WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y){y -> S(y)} =
            f(x,S(y)) ->^+ f(S(x),y)
              = C[f(S(x),y) = f(x,y){x -> S(x)}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + 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,g}
        TcT has computed the following interpretation:
          p(0) = 0          
          p(S) = x1         
          p(f) = 8 + 8*x1   
          p(g) = 8*x1 + 8*x2
        
        Following rules are strictly oriented:
        f(x1,0()) = 8 + 8*x1 
                  > 8*x1     
                  = g(x1,0())
        
        
        Following rules are (at-least) weakly oriented:
        f(y,S(x)) =  8 + 8*y  
                  >= 8 + 8*y  
                  =  f(S(y),x)
        
        g(0(),x2) =  8*x2     
                  >= x2       
                  =  x2       
        
        g(S(x),y) =  8*x + 8*y
                  >= 8*x + 8*y
                  =  g(x,S(y))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Weak TRS:
            f(x1,0()) -> g(x1,0())
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + 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,g}
        TcT has computed the following interpretation:
          p(0) = 0               
          p(S) = 2 + x1          
          p(f) = 2 + 8*x1 + 12*x2
          p(g) = 2 + 8*x1 + 8*x2 
        
        Following rules are strictly oriented:
        f(y,S(x)) = 26 + 12*x + 8*y
                  > 18 + 12*x + 8*y
                  = f(S(y),x)      
        
        g(0(),x2) = 2 + 8*x2       
                  > x2             
                  = x2             
        
        
        Following rules are (at-least) weakly oriented:
        f(x1,0()) =  2 + 8*x1      
                  >= 2 + 8*x1      
                  =  g(x1,0())     
        
        g(S(x),y) =  18 + 8*x + 8*y
                  >= 18 + 8*x + 8*y
                  =  g(x,S(y))     
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(S(x),y) -> g(x,S(y))
        - Weak TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + 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,g}
        TcT has computed the following interpretation:
          p(0) = 0            
          p(S) = 1 + x1       
          p(f) = 14*x1 + 14*x2
          p(g) = 14*x1 + 4*x2 
        
        Following rules are strictly oriented:
        g(S(x),y) = 14 + 14*x + 4*y
                  > 4 + 14*x + 4*y 
                  = g(x,S(y))      
        
        
        Following rules are (at-least) weakly oriented:
        f(x1,0()) =  14*x1           
                  >= 14*x1           
                  =  g(x1,0())       
        
        f(y,S(x)) =  14 + 14*x + 14*y
                  >= 14 + 14*x + 14*y
                  =  f(S(y),x)       
        
        g(0(),x2) =  4*x2            
                  >= x2              
                  =  x2              
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x1,0()) -> g(x1,0())
            f(y,S(x)) -> f(S(y),x)
            g(0(),x2) -> x2
            g(S(x),y) -> g(x,S(y))
        - Signature:
            {f/2,g/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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