1]         
                                  
              [c_4] = [0]         
                                  
   [activate^#](x1) = [1] x1 + [1]
                                  
              [c_5] = [0]         
                                  
          [c_6](x1) = [1] x1 + [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(1),O(1)).

Strict DPs:
  { f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
  , c^#() -> c_3(b^#())
  , activate^#(n__b()) -> c_6(b^#()) }
Weak DPs:
  { c^#() -> c_2()
  , b^#() -> c_4()
  , activate^#(X) -> c_5() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {1,2,3} by applications of
Pre({1,2,3}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
    , 2: c^#() -> c_3(b^#())
    , 3: activate^#(n__b()) -> c_6(b^#())
    , 4: c^#() -> c_2()
    , 5: b^#() -> c_4()
    , 6: activate^#(X) -> c_5() }

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

Weak DPs:
  { f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ f^#(a(), n__b(), X) -> c_1(f^#(X, X, X))
, c^#() -> c_2()
, c^#() -> c_3(b^#())
, b^#() -> c_4()
, activate^#(X) -> c_5()
, activate^#(n__b()) -> c_6(b^#()) }

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

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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