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(+) = [6] x1 + [0]
                 p(0) = [1]         
            p(double) = [6] x1 + [8]
                 p(s) = [1] x1 + [3]
          
          Following rules are strictly oriented:
             +(s(x),y) = [6] x + [18]   
                       > [6] x + [3]    
                       = s(+(x,y))      
          
             double(x) = [6] x + [8]    
                       > [6] x + [0]    
                       = +(x,x)         
          
           double(0()) = [14]           
                       > [1]            
                       = 0()            
          
          double(s(x)) = [6] x + [26]   
                       > [6] x + [14]   
                       = s(s(double(x)))
          
          
          Following rules are (at-least) weakly oriented:
           +(x,0()) =  [6] x + [0]
                    >= [1] x + [0]
                    =  x          
          
          +(x,s(y)) =  [6] x + [0]
                    >= [6] x + [3]
                    =  s(+(x,y))  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Weak TRS:
            +(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(+) = [3] x1 + [2] x2 + [7]
                 p(0) = [1]                  
            p(double) = [5] x1 + [9]         
                 p(s) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
           +(x,0()) = [3] x + [9]        
                    > [1] x + [0]        
                    = x                  
          
          +(x,s(y)) = [3] x + [2] y + [9]
                    > [3] x + [2] y + [8]
                    = s(+(x,y))          
          
          
          Following rules are (at-least) weakly oriented:
             +(s(x),y) =  [3] x + [2] y + [10]
                       >= [3] x + [2] y + [8] 
                       =  s(+(x,y))           
          
             double(x) =  [5] x + [9]         
                       >= [5] x + [7]         
                       =  +(x,x)              
          
           double(0()) =  [14]                
                       >= [1]                 
                       =  0()                 
          
          double(s(x)) =  [5] x + [14]        
                       >= [5] x + [11]        
                       =  s(s(double(x)))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: 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))