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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2} / {a/0,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,g}
    + 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:
          none
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(a) = 2              
          p(f) = 2 + 8*x1 + 8*x2
          p(g) = 1 + x1         
        
        Following rules are strictly oriented:
        f(x,a()) = 18 + 8*x
                 > x       
                 = x       
        
        
        Following rules are (at-least) weakly oriented:
        f(x,g(y)) =  10 + 8*x + 8*y
                  >= 10 + 8*x + 8*y
                  =  f(g(x),y)     
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,g(y)) -> f(g(x),y)
        - Weak TRS:
            f(x,a()) -> x
        - Signature:
            {f/2} / {a/0,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,g}
    + 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:
          none
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(a) = 0           
          p(f) = 8*x1 + 13*x2
          p(g) = 2 + x1      
        
        Following rules are strictly oriented:
        f(x,g(y)) = 26 + 8*x + 13*y
                  > 16 + 8*x + 13*y
                  = f(g(x),y)      
        
        
        Following rules are (at-least) weakly oriented:
        f(x,a()) =  8*x
                 >= x  
                 =  x  
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2} / {a/0,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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