WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + 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(s) = {1}
        
        Following symbols are considered usable:
          {+,double}
        TcT has computed the following interpretation:
               p(+) = 3 + x1
               p(0) = 1     
          p(double) = 6 + x1
               p(s) = x1    
        
        Following rules are strictly oriented:
           +(x,0()) = 3 + x 
                    > x     
                    = x     
        
          double(x) = 6 + x 
                    > 3 + x 
                    = +(x,x)
        
        double(0()) = 7     
                    > 1     
                    = 0()   
        
        
        Following rules are (at-least) weakly oriented:
           +(x,s(y)) =  3 + x          
                     >= 3 + x          
                     =  s(+(x,y))      
        
           +(s(x),y) =  3 + x          
                     >= 3 + x          
                     =  s(+(x,y))      
        
        double(s(x)) =  6 + x          
                     >= 6 + x          
                     =  s(s(double(x)))
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(s(x)) -> s(s(double(x)))
        - Weak TRS:
            +(x,0()) -> x
            double(x) -> +(x,x)
            double(0()) -> 0()
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + 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(s) = {1}
        
        Following symbols are considered usable:
          {+,double}
        TcT has computed the following interpretation:
               p(+) = [1] x1 + [1] x2 + [0]
               p(0) = [2]                  
          p(double) = [4] x1 + [0]         
               p(s) = [1] x1 + [4]         
        
        Following rules are strictly oriented:
        double(s(x)) = [4] x + [16]   
                     > [4] x + [8]    
                     = s(s(double(x)))
        
        
        Following rules are (at-least) weakly oriented:
           +(x,0()) =  [1] x + [2]        
                    >= [1] x + [0]        
                    =  x                  
        
          +(x,s(y)) =  [1] x + [1] y + [4]
                    >= [1] x + [1] y + [4]
                    =  s(+(x,y))          
        
          +(s(x),y) =  [1] x + [1] y + [4]
                    >= [1] x + [1] y + [4]
                    =  s(+(x,y))          
        
          double(x) =  [4] x + [0]        
                    >= [2] x + [0]        
                    =  +(x,x)             
        
        double(0()) =  [8]                
                    >= [2]                
                    =  0()                
        
** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + 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(s) = {1}
        
        Following symbols are considered usable:
          {+,double}
        TcT has computed the following interpretation:
               p(+) = [2] x1 + [2] x2 + [0]
               p(0) = [0]                  
          p(double) = [5] x1 + [1]         
               p(s) = [1] x1 + [4]         
        
        Following rules are strictly oriented:
        +(x,s(y)) = [2] x + [2] y + [8]
                  > [2] x + [2] y + [4]
                  = s(+(x,y))          
        
        +(s(x),y) = [2] x + [2] y + [8]
                  > [2] x + [2] y + [4]
                  = s(+(x,y))          
        
        
        Following rules are (at-least) weakly oriented:
            +(x,0()) =  [2] x + [0]    
                     >= [1] x + [0]    
                     =  x              
        
           double(x) =  [5] x + [1]    
                     >= [4] x + [0]    
                     =  +(x,x)         
        
         double(0()) =  [1]            
                     >= [0]            
                     =  0()            
        
        double(s(x)) =  [5] x + [21]   
                     >= [5] x + [9]    
                     =  s(s(double(x)))
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(s(x),y) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
        - Signature:
            {+/2,double/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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