WORST_CASE(?,O(n^1))
* Step 1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            flatten(dd(x,xs)) -> @(x,flatten(xs))
            flatten(nil()) -> nil()
        - Signature:
            {@/2,flatten/1} / {dd/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,flatten} and constructors {dd,nil}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(@) = {2},
          uargs(dd) = {2}
        
        Following symbols are considered usable:
          {@,flatten}
        TcT has computed the following interpretation:
                p(@) = x2  
               p(dd) = x2  
          p(flatten) = 2*x1
              p(nil) = 2   
        
        Following rules are strictly oriented:
        flatten(nil()) = 4    
                       > 2    
                       = nil()
        
        
        Following rules are (at-least) weakly oriented:
           @(dd(x,xs),ys) =  ys              
                          >= ys              
                          =  dd(x,@(xs,ys))  
        
              @(nil(),xs) =  xs              
                          >= xs              
                          =  xs              
        
        flatten(dd(x,xs)) =  2*xs            
                          >= 2*xs            
                          =  @(x,flatten(xs))
        
* Step 2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            flatten(dd(x,xs)) -> @(x,flatten(xs))
        - Weak TRS:
            flatten(nil()) -> nil()
        - Signature:
            {@/2,flatten/1} / {dd/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,flatten} and constructors {dd,nil}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(@) = {2},
          uargs(dd) = {2}
        
        Following symbols are considered usable:
          {@,flatten}
        TcT has computed the following interpretation:
                p(@) = x1 + x2
               p(dd) = x1 + x2
          p(flatten) = 3 + x1 
              p(nil) = 2      
        
        Following rules are strictly oriented:
        @(nil(),xs) = 2 + xs
                    > xs    
                    = xs    
        
        
        Following rules are (at-least) weakly oriented:
           @(dd(x,xs),ys) =  x + xs + ys     
                          >= x + xs + ys     
                          =  dd(x,@(xs,ys))  
        
        flatten(dd(x,xs)) =  3 + x + xs      
                          >= 3 + x + xs      
                          =  @(x,flatten(xs))
        
           flatten(nil()) =  5               
                          >= 2               
                          =  nil()           
        
* Step 3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            flatten(dd(x,xs)) -> @(x,flatten(xs))
        - Weak TRS:
            @(nil(),xs) -> xs
            flatten(nil()) -> nil()
        - Signature:
            {@/2,flatten/1} / {dd/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,flatten} and constructors {dd,nil}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(@) = {2},
          uargs(dd) = {2}
        
        Following symbols are considered usable:
          {@,flatten}
        TcT has computed the following interpretation:
                p(@) = 5 + 2*x1 + x2
               p(dd) = 3 + x1 + x2  
          p(flatten) = 1 + 2*x1     
              p(nil) = 0            
        
        Following rules are strictly oriented:
           @(dd(x,xs),ys) = 11 + 2*x + 2*xs + ys
                          > 8 + x + 2*xs + ys   
                          = dd(x,@(xs,ys))      
        
        flatten(dd(x,xs)) = 7 + 2*x + 2*xs      
                          > 6 + 2*x + 2*xs      
                          = @(x,flatten(xs))    
        
        
        Following rules are (at-least) weakly oriented:
           @(nil(),xs) =  5 + xs
                       >= xs    
                       =  xs    
        
        flatten(nil()) =  1     
                       >= 0     
                       =  nil() 
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            @(dd(x,xs),ys) -> dd(x,@(xs,ys))
            @(nil(),xs) -> xs
            flatten(dd(x,xs)) -> @(x,flatten(xs))
            flatten(nil()) -> nil()
        - Signature:
            {@/2,flatten/1} / {dd/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,flatten} and constructors {dd,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))