*** 1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {app/2,goal/1,naiverev/1,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0}
      Obligation:
        Innermost
        basic terms: {app,goal,naiverev,notEmpty}/{Cons,False,Nil,True}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        app#(Nil(),ys) -> c_2()
        goal#(xs) -> c_3(naiverev#(xs))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        naiverev#(Nil()) -> c_5()
        notEmpty#(Cons(x,xs)) -> c_6()
        notEmpty#(Nil()) -> c_7()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        app#(Nil(),ys) -> c_2()
        goal#(xs) -> c_3(naiverev#(xs))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        naiverev#(Nil()) -> c_5()
        notEmpty#(Cons(x,xs)) -> c_6()
        notEmpty#(Nil()) -> c_7()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
      Obligation:
        Innermost
        basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        app(Nil(),ys) -> ys
        naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
        naiverev(Nil()) -> Nil()
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        app#(Nil(),ys) -> c_2()
        goal#(xs) -> c_3(naiverev#(xs))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        naiverev#(Nil()) -> c_5()
        notEmpty#(Cons(x,xs)) -> c_6()
        notEmpty#(Nil()) -> c_7()
*** 1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        app#(Nil(),ys) -> c_2()
        goal#(xs) -> c_3(naiverev#(xs))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        naiverev#(Nil()) -> c_5()
        notEmpty#(Cons(x,xs)) -> c_6()
        notEmpty#(Nil()) -> c_7()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        app(Nil(),ys) -> ys
        naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
        naiverev(Nil()) -> Nil()
      Signature:
        {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
      Obligation:
        Innermost
        basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {2,5,6,7}
      by application of
        Pre({2,5,6,7}) = {1,3,4}.
      Here rules are labelled as follows:
        1: app#(Cons(x,xs),ys) ->         
             c_1(app#(xs,ys))             
        2: app#(Nil(),ys) -> c_2()        
        3: goal#(xs) -> c_3(naiverev#(xs))
        4: naiverev#(Cons(x,xs)) ->       
             c_4(app#(naiverev(xs)        
                     ,Cons(x,Nil()))      
                ,naiverev#(xs))           
        5: naiverev#(Nil()) -> c_5()      
        6: notEmpty#(Cons(x,xs)) -> c_6() 
        7: notEmpty#(Nil()) -> c_7()      
*** 1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        goal#(xs) -> c_3(naiverev#(xs))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
      Strict TRS Rules:
        
      Weak DP Rules:
        app#(Nil(),ys) -> c_2()
        naiverev#(Nil()) -> c_5()
        notEmpty#(Cons(x,xs)) -> c_6()
        notEmpty#(Nil()) -> c_7()
      Weak TRS Rules:
        app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        app(Nil(),ys) -> ys
        naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
        naiverev(Nil()) -> Nil()
      Signature:
        {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
      Obligation:
        Innermost
        basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
           -->_1 app#(Nil(),ys) -> c_2():4
           -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
        
        2:S:goal#(xs) -> c_3(naiverev#(xs))
           -->_1 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):3
           -->_1 naiverev#(Nil()) -> c_5():5
        
        3:S:naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
           -->_2 naiverev#(Nil()) -> c_5():5
           -->_1 app#(Nil(),ys) -> c_2():4
           -->_2 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):3
           -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
        
        4:W:app#(Nil(),ys) -> c_2()
           
        
        5:W:naiverev#(Nil()) -> c_5()
           
        
        6:W:notEmpty#(Cons(x,xs)) -> c_6()
           
        
        7:W:notEmpty#(Nil()) -> c_7()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        7: notEmpty#(Nil()) -> c_7()     
        6: notEmpty#(Cons(x,xs)) -> c_6()
        5: naiverev#(Nil()) -> c_5()     
        4: app#(Nil(),ys) -> c_2()       
*** 1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        goal#(xs) -> c_3(naiverev#(xs))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        app(Nil(),ys) -> ys
        naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
        naiverev(Nil()) -> Nil()
      Signature:
        {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
      Obligation:
        Innermost
        basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
    Applied Processor:
      RemoveHeads
    Proof:
      Consider the dependency graph
      
      1:S:app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
         -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
      
      2:S:goal#(xs) -> c_3(naiverev#(xs))
         -->_1 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):3
      
      3:S:naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
         -->_2 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):3
         -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
      
      
      Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
      
      [(2,goal#(xs) -> c_3(naiverev#(xs)))]
*** 1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
        app(Nil(),ys) -> ys
        naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
        naiverev(Nil()) -> Nil()
      Signature:
        {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
      Obligation:
        Innermost
        basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
    Applied Processor:
      Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    Proof:
      We analyse the complexity of following sub-problems (R) and (S).
      Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
      
      Problem (R)
        Strict DP Rules:
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        Strict TRS Rules:
          
        Weak DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        Weak TRS Rules:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
          naiverev(Nil()) -> Nil()
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
      
      Problem (S)
        Strict DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        Strict TRS Rules:
          
        Weak DP Rules:
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        Weak TRS Rules:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
          naiverev(Nil()) -> Nil()
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        Strict TRS Rules:
          
        Weak DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        Weak TRS Rules:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
          naiverev(Nil()) -> Nil()
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
      Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: app#(Cons(x,xs),ys) ->
               c_1(app#(xs,ys))    
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
        Considered Problem:
          Strict DP Rules:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          Strict TRS Rules:
            
          Weak DP Rules:
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
          Weak TRS Rules:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
        Proof:
          We apply a polynomial interpretation of kind constructor-based(mixed(2)):
          The following argument positions are considered usable:
            uargs(c_1) = {1},
            uargs(c_4) = {1,2}
          
          Following symbols are considered usable:
            {app,naiverev,app#,goal#,naiverev#,notEmpty#}
          TcT has computed the following interpretation:
                 p(Cons) = 1 + x2         
                p(False) = 0              
                  p(Nil) = 0              
                 p(True) = 0              
                  p(app) = x1 + x2        
                 p(goal) = 1              
             p(naiverev) = x1             
             p(notEmpty) = 0              
                 p(app#) = x1 + 7*x1*x2   
                p(goal#) = 2*x1           
            p(naiverev#) = 1 + x1 + 5*x1^2
            p(notEmpty#) = 1 + x1^2       
                  p(c_1) = x1             
                  p(c_2) = 1              
                  p(c_3) = 0              
                  p(c_4) = 1 + x1 + x2    
                  p(c_5) = 1              
                  p(c_6) = 1              
                  p(c_7) = 1              
          
          Following rules are strictly oriented:
          app#(Cons(x,xs),ys) = 1 + xs + 7*xs*ys + 7*ys
                              > xs + 7*xs*ys           
                              = c_1(app#(xs,ys))       
          
          
          Following rules are (at-least) weakly oriented:
          naiverev#(Cons(x,xs)) =  7 + 11*xs + 5*xs^2             
                                >= 2 + 9*xs + 5*xs^2              
                                =  c_4(app#(naiverev(xs)          
                                           ,Cons(x,Nil()))        
                                      ,naiverev#(xs))             
          
             app(Cons(x,xs),ys) =  1 + xs + ys                    
                                >= 1 + xs + ys                    
                                =  Cons(x,app(xs,ys))             
          
                  app(Nil(),ys) =  ys                             
                                >= ys                             
                                =  ys                             
          
           naiverev(Cons(x,xs)) =  1 + xs                         
                                >= 1 + xs                         
                                =  app(naiverev(xs),Cons(x,Nil()))
          
                naiverev(Nil()) =  0                              
                                >= 0                              
                                =  Nil()                          
          
    *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
          Weak TRS Rules:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
          Weak TRS Rules:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
               -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
            
            2:W:naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
               -->_2 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):2
               -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            2: naiverev#(Cons(x,xs)) -> 
                 c_4(app#(naiverev(xs)  
                         ,Cons(x,Nil()))
                    ,naiverev#(xs))     
            1: app#(Cons(x,xs),ys) ->   
                 c_1(app#(xs,ys))       
    *** 1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
            naiverev(Nil()) -> Nil()
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).
    
  *** 1.1.1.1.1.1.2 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        Strict TRS Rules:
          
        Weak DP Rules:
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        Weak TRS Rules:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
          naiverev(Nil()) -> Nil()
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:S:naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):2
             -->_2 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):1
          
          2:W:app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: app#(Cons(x,xs),ys) ->
               c_1(app#(xs,ys))    
  *** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
          naiverev(Nil()) -> Nil()
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
      Applied Processor:
        SimplifyRHS
      Proof:
        Consider the dependency graph
          1:S:naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
             -->_2 naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
  *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          naiverev(Cons(x,xs)) -> app(naiverev(xs),Cons(x,Nil()))
          naiverev(Nil()) -> Nil()
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
      Applied Processor:
        UsableRules
      Proof:
        We replace rewrite rules by usable rules:
          naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
  *** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0}
        Obligation:
          Innermost
          basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
      Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: naiverev#(Cons(x,xs)) ->
               c_4(naiverev#(xs))    
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
        Proof:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            {app#,goal#,naiverev#,notEmpty#}
          TcT has computed the following interpretation:
                 p(Cons) = [1] x2 + [5] 
                p(False) = [2]          
                  p(Nil) = [0]          
                 p(True) = [0]          
                  p(app) = [8] x2 + [0] 
                 p(goal) = [1] x1 + [0] 
             p(naiverev) = [1] x1 + [0] 
             p(notEmpty) = [1] x1 + [1] 
                 p(app#) = [2] x2 + [0] 
                p(goal#) = [1]          
            p(naiverev#) = [4] x1 + [1] 
            p(notEmpty#) = [2] x1 + [2] 
                  p(c_1) = [1] x1 + [1] 
                  p(c_2) = [1]          
                  p(c_3) = [1] x1 + [2] 
                  p(c_4) = [1] x1 + [15]
                  p(c_5) = [1]          
                  p(c_6) = [1]          
                  p(c_7) = [1]          
          
          Following rules are strictly oriented:
          naiverev#(Cons(x,xs)) = [4] xs + [21]     
                                > [4] xs + [16]     
                                = c_4(naiverev#(xs))
          
          
          Following rules are (at-least) weakly oriented:
          
    *** 1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
          Weak TRS Rules:
            
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
          Weak TRS Rules:
            
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
               -->_1 naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs)):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: naiverev#(Cons(x,xs)) ->
                 c_4(naiverev#(xs))    
    *** 1.1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            
          Signature:
            {app/2,goal/1,naiverev/1,notEmpty/1,app#/2,goal#/1,naiverev#/1,notEmpty#/1} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0}
          Obligation:
            Innermost
            basic terms: {app#,goal#,naiverev#,notEmpty#}/{Cons,False,Nil,True}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).