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

Strict Trs:
  { f(a()) -> f(c(a()))
  , f(a()) -> f(d(a()))
  , f(c(X)) -> X
  , f(c(a())) -> f(d(b()))
  , f(c(b())) -> f(d(a()))
  , f(d(X)) -> X
  , e(g(X)) -> e(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following weak dependency pairs:

Strict DPs:
  { f^#(a()) -> c_1(f^#(c(a())))
  , f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(X)) -> c_3()
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a())))
  , f^#(d(X)) -> c_6()
  , e^#(g(X)) -> c_7(e^#(X)) }

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^#(a()) -> c_1(f^#(c(a())))
  , f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(X)) -> c_3()
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a())))
  , f^#(d(X)) -> c_6()
  , e^#(g(X)) -> c_7(e^#(X)) }
Strict Trs:
  { f(a()) -> f(c(a()))
  , f(a()) -> f(d(a()))
  , f(c(X)) -> X
  , f(c(a())) -> f(d(b()))
  , f(c(b())) -> f(d(a()))
  , f(d(X)) -> X
  , e(g(X)) -> e(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^#(a()) -> c_1(f^#(c(a())))
  , f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(X)) -> c_3()
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a())))
  , f^#(d(X)) -> c_6()
  , e^#(g(X)) -> c_7(e^#(X)) }
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_2) = {1}, Uargs(c_4) = {1},
  Uargs(c_5) = {1}, Uargs(c_7) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

        [a] = [0]           
              [0]           
                            
    [c](x1) = [0]           
              [0]           
                            
    [d](x1) = [0]           
              [0]           
                            
        [b] = [0]           
              [0]           
                            
    [g](x1) = [1 0] x1 + [0]
              [0 1]      [1]
                            
  [f^#](x1) = [0]           
              [0]           
                            
  [c_1](x1) = [1 0] x1 + [2]
              [0 1]      [2]
                            
  [c_2](x1) = [1 0] x1 + [2]
              [0 1]      [2]
                            
      [c_3] = [0]           
              [0]           
                            
  [c_4](x1) = [1 0] x1 + [2]
              [0 1]      [0]
                            
  [c_5](x1) = [1 0] x1 + [2]
              [0 1]      [2]
                            
      [c_6] = [0]           
              [0]           
                            
  [e^#](x1) = [1 1] x1 + [0]
              [0 0]      [0]
                            
  [c_7](x1) = [1 0] x1 + [0]
              [0 1]      [0]

The order satisfies the following ordering constraints:

     [f^#(a())] =  [0]               
                   [0]               
                ?  [2]               
                   [2]               
                =  [c_1(f^#(c(a())))]
                                     
     [f^#(a())] =  [0]               
                   [0]               
                ?  [2]               
                   [2]               
                =  [c_2(f^#(d(a())))]
                                     
    [f^#(c(X))] =  [0]               
                   [0]               
                >= [0]               
                   [0]               
                =  [c_3()]           
                                     
  [f^#(c(a()))] =  [0]               
                   [0]               
                ?  [2]               
                   [0]               
                =  [c_4(f^#(d(b())))]
                                     
  [f^#(c(b()))] =  [0]               
                   [0]               
                ?  [2]               
                   [2]               
                =  [c_5(f^#(d(a())))]
                                     
    [f^#(d(X))] =  [0]               
                   [0]               
                >= [0]               
                   [0]               
                =  [c_6()]           
                                     
    [e^#(g(X))] =  [1 1] X + [1]     
                   [0 0]     [0]     
                >  [1 1] X + [0]     
                   [0 0]     [0]     
                =  [c_7(e^#(X))]     
                                     

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()) -> c_1(f^#(c(a())))
  , f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(X)) -> c_3()
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a())))
  , f^#(d(X)) -> c_6() }
Weak DPs: { e^#(g(X)) -> c_7(e^#(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {3,6} by applications of
Pre({3,6}) = {1,2,4,5}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(a()) -> c_1(f^#(c(a())))
    , 2: f^#(a()) -> c_2(f^#(d(a())))
    , 3: f^#(c(X)) -> c_3()
    , 4: f^#(c(a())) -> c_4(f^#(d(b())))
    , 5: f^#(c(b())) -> c_5(f^#(d(a())))
    , 6: f^#(d(X)) -> c_6()
    , 7: e^#(g(X)) -> c_7(e^#(X)) }

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

Strict DPs:
  { f^#(a()) -> c_1(f^#(c(a())))
  , f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a()))) }
Weak DPs:
  { f^#(c(X)) -> c_3()
  , f^#(d(X)) -> c_6()
  , e^#(g(X)) -> c_7(e^#(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {2,3,4} by applications of
Pre({2,3,4}) = {1}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(a()) -> c_1(f^#(c(a())))
    , 2: f^#(a()) -> c_2(f^#(d(a())))
    , 3: f^#(c(a())) -> c_4(f^#(d(b())))
    , 4: f^#(c(b())) -> c_5(f^#(d(a())))
    , 5: f^#(c(X)) -> c_3()
    , 6: f^#(d(X)) -> c_6()
    , 7: e^#(g(X)) -> c_7(e^#(X)) }

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

Strict DPs: { f^#(a()) -> c_1(f^#(c(a()))) }
Weak DPs:
  { f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(X)) -> c_3()
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a())))
  , f^#(d(X)) -> c_6()
  , e^#(g(X)) -> c_7(e^#(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: f^#(a()) -> c_1(f^#(c(a())))
    , 2: f^#(a()) -> c_2(f^#(d(a())))
    , 3: f^#(c(X)) -> c_3()
    , 4: f^#(c(a())) -> c_4(f^#(d(b())))
    , 5: f^#(c(b())) -> c_5(f^#(d(a())))
    , 6: f^#(d(X)) -> c_6()
    , 7: e^#(g(X)) -> c_7(e^#(X)) }

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

Weak DPs:
  { f^#(a()) -> c_1(f^#(c(a())))
  , f^#(a()) -> c_2(f^#(d(a())))
  , f^#(c(X)) -> c_3()
  , f^#(c(a())) -> c_4(f^#(d(b())))
  , f^#(c(b())) -> c_5(f^#(d(a())))
  , f^#(d(X)) -> c_6()
  , e^#(g(X)) -> c_7(e^#(X)) }
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()) -> c_1(f^#(c(a())))
, f^#(a()) -> c_2(f^#(d(a())))
, f^#(c(X)) -> c_3()
, f^#(c(a())) -> c_4(f^#(d(b())))
, f^#(c(b())) -> c_5(f^#(d(a())))
, f^#(d(X)) -> c_6()
, e^#(g(X)) -> c_7(e^#(X)) }

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(n^1))