MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(activate(X))
            f(X) -> cons(X,n__f(n__g(X)))
            f(X) -> n__f(X)
            g(X) -> n__g(X)
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,f/1,g/1,sel/2} / {0/0,cons/2,n__f/1,n__g/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,g,sel} and constructors {0,cons,n__f,n__g,s}
    + 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#(activate(X)),activate#(X))
          activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
          f#(X) -> c_4()
          f#(X) -> c_5()
          g#(X) -> c_6()
          g#(0()) -> c_7()
          g#(s(X)) -> c_8(g#(X))
          sel#(0(),cons(X,Y)) -> c_9()
          sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
            f#(X) -> c_4()
            f#(X) -> c_5()
            g#(X) -> c_6()
            g#(0()) -> c_7()
            g#(s(X)) -> c_8(g#(X))
            sel#(0(),cons(X,Y)) -> c_9()
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(activate(X))
            f(X) -> cons(X,n__f(n__g(X)))
            f(X) -> n__f(X)
            g(X) -> n__g(X)
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,n__g/1,s/1,c_1/0,c_2/2,c_3/2
            ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g
            ,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__f(X)) -> f(activate(X))
          activate(n__g(X)) -> g(activate(X))
          f(X) -> cons(X,n__f(n__g(X)))
          f(X) -> n__f(X)
          g(X) -> n__g(X)
          g(0()) -> s(0())
          g(s(X)) -> s(s(g(X)))
          activate#(X) -> c_1()
          activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
          activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
          f#(X) -> c_4()
          f#(X) -> c_5()
          g#(X) -> c_6()
          g#(0()) -> c_7()
          g#(s(X)) -> c_8(g#(X))
          sel#(0(),cons(X,Y)) -> c_9()
          sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
            f#(X) -> c_4()
            f#(X) -> c_5()
            g#(X) -> c_6()
            g#(0()) -> c_7()
            g#(s(X)) -> c_8(g#(X))
            sel#(0(),cons(X,Y)) -> c_9()
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(activate(X))
            f(X) -> cons(X,n__f(n__g(X)))
            f(X) -> n__f(X)
            g(X) -> n__g(X)
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,n__g/1,s/1,c_1/0,c_2/2,c_3/2
            ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g
            ,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,6,7,9}
        by application of
          Pre({1,4,5,6,7,9}) = {2,3,8,10}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
          3: activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
          4: f#(X) -> c_4()
          5: f#(X) -> c_5()
          6: g#(X) -> c_6()
          7: g#(0()) -> c_7()
          8: g#(s(X)) -> c_8(g#(X))
          9: sel#(0(),cons(X,Y)) -> c_9()
          10: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
            g#(s(X)) -> c_8(g#(X))
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
        - Weak DPs:
            activate#(X) -> c_1()
            f#(X) -> c_4()
            f#(X) -> c_5()
            g#(X) -> c_6()
            g#(0()) -> c_7()
            sel#(0(),cons(X,Y)) -> c_9()
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(activate(X))
            f(X) -> cons(X,n__f(n__g(X)))
            f(X) -> n__f(X)
            g(X) -> n__g(X)
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,n__g/1,s/1,c_1/0,c_2/2,c_3/2
            ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g
            ,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_1 f#(X) -> c_5():7
             -->_1 f#(X) -> c_4():6
             -->_2 activate#(X) -> c_1():5
             -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1
          
          2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
             -->_1 g#(s(X)) -> c_8(g#(X)):3
             -->_1 g#(0()) -> c_7():9
             -->_1 g#(X) -> c_6():8
             -->_2 activate#(X) -> c_1():5
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1
          
          3:S:g#(s(X)) -> c_8(g#(X))
             -->_1 g#(0()) -> c_7():9
             -->_1 g#(X) -> c_6():8
             -->_1 g#(s(X)) -> c_8(g#(X)):3
          
          4:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
             -->_1 sel#(0(),cons(X,Y)) -> c_9():10
             -->_2 activate#(X) -> c_1():5
             -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):4
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1
          
          5:W:activate#(X) -> c_1()
             
          
          6:W:f#(X) -> c_4()
             
          
          7:W:f#(X) -> c_5()
             
          
          8:W:g#(X) -> c_6()
             
          
          9:W:g#(0()) -> c_7()
             
          
          10:W:sel#(0(),cons(X,Y)) -> c_9()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: sel#(0(),cons(X,Y)) -> c_9()
          6: f#(X) -> c_4()
          7: f#(X) -> c_5()
          5: activate#(X) -> c_1()
          8: g#(X) -> c_6()
          9: g#(0()) -> c_7()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
            g#(s(X)) -> c_8(g#(X))
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(activate(X))
            f(X) -> cons(X,n__f(n__g(X)))
            f(X) -> n__f(X)
            g(X) -> n__g(X)
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,n__g/1,s/1,c_1/0,c_2/2,c_3/2
            ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g
            ,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X))
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1
          
          2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
             -->_1 g#(s(X)) -> c_8(g#(X)):3
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1
          
          3:S:g#(s(X)) -> c_8(g#(X))
             -->_1 g#(s(X)) -> c_8(g#(X)):3
          
          4:S:sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
             -->_1 sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z)):4
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X)) -> c_2(f#(activate(X)),activate#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          activate#(n__f(X)) -> c_2(activate#(X))
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          activate#(n__f(X)) -> c_2(activate#(X))
          activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
          g#(s(X)) -> c_8(g#(X))
          sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)),activate#(Z))
      - Weak TRS:
          activate(X) -> X
          activate(n__f(X)) -> f(activate(X))
          activate(n__g(X)) -> g(activate(X))
          f(X) -> cons(X,n__f(n__g(X)))
          f(X) -> n__f(X)
          g(X) -> n__g(X)
          g(0()) -> s(0())
          g(s(X)) -> s(s(g(X)))
      - Signature:
          {activate/1,f/1,g/1,sel/2,activate#/1,f#/1,g#/1,sel#/2} / {0/0,cons/2,n__f/1,n__g/1,s/1,c_1/0,c_2/1,c_3/2
          ,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {activate#,f#,g#,sel#} and constructors {0,cons,n__f,n__g
          ,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE