WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + 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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: NaturalPI 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:
        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(g) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+,f,g}
        TcT has computed the following interpretation:
          p(+) = x1             
          p(0) = 2              
          p(1) = 14             
          p(f) = 8 + 10*x1      
          p(g) = 1 + 2*x1 + 4*x2
          p(s) = x1             
        
        Following rules are strictly oriented:
          f(0()) = 28       
                 > 14       
                 = 1()      
        
         f(s(x)) = 8 + 10*x 
                 > 1 + 6*x  
                 = g(x,s(x))
        
        g(0(),y) = 5 + 4*y  
                 > y        
                 = y        
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  x             
                  >= x             
                  =  x             
        
        +(x,s(y)) =  x             
                  >= x             
                  =  s(+(x,y))     
        
        g(s(x),y) =  1 + 2*x + 4*y 
                  >= 1 + 2*x + 4*y 
                  =  g(x,+(y,s(x)))
        
        g(s(x),y) =  1 + 2*x + 4*y 
                  >= 1 + 2*x + 4*y 
                  =  g(x,s(+(y,x)))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Weak TRS:
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
        - 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(+) = x1 + x2                 
          p(0) = 1                       
          p(1) = 7                       
          p(f) = 7*x1^2                  
          p(g) = 5 + 6*x1 + 2*x1^2 + 2*x2
          p(s) = 1 + x1                  
        
        Following rules are strictly oriented:
         +(x,0()) = 1 + x                  
                  > x                      
                  = x                      
        
        g(s(x),y) = 13 + 10*x + 2*x^2 + 2*y
                  > 7 + 8*x + 2*x^2 + 2*y  
                  = g(x,+(y,s(x)))         
        
        g(s(x),y) = 13 + 10*x + 2*x^2 + 2*y
                  > 7 + 8*x + 2*x^2 + 2*y  
                  = g(x,s(+(y,x)))         
        
        
        Following rules are (at-least) weakly oriented:
        +(x,s(y)) =  1 + x + y       
                  >= 1 + x + y       
                  =  s(+(x,y))       
        
           f(0()) =  7               
                  >= 7               
                  =  1()             
        
          f(s(x)) =  7 + 14*x + 7*x^2
                  >= 7 + 8*x + 2*x^2 
                  =  g(x,s(x))       
        
         g(0(),y) =  13 + 2*y        
                  >= y               
                  =  y               
        
** Step 1.b:3: 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 {araHeuristics = Heuristics, minDegree = 2, maxDegree = 2, araTimeout = 3, araRuleShifting = Nothing}
    + Details:
        Signatures used:
        ----------------
          + :: ["A"(0, 0) x "A"(1, 0)] -(0)-> "A"(0, 0)
          0 :: [] -(0)-> "A"(1, 0)
          0 :: [] -(0)-> "A"(2, 2)
          0 :: [] -(0)-> "A"(1, 1)
          1 :: [] -(0)-> "A"(0, 0)
          f :: ["A"(2, 2)] -(0)-> "A"(0, 0)
          g :: ["A"(1, 1) x "A"(0, 0)] -(0)-> "A"(0, 0)
          s :: ["A"(1, 0)] -(1)-> "A"(1, 0)
          s :: ["A"(4, 2)] -(2)-> "A"(2, 2)
          s :: ["A"(2, 1)] -(1)-> "A"(1, 1)
          s :: ["A"(0, 0)] -(0)-> "A"(0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        

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