WORST_CASE(?,O(1))
* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(f(a())) -> c(n__f(g(f(a()))))
        - Signature:
            {activate/1,f/1} / {a/0,c/1,g/1,n__f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f} and constructors {a,c,g,n__f}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__f(X)) -> c_2(f#(X))
          f#(X) -> c_3()
          f#(f(a())) -> c_4(f#(a()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__f(X)) -> c_2(f#(X))
            f#(X) -> c_3()
            f#(f(a())) -> c_4(f#(a()))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(f(a())) -> c(n__f(g(f(a()))))
        - Signature:
            {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#} and constructors {a,c,g,n__f}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4}
        by application of
          Pre({1,3,4}) = {2}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__f(X)) -> c_2(f#(X))
          3: f#(X) -> c_3()
          4: f#(f(a())) -> c_4(f#(a()))
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            f#(X) -> c_3()
            f#(f(a())) -> c_4(f#(a()))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(f(a())) -> c(n__f(g(f(a()))))
        - Signature:
            {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#} and constructors {a,c,g,n__f}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: activate#(n__f(X)) -> c_2(f#(X))
          2: activate#(X) -> c_1()
          3: f#(X) -> c_3()
          4: f#(f(a())) -> c_4(f#(a()))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__f(X)) -> c_2(f#(X))
            f#(X) -> c_3()
            f#(f(a())) -> c_4(f#(a()))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(f(a())) -> c(n__f(g(f(a()))))
        - Signature:
            {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#} and constructors {a,c,g,n__f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(X) -> c_1()
             
          
          2:W:activate#(n__f(X)) -> c_2(f#(X))
             -->_1 f#(X) -> c_3():3
          
          3:W:f#(X) -> c_3()
             
          
          4:W:f#(f(a())) -> c_4(f#(a()))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: f#(f(a())) -> c_4(f#(a()))
          2: activate#(n__f(X)) -> c_2(f#(X))
          3: f#(X) -> c_3()
          1: activate#(X) -> c_1()
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(f(a())) -> c(n__f(g(f(a()))))
        - Signature:
            {activate/1,f/1,activate#/1,f#/1} / {a/0,c/1,g/1,n__f/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#} and constructors {a,c,g,n__f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))