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: Sum.  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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: 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: NaturalMI.  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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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) = [1] x2 + [0]
          p(g) = [5] x2 + [6]
          p(h) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        g(x,h(y,z)) = [5] y + [11]
                    > [5] y + [7] 
                    = h(g(x,y),z) 
        
        
        Following rules are (at-least) weakly oriented:
        g(f(x,y),z) =  [5] z + [6]
                    >= [5] z + [6]
                    =  f(x,g(y,z))
        
        g(h(x,y),z) =  [5] z + [6]
                    >= [5] z + [6]
                    =  g(x,f(y,z))
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(f(x,y),z) -> f(x,g(y,z))
            g(h(x,y),z) -> g(x,f(y,z))
        - Weak TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {f,h}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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) = [1] x2 + [8]         
          p(g) = [2] x1 + [1] x2 + [0]
          p(h) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
        g(f(x,y),z) = [2] y + [1] z + [16]
                    > [2] y + [1] z + [8] 
                    = f(x,g(y,z))         
        
        g(h(x,y),z) = [2] x + [1] z + [16]
                    > [2] x + [1] z + [8] 
                    = g(x,f(y,z))         
        
        
        Following rules are (at-least) weakly oriented:
        g(x,h(y,z)) =  [2] x + [1] y + [8]
                    >= [2] x + [1] y + [8]
                    =  h(g(x,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))