WORST_CASE(?,O(n^1))
* Step 1: WeightGap 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:
        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(+) = [7] x1 + [0]
                 p(0) = [0]         
            p(double) = [7] x1 + [0]
                 p(s) = [1] x1 + [1]
          
          Following rules are strictly oriented:
             +(s(x),y) = [7] x + [7]    
                       > [7] x + [1]    
                       = s(+(x,y))      
          
          double(s(x)) = [7] x + [7]    
                       > [7] x + [2]    
                       = s(s(double(x)))
          
          
          Following rules are (at-least) weakly oriented:
             +(x,0()) =  [7] x + [0]
                      >= [1] x + [0]
                      =  x          
          
            +(x,s(y)) =  [7] x + [0]
                      >= [7] x + [1]
                      =  s(+(x,y))  
          
            double(x) =  [7] x + [0]
                      >= [7] x + [0]
                      =  +(x,x)     
          
          double(0()) =  [0]        
                      >= [0]        
                      =  0()        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(x) -> +(x,x)
            double(0()) -> 0()
        - Weak TRS:
            +(s(x),y) -> s(+(x,y))
            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 + [9] x2 + [0]
               p(0) = [0]                  
          p(double) = [12] x1 + [4]        
               p(s) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
          +(x,s(y)) = [2] x + [9] y + [18]
                    > [2] x + [9] y + [2] 
                    = s(+(x,y))           
        
          double(x) = [12] x + [4]        
                    > [11] x + [0]        
                    = +(x,x)              
        
        double(0()) = [4]                 
                    > [0]                 
                    = 0()                 
        
        
        Following rules are (at-least) weakly oriented:
            +(x,0()) =  [2] x + [0]        
                     >= [1] x + [0]        
                     =  x                  
        
           +(s(x),y) =  [2] x + [9] y + [4]
                     >= [2] x + [9] y + [2]
                     =  s(+(x,y))          
        
        double(s(x)) =  [12] x + [28]      
                     >= [12] x + [8]       
                     =  s(s(double(x)))    
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
        - Weak TRS:
            +(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(+) = [1] x1 + [8]
               p(0) = [0]         
          p(double) = [1] x1 + [8]
               p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        +(x,0()) = [1] x + [8]
                 > [1] x + [0]
                 = x          
        
        
        Following rules are (at-least) weakly oriented:
           +(x,s(y)) =  [1] x + [8]    
                     >= [1] x + [8]    
                     =  s(+(x,y))      
        
           +(s(x),y) =  [1] x + [8]    
                     >= [1] x + [8]    
                     =  s(+(x,y))      
        
           double(x) =  [1] x + [8]    
                     >= [1] x + [8]    
                     =  +(x,x)         
        
         double(0()) =  [8]            
                     >= [0]            
                     =  0()            
        
        double(s(x)) =  [1] x + [8]    
                     >= [1] x + [8]    
                     =  s(s(double(x)))
        
* 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))