/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),O(n^2))
* 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: Sum.  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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: 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: 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
            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:
        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:3: 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
            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:
        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:4: 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
            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:
        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:5: UsableRules.  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
            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))
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
** Step 1.b:6: DecomposeDG.  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:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          naiverev#(Cons(x,xs)) -> c_4(app#(naiverev(xs),Cons(x,Nil())),naiverev#(xs))
        and a lower component
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        Further, following extension rules are added to the lower component.
          naiverev#(Cons(x,xs)) -> app#(naiverev(xs),Cons(x,Nil()))
          naiverev#(Cons(x,xs)) -> naiverev#(xs)
*** Step 1.b:6.a:1: 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.a:2: 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.a:3: 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 any 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.a:4: EmptyProcessor.  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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:6.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak DPs:
            naiverev#(Cons(x,xs)) -> app#(naiverev(xs),Cons(x,Nil()))
            naiverev#(Cons(x,xs)) -> 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:
        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(c_1) = {1}
        
        Following symbols are considered usable:
          {app,naiverev,app#,goal#,naiverev#,notEmpty#}
        TcT has computed the following interpretation:
               p(Cons) = [1] x1 + [1] x2 + [2]
              p(False) = [1]                  
                p(Nil) = [0]                  
               p(True) = [1]                  
                p(app) = [1] x1 + [1] x2 + [0]
               p(goal) = [0]                  
           p(naiverev) = [1] x1 + [0]         
           p(notEmpty) = [2]                  
               p(app#) = [8] x1 + [0]         
              p(goal#) = [1]                  
          p(naiverev#) = [12] x1 + [5]        
          p(notEmpty#) = [1] x1 + [4]         
                p(c_1) = [1] x1 + [0]         
                p(c_2) = [0]                  
                p(c_3) = [2] x1 + [8]         
                p(c_4) = [4] x1 + [2] x2 + [0]
                p(c_5) = [8]                  
                p(c_6) = [2]                  
                p(c_7) = [1]                  
        
        Following rules are strictly oriented:
        app#(Cons(x,xs),ys) = [8] x + [8] xs + [16]
                            > [8] xs + [0]         
                            = c_1(app#(xs,ys))     
        
        
        Following rules are (at-least) weakly oriented:
        naiverev#(Cons(x,xs)) =  [12] x + [12] xs + [29]         
                              >= [8] xs + [0]                    
                              =  app#(naiverev(xs),Cons(x,Nil()))
        
        naiverev#(Cons(x,xs)) =  [12] x + [12] xs + [29]         
                              >= [12] xs + [5]                   
                              =  naiverev#(xs)                   
        
           app(Cons(x,xs),ys) =  [1] x + [1] xs + [1] ys + [2]   
                              >= [1] x + [1] xs + [1] ys + [2]   
                              =  Cons(x,app(xs,ys))              
        
                app(Nil(),ys) =  [1] ys + [0]                    
                              >= [1] ys + [0]                    
                              =  ys                              
        
         naiverev(Cons(x,xs)) =  [1] x + [1] xs + [2]            
                              >= [1] x + [1] xs + [2]            
                              =  app(naiverev(xs),Cons(x,Nil())) 
        
              naiverev(Nil()) =  [0]                             
                              >= [0]                             
                              =  Nil()                           
        
*** Step 1.b:6.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            naiverev#(Cons(x,xs)) -> app#(naiverev(xs),Cons(x,Nil()))
            naiverev#(Cons(x,xs)) -> 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))