MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> subsets(xs)
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
        - Weak TRS:
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2} / {Cons/2,False/0,Nil/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal,mapconsapp,notEmpty,subsets
            ,subsets[Ite][True][Let]} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          goal#(xs) -> c_1(subsets#(xs))
          mapconsapp#(x,Nil(),rest) -> c_2()
          mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
          notEmpty#(Cons(x,xs)) -> c_4()
          notEmpty#(Nil()) -> c_5()
          subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          subsets#(Nil()) -> c_7()
        Weak DPs
          subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(subsets#(xs))
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            notEmpty#(Cons(x,xs)) -> c_4()
            notEmpty#(Nil()) -> c_5()
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets#(Nil()) -> c_7()
        - Weak DPs:
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            goal(xs) -> subsets(xs)
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mapconsapp(x,Nil(),rest) -> rest
          mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
          subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
          subsets(Nil()) -> Cons(Nil(),Nil())
          subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
          goal#(xs) -> c_1(subsets#(xs))
          mapconsapp#(x,Nil(),rest) -> c_2()
          mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
          notEmpty#(Cons(x,xs)) -> c_4()
          notEmpty#(Nil()) -> c_5()
          subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          subsets#(Nil()) -> c_7()
          subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(subsets#(xs))
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            notEmpty#(Cons(x,xs)) -> c_4()
            notEmpty#(Nil()) -> c_5()
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets#(Nil()) -> c_7()
        - Weak DPs:
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,5,7}
        by application of
          Pre({4,5,7}) = {1,6}.
        Here rules are labelled as follows:
          1: goal#(xs) -> c_1(subsets#(xs))
          2: mapconsapp#(x,Nil(),rest) -> c_2()
          3: mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
          4: notEmpty#(Cons(x,xs)) -> c_4()
          5: notEmpty#(Nil()) -> c_5()
          6: subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          7: subsets#(Nil()) -> c_7()
          8: subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(subsets#(xs))
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak DPs:
            notEmpty#(Cons(x,xs)) -> c_4()
            notEmpty#(Nil()) -> c_5()
            subsets#(Nil()) -> c_7()
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:goal#(xs) -> c_1(subsets#(xs))
             -->_1 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):4
             -->_1 subsets#(Nil()) -> c_7():7
          
          2:S:mapconsapp#(x,Nil(),rest) -> c_2()
             
          
          3:S:mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):3
             -->_1 mapconsapp#(x,Nil(),rest) -> c_2():2
          
          4:S:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
             -->_1 subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs)):8
             -->_2 subsets#(Nil()) -> c_7():7
             -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):4
          
          5:W:notEmpty#(Cons(x,xs)) -> c_4()
             
          
          6:W:notEmpty#(Nil()) -> c_5()
             
          
          7:W:subsets#(Nil()) -> c_7()
             
          
          8:W:subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):3
             -->_1 mapconsapp#(x,Nil(),rest) -> c_2():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: notEmpty#(Nil()) -> c_5()
          5: notEmpty#(Cons(x,xs)) -> c_4()
          7: subsets#(Nil()) -> c_7()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(subsets#(xs))
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak DPs:
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:goal#(xs) -> c_1(subsets#(xs))
           -->_1 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):4
        
        2:S:mapconsapp#(x,Nil(),rest) -> c_2()
           
        
        3:S:mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
           -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):3
           -->_1 mapconsapp#(x,Nil(),rest) -> c_2():2
        
        4:S:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
           -->_1 subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs)):8
           -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):4
        
        8:W:subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
           -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):3
           -->_1 mapconsapp#(x,Nil(),rest) -> c_2():2
        
        
        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).
        
        [(1,goal#(xs) -> c_1(subsets#(xs)))]
* Step 6: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak DPs:
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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:
              mapconsapp#(x,Nil(),rest) -> c_2()
          - Weak DPs:
              mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
              subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
              subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
          - Weak TRS:
              mapconsapp(x,Nil(),rest) -> rest
              mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
              subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
              subsets(Nil()) -> Cons(Nil(),Nil())
              subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
          - Signature:
              {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
              ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
              ,c_7/0,c_8/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
              ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
        
        Problem (S)
          - Strict DPs:
              mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
              subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          - Weak DPs:
              mapconsapp#(x,Nil(),rest) -> c_2()
              subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
          - Weak TRS:
              mapconsapp(x,Nil(),rest) -> rest
              mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
              subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
              subsets(Nil()) -> Cons(Nil(),Nil())
              subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
          - Signature:
              {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
              ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
              ,c_7/0,c_8/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
              ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mapconsapp#(x,Nil(),rest) -> c_2()
        - Weak DPs:
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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:
          2: mapconsapp#(x,Nil(),rest) -> c_2()
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mapconsapp#(x,Nil(),rest) -> c_2()
        - Weak DPs:
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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_3) = {1},
          uargs(c_6) = {1,2},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {goal#,mapconsapp#,notEmpty#,subsets#,subsets[Ite][True][Let]#}
        TcT has computed the following interpretation:
                              p(Cons) = [1] x2 + [8]         
                             p(False) = [0]                  
                               p(Nil) = [2]                  
                              p(True) = [4]                  
                              p(goal) = [1] x1 + [1]         
                        p(mapconsapp) = [2] x2 + [7]         
                          p(notEmpty) = [1] x1 + [0]         
                           p(subsets) = [0]                  
           p(subsets[Ite][True][Let]) = [0]                  
                             p(goal#) = [0]                  
                       p(mapconsapp#) = [4]                  
                         p(notEmpty#) = [1] x1 + [0]         
                          p(subsets#) = [2] x1 + [0]         
          p(subsets[Ite][True][Let]#) = [8]                  
                               p(c_1) = [1] x1 + [2]         
                               p(c_2) = [2]                  
                               p(c_3) = [1] x1 + [0]         
                               p(c_4) = [2]                  
                               p(c_5) = [1]                  
                               p(c_6) = [2] x1 + [1] x2 + [0]
                               p(c_7) = [1]                  
                               p(c_8) = [2] x1 + [0]         
        
        Following rules are strictly oriented:
        mapconsapp#(x,Nil(),rest) = [4]  
                                  > [2]  
                                  = c_2()
        
        
        Following rules are (at-least) weakly oriented:
                  mapconsapp#(x',Cons(x,xs),rest) =  [4]                                                               
                                                  >= [4]                                                               
                                                  =  c_3(mapconsapp#(x',xs,rest))                                      
        
                             subsets#(Cons(x,xs)) =  [2] xs + [16]                                                     
                                                  >= [2] xs + [16]                                                     
                                                  =  c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        
        subsets[Ite][True][Let]#(Cons(x,xs),subs) =  [8]                                                               
                                                  >= [8]                                                               
                                                  =  c_8(mapconsapp#(x,subs,subs))                                     
        
*** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mapconsapp#(x,Nil(),rest) -> c_2()
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mapconsapp#(x,Nil(),rest) -> c_2()
             
          
          2:W:mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):2
             -->_1 mapconsapp#(x,Nil(),rest) -> c_2():1
          
          3:W:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
             -->_1 subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs)):4
             -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):3
          
          4:W:subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):2
             -->_1 mapconsapp#(x,Nil(),rest) -> c_2():1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          4: subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
          2: mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
          1: mapconsapp#(x,Nil(),rest) -> c_2()
*** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak DPs:
            mapconsapp#(x,Nil(),rest) -> c_2()
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
             -->_1 mapconsapp#(x,Nil(),rest) -> c_2():3
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):1
          
          2:S:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
             -->_1 subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs)):4
             -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):2
          
          3:W:mapconsapp#(x,Nil(),rest) -> c_2()
             
          
          4:W:subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
             -->_1 mapconsapp#(x,Nil(),rest) -> c_2():3
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: mapconsapp#(x,Nil(),rest) -> c_2()
** Step 6.b:2: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak DPs:
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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:
              mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
          - Weak DPs:
              subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
              subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
          - Weak TRS:
              mapconsapp(x,Nil(),rest) -> rest
              mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
              subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
              subsets(Nil()) -> Cons(Nil(),Nil())
              subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
          - Signature:
              {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
              ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
              ,c_7/0,c_8/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
              ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
        
        Problem (S)
          - Strict DPs:
              subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          - Weak DPs:
              mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
              subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
          - Weak TRS:
              mapconsapp(x,Nil(),rest) -> rest
              mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
              subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
              subsets(Nil()) -> Cons(Nil(),Nil())
              subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
          - Signature:
              {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
              ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
              ,c_7/0,c_8/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
              ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
*** Step 6.b:2.a:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
        - Weak DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        and a lower component
          mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
          subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        Further, following extension rules are added to the lower component.
          subsets#(Cons(x,xs)) -> subsets#(xs)
          subsets#(Cons(x,xs)) -> subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs))
**** Step 6.b:2.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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: subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
          
        The strictly oriented rules are moved into the weak component.
***** Step 6.b:2.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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_6) = {1,2}
        
        Following symbols are considered usable:
          {goal#,mapconsapp#,notEmpty#,subsets#,subsets[Ite][True][Let]#}
        TcT has computed the following interpretation:
                              p(Cons) = [1] x1 + [1] x2 + [4]
                             p(False) = [0]                  
                               p(Nil) = [0]                  
                              p(True) = [0]                  
                              p(goal) = [0]                  
                        p(mapconsapp) = [0]                  
                          p(notEmpty) = [2]                  
                           p(subsets) = [4]                  
           p(subsets[Ite][True][Let]) = [7] x1 + [0]         
                             p(goal#) = [1]                  
                       p(mapconsapp#) = [1] x1 + [2]         
                         p(notEmpty#) = [1] x1 + [2]         
                          p(subsets#) = [1] x1 + [0]         
          p(subsets[Ite][True][Let]#) = [0]                  
                               p(c_1) = [2]                  
                               p(c_2) = [4]                  
                               p(c_3) = [0]                  
                               p(c_4) = [1]                  
                               p(c_5) = [2]                  
                               p(c_6) = [8] x1 + [1] x2 + [0]
                               p(c_7) = [0]                  
                               p(c_8) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        subsets#(Cons(x,xs)) = [1] x + [1] xs + [4]                                              
                             > [1] xs + [0]                                                      
                             = c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        
        
        Following rules are (at-least) weakly oriented:
        
***** Step 6.b:2.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 6.b:2.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
             -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
***** Step 6.b:2.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 6.b:2.a:1.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
      - Weak DPs:
          subsets#(Cons(x,xs)) -> subsets#(xs)
          subsets#(Cons(x,xs)) -> subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs))
          subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
      - Weak TRS:
          mapconsapp(x,Nil(),rest) -> rest
          mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
          subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
          subsets(Nil()) -> Cons(Nil(),Nil())
          subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
      - Signature:
          {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
          ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
          ,c_7/0,c_8/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
          ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 6.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak DPs:
            mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
            subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
             -->_1 subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs)):3
             -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):1
          
          2:W:mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):2
          
          3:W:subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
             -->_1 mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: subsets[Ite][True][Let]#(Cons(x,xs),subs) -> c_8(mapconsapp#(x,subs,subs))
          2: mapconsapp#(x',Cons(x,xs),rest) -> c_3(mapconsapp#(x',xs,rest))
*** Step 6.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs))
             -->_2 subsets#(Cons(x,xs)) -> c_6(subsets[Ite][True][Let]#(Cons(x,xs),subsets(xs)),subsets#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
*** Step 6.b:2.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
        - Weak TRS:
            mapconsapp(x,Nil(),rest) -> rest
            mapconsapp(x',Cons(x,xs),rest) -> Cons(Cons(x',x),mapconsapp(x',xs,rest))
            subsets(Cons(x,xs)) -> subsets[Ite][True][Let](Cons(x,xs),subsets(xs))
            subsets(Nil()) -> Cons(Nil(),Nil())
            subsets[Ite][True][Let](Cons(x,xs),subs) -> mapconsapp(x,subs,subs)
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
*** Step 6.b:2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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: subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.b:2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} 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_6) = {1}
        
        Following symbols are considered usable:
          {goal#,mapconsapp#,notEmpty#,subsets#,subsets[Ite][True][Let]#}
        TcT has computed the following interpretation:
                              p(Cons) = [1] x2 + [8]         
                             p(False) = [4]                  
                               p(Nil) = [1]                  
                              p(True) = [1]                  
                              p(goal) = [2] x1 + [8]         
                        p(mapconsapp) = [1] x2 + [1] x3 + [1]
                          p(notEmpty) = [8]                  
                           p(subsets) = [1]                  
           p(subsets[Ite][True][Let]) = [1] x1 + [2]         
                             p(goal#) = [1] x1 + [1]         
                       p(mapconsapp#) = [2] x2 + [1] x3 + [1]
                         p(notEmpty#) = [0]                  
                          p(subsets#) = [1] x1 + [2]         
          p(subsets[Ite][True][Let]#) = [2] x1 + [1] x2 + [0]
                               p(c_1) = [1] x1 + [4]         
                               p(c_2) = [1]                  
                               p(c_3) = [1] x1 + [1]         
                               p(c_4) = [2]                  
                               p(c_5) = [1]                  
                               p(c_6) = [1] x1 + [0]         
                               p(c_7) = [1]                  
                               p(c_8) = [1]                  
        
        Following rules are strictly oriented:
        subsets#(Cons(x,xs)) = [1] xs + [10]    
                             > [1] xs + [2]     
                             = c_6(subsets#(xs))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 6.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.b:2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
             -->_1 subsets#(Cons(x,xs)) -> c_6(subsets#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: subsets#(Cons(x,xs)) -> c_6(subsets#(xs))
**** Step 6.b:2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {goal/1,mapconsapp/3,notEmpty/1,subsets/1,subsets[Ite][True][Let]/2,goal#/1,mapconsapp#/3,notEmpty#/1
            ,subsets#/1,subsets[Ite][True][Let]#/2} / {Cons/2,False/0,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {goal#,mapconsapp#,notEmpty#,subsets#
            ,subsets[Ite][True][Let]#} and constructors {Cons,False,Nil,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE