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
            -(s(x),s(y)) -> -(x,y)
            f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
            f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
            p(s(x)) -> x
        - Signature:
            {-/2,f/2,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,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
            -(s(x),s(y)) -> -(x,y)
            f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
            f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
            p(s(x)) -> x
        - Signature:
            {-/2,f/2,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          -(x,y){x -> s(x),y -> s(y)} =
            -(s(x),s(y)) ->^+ -(x,y)
              = C[-(x,y) = -(x,y){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(s(x),s(y)) -> -(x,y)
            f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
            f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
            p(s(x)) -> x
        - Signature:
            {-/2,f/2,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,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(f) = {1,2},
          uargs(p) = {1}
        
        Following symbols are considered usable:
          {-,f,p}
        TcT has computed the following interpretation:
          p(-) = x1           
          p(0) = 0            
          p(f) = 1 + x1 + 2*x2
          p(p) = x1           
          p(s) = 8 + x1       
        
        Following rules are strictly oriented:
        -(s(x),s(y)) = 8 + x 
                     > x     
                     = -(x,y)
        
             p(s(x)) = 8 + x 
                     > x     
                     = x     
        
        
        Following rules are (at-least) weakly oriented:
         -(x,0()) =  x                           
                  >= x                           
                  =  x                           
        
        f(x,s(y)) =  17 + x + 2*y                
                  >= 17 + x + 2*y                
                  =  f(p(-(x,s(y))),p(-(s(y),x)))
        
        f(s(x),y) =  9 + x + 2*y                 
                  >= 9 + x + 2*y                 
                  =  f(p(-(s(x),y)),p(-(y,s(x))))
        
** Step 1.b:2: Ara WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x)))
            f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x))))
        - Weak TRS:
            -(s(x),s(y)) -> -(x,y)
            p(s(x)) -> x
        - Signature:
            {-/2,f/2,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,s}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 5, araRuleShifting = Nothing}
    + Details:
        Signatures used:
        ----------------
          - :: ["A"(3, 3) x "A"(0, 0)] -(1)-> "A"(3, 3)
          0 :: [] -(0)-> "A"(0, 0)
          f :: ["A"(6, 3) x "A"(6, 3)] -(0)-> "A"(0, 0)
          p :: ["A"(3, 3)] -(0)-> "A"(6, 3)
          s :: ["A"(9, 3)] -(6)-> "A"(6, 3)
          s :: ["A"(6, 3)] -(3)-> "A"(3, 3)
          s :: ["A"(0, 0)] -(0)-> "A"(0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        

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