* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} 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(+) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(+) = [1] x1 + [5]
              p(0) = [0]         
              p(s) = [1] x1 + [8]
            p(sum) = [2] x1 + [0]
          
          Following rules are strictly oriented:
           +(x,0()) = [1] x + [5]   
                    > [1] x + [0]   
                    = x             
          
          sum(s(x)) = [2] x + [16]  
                    > [2] x + [5]   
                    = +(sum(x),s(x))
          
          
          Following rules are (at-least) weakly oriented:
          +(x,s(y)) =  [1] x + [5] 
                    >= [1] x + [13]
                    =  s(+(x,y))   
          
           sum(0()) =  [0]         
                    >= [0]         
                    =  0()         
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
        - Weak TRS:
            +(x,0()) -> x
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,sum}
        TcT has computed the following interpretation:
            p(+) = [1] x_1 + [0]
            p(0) = [0]          
            p(s) = [1] x_1 + [0]
          p(sum) = [1]          
        
        Following rules are strictly oriented:
        sum(0()) = [1]
                 > [0]
                 = 0()
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  [1] x + [0]   
                  >= [1] x + [0]   
                  =  x             
        
        +(x,s(y)) =  [1] x + [0]   
                  >= [1] x + [0]   
                  =  s(+(x,y))     
        
        sum(s(x)) =  [1]           
                  >= [1]           
                  =  +(sum(x),s(x))
        
* Step 3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,sum}
        TcT has computed the following interpretation:
            p(+) = x1 + 2*x2    
            p(0) = 0            
            p(s) = 1 + x1       
          p(sum) = 2*x1 + 2*x1^2
        
        Following rules are strictly oriented:
        +(x,s(y)) = 2 + x + 2*y
                  > 1 + x + 2*y
                  = s(+(x,y))  
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  x              
                  >= x              
                  =  x              
        
         sum(0()) =  0              
                  >= 0              
                  =  0()            
        
        sum(s(x)) =  4 + 6*x + 2*x^2
                  >= 2 + 4*x + 2*x^2
                  =  +(sum(x),s(x)) 
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))