WORST_CASE(?,O(n^1))
* Step 1: NaturalMI 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:
        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 + [4] x2 + [0]
               p(0) = [4]                  
          p(double) = [6] x1 + [2]         
               p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
           +(x,0()) = [2] x + [16]
                    > [1] x + [0] 
                    = x           
        
          double(x) = [6] x + [2] 
                    > [6] x + [0] 
                    = +(x,x)      
        
        double(0()) = [26]        
                    > [4]         
                    = 0()         
        
        
        Following rules are (at-least) weakly oriented:
           +(x,s(y)) =  [2] x + [4] y + [0]
                     >= [2] x + [4] y + [0]
                     =  s(+(x,y))          
        
           +(s(x),y) =  [2] x + [4] y + [0]
                     >= [2] x + [4] y + [0]
                     =  s(+(x,y))          
        
        double(s(x)) =  [6] x + [2]        
                     >= [6] x + [2]        
                     =  s(s(double(x)))    
        
* Step 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 3: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
                 p(+) = [4] x1 + [2] x2 + [2]
                 p(0) = [4]                  
            p(double) = [6] x1 + [2]         
                 p(s) = [1] x1 + [2]         
          
          Following rules are strictly oriented:
          +(x,s(y)) = [4] x + [2] y + [6] 
                    > [4] x + [2] y + [4] 
                    = s(+(x,y))           
          
          +(s(x),y) = [4] x + [2] y + [10]
                    > [4] x + [2] y + [4] 
                    = s(+(x,y))           
          
          
          Following rules are (at-least) weakly oriented:
              +(x,0()) =  [4] x + [10]   
                       >= [1] x + [0]    
                       =  x              
          
             double(x) =  [6] x + [2]    
                       >= [6] x + [2]    
                       =  +(x,x)         
          
           double(0()) =  [26]           
                       >= [4]            
                       =  0()            
          
          double(s(x)) =  [6] x + [14]   
                       >= [6] x + [6]    
                       =  s(s(double(x)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 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(?,O(n^1))