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(+) = [1] x1 + [14] x2 + [1]
               p(0) = [0]                   
          p(double) = [15] x1 + [4]         
               p(s) = [1] x1 + [1]          
        
        Following rules are strictly oriented:
            +(x,0()) = [1] x + [1]          
                     > [1] x + [0]          
                     = x                    
        
           +(x,s(y)) = [1] x + [14] y + [15]
                     > [1] x + [14] y + [2] 
                     = s(+(x,y))            
        
           double(x) = [15] x + [4]         
                     > [15] x + [1]         
                     = +(x,x)               
        
         double(0()) = [4]                  
                     > [0]                  
                     = 0()                  
        
        double(s(x)) = [15] x + [19]        
                     > [15] x + [6]         
                     = s(s(double(x)))      
        
        
        Following rules are (at-least) weakly oriented:
        +(s(x),y) =  [1] x + [14] y + [2]
                  >= [1] x + [14] y + [2]
                  =  s(+(x,y))           
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(s(x),y) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(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 + [1] x2 + [2]
               p(0) = [0]                  
          p(double) = [3] x1 + [4]         
               p(s) = [1] x1 + [9]         
        
        Following rules are strictly oriented:
        +(s(x),y) = [2] x + [1] y + [20]
                  > [2] x + [1] y + [11]
                  = s(+(x,y))           
        
        
        Following rules are (at-least) weakly oriented:
            +(x,0()) =  [2] x + [2]         
                     >= [1] x + [0]         
                     =  x                   
        
           +(x,s(y)) =  [2] x + [1] y + [11]
                     >= [2] x + [1] y + [11]
                     =  s(+(x,y))           
        
           double(x) =  [3] x + [4]         
                     >= [3] x + [2]         
                     =  +(x,x)              
        
         double(0()) =  [4]                 
                     >= [0]                 
                     =  0()                 
        
        double(s(x)) =  [3] x + [31]        
                     >= [3] x + [22]        
                     =  s(s(double(x)))     
        
* 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))