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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(x)) -> g(g(f(x)))
            f(g(x)) -> g(g(g(x)))
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {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:
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(f) = 8 + x1
          p(g) = x1    
        
        Following rules are strictly oriented:
        f(g(x)) = 8 + x     
                > x         
                = g(g(g(x)))
        
        
        Following rules are (at-least) weakly oriented:
        f(g(x)) =  8 + x     
                >= 8 + x     
                =  g(g(f(x)))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(x)) -> g(g(f(x)))
        - Weak TRS:
            f(g(x)) -> g(g(g(x)))
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {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:
          uargs(g) = {1}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(f) = 2 + 8*x1
          p(g) = 1 + x1  
        
        Following rules are strictly oriented:
        f(g(x)) = 10 + 8*x  
                > 4 + 8*x   
                = g(g(f(x)))
        
        
        Following rules are (at-least) weakly oriented:
        f(g(x)) =  10 + 8*x  
                >= 3 + x     
                =  g(g(g(x)))
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(g(x)) -> g(g(f(x)))
            f(g(x)) -> g(g(g(x)))
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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