WORST_CASE(?,O(n^2))
* Step 1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            goal(xs) -> naiverev(xs)
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(Cons) = {2},
          uargs(app) = {1}
        
        Following symbols are considered usable:
          {app,goal,naiverev,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = [1] x2 + [0]         
             p(False) = [0]                  
               p(Nil) = [0]                  
              p(True) = [4]                  
               p(app) = [2] x1 + [1] x2 + [0]
              p(goal) = [0]                  
          p(naiverev) = [0]                  
          p(notEmpty) = [4]                  
        
        Following rules are strictly oriented:
        notEmpty(Nil()) = [4]    
                        > [0]    
                        = False()
        
        
        Following rules are (at-least) weakly oriented:
          app(Cons(x,xs),ys) =  [2] xs + [1] ys + [0]          
                             >= [2] xs + [1] ys + [0]          
                             =  Cons(x,app(xs,ys))             
        
               app(Nil(),ys) =  [1] ys + [0]                   
                             >= [1] ys + [0]                   
                             =  ys                             
        
                    goal(xs) =  [0]                            
                             >= [0]                            
                             =  naiverev(xs)                   
        
        naiverev(Cons(x,xs)) =  [0]                            
                             >= [0]                            
                             =  app(naiverev(xs),Cons(x,Nil()))
        
             naiverev(Nil()) =  [0]                            
                             >= [0]                            
                             =  Nil()                          
        
        notEmpty(Cons(x,xs)) =  [4]                            
                             >= [4]                            
                             =  True()                         
        
* Step 2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            goal(xs) -> naiverev(xs)
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
            notEmpty(Cons(x,xs)) -> True()
        - Weak TRS:
            notEmpty(Nil()) -> False()
        - Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(Cons) = {2},
            uargs(app) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [0]         
               p(False) = [1]                  
                 p(Nil) = [0]                  
                p(True) = [0]                  
                 p(app) = [1] x1 + [2] x2 + [0]
                p(goal) = [8]                  
            p(naiverev) = [2]                  
            p(notEmpty) = [1]                  
          
          Following rules are strictly oriented:
                      goal(xs) = [8]         
                               > [2]         
                               = naiverev(xs)
          
               naiverev(Nil()) = [2]         
                               > [0]         
                               = Nil()       
          
          notEmpty(Cons(x,xs)) = [1]         
                               > [0]         
                               = True()      
          
          
          Following rules are (at-least) weakly oriented:
            app(Cons(x,xs),ys) =  [1] xs + [2] ys + [0]          
                               >= [1] xs + [2] ys + [0]          
                               =  Cons(x,app(xs,ys))             
          
                 app(Nil(),ys) =  [2] ys + [0]                   
                               >= [1] ys + [0]                   
                               =  ys                             
          
          naiverev(Cons(x,xs)) =  [2]                            
                               >= [2]                            
                               =  app(naiverev(xs),Cons(x,Nil()))
          
               notEmpty(Nil()) =  [1]                            
                               >= [1]                            
                               =  False()                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
        - Weak TRS:
            goal(xs) -> naiverev(xs)
            naiverev(Nil()) -> Nil()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(Cons) = {2},
          uargs(app) = {1}
        
        Following symbols are considered usable:
          {app,goal,naiverev,notEmpty}
        TcT has computed the following interpretation:
              p(Cons) = [1] x2 + [8]         
             p(False) = [0]                  
               p(Nil) = [3]                  
              p(True) = [0]                  
               p(app) = [1] x1 + [2] x2 + [1]
              p(goal) = [4] x1 + [5]         
          p(naiverev) = [3] x1 + [4]         
          p(notEmpty) = [4]                  
        
        Following rules are strictly oriented:
               app(Nil(),ys) = [2] ys + [4]                   
                             > [1] ys + [0]                   
                             = ys                             
        
        naiverev(Cons(x,xs)) = [3] xs + [28]                  
                             > [3] xs + [27]                  
                             = app(naiverev(xs),Cons(x,Nil()))
        
        
        Following rules are (at-least) weakly oriented:
          app(Cons(x,xs),ys) =  [1] xs + [2] ys + [9]
                             >= [1] xs + [2] ys + [9]
                             =  Cons(x,app(xs,ys))   
        
                    goal(xs) =  [4] xs + [5]         
                             >= [3] xs + [4]         
                             =  naiverev(xs)         
        
             naiverev(Nil()) =  [13]                 
                             >= [3]                  
                             =  Nil()                
        
        notEmpty(Cons(x,xs)) =  [4]                  
                             >= [0]                  
                             =  True()               
        
             notEmpty(Nil()) =  [4]                  
                             >= [0]                  
                             =  False()              
        
* Step 4: Ara WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        - Weak TRS:
            app(Nil(),ys) -> ys
            goal(xs) -> naiverev(xs)
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1}
    + Details:
        Signatures used:
        ----------------
          Cons :: [A(0, 0) x A(1, 0)] -(1)-> A(1, 0)
          Cons :: [A(5, 5) x A(5, 5)] -(5)-> A(0, 5)
          Cons :: [A(8, 8) x A(8, 8)] -(8)-> A(0, 8)
          Cons :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0)
          False :: [] -(0)-> A(14, 14)
          Nil :: [] -(0)-> A(1, 0)
          Nil :: [] -(0)-> A(0, 5)
          Nil :: [] -(0)-> A(0, 8)
          Nil :: [] -(0)-> A(7, 13)
          Nil :: [] -(0)-> A(7, 7)
          True :: [] -(0)-> A(14, 14)
          app :: [A(1, 0) x A(0, 0)] -(4)-> A(0, 0)
          goal :: [A(14, 11)] -(15)-> A(0, 0)
          naiverev :: [A(0, 5)] -(1)-> A(0, 0)
          notEmpty :: [A(0, 8)] -(15)-> A(0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
          Cons :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
          Cons :: [A_cf(0, 0) x A_cf(5, 0)] -(5)-> A_cf(5, 0)
          Cons :: [A_cf(0, 0) x A_cf(2, 0)] -(2)-> A_cf(2, 0)
          Nil :: [] -(0)-> A_cf(0, 0)
          Nil :: [] -(0)-> A_cf(5, 0)
          Nil :: [] -(0)-> A_cf(2, 0)
          Nil :: [] -(0)-> A_cf(11, 11)
          Nil :: [] -(0)-> A_cf(3, 2)
          app :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
          app :: [A_cf(2, 0) x A_cf(2, 0)] -(0)-> A_cf(2, 0)
          naiverev :: [A_cf(5, 0)] -(1)-> A_cf(2, 0)
        
        
        Base Constructor Signatures used:
        ---------------------------------
          Cons_A :: [A(0, 0) x A(1, 0)] -(1)-> A(1, 0)
          Cons_A :: [A(1, 1) x A(1, 1)] -(1)-> A(0, 1)
          False_A :: [] -(0)-> A(1, 0)
          False_A :: [] -(0)-> A(0, 1)
          Nil_A :: [] -(0)-> A(1, 0)
          Nil_A :: [] -(0)-> A(0, 1)
          True_A :: [] -(0)-> A(1, 0)
          True_A :: [] -(0)-> A(0, 1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        2. Weak:
          
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            goal(xs) -> naiverev(xs)
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
        - Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,goal,naiverev,notEmpty} and constructors {Cons,False
            ,Nil,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))