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(X)) -> f(activate(X))
            activate(n__h(X)) -> h(activate(X))
            f(X) -> g(n__h(n__f(X)))
            f(X) -> n__f(X)
            h(X) -> n__h(X)
        - Signature:
            {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,h} and constructors {g,n__f,n__h}
    + 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(X)) -> f(activate(X))
            activate(n__h(X)) -> h(activate(X))
            f(X) -> g(n__h(n__f(X)))
            f(X) -> n__f(X)
            h(X) -> n__h(X)
        - Signature:
            {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,h} and constructors {g,n__f,n__h}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          activate(x){x -> n__f(x)} =
            activate(n__f(x)) ->^+ f(activate(x))
              = C[activate(x) = activate(x){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__h(X)) -> h(activate(X))
            f(X) -> g(n__h(n__f(X)))
            f(X) -> n__f(X)
            h(X) -> n__h(X)
        - Signature:
            {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,h} and constructors {g,n__f,n__h}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(f) = {1},
          uargs(h) = {1}
        
        Following symbols are considered usable:
          {activate,f,h}
        TcT has computed the following interpretation:
          p(activate) = 8*x1  
                 p(f) = 8 + x1
                 p(g) = 1 + x1
                 p(h) = 8 + x1
              p(n__f) = 2 + x1
              p(n__h) = 1 + x1
        
        Following rules are strictly oriented:
        activate(n__f(X)) = 16 + 8*X        
                          > 8 + 8*X         
                          = f(activate(X))  
        
                     f(X) = 8 + X           
                          > 4 + X           
                          = g(n__h(n__f(X)))
        
                     f(X) = 8 + X           
                          > 2 + X           
                          = n__f(X)         
        
                     h(X) = 8 + X           
                          > 1 + X           
                          = n__h(X)         
        
        
        Following rules are (at-least) weakly oriented:
              activate(X) =  8*X           
                          >= X             
                          =  X             
        
        activate(n__h(X)) =  8 + 8*X       
                          >= 8 + 8*X       
                          =  h(activate(X))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__h(X)) -> h(activate(X))
        - Weak TRS:
            activate(n__f(X)) -> f(activate(X))
            f(X) -> g(n__h(n__f(X)))
            f(X) -> n__f(X)
            h(X) -> n__h(X)
        - Signature:
            {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,h} and constructors {g,n__f,n__h}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(f) = {1},
          uargs(h) = {1}
        
        Following symbols are considered usable:
          {activate,f,h}
        TcT has computed the following interpretation:
          p(activate) = 2 + 4*x1
                 p(f) = 12 + x1 
                 p(g) = x1      
                 p(h) = 9 + x1  
              p(n__f) = 4 + x1  
              p(n__h) = 4 + x1  
        
        Following rules are strictly oriented:
              activate(X) = 2 + 4*X       
                          > X             
                          = X             
        
        activate(n__h(X)) = 18 + 4*X      
                          > 11 + 4*X      
                          = h(activate(X))
        
        
        Following rules are (at-least) weakly oriented:
        activate(n__f(X)) =  18 + 4*X        
                          >= 14 + 4*X        
                          =  f(activate(X))  
        
                     f(X) =  12 + X          
                          >= 8 + X           
                          =  g(n__h(n__f(X)))
        
                     f(X) =  12 + X          
                          >= 4 + X           
                          =  n__f(X)         
        
                     h(X) =  9 + X           
                          >= 4 + X           
                          =  n__h(X)         
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(activate(X))
            activate(n__h(X)) -> h(activate(X))
            f(X) -> g(n__h(n__f(X)))
            f(X) -> n__f(X)
            h(X) -> n__h(X)
        - Signature:
            {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,h} and constructors {g,n__f,n__h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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