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: Failure 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:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE