WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1} / {n__f/2,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,g} and constructors {n__f,n__g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1} / {n__f/2,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,g} and constructors {n__f,n__g}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          activate(x){x -> n__f(x,y)} =
            activate(n__f(x,y)) ->^+ f(activate(x),y)
              = C[activate(x) = activate(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1} / {n__f/2,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,g} and constructors {n__f,n__g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
          activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
          f#(X1,X2) -> c_4()
          f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y))
          g#(X) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
            f#(X1,X2) -> c_4()
            f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y))
            g#(X) -> c_6()
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,6}
        by application of
          Pre({1,4,5,6}) = {2,3}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
          3: activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
          4: f#(X1,X2) -> c_4()
          5: f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y))
          6: g#(X) -> c_6()
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            f#(X1,X2) -> c_4()
            f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y))
            g#(X) -> c_6()
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_1 f#(X1,X2) -> c_4():4
             -->_2 activate#(X) -> c_1():3
             -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1
          
          2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
             -->_1 g#(X) -> c_6():6
             -->_2 activate#(X) -> c_1():3
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1
          
          3:W:activate#(X) -> c_1()
             
          
          4:W:f#(X1,X2) -> c_4()
             
          
          5:W:f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y))
             
          
          6:W:g#(X) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: f#(g(X),Y) -> c_5(f#(X,n__f(n__g(X),activate(Y))),activate#(Y))
          4: f#(X1,X2) -> c_4()
          3: activate#(X) -> c_1()
          6: g#(X) -> c_6()
** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
            activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1))
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1
          
          2:S:activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X))
             -->_2 activate#(n__g(X)) -> c_3(g#(activate(X)),activate#(X)):2
             -->_2 activate#(n__f(X1,X2)) -> c_2(f#(activate(X1),X2),activate#(X1)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          activate#(n__f(X1,X2)) -> c_2(activate#(X1))
          activate#(n__g(X)) -> c_3(activate#(X))
** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X1,X2)) -> c_2(activate#(X1))
            activate#(n__g(X)) -> c_3(activate#(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X1,X2)) -> f(activate(X1),X2)
            activate(n__g(X)) -> g(activate(X))
            f(X1,X2) -> n__f(X1,X2)
            f(g(X),Y) -> f(X,n__f(n__g(X),activate(Y)))
            g(X) -> n__g(X)
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(n__f(X1,X2)) -> c_2(activate#(X1))
          activate#(n__g(X)) -> c_3(activate#(X))
** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X1,X2)) -> c_2(activate#(X1))
            activate#(n__g(X)) -> c_3(activate#(X))
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {activate#,f#,g#}
        TcT has computed the following interpretation:
           p(activate) = [2]                  
                  p(f) = [1] x1 + [0]         
                  p(g) = [1] x1 + [2]         
               p(n__f) = [1] x1 + [1]         
               p(n__g) = [1] x1 + [2]         
          p(activate#) = [8] x1 + [2]         
                 p(f#) = [1] x1 + [1] x2 + [2]
                 p(g#) = [1] x1 + [0]         
                p(c_1) = [1]                  
                p(c_2) = [1] x1 + [8]         
                p(c_3) = [1] x1 + [3]         
                p(c_4) = [1]                  
                p(c_5) = [1]                  
                p(c_6) = [1]                  
        
        Following rules are strictly oriented:
        activate#(n__g(X)) = [8] X + [18]     
                           > [8] X + [5]      
                           = c_3(activate#(X))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__f(X1,X2)) =  [8] X1 + [10]     
                               >= [8] X1 + [10]     
                               =  c_2(activate#(X1))
        
** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X1,X2)) -> c_2(activate#(X1))
        - Weak DPs:
            activate#(n__g(X)) -> c_3(activate#(X))
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {activate#,f#,g#}
        TcT has computed the following interpretation:
           p(activate) = [1] x1 + [1] 
                  p(f) = [1]          
                  p(g) = [1] x1 + [0] 
               p(n__f) = [1] x1 + [1] 
               p(n__g) = [1] x1 + [11]
          p(activate#) = [1] x1 + [2] 
                 p(f#) = [0]          
                 p(g#) = [2] x1 + [1] 
                p(c_1) = [0]          
                p(c_2) = [1] x1 + [0] 
                p(c_3) = [1] x1 + [4] 
                p(c_4) = [4]          
                p(c_5) = [1] x1 + [0] 
                p(c_6) = [1]          
        
        Following rules are strictly oriented:
        activate#(n__f(X1,X2)) = [1] X1 + [3]      
                               > [1] X1 + [2]      
                               = c_2(activate#(X1))
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__g(X)) =  [1] X + [13]     
                           >= [1] X + [6]      
                           =  c_3(activate#(X))
        
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__f(X1,X2)) -> c_2(activate#(X1))
            activate#(n__g(X)) -> c_3(activate#(X))
        - Signature:
            {activate/1,f/2,g/1,activate#/1,f#/2,g#/1} / {n__f/2,n__g/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,g#} and constructors {n__f,n__g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))