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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(c(s(x),y)) -> f(c(x,s(y)))
            g(c(x,s(y))) -> g(c(s(x),y))
            g(s(f(x))) -> g(f(x))
        - Signature:
            {f/1,g/1} / {c/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {c,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(c) = x2       
          p(f) = 1        
          p(g) = 12 + 2*x1
          p(s) = 8 + x1   
        
        Following rules are strictly oriented:
        g(c(x,s(y))) = 28 + 2*y    
                     > 12 + 2*y    
                     = g(c(s(x),y))
        
          g(s(f(x))) = 30          
                     > 14          
                     = g(f(x))     
        
        
        Following rules are (at-least) weakly oriented:
        f(c(s(x),y)) =  1           
                     >= 1           
                     =  f(c(x,s(y)))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(c(s(x),y)) -> f(c(x,s(y)))
        - Weak TRS:
            g(c(x,s(y))) -> g(c(s(x),y))
            g(s(f(x))) -> g(f(x))
        - Signature:
            {f/1,g/1} / {c/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {c,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(c) = x1    
          p(f) = 2 + x1
          p(g) = 1     
          p(s) = 1 + x1
        
        Following rules are strictly oriented:
        f(c(s(x),y)) = 3 + x       
                     > 2 + x       
                     = f(c(x,s(y)))
        
        
        Following rules are (at-least) weakly oriented:
        g(c(x,s(y))) =  1           
                     >= 1           
                     =  g(c(s(x),y))
        
          g(s(f(x))) =  1           
                     >= 1           
                     =  g(f(x))     
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(c(s(x),y)) -> f(c(x,s(y)))
            g(c(x,s(y))) -> g(c(s(x),y))
            g(s(f(x))) -> g(f(x))
        - Signature:
            {f/1,g/1} / {c/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {c,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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