WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
            select(Cons(x,xs)) -> selects(x,Nil(),xs)
            select(Nil()) -> Nil()
            selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil())
            selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs)))
                                                    ,selects(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp,select,selects} and constructors {Cons,Nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
          revapp#(Nil(),rest) -> c_2()
          select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
          select#(Nil()) -> c_4()
          selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
          selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
            revapp#(Nil(),rest) -> c_2()
            select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
            select#(Nil()) -> c_4()
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
            selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        - Weak TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
            select(Cons(x,xs)) -> selects(x,Nil(),xs)
            select(Nil()) -> Nil()
            selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil())
            selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs)))
                                                    ,selects(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,5,6}.
        Here rules are labelled as follows:
          1: revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
          2: revapp#(Nil(),rest) -> c_2()
          3: select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
          4: select#(Nil()) -> c_4()
          5: selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
          6: selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
            select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
            selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        - Weak DPs:
            revapp#(Nil(),rest) -> c_2()
            select#(Nil()) -> c_4()
        - Weak TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
            select(Cons(x,xs)) -> selects(x,Nil(),xs)
            select(Nil()) -> Nil()
            selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil())
            selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs)))
                                                    ,selects(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
             -->_1 revapp#(Nil(),rest) -> c_2():5
             -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1
          
          2:S:select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
             -->_1 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                           ,selects#(x,Cons(x',revprefix),xs)):4
             -->_1 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3
          
          3:S:selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
             -->_1 revapp#(Nil(),rest) -> c_2():5
             -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1
          
          4:S:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                      ,selects#(x,Cons(x',revprefix),xs))
             -->_1 revapp#(Nil(),rest) -> c_2():5
             -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                           ,selects#(x,Cons(x',revprefix),xs)):4
             -->_2 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3
             -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1
          
          5:W:revapp#(Nil(),rest) -> c_2()
             
          
          6:W:select#(Nil()) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: select#(Nil()) -> c_4()
          5: revapp#(Nil(),rest) -> c_2()
* Step 4: RemoveHeads WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
            select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
            selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        - Weak TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
            select(Cons(x,xs)) -> selects(x,Nil(),xs)
            select(Nil()) -> Nil()
            selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil())
            selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs)))
                                                    ,selects(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
           -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1
        
        2:S:select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs))
           -->_1 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                         ,selects#(x,Cons(x',revprefix),xs)):4
           -->_1 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3
        
        3:S:selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
           -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):1
        
        4:S:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                    ,selects#(x,Cons(x',revprefix),xs))
           -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                         ,selects#(x,Cons(x',revprefix),xs)):4
           -->_2 selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil())):3
           -->_1 revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest))):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,select#(Cons(x,xs)) -> c_3(selects#(x,Nil(),xs)))]
* Step 5: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
            selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        - Weak TRS:
            revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest))
            revapp(Nil(),rest) -> rest
            select(Cons(x,xs)) -> selects(x,Nil(),xs)
            select(Nil()) -> Nil()
            selects(x,revprefix,Nil()) -> Cons(Cons(x,revapp(revprefix,Nil())),Nil())
            selects(x',revprefix,Cons(x,xs)) -> Cons(Cons(x',revapp(revprefix,Cons(x,xs)))
                                                    ,selects(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
          selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
          selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
            selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + 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
          selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        and a lower component
          revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
          selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
        Further, following extension rules are added to the lower component.
          selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs))
          selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs)
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs)),selects#(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                      ,selects#(x,Cons(x',revprefix),xs))
             -->_2 selects#(x',revprefix,Cons(x,xs)) -> c_6(revapp#(revprefix,Cons(x,xs))
                                                           ,selects#(x,Cons(x',revprefix),xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs))
** Step 6.a:2: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs))
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          Cons :: ["A"(14) x "A"(14)] -(14)-> "A"(14)
          Cons :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          selects# :: ["A"(0) x "A"(0) x "A"(14)] -(0)-> "A"(0)
          c_6 :: ["A"(0)] -(0)-> "A"(14)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "Cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1)
          "c_6_A" :: ["A"(0)] -(0)-> "A"(1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          selects#(x',revprefix,Cons(x,xs)) -> c_6(selects#(x,Cons(x',revprefix),xs))
        2. Weak:
          

** Step 6.b:1: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
        - Weak DPs:
            selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs))
            selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs)
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          Cons :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          Nil :: [] -(0)-> "A"(0)
          revapp# :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          selects# :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0)
          c_1 :: ["A"(0)] -(0)-> "A"(0)
          c_5 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
        2. Weak:
          revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
** Step 6.b:2: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
        - Weak DPs:
            selects#(x,revprefix,Nil()) -> c_5(revapp#(revprefix,Nil()))
            selects#(x',revprefix,Cons(x,xs)) -> revapp#(revprefix,Cons(x,xs))
            selects#(x',revprefix,Cons(x,xs)) -> selects#(x,Cons(x',revprefix),xs)
        - Signature:
            {revapp/2,select/1,selects/3,revapp#/2,select#/1,selects#/3} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1
            ,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {revapp#,select#,selects#} and constructors {Cons,Nil}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          Cons :: ["A"(0) x "A"(13)] -(13)-> "A"(13)
          Cons :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          Nil :: [] -(0)-> "A"(13)
          Nil :: [] -(0)-> "A"(14)
          revapp# :: ["A"(13) x "A"(0)] -(3)-> "A"(0)
          selects# :: ["A"(0) x "A"(13) x "A"(13)] -(11)-> "A"(0)
          c_1 :: ["A"(0)] -(0)-> "A"(2)
          c_5 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          "Nil_A" :: [] -(0)-> "A"(1)
          "c_1_A" :: ["A"(0)] -(0)-> "A"(1)
          "c_5_A" :: ["A"(0)] -(0)-> "A"(1)
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          revapp#(Cons(x,xs),rest) -> c_1(revapp#(xs,Cons(x,rest)))
        2. Weak:
          

WORST_CASE(?,O(n^2))