WORST_CASE(?,O(n^1))
* Step 1: Sum WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> if(X,c(),n__f(true()))
            f(X) -> n__f(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> X
        - Signature:
            {activate/1,f/1,if/3} / {c/0,false/0,n__f/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,if} and constructors {c,false,n__f,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> if(X,c(),n__f(true()))
            f(X) -> n__f(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> X
        - Signature:
            {activate/1,f/1,if/3} / {c/0,false/0,n__f/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,if} and constructors {c,false,n__f,true}
    + 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:
          none
        
        Following symbols are considered usable:
          {activate,f,if}
        TcT has computed the following interpretation:
          p(activate) = 8*x1       
                 p(c) = 0          
                 p(f) = 8          
             p(false) = 0          
                p(if) = 8*x2 + 8*x3
              p(n__f) = 1          
              p(true) = 0          
        
        Following rules are strictly oriented:
        f(X) = 8      
             > 1      
             = n__f(X)
        
        
        Following rules are (at-least) weakly oriented:
              activate(X) =  8*X                   
                          >= X                     
                          =  X                     
        
        activate(n__f(X)) =  8                     
                          >= 8                     
                          =  f(X)                  
        
                     f(X) =  8                     
                          >= 8                     
                          =  if(X,c(),n__f(true()))
        
          if(false(),X,Y) =  8*X + 8*Y             
                          >= 8*Y                   
                          =  activate(Y)           
        
           if(true(),X,Y) =  8*X + 8*Y             
                          >= X                     
                          =  X                     
        
* Step 3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> if(X,c(),n__f(true()))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> X
        - Weak TRS:
            f(X) -> n__f(X)
        - Signature:
            {activate/1,f/1,if/3} / {c/0,false/0,n__f/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,if} and constructors {c,false,n__f,true}
    + 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:
          none
        
        Following symbols are considered usable:
          {activate,f,if}
        TcT has computed the following interpretation:
          p(activate) = 2 + 3*x1          
                 p(c) = 0                 
                 p(f) = 1 + 3*x1          
             p(false) = 10                
                p(if) = 2*x1 + 8*x2 + 3*x3
              p(n__f) = x1                
              p(true) = 0                 
        
        Following rules are strictly oriented:
              activate(X) = 2 + 3*X               
                          > X                     
                          = X                     
        
        activate(n__f(X)) = 2 + 3*X               
                          > 1 + 3*X               
                          = f(X)                  
        
                     f(X) = 1 + 3*X               
                          > 2*X                   
                          = if(X,c(),n__f(true()))
        
          if(false(),X,Y) = 20 + 8*X + 3*Y        
                          > 2 + 3*Y               
                          = activate(Y)           
        
        
        Following rules are (at-least) weakly oriented:
                  f(X) =  1 + 3*X  
                       >= X        
                       =  n__f(X)  
        
        if(true(),X,Y) =  8*X + 3*Y
                       >= X        
                       =  X        
        
* Step 4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            if(true(),X,Y) -> X
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> if(X,c(),n__f(true()))
            f(X) -> n__f(X)
            if(false(),X,Y) -> activate(Y)
        - Signature:
            {activate/1,f/1,if/3} / {c/0,false/0,n__f/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,if} and constructors {c,false,n__f,true}
    + 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:
          none
        
        Following symbols are considered usable:
          {activate,f,if}
        TcT has computed the following interpretation:
          p(activate) = 12 + 4*x1           
                 p(c) = 1                   
                 p(f) = 14 + 4*x1           
             p(false) = 10                  
                p(if) = 6 + x1 + 4*x2 + 4*x3
              p(n__f) = 1 + x1              
              p(true) = 0                   
        
        Following rules are strictly oriented:
        if(true(),X,Y) = 6 + 4*X + 4*Y
                       > X            
                       = X            
        
        
        Following rules are (at-least) weakly oriented:
              activate(X) =  12 + 4*X              
                          >= X                     
                          =  X                     
        
        activate(n__f(X)) =  16 + 4*X              
                          >= 14 + 4*X              
                          =  f(X)                  
        
                     f(X) =  14 + 4*X              
                          >= 14 + X                
                          =  if(X,c(),n__f(true()))
        
                     f(X) =  14 + 4*X              
                          >= 1 + X                 
                          =  n__f(X)               
        
          if(false(),X,Y) =  16 + 4*X + 4*Y        
                          >= 12 + 4*Y              
                          =  activate(Y)           
        
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> if(X,c(),n__f(true()))
            f(X) -> n__f(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> X
        - Signature:
            {activate/1,f/1,if/3} / {c/0,false/0,n__f/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,if} and constructors {c,false,n__f,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))