WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            max(L(x)) -> x
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {max} and constructors {0,L,N,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            max(L(x)) -> x
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {max} and constructors {0,L,N,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          max(N(L(x),L(y))){x -> s(x),y -> s(y)} =
            max(N(L(s(x)),L(s(y)))) ->^+ s(max(N(L(x),L(y))))
              = C[max(N(L(x),L(y))) = max(N(L(x),L(y))){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            max(L(x)) -> x
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {max} and constructors {0,L,N,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(L) = {1},
          uargs(N) = {2},
          uargs(max) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {max}
        TcT has computed the following interpretation:
            p(0) = 8          
            p(L) = x1         
            p(N) = 4 + x1 + x2
          p(max) = x1         
            p(s) = 2 + x1     
        
        Following rules are strictly oriented:
            max(N(L(0()),L(y))) = 12 + y              
                                > y                   
                                = y                   
        
        max(N(L(s(x)),L(s(y)))) = 8 + x + y           
                                > 6 + x + y           
                                = s(max(N(L(x),L(y))))
        
        
        Following rules are (at-least) weakly oriented:
                  max(L(x)) =  x                          
                            >= x                          
                            =  x                          
        
        max(N(L(x),N(y,z))) =  8 + x + y + z              
                            >= 8 + x + y + z              
                            =  max(N(L(x),L(max(N(y,z)))))
        
** Step 1.b:2: Ara WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            max(L(x)) -> x
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        - Weak TRS:
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {max} and constructors {0,L,N,s}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 5, araRuleShifting = Nothing}
    + Details:
        Signatures used:
        ----------------
          0 :: [] -(0)-> "A"(0, 0)
          L :: ["A"(0, 0)] -(0)-> "A"(0, 13)
          L :: ["A"(0, 0)] -(0)-> "A"(13, 0)
          L :: ["A"(0, 0)] -(0)-> "A"(13, 13)
          L :: ["A"(0, 0)] -(0)-> "A"(14, 1)
          L :: ["A"(0, 0)] -(0)-> "A"(13, 15)
          L :: ["A"(0, 0)] -(0)-> "A"(15, 7)
          L :: ["A"(0, 0)] -(0)-> "A"(15, 15)
          N :: ["A"(13, 0) x "A"(13, 13)] -(0)-> "A"(0, 13)
          N :: ["A"(13, 0) x "A"(13, 13)] -(13)-> "A"(13, 13)
          N :: ["A"(13, 0) x "A"(13, 13)] -(1)-> "A"(1, 13)
          N :: ["A"(13, 0) x "A"(13, 13)] -(2)-> "A"(2, 13)
          N :: ["A"(14, 0) x "A"(14, 14)] -(0)-> "A"(0, 14)
          max :: ["A"(0, 13)] -(1)-> "A"(0, 0)
          s :: ["A"(0, 0)] -(0)-> "A"(0, 0)
          s :: ["A"(0, 0)] -(0)-> "A"(4, 15)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "0_A" :: [] -(0)-> "A"(1, 0)
          "0_A" :: [] -(0)-> "A"(0, 1)
          "L_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0)
          "L_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1)
          "N_A" :: ["A"(0, 0) x "A"(0, 0)] -(1)-> "A"(1, 0)
          "N_A" :: ["A"(1, 0) x "A"(1, 1)] -(0)-> "A"(0, 1)
          "s_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0)
          "s_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1)
        

WORST_CASE(Omega(n^1),O(n^2))