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__d(X)) -> d(X)
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(X)
            c(X) -> d(activate(X))
            d(X) -> n__d(X)
            f(X) -> n__f(X)
            f(f(X)) -> c(n__f(n__g(n__f(X))))
            g(X) -> n__g(X)
            h(X) -> c(n__d(X))
        - Signature:
            {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,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__d(X)) -> d(X)
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(X)
            c(X) -> d(activate(X))
            d(X) -> n__d(X)
            f(X) -> n__f(X)
            f(f(X)) -> c(n__f(n__g(n__f(X))))
            g(X) -> n__g(X)
            h(X) -> c(n__d(X))
        - Signature:
            {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
    + 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__d(X)) -> d(X)
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(X)
            c(X) -> d(activate(X))
            d(X) -> n__d(X)
            f(X) -> n__f(X)
            f(f(X)) -> c(n__f(n__g(n__f(X))))
            g(X) -> n__g(X)
            h(X) -> c(n__d(X))
        - Signature:
            {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
    + 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(d) = {1},
          uargs(f) = {1}
        
        Following symbols are considered usable:
          {activate,c,d,f,g,h}
        TcT has computed the following interpretation:
          p(activate) = 2*x1      
                 p(c) = 8*x1      
                 p(d) = 2*x1      
                 p(f) = x1        
                 p(g) = 0         
                 p(h) = 10 + 14*x1
              p(n__d) = x1        
              p(n__f) = x1        
              p(n__g) = 0         
        
        Following rules are strictly oriented:
        h(X) = 10 + 14*X 
             > 8*X       
             = c(n__d(X))
        
        
        Following rules are (at-least) weakly oriented:
              activate(X) =  2*X                   
                          >= X                     
                          =  X                     
        
        activate(n__d(X)) =  2*X                   
                          >= 2*X                   
                          =  d(X)                  
        
        activate(n__f(X)) =  2*X                   
                          >= 2*X                   
                          =  f(activate(X))        
        
        activate(n__g(X)) =  0                     
                          >= 0                     
                          =  g(X)                  
        
                     c(X) =  8*X                   
                          >= 4*X                   
                          =  d(activate(X))        
        
                     d(X) =  2*X                   
                          >= X                     
                          =  n__d(X)               
        
                     f(X) =  X                     
                          >= X                     
                          =  n__f(X)               
        
                  f(f(X)) =  X                     
                          >= 0                     
                          =  c(n__f(n__g(n__f(X))))
        
                     g(X) =  0                     
                          >= 0                     
                          =  n__g(X)               
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__d(X)) -> d(X)
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(X)
            c(X) -> d(activate(X))
            d(X) -> n__d(X)
            f(X) -> n__f(X)
            f(f(X)) -> c(n__f(n__g(n__f(X))))
            g(X) -> n__g(X)
        - Weak TRS:
            h(X) -> c(n__d(X))
        - Signature:
            {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
    + 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(d) = {1},
          uargs(f) = {1}
        
        Following symbols are considered usable:
          {activate,c,d,f,g,h}
        TcT has computed the following interpretation:
          p(activate) = 5 + 9*x1  
                 p(c) = 12 + 9*x1 
                 p(d) = 4 + x1    
                 p(f) = 15 + x1   
                 p(g) = 0         
                 p(h) = 14 + 12*x1
              p(n__d) = x1        
              p(n__f) = 2 + x1    
              p(n__g) = 0         
        
        Following rules are strictly oriented:
              activate(X) = 5 + 9*X       
                          > X             
                          = X             
        
        activate(n__d(X)) = 5 + 9*X       
                          > 4 + X         
                          = d(X)          
        
        activate(n__f(X)) = 23 + 9*X      
                          > 20 + 9*X      
                          = f(activate(X))
        
        activate(n__g(X)) = 5             
                          > 0             
                          = g(X)          
        
                     c(X) = 12 + 9*X      
                          > 9 + 9*X       
                          = d(activate(X))
        
                     d(X) = 4 + X         
                          > X             
                          = n__d(X)       
        
                     f(X) = 15 + X        
                          > 2 + X         
                          = n__f(X)       
        
        
        Following rules are (at-least) weakly oriented:
        f(f(X)) =  30 + X                
                >= 30                    
                =  c(n__f(n__g(n__f(X))))
        
           g(X) =  0                     
                >= 0                     
                =  n__g(X)               
        
           h(X) =  14 + 12*X             
                >= 12 + 9*X              
                =  c(n__d(X))            
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(X)) -> c(n__f(n__g(n__f(X))))
            g(X) -> n__g(X)
        - Weak TRS:
            activate(X) -> X
            activate(n__d(X)) -> d(X)
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(X)
            c(X) -> d(activate(X))
            d(X) -> n__d(X)
            f(X) -> n__f(X)
            h(X) -> c(n__d(X))
        - Signature:
            {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
    + 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(d) = {1},
          uargs(f) = {1}
        
        Following symbols are considered usable:
          {activate,c,d,f,g,h}
        TcT has computed the following interpretation:
          p(activate) = 9 + 2*x1 
                 p(c) = 10 + 2*x1
                 p(d) = x1       
                 p(f) = 15 + x1  
                 p(g) = 9        
                 p(h) = 10 + 8*x1
              p(n__d) = x1       
              p(n__f) = 8 + x1   
              p(n__g) = 0        
        
        Following rules are strictly oriented:
        f(f(X)) = 30 + X                
                > 26                    
                = c(n__f(n__g(n__f(X))))
        
           g(X) = 9                     
                > 0                     
                = n__g(X)               
        
        
        Following rules are (at-least) weakly oriented:
              activate(X) =  9 + 2*X       
                          >= X             
                          =  X             
        
        activate(n__d(X)) =  9 + 2*X       
                          >= X             
                          =  d(X)          
        
        activate(n__f(X)) =  25 + 2*X      
                          >= 24 + 2*X      
                          =  f(activate(X))
        
        activate(n__g(X)) =  9             
                          >= 9             
                          =  g(X)          
        
                     c(X) =  10 + 2*X      
                          >= 9 + 2*X       
                          =  d(activate(X))
        
                     d(X) =  X             
                          >= X             
                          =  n__d(X)       
        
                     f(X) =  15 + X        
                          >= 8 + X         
                          =  n__f(X)       
        
                     h(X) =  10 + 8*X      
                          >= 10 + 2*X      
                          =  c(n__d(X))    
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__d(X)) -> d(X)
            activate(n__f(X)) -> f(activate(X))
            activate(n__g(X)) -> g(X)
            c(X) -> d(activate(X))
            d(X) -> n__d(X)
            f(X) -> n__f(X)
            f(f(X)) -> c(n__f(n__g(n__f(X))))
            g(X) -> n__g(X)
            h(X) -> c(n__d(X))
        - Signature:
            {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,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))