WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,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(g) = {2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(+) = [1] x1 + [6]         
            p(0) = [0]                  
            p(1) = [0]                  
            p(f) = [7] x1 + [1]         
            p(g) = [5] x1 + [1] x2 + [1]
            p(s) = [1] x1 + [4]         
          
          Following rules are strictly oriented:
           +(x,0()) = [1] x + [6]         
                    > [1] x + [0]         
                    = x                   
          
             f(0()) = [1]                 
                    > [0]                 
                    = 1()                 
          
            f(s(x)) = [7] x + [29]        
                    > [6] x + [5]         
                    = g(x,s(x))           
          
           g(0(),y) = [1] y + [1]         
                    > [1] y + [0]         
                    = y                   
          
          g(s(x),y) = [5] x + [1] y + [21]
                    > [5] x + [1] y + [7] 
                    = g(x,+(y,s(x)))      
          
          g(s(x),y) = [5] x + [1] y + [21]
                    > [5] x + [1] y + [11]
                    = g(x,s(+(y,x)))      
          
          
          Following rules are (at-least) weakly oriented:
          +(x,s(y)) =  [1] x + [6] 
                    >= [1] x + [10]
                    =  s(+(x,y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,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(g) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,f,g}
        TcT has computed the following interpretation:
          p(+) = 1 + x1 + 2*x2       
          p(0) = 1                   
          p(1) = 4                   
          p(f) = 3 + x1 + 4*x1^2     
          p(g) = 4*x1 + 2*x1^2 + 2*x2
          p(s) = 1 + x1              
        
        Following rules are strictly oriented:
        +(x,s(y)) = 3 + x + 2*y
                  > 2 + x + 2*y
                  = s(+(x,y))  
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  3 + x                
                  >= x                    
                  =  x                    
        
           f(0()) =  8                    
                  >= 4                    
                  =  1()                  
        
          f(s(x)) =  8 + 9*x + 4*x^2      
                  >= 2 + 6*x + 2*x^2      
                  =  g(x,s(x))            
        
         g(0(),y) =  6 + 2*y              
                  >= y                    
                  =  y                    
        
        g(s(x),y) =  6 + 8*x + 2*x^2 + 2*y
                  >= 6 + 8*x + 2*x^2 + 2*y
                  =  g(x,+(y,s(x)))       
        
        g(s(x),y) =  6 + 8*x + 2*x^2 + 2*y
                  >= 4 + 8*x + 2*x^2 + 2*y
                  =  g(x,s(+(y,x)))       
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))