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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {append,goal}
        TcT has computed the following interpretation:
            p(Cons) = x1 + x2          
             p(Nil) = 2                
          p(append) = 1 + 8*x1 + 8*x2  
            p(goal) = 8 + 11*x1 + 12*x2
        
        Following rules are strictly oriented:
        append(Nil(),ys) = 17 + 8*ys      
                         > ys             
                         = ys             
        
               goal(x,y) = 8 + 11*x + 12*y
                         > 1 + 8*x + 8*y  
                         = append(x,y)    
        
        
        Following rules are (at-least) weakly oriented:
        append(Cons(x,xs),ys) =  1 + 8*x + 8*xs + 8*ys
                              >= 1 + x + 8*xs + 8*ys  
                              =  Cons(x,append(xs,ys))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
        - Weak TRS:
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {append,goal}
        TcT has computed the following interpretation:
            p(Cons) = 7 + x1 + x2 
             p(Nil) = 0           
          p(append) = 2*x1 + 12*x2
            p(goal) = 2*x1 + 12*x2
        
        Following rules are strictly oriented:
        append(Cons(x,xs),ys) = 14 + 2*x + 2*xs + 12*ys
                              > 7 + x + 2*xs + 12*ys   
                              = Cons(x,append(xs,ys))  
        
        
        Following rules are (at-least) weakly oriented:
        append(Nil(),ys) =  12*ys      
                         >= ys         
                         =  ys         
        
               goal(x,y) =  2*x + 12*y 
                         >= 2*x + 12*y 
                         =  append(x,y)
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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