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(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - 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(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - 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)){y -> s(y)} =
            f(c(x,s(y))) ->^+ f(c(s(x),y))
              = C[f(c(s(x),y)) = f(c(x,y)){x -> s(x)}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(c(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - 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) = 0 
          p(f) = 4 
          p(g) = 12
          p(s) = 1 
        
        Following rules are strictly oriented:
        g(c(s(X),Y)) = 12          
                     > 4           
                     = f(c(X,s(Y)))
        
        
        Following rules are (at-least) weakly oriented:
        f(c(X,s(Y))) =  4           
                     >= 4           
                     =  f(c(s(X),Y))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(c(X,s(Y))) -> f(c(s(X),Y))
        - Weak TRS:
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - 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) = 2 + x2   
          p(f) = x1       
          p(g) = 13 + 8*x1
          p(s) = 14 + x1  
        
        Following rules are strictly oriented:
        f(c(X,s(Y))) = 16 + Y      
                     > 2 + Y       
                     = f(c(s(X),Y))
        
        
        Following rules are (at-least) weakly oriented:
        g(c(s(X),Y)) =  29 + 8*Y    
                     >= 16 + Y      
                     =  f(c(X,s(Y)))
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(c(X,s(Y))) -> f(c(s(X),Y))
            g(c(s(X),Y)) -> f(c(X,s(Y)))
        - 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))