* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + 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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          app(y,z){y -> Cons(x,y)} =
            app(Cons(x,y),z) ->^+ Cons(x,app(y,z))
              = C[app(y,z) = app(y,z){}]

** Step 1.b:1: DependencyPairs 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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        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.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - 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 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,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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        UsableRules
    + Details:
        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()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - 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 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        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()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak DPs:
            app#(Nil(),ys) -> c_2()
            naiverev#(Nil()) -> c_5()
            notEmpty#(Cons(x,xs)) -> c_6()
            notEmpty#(Nil()) -> c_7()
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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()
** Step 1.b:5: RemoveHeads WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        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)))]
** Step 1.b:6: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            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:
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        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 DPs:
              app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          - Weak DPs:
              naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
          - Weak 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()))
              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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
              ,False,Nil,True}
        
        Problem (S)
          - Strict DPs:
              naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
          - Weak DPs:
              app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          - Weak 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()))
              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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
              ,False,Nil,True}
*** Step 1.b:6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak DPs:
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {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}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} 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.
**** Step 1.b:6.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak DPs:
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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) = 1                
               p(True) = 1                
                p(app) = x1 + x2          
               p(goal) = 1 + x1^2         
           p(naiverev) = 2*x1             
           p(notEmpty) = 1 + x1 + x1^2    
               p(app#) = 2*x1 + x1*x2     
              p(goal#) = 2                
          p(naiverev#) = 4 + 3*x1 + 5*x1^2
          p(notEmpty#) = 2 + x1 + x1^2    
                p(c_1) = x1               
                p(c_2) = 0                
                p(c_3) = 0                
                p(c_4) = 1 + x1 + x2      
                p(c_5) = 0                
                p(c_6) = 0                
                p(c_7) = 0                
        
        Following rules are strictly oriented:
        app#(Cons(x,xs),ys) = 2 + 2*xs + xs*ys + ys
                            > 2*xs + xs*ys         
                            = c_1(app#(xs,ys))     
        
        
        Following rules are (at-least) weakly oriented:
        naiverev#(Cons(x,xs)) =  12 + 13*xs + 5*xs^2                                
                              >= 5 + 11*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) =  1 + ys                                             
                              >= ys                                                 
                              =  ys                                                 
        
         naiverev(Cons(x,xs)) =  2 + 2*xs                                           
                              >= 2 + 2*xs                                           
                              =  app(naiverev(xs),Cons(x,Nil()))                    
        
              naiverev(Nil()) =  2                                                  
                              >= 1                                                  
                              =  Nil()                                              
        
**** Step 1.b:6.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))
**** Step 1.b:6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak 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()))
            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 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).

*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        - Weak DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        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))
*** Step 1.b:6.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        - Weak 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()))
            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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
*** Step 1.b:6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        - 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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        - 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 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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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] x1 + [1] x2 + [8]
              p(False) = [0]                  
                p(Nil) = [0]                  
               p(True) = [2]                  
                p(app) = [1] x1 + [1] x2 + [0]
               p(goal) = [1] x1 + [8]         
           p(naiverev) = [8] x1 + [1]         
           p(notEmpty) = [1] x1 + [1]         
               p(app#) = [1] x1 + [2]         
              p(goal#) = [1] x1 + [1]         
          p(naiverev#) = [2] x1 + [0]         
          p(notEmpty#) = [8] x1 + [1]         
                p(c_1) = [2] x1 + [1]         
                p(c_2) = [0]                  
                p(c_3) = [1] x1 + [4]         
                p(c_4) = [1] x1 + [9]         
                p(c_5) = [0]                  
                p(c_6) = [4]                  
                p(c_7) = [1]                  
        
        Following rules are strictly oriented:
        naiverev#(Cons(x,xs)) = [2] x + [2] xs + [16]
                              > [2] xs + [9]         
                              = c_4(naiverev#(xs))   
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 1.b:6.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        - 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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            naiverev#(Cons(x,xs)) -> c_4(naiverev#(xs))
        - 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 runtime complexity wrt. defined symbols {app#,goal#,naiverev#,notEmpty#} and constructors {Cons
            ,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))
**** Step 1.b:6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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 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(Omega(n^1),O(n^2))