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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
            g(f(x,y),z) -> f(x,g(y,z))
            g(h(x,y),z) -> g(x,f(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {f,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:
          uargs(f) = {2},
          uargs(h) = {1}
        
        Following symbols are considered usable:
          {g}
        TcT has computed the following interpretation:
          p(f) = 5 + x2  
          p(g) = 9 + 4*x1
          p(h) = x1      
        
        Following rules are strictly oriented:
        g(f(x,y),z) = 29 + 4*y   
                    > 14 + 4*y   
                    = f(x,g(y,z))
        
        
        Following rules are (at-least) weakly oriented:
        g(x,h(y,z)) =  9 + 4*x    
                    >= 9 + 4*x    
                    =  h(g(x,y),z)
        
        g(h(x,y),z) =  9 + 4*x    
                    >= 9 + 4*x    
                    =  g(x,f(y,z))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
            g(h(x,y),z) -> g(x,f(y,z))
        - Weak TRS:
            g(f(x,y),z) -> f(x,g(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {f,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:
          uargs(f) = {2},
          uargs(h) = {1}
        
        Following symbols are considered usable:
          {g}
        TcT has computed the following interpretation:
          p(f) = 5 + x2     
          p(g) = 4*x1 + 4*x2
          p(h) = 6 + x1     
        
        Following rules are strictly oriented:
        g(x,h(y,z)) = 24 + 4*x + 4*y
                    > 6 + 4*x + 4*y 
                    = h(g(x,y),z)   
        
        g(h(x,y),z) = 24 + 4*x + 4*z
                    > 20 + 4*x + 4*z
                    = g(x,f(y,z))   
        
        
        Following rules are (at-least) weakly oriented:
        g(f(x,y),z) =  20 + 4*y + 4*z
                    >= 5 + 4*y + 4*z 
                    =  f(x,g(y,z))   
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
            g(f(x,y),z) -> f(x,g(y,z))
            g(h(x,y),z) -> g(x,f(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {f,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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