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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(x,s(y)) -> g(f(x,y),0())
            g(0(),f(x,x)) -> x
            g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
            g(s(x),y) -> g(f(x,y),0())
        - Signature:
            {g/2} / {0/0,f/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,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:
          uargs(f) = {1,2}
        
        Following symbols are considered usable:
          {g}
        TcT has computed the following interpretation:
          p(0) = 0      
          p(f) = x1 + x2
          p(g) = x2     
          p(s) = 14     
        
        Following rules are strictly oriented:
        g(x,s(y)) = 14           
                  > 0            
                  = g(f(x,y),0())
        
        
        Following rules are (at-least) weakly oriented:
        g(0(),f(x,x)) =  2*x                 
                      >= x                   
                      =  x                   
        
        g(f(x,y),0()) =  0                   
                      >= 0                   
                      =  f(g(x,0()),g(y,0()))
        
            g(s(x),y) =  y                   
                      >= 0                   
                      =  g(f(x,y),0())       
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(0(),f(x,x)) -> x
            g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
            g(s(x),y) -> g(f(x,y),0())
        - Weak TRS:
            g(x,s(y)) -> g(f(x,y),0())
        - Signature:
            {g/2} / {0/0,f/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,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:
          uargs(f) = {1,2}
        
        Following symbols are considered usable:
          {g}
        TcT has computed the following interpretation:
          p(0) = 1          
          p(f) = 7 + x1 + x2
          p(g) = 2*x1 + 2*x2
          p(s) = 9 + x1     
        
        Following rules are strictly oriented:
        g(0(),f(x,x)) = 16 + 4*x            
                      > x                   
                      = x                   
        
        g(f(x,y),0()) = 16 + 2*x + 2*y      
                      > 11 + 2*x + 2*y      
                      = f(g(x,0()),g(y,0()))
        
            g(s(x),y) = 18 + 2*x + 2*y      
                      > 16 + 2*x + 2*y      
                      = g(f(x,y),0())       
        
        
        Following rules are (at-least) weakly oriented:
        g(x,s(y)) =  18 + 2*x + 2*y
                  >= 16 + 2*x + 2*y
                  =  g(f(x,y),0()) 
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(x,s(y)) -> g(f(x,y),0())
            g(0(),f(x,x)) -> x
            g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
            g(s(x),y) -> g(f(x,y),0())
        - Signature:
            {g/2} / {0/0,f/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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