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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(r(x)) -> r(b(x))
            b(w(x)) -> w(b(x))
            w(r(x)) -> r(w(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b,w} and constructors {r}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(r) = {1},
          uargs(w) = {1}
        
        Following symbols are considered usable:
          {b,w}
        TcT has computed the following interpretation:
          p(b) = 2*x1  
          p(r) = 2 + x1
          p(w) = x1    
        
        Following rules are strictly oriented:
        b(r(x)) = 4 + 2*x
                > 2 + 2*x
                = r(b(x))
        
        
        Following rules are (at-least) weakly oriented:
        b(w(x)) =  2*x    
                >= 2*x    
                =  w(b(x))
        
        w(r(x)) =  2 + x  
                >= 2 + x  
                =  r(w(x))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(w(x)) -> w(b(x))
            w(r(x)) -> r(w(x))
        - Weak TRS:
            b(r(x)) -> r(b(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b,w} and constructors {r}
    + 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(r) = {1},
          uargs(w) = {1}
        
        Following symbols are considered usable:
          {b,w}
        TcT has computed the following interpretation:
          p(b) = 8 + 2*x1
          p(r) = 8 + x1  
          p(w) = 2 + x1  
        
        Following rules are strictly oriented:
        b(w(x)) = 12 + 2*x
                > 10 + 2*x
                = w(b(x)) 
        
        
        Following rules are (at-least) weakly oriented:
        b(r(x)) =  24 + 2*x
                >= 16 + 2*x
                =  r(b(x)) 
        
        w(r(x)) =  10 + x  
                >= 10 + x  
                =  r(w(x)) 
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            w(r(x)) -> r(w(x))
        - Weak TRS:
            b(r(x)) -> r(b(x))
            b(w(x)) -> w(b(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b,w} and constructors {r}
    + 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(r) = {1},
          uargs(w) = {1}
        
        Following symbols are considered usable:
          {b,w}
        TcT has computed the following interpretation:
          p(b) = 2*x1     
          p(r) = 8 + x1   
          p(w) = 12 + 2*x1
        
        Following rules are strictly oriented:
        w(r(x)) = 28 + 2*x
                > 20 + 2*x
                = r(w(x)) 
        
        
        Following rules are (at-least) weakly oriented:
        b(r(x)) =  16 + 2*x
                >= 8 + 2*x 
                =  r(b(x)) 
        
        b(w(x)) =  24 + 4*x
                >= 12 + 4*x
                =  w(b(x)) 
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            b(r(x)) -> r(b(x))
            b(w(x)) -> w(b(x))
            w(r(x)) -> r(w(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {b,w} and constructors {r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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