* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(X)
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#() -> c_1()
          activate#(X) -> c_2()
          activate#(n__a()) -> c_3(a#())
          activate#(n__f(X)) -> c_4(f#(X))
          activate#(n__g(X)) -> c_5(g#(X))
          f#(X) -> c_6()
          f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
          g#(X) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            a#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__a()) -> c_3(a#())
            activate#(n__f(X)) -> c_4(f#(X))
            activate#(n__g(X)) -> c_5(g#(X))
            f#(X) -> c_6()
            f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
            g#(X) -> c_8()
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(X)
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f(X) -> n__f(X)
          a#() -> c_1()
          activate#(X) -> c_2()
          activate#(n__a()) -> c_3(a#())
          activate#(n__f(X)) -> c_4(f#(X))
          activate#(n__g(X)) -> c_5(g#(X))
          f#(X) -> c_6()
          f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
          g#(X) -> c_8()
* Step 3: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            a#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__a()) -> c_3(a#())
            activate#(n__f(X)) -> c_4(f#(X))
            activate#(n__g(X)) -> c_5(g#(X))
            f#(X) -> c_6()
            f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
            g#(X) -> c_8()
        - Weak TRS:
            f(X) -> n__f(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:a#() -> c_1()
             
          
          2:S:activate#(X) -> c_2()
             
          
          3:S:activate#(n__a()) -> c_3(a#())
             -->_1 a#() -> c_1():1
          
          4:S:activate#(n__f(X)) -> c_4(f#(X))
             -->_1 f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a())):7
             -->_1 f#(X) -> c_6():6
          
          5:S:activate#(n__g(X)) -> c_5(g#(X))
             -->_1 g#(X) -> c_8():8
          
          6:S:f#(X) -> c_6()
             
          
          7:S:f#(n__f(n__a())) -> c_7(f#(n__g(f(n__a()))),f#(n__a()))
             -->_2 f#(X) -> c_6():6
             -->_1 f#(X) -> c_6():6
          
          8:S:g#(X) -> c_8()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(X) -> n__f(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/2,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))