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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,g(x)) -> x
            f(x,h(y)) -> f(h(x),y)
        - Signature:
            {f/2} / {g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g,h}
    + 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(f) = 2 + 3*x1 + 12*x2
          p(g) = 1 + x1          
          p(h) = x1              
        
        Following rules are strictly oriented:
        f(x,g(x)) = 14 + 15*x
                  > x        
                  = x        
        
        
        Following rules are (at-least) weakly oriented:
        f(x,h(y)) =  2 + 3*x + 12*y
                  >= 2 + 3*x + 12*y
                  =  f(h(x),y)     
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,h(y)) -> f(h(x),y)
        - Weak TRS:
            f(x,g(x)) -> x
        - Signature:
            {f/2} / {g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g,h}
    + 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(f) = 2 + 4*x1 + 8*x2
          p(g) = 0              
          p(h) = 2 + x1         
        
        Following rules are strictly oriented:
        f(x,h(y)) = 18 + 4*x + 8*y
                  > 10 + 4*x + 8*y
                  = f(h(x),y)     
        
        
        Following rules are (at-least) weakly oriented:
        f(x,g(x)) =  2 + 4*x
                  >= x      
                  =  x      
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,g(x)) -> x
            f(x,h(y)) -> f(h(x),y)
        - Signature:
            {f/2} / {g/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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