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: Ara 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:
        Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1}
    + Details:
        Signatures used:
        ----------------
          + :: [A(0, 0) x A(8, 0)] -(2)-> A(0, 0)
          0 :: [] -(0)-> A(8, 0)
          0 :: [] -(0)-> A(0, 14)
          1 :: [] -(0)-> A(6, 6)
          f :: [A(0, 14)] -(0)-> A(0, 0)
          g :: [A(0, 14) x A(0, 0)] -(0)-> A(0, 0)
          s :: [A(8, 0)] -(8)-> A(8, 0)
          s :: [A(14, 14)] -(14)-> A(0, 14)
          s :: [A(0, 0)] -(0)-> A(0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
          + :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
          0 :: [] -(0)-> A_cf(0, 0)
          0 :: [] -(0)-> A_cf(1, 0)
          g :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
          g :: [A_cf(1, 0) x A_cf(0, 0)] -(3)-> A_cf(0, 0)
          s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0)
          s :: [A_cf(1, 0)] -(1)-> A_cf(1, 0)
        
        
        Base Constructor Signatures used:
        ---------------------------------
          0_A :: [] -(0)-> A(1, 0)
          0_A :: [] -(0)-> A(0, 1)
          1_A :: [] -(0)-> A(1, 0)
          1_A :: [] -(0)-> A(0, 1)
          s_A :: [A(1, 0)] -(1)-> A(1, 0)
          s_A :: [A(1, 1)] -(1)-> A(0, 1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          +(x,s(y)) -> s(+(x,y))
        2. Weak:
          
* 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))