* 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
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,>,app,notEmpty,part,part[False][Ite],part[Ite],qs
            ,quicksort} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,>,app,notEmpty,part,part[False][Ite],part[Ite],qs
            ,quicksort} and constructors {0,Cons,False,Nil,S,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
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1} / {0/0,Cons/2,False/0
            ,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,>,app,notEmpty,part,part[False][Ite],part[Ite],qs
            ,quicksort} and constructors {0,Cons,False,Nil,S,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()
          notEmpty#(Cons(x,xs)) -> c_3()
          notEmpty#(Nil()) -> c_4()
          part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
          part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
          qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
          quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                ,part#(x,Cons(x',xs),Nil(),Nil()))
          quicksort#(Cons(x,Nil())) -> c_9()
          quicksort#(Nil()) -> c_10()
        Weak DPs
          <#(x,0()) -> c_11()
          <#(0(),S(y)) -> c_12()
          <#(S(x),S(y)) -> c_13(<#(x,y))
          >#(0(),y) -> c_14()
          >#(S(x),0()) -> c_15()
          >#(S(x),S(y)) -> c_16(>#(x,y))
          part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2),<#(x',x))
          part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            app#(Nil(),ys) -> c_2()
            notEmpty#(Cons(x,xs)) -> c_3()
            notEmpty#(Nil()) -> c_4()
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
            quicksort#(Cons(x,Nil())) -> c_9()
            quicksort#(Nil()) -> c_10()
        - Weak DPs:
            <#(x,0()) -> c_11()
            <#(0(),S(y)) -> c_12()
            <#(S(x),S(y)) -> c_13(<#(x,y))
            >#(0(),y) -> c_14()
            >#(S(x),0()) -> c_15()
            >#(S(x),S(y)) -> c_16(>#(x,y))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2),<#(x',x))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/2,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          <(x,0()) -> False()
          <(0(),S(y)) -> True()
          <(S(x),S(y)) -> <(x,y)
          >(0(),y) -> False()
          >(S(x),0()) -> True()
          >(S(x),S(y)) -> >(x,y)
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
          part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
          part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
          part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
          part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
          part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
          qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
          quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
          quicksort(Cons(x,Nil())) -> Cons(x,Nil())
          quicksort(Nil()) -> Nil()
          <#(x,0()) -> c_11()
          <#(0(),S(y)) -> c_12()
          <#(S(x),S(y)) -> c_13(<#(x,y))
          >#(0(),y) -> c_14()
          >#(S(x),0()) -> c_15()
          >#(S(x),S(y)) -> c_16(>#(x,y))
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          app#(Nil(),ys) -> c_2()
          notEmpty#(Cons(x,xs)) -> c_3()
          notEmpty#(Nil()) -> c_4()
          part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
          part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
          part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2),<#(x',x))
          part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
          quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                ,part#(x,Cons(x',xs),Nil(),Nil()))
          quicksort#(Cons(x,Nil())) -> c_9()
          quicksort#(Nil()) -> c_10()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            app#(Nil(),ys) -> c_2()
            notEmpty#(Cons(x,xs)) -> c_3()
            notEmpty#(Nil()) -> c_4()
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
            quicksort#(Cons(x,Nil())) -> c_9()
            quicksort#(Nil()) -> c_10()
        - Weak DPs:
            <#(x,0()) -> c_11()
            <#(0(),S(y)) -> c_12()
            <#(S(x),S(y)) -> c_13(<#(x,y))
            >#(0(),y) -> c_14()
            >#(S(x),0()) -> c_15()
            >#(S(x),S(y)) -> c_16(>#(x,y))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2),<#(x',x))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/2,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,9,10}
        by application of
          Pre({2,3,4,9,10}) = {1,5,7}.
        Here rules are labelled as follows:
          1: app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          2: app#(Nil(),ys) -> c_2()
          3: notEmpty#(Cons(x,xs)) -> c_3()
          4: notEmpty#(Nil()) -> c_4()
          5: part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
          6: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
          7: qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
          8: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
          9: quicksort#(Cons(x,Nil())) -> c_9()
          10: quicksort#(Nil()) -> c_10()
          11: <#(x,0()) -> c_11()
          12: <#(0(),S(y)) -> c_12()
          13: <#(S(x),S(y)) -> c_13(<#(x,y))
          14: >#(0(),y) -> c_14()
          15: >#(S(x),0()) -> c_15()
          16: >#(S(x),S(y)) -> c_16(>#(x,y))
          17: part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          18: part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          19: part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)
                                                               ,<#(x',x))
          20: part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            <#(x,0()) -> c_11()
            <#(0(),S(y)) -> c_12()
            <#(S(x),S(y)) -> c_13(<#(x,y))
            >#(0(),y) -> c_14()
            >#(S(x),0()) -> c_15()
            >#(S(x),S(y)) -> c_16(>#(x,y))
            app#(Nil(),ys) -> c_2()
            notEmpty#(Cons(x,xs)) -> c_3()
            notEmpty#(Nil()) -> c_4()
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2),<#(x',x))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            quicksort#(Cons(x,Nil())) -> c_9()
            quicksort#(Nil()) -> c_10()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/2,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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():12
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          2:S:part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
             -->_1 app#(Nil(),ys) -> c_2():12
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          3:S:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):18
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)
                                                                    ,<#(x',x)):17
             -->_2 >#(S(x),S(y)) -> c_16(>#(x,y)):11
             -->_2 >#(S(x),0()) -> c_15():10
             -->_2 >#(0(),y) -> c_14():9
          
          4:S:qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
             -->_2 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):5
             -->_2 quicksort#(Nil()) -> c_10():20
             -->_2 quicksort#(Cons(x,Nil())) -> c_9():19
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          5:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs)):4
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
          
          6:W:<#(x,0()) -> c_11()
             
          
          7:W:<#(0(),S(y)) -> c_12()
             
          
          8:W:<#(S(x),S(y)) -> c_13(<#(x,y))
             -->_1 <#(S(x),S(y)) -> c_13(<#(x,y)):8
             -->_1 <#(0(),S(y)) -> c_12():7
             -->_1 <#(x,0()) -> c_11():6
          
          9:W:>#(0(),y) -> c_14()
             
          
          10:W:>#(S(x),0()) -> c_15()
             
          
          11:W:>#(S(x),S(y)) -> c_16(>#(x,y))
             -->_1 >#(S(x),S(y)) -> c_16(>#(x,y)):11
             -->_1 >#(S(x),0()) -> c_15():10
             -->_1 >#(0(),y) -> c_14():9
          
          12:W:app#(Nil(),ys) -> c_2()
             
          
          13:W:notEmpty#(Cons(x,xs)) -> c_3()
             
          
          14:W:notEmpty#(Nil()) -> c_4()
             
          
          15:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          16:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          17:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)
                                                                ,<#(x',x))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):16
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):15
             -->_2 <#(S(x),S(y)) -> c_13(<#(x,y)):8
             -->_2 <#(0(),S(y)) -> c_12():7
             -->_2 <#(x,0()) -> c_11():6
          
          18:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          19:W:quicksort#(Cons(x,Nil())) -> c_9()
             
          
          20:W:quicksort#(Nil()) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          14: notEmpty#(Nil()) -> c_4()
          13: notEmpty#(Cons(x,xs)) -> c_3()
          19: quicksort#(Cons(x,Nil())) -> c_9()
          20: quicksort#(Nil()) -> c_10()
          11: >#(S(x),S(y)) -> c_16(>#(x,y))
          9: >#(0(),y) -> c_14()
          10: >#(S(x),0()) -> c_15()
          8: <#(S(x),S(y)) -> c_13(<#(x,y))
          6: <#(x,0()) -> c_11()
          7: <#(0(),S(y)) -> c_12()
          12: app#(Nil(),ys) -> c_2()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2),<#(x',x))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/2,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/2,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + 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:part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          3:S:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):18
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)
                                                                    ,<#(x',x)):17
          
          4:S:qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
             -->_2 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):5
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          5:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs)):4
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
          
          15:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          16:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          17:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)
                                                                ,<#(x',x))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):16
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):15
          
          18:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2),>#(x',x)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
** Step 1.b:6: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          - Weak DPs:
              part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
              part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
              part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
              part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
              part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
              part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
              qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
              quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              >(0(),y) -> False()
              >(S(x),0()) -> True()
              >(S(x),S(y)) -> >(x,y)
              app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
              app(Nil(),ys) -> ys
              part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
              part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
              part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
              part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
              part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
              part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
              qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
              quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
              quicksort(Cons(x,Nil())) -> Cons(x,Nil())
              quicksort(Nil()) -> Nil()
          - Signature:
              {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
              ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
              ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
              ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
              ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
              part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
              qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
              quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
          - Weak DPs:
              app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
              part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
              part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
              part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
              part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              >(0(),y) -> False()
              >(S(x),0()) -> True()
              >(S(x),S(y)) -> >(x,y)
              app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
              app(Nil(),ys) -> ys
              part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
              part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
              part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
              part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
              part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
              part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
              qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
              quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
              quicksort(Cons(x,Nil())) -> Cons(x,Nil())
              quicksort(Nil()) -> Nil()
          - Signature:
              {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
              ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
              ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
              ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
              ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
*** Step 1.b:6.a:1: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak DPs:
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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
          qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
          quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                ,part#(x,Cons(x',xs),Nil(),Nil()))
        and a lower component
          app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
          part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        Further, following extension rules are added to the lower component.
          qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
          qs#(x',Cons(x,xs)) -> quicksort#(xs)
          quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
          quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
**** Step 1.b:6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
          
        Consider the set of all dependency pairs
          1: qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
          2: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
***** Step 1.b:6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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_7) = {1,2},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {app,part,part[False][Ite],part[Ite],<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#,qs#
          ,quicksort#}
        TcT has computed the following interpretation:
                          p(0) = [0]                           
                          p(<) = [0]                           
                          p(>) = [0]                           
                       p(Cons) = [1] x1 + [1] x2 + [2]         
                      p(False) = [0]                           
                        p(Nil) = [0]                           
                          p(S) = [4]                           
                       p(True) = [0]                           
                        p(app) = [1] x1 + [1] x2 + [0]         
                   p(notEmpty) = [1]                           
                       p(part) = [1] x2 + [1] x3 + [1] x4 + [0]
           p(part[False][Ite]) = [1] x3 + [1] x4 + [1] x5 + [0]
                  p(part[Ite]) = [1] x3 + [1] x4 + [1] x5 + [0]
                         p(qs) = [5]                           
                  p(quicksort) = [0]                           
                         p(<#) = [2] x2 + [1]                  
                         p(>#) = [0]                           
                       p(app#) = [1]                           
                  p(notEmpty#) = [4]                           
                      p(part#) = [4] x3 + [0]                  
          p(part[False][Ite]#) = [1] x2 + [1] x4 + [1] x5 + [1]
                 p(part[Ite]#) = [4] x2 + [2] x3 + [0]         
                        p(qs#) = [1] x2 + [0]                  
                 p(quicksort#) = [1] x1 + [1]                  
                        p(c_1) = [4]                           
                        p(c_2) = [1]                           
                        p(c_3) = [0]                           
                        p(c_4) = [0]                           
                        p(c_5) = [4]                           
                        p(c_6) = [0]                           
                        p(c_7) = [1] x1 + [1] x2 + [0]         
                        p(c_8) = [1] x1 + [2]                  
                        p(c_9) = [0]                           
                       p(c_10) = [1]                           
                       p(c_11) = [4]                           
                       p(c_12) = [1]                           
                       p(c_13) = [0]                           
                       p(c_14) = [1]                           
                       p(c_15) = [0]                           
                       p(c_16) = [1] x1 + [0]                  
                       p(c_17) = [1] x1 + [0]                  
                       p(c_18) = [4] x1 + [0]                  
                       p(c_19) = [4] x1 + [1]                  
                       p(c_20) = [4] x1 + [1]                  
        
        Following rules are strictly oriented:
        quicksort#(Cons(x,Cons(x',xs))) = [1] x + [1] x' + [1] xs + [5]                                               
                                        > [1] x' + [1] xs + [4]                                                       
                                        = c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())),part#(x,Cons(x',xs),Nil(),Nil()))
        
        
        Following rules are (at-least) weakly oriented:
                                     qs#(x',Cons(x,xs)) =  [1] x + [1] xs + [2]                                          
                                                        >= [1] xs + [2]                                                  
                                                        =  c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(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                                                            
        
                                  part(x,Nil(),xs1,xs2) =  [1] xs1 + [1] xs2 + [0]                                       
                                                        >= [1] xs1 + [1] xs2 + [0]                                       
                                                        =  app(xs1,xs2)                                                  
        
                            part(x',Cons(x,xs),xs1,xs2) =  [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        >= [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        =  part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)                      
        
        part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [0]                              
                                                        =  part(x',xs,xs1,xs2)                                           
        
         part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        >= [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        =  part(x',xs,xs1,Cons(x,xs2))                                   
        
               part[Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        >= [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        =  part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)               
        
                part[Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        >= [1] x + [1] xs + [1] xs1 + [1] xs2 + [2]                      
                                                        =  part(x',xs,Cons(x,xs1),xs2)                                   
        
***** Step 1.b:6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
        - Weak DPs:
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
             -->_2 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):2
          
          2:W:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
          2: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
***** Step 1.b:6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak DPs:
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
        - Weak DPs:
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_5) = {1},
          uargs(c_6) = {1},
          uargs(c_17) = {1},
          uargs(c_18) = {1},
          uargs(c_19) = {1},
          uargs(c_20) = {1}
        
        Following symbols are considered usable:
          {app,part,part[False][Ite],part[Ite],<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#,qs#
          ,quicksort#}
        TcT has computed the following interpretation:
                          p(0) = [0]                                 
                                 [1]                                 
                          p(<) = [0 1] x1 + [1 0] x2 + [1]           
                                 [0 0]      [0 0]      [0]           
                          p(>) = [0 1] x2 + [0]                      
                                 [0 0]      [0]                      
                       p(Cons) = [0 0] x2 + [0]                      
                                 [0 1]      [1]                      
                      p(False) = [0]                                 
                                 [0]                                 
                        p(Nil) = [0]                                 
                                 [0]                                 
                          p(S) = [0 1] x1 + [0]                      
                                 [0 0]      [1]                      
                       p(True) = [0]                                 
                                 [0]                                 
                        p(app) = [0 0] x1 + [1 0] x2 + [0]           
                                 [0 1]      [0 1]      [0]           
                   p(notEmpty) = [0]                                 
                                 [0]                                 
                       p(part) = [0 0] x2 + [0 0] x3 + [1 0] x4 + [0]
                                 [0 1]      [0 1]      [0 1]      [0]
           p(part[False][Ite]) = [0 0] x3 + [0 0] x4 + [1 0] x5 + [0]
                                 [0 1]      [0 1]      [0 1]      [0]
                  p(part[Ite]) = [0 0] x3 + [0 0] x4 + [1 0] x5 + [0]
                                 [0 1]      [0 1]      [0 1]      [0]
                         p(qs) = [0]                                 
                                 [0]                                 
                  p(quicksort) = [1]                                 
                                 [0]                                 
                         p(<#) = [0]                                 
                                 [0]                                 
                         p(>#) = [0]                                 
                                 [0]                                 
                       p(app#) = [0 1] x1 + [0]                      
                                 [0 1]      [1]                      
                  p(notEmpty#) = [0]                                 
                                 [0]                                 
                      p(part#) = [0 1] x2 + [0 1] x3 + [0 0] x4 + [0]
                                 [0 0]      [1 0]      [1 0]      [0]
          p(part[False][Ite]#) = [0 1] x3 + [0 1] x4 + [0]           
                                 [0 1]      [0 1]      [0]           
                 p(part[Ite]#) = [0 1] x3 + [0 1] x4 + [0]           
                                 [0 1]      [0 1]      [1]           
                        p(qs#) = [0 1] x2 + [0]                      
                                 [0 1]      [1]                      
                 p(quicksort#) = [0 1] x1 + [0]                      
                                 [0 1]      [1]                      
                        p(c_1) = [1 0] x1 + [0]                      
                                 [0 0]      [0]                      
                        p(c_2) = [0]                                 
                                 [0]                                 
                        p(c_3) = [0]                                 
                                 [0]                                 
                        p(c_4) = [0]                                 
                                 [0]                                 
                        p(c_5) = [1 0] x1 + [0]                      
                                 [0 0]      [0]                      
                        p(c_6) = [1 0] x1 + [0]                      
                                 [0 0]      [0]                      
                        p(c_7) = [0]                                 
                                 [0]                                 
                        p(c_8) = [0]                                 
                                 [0]                                 
                        p(c_9) = [0]                                 
                                 [0]                                 
                       p(c_10) = [0]                                 
                                 [0]                                 
                       p(c_11) = [0]                                 
                                 [0]                                 
                       p(c_12) = [0]                                 
                                 [0]                                 
                       p(c_13) = [0]                                 
                                 [0]                                 
                       p(c_14) = [0]                                 
                                 [0]                                 
                       p(c_15) = [0]                                 
                                 [0]                                 
                       p(c_16) = [0]                                 
                                 [0]                                 
                       p(c_17) = [1 0] x1 + [1]                      
                                 [1 0]      [1]                      
                       p(c_18) = [1 0] x1 + [0]                      
                                 [1 0]      [0]                      
                       p(c_19) = [1 0] x1 + [0]                      
                                 [0 1]      [0]                      
                       p(c_20) = [1 0] x1 + [0]                      
                                 [1 0]      [1]                      
        
        Following rules are strictly oriented:
        app#(Cons(x,xs),ys) = [0 1] xs + [1]  
                              [0 1]      [2]  
                            > [0 1] xs + [0]  
                              [0 0]      [0]  
                            = c_1(app#(xs,ys))
        
        
        Following rules are (at-least) weakly oriented:
                                  part#(x,Nil(),xs1,xs2) =  [0 1] xs1 + [0 0] xs2 + [0]                           
                                                            [1 0]       [1 0]       [0]                           
                                                         >= [0 1] xs1 + [0]                                       
                                                            [0 0]       [0]                                       
                                                         =  c_5(app#(xs1,xs2))                                    
        
                            part#(x',Cons(x,xs),xs1,xs2) =  [0 1] xs + [0 1] xs1 + [0 0] xs2 + [1]                
                                                            [0 0]      [1 0]       [1 0]       [0]                
                                                         >= [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 0]      [0 0]       [0]                            
                                                         =  c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))        
        
        part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) =  [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [1]                            
                                                         >= [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [1]                            
                                                         =  c_17(part#(x',xs,xs1,xs2))                            
        
         part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) =  [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [1]                            
                                                         >= [0 1] xs + [0 1] xs1 + [0]                            
                                                            [0 1]      [0 1]       [0]                            
                                                         =  c_18(part#(x',xs,xs1,Cons(x,xs2)))                    
        
               part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) =  [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [2]                            
                                                         >= [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [1]                            
                                                         =  c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
        
                part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) =  [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [2]                            
                                                         >= [0 1] xs + [0 1] xs1 + [1]                            
                                                            [0 1]      [0 1]       [2]                            
                                                         =  c_20(part#(x',xs,Cons(x,xs1),xs2))                    
        
                                      qs#(x',Cons(x,xs)) =  [0 1] xs + [1]                                        
                                                            [0 1]      [2]                                        
                                                         >= [1]                                                   
                                                            [2]                                                   
                                                         =  app#(Cons(x,Nil()),Cons(x',quicksort(xs)))            
        
                                      qs#(x',Cons(x,xs)) =  [0 1] xs + [1]                                        
                                                            [0 1]      [2]                                        
                                                         >= [0 1] xs + [0]                                        
                                                            [0 1]      [1]                                        
                                                         =  quicksort#(xs)                                        
        
                         quicksort#(Cons(x,Cons(x',xs))) =  [0 1] xs + [2]                                        
                                                            [0 1]      [3]                                        
                                                         >= [0 1] xs + [1]                                        
                                                            [0 0]      [0]                                        
                                                         =  part#(x,Cons(x',xs),Nil(),Nil())                      
        
                         quicksort#(Cons(x,Cons(x',xs))) =  [0 1] xs + [2]                                        
                                                            [0 1]      [3]                                        
                                                         >= [0 1] xs + [1]                                        
                                                            [0 1]      [2]                                        
                                                         =  qs#(x,part(x,Cons(x',xs),Nil(),Nil()))                
        
                                      app(Cons(x,xs),ys) =  [0 0] xs + [1 0] ys + [0]                             
                                                            [0 1]      [0 1]      [1]                             
                                                         >= [0 0] xs + [0 0] ys + [0]                             
                                                            [0 1]      [0 1]      [1]                             
                                                         =  Cons(x,app(xs,ys))                                    
        
                                           app(Nil(),ys) =  [1 0] ys + [0]                                        
                                                            [0 1]      [0]                                        
                                                         >= [1 0] ys + [0]                                        
                                                            [0 1]      [0]                                        
                                                         =  ys                                                    
        
                                   part(x,Nil(),xs1,xs2) =  [0 0] xs1 + [1 0] xs2 + [0]                           
                                                            [0 1]       [0 1]       [0]                           
                                                         >= [0 0] xs1 + [1 0] xs2 + [0]                           
                                                            [0 1]       [0 1]       [0]                           
                                                         =  app(xs1,xs2)                                          
        
                             part(x',Cons(x,xs),xs1,xs2) =  [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         >= [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         =  part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)              
        
         part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) =  [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         >= [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [0]                
                                                         =  part(x',xs,xs1,xs2)                                   
        
          part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) =  [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         >= [0 0] xs + [0 0] xs1 + [0 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         =  part(x',xs,xs1,Cons(x,xs2))                           
        
                part[Ite](False(),x',Cons(x,xs),xs1,xs2) =  [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         >= [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         =  part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)       
        
                 part[Ite](True(),x',Cons(x,xs),xs1,xs2) =  [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         >= [0 0] xs + [0 0] xs1 + [1 0] xs2 + [0]                
                                                            [0 1]      [0 1]       [0 1]       [1]                
                                                         =  part(x',xs,Cons(x,xs1),xs2)                           
        
***** Step 1.b:6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          2:W:part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          3:W:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):7
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):6
          
          4:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          5:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          6:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):5
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):4
          
          7:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):2
          
          8:W:qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):1
          
          9:W:qs#(x',Cons(x,xs)) -> quicksort#(xs)
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil())):11
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil()):10
          
          10:W:quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
          
          11:W:quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> quicksort#(xs):9
             -->_1 qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs))):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: qs#(x',Cons(x,xs)) -> quicksort#(xs)
          11: quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
          10: quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
          8: qs#(x',Cons(x,xs)) -> app#(Cons(x,Nil()),Cons(x',quicksort(xs)))
          3: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          7: part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          5: part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          6: part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          4: part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          2: part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
          1: app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
***** Step 1.b:6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):5
          
          2:S:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):9
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):8
          
          3:S:qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):5
             -->_2 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):4
          
          4:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs)):3
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
          
          5:W:app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
             -->_1 app#(Cons(x,xs),ys) -> c_1(app#(xs,ys)):5
          
          6:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):1
          
          7:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):1
          
          8:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):7
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):6
          
          9:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: app#(Cons(x,xs),ys) -> c_1(app#(xs,ys))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/2,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2))
             
          
          2:S:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):9
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):8
          
          3:S:qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs))
             -->_2 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):4
          
          4:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(app#(Cons(x,Nil()),Cons(x',quicksort(xs))),quicksort#(xs)):3
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
          
          6:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):1
          
          7:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):1
          
          8:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):7
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):6
          
          9:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5(app#(xs1,xs2)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          part#(x,Nil(),xs1,xs2) -> c_5()
          qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
*** Step 1.b:6.b:3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
            qs(x',Cons(x,xs)) -> app(Cons(x,Nil()),Cons(x',quicksort(xs)))
            quicksort(Cons(x,Cons(x',xs))) -> qs(x,part(x,Cons(x',xs),Nil(),Nil()))
            quicksort(Cons(x,Nil())) -> Cons(x,Nil())
            quicksort(Nil()) -> Nil()
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          <(x,0()) -> False()
          <(0(),S(y)) -> True()
          <(S(x),S(y)) -> <(x,y)
          >(0(),y) -> False()
          >(S(x),0()) -> True()
          >(S(x),S(y)) -> >(x,y)
          app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
          app(Nil(),ys) -> ys
          part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
          part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
          part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
          part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
          part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
          part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
          part#(x,Nil(),xs1,xs2) -> c_5()
          part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                ,part#(x,Cons(x',xs),Nil(),Nil()))
*** Step 1.b:6.b:4: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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:
              part#(x,Nil(),xs1,xs2) -> c_5()
          - Weak DPs:
              part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
              part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
              part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
              part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
              part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
              qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
              quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              >(0(),y) -> False()
              >(S(x),0()) -> True()
              >(S(x),S(y)) -> >(x,y)
              app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
              app(Nil(),ys) -> ys
              part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
              part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
              part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
              part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
              part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
              part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
          - Signature:
              {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
              ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
              ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
              ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
              ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
              qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
              quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
          - Weak DPs:
              part#(x,Nil(),xs1,xs2) -> c_5()
              part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
              part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
              part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
              part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              >(0(),y) -> False()
              >(S(x),0()) -> True()
              >(S(x),S(y)) -> >(x,y)
              app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
              app(Nil(),ys) -> ys
              part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
              part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
              part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
              part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
              part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
              part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
          - Signature:
              {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
              ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
              ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
              ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
              ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
**** Step 1.b:6.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
        - Weak DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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: part#(x,Nil(),xs1,xs2) -> c_5()
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
        - Weak DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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},
          uargs(c_7) = {1},
          uargs(c_8) = {1,2},
          uargs(c_17) = {1},
          uargs(c_18) = {1},
          uargs(c_19) = {1},
          uargs(c_20) = {1}
        
        Following symbols are considered usable:
          {app,part,part[False][Ite],part[Ite],<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#,qs#
          ,quicksort#}
        TcT has computed the following interpretation:
                          p(0) = [0]                           
                          p(<) = [4] x1 + [0]                  
                          p(>) = [0]                           
                       p(Cons) = [1] x2 + [4]                  
                      p(False) = [0]                           
                        p(Nil) = [3]                           
                          p(S) = [1]                           
                       p(True) = [0]                           
                        p(app) = [1] x1 + [1] x2 + [0]         
                   p(notEmpty) = [1]                           
                       p(part) = [1] x2 + [1] x3 + [1] x4 + [1]
           p(part[False][Ite]) = [1] x3 + [1] x4 + [1] x5 + [1]
                  p(part[Ite]) = [1] x3 + [1] x4 + [1] x5 + [1]
                         p(qs) = [4] x2 + [0]                  
                  p(quicksort) = [0]                           
                         p(<#) = [2] x1 + [1]                  
                         p(>#) = [1] x1 + [2] x2 + [0]         
                       p(app#) = [4] x1 + [1] x2 + [1]         
                  p(notEmpty#) = [2] x1 + [2]                  
                      p(part#) = [1]                           
          p(part[False][Ite]#) = [1]                           
                 p(part[Ite]#) = [1]                           
                        p(qs#) = [1] x2 + [0]                  
                 p(quicksort#) = [1] x1 + [4]                  
                        p(c_1) = [2] x1 + [4]                  
                        p(c_2) = [1]                           
                        p(c_3) = [2]                           
                        p(c_4) = [4]                           
                        p(c_5) = [0]                           
                        p(c_6) = [1] x1 + [0]                  
                        p(c_7) = [1] x1 + [0]                  
                        p(c_8) = [1] x1 + [1] x2 + [0]         
                        p(c_9) = [1]                           
                       p(c_10) = [0]                           
                       p(c_11) = [1]                           
                       p(c_12) = [1]                           
                       p(c_13) = [0]                           
                       p(c_14) = [0]                           
                       p(c_15) = [1]                           
                       p(c_16) = [1]                           
                       p(c_17) = [1] x1 + [0]                  
                       p(c_18) = [1] x1 + [0]                  
                       p(c_19) = [1] x1 + [0]                  
                       p(c_20) = [1] x1 + [0]                  
        
        Following rules are strictly oriented:
        part#(x,Nil(),xs1,xs2) = [1]  
                               > [0]  
                               = c_5()
        
        
        Following rules are (at-least) weakly oriented:
                            part#(x',Cons(x,xs),xs1,xs2) =  [1]                                                                         
                                                         >= [1]                                                                         
                                                         =  c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))                              
        
        part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) =  [1]                                                                         
                                                         >= [1]                                                                         
                                                         =  c_17(part#(x',xs,xs1,xs2))                                                  
        
         part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) =  [1]                                                                         
                                                         >= [1]                                                                         
                                                         =  c_18(part#(x',xs,xs1,Cons(x,xs2)))                                          
        
               part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) =  [1]                                                                         
                                                         >= [1]                                                                         
                                                         =  c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))                      
        
                part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) =  [1]                                                                         
                                                         >= [1]                                                                         
                                                         =  c_20(part#(x',xs,Cons(x,xs1),xs2))                                          
        
                                      qs#(x',Cons(x,xs)) =  [1] xs + [4]                                                                
                                                         >= [1] xs + [4]                                                                
                                                         =  c_7(quicksort#(xs))                                                         
        
                         quicksort#(Cons(x,Cons(x',xs))) =  [1] xs + [12]                                                               
                                                         >= [1] xs + [12]                                                               
                                                         =  c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())),part#(x,Cons(x',xs),Nil(),Nil()))
        
                                      app(Cons(x,xs),ys) =  [1] xs + [1] ys + [4]                                                       
                                                         >= [1] xs + [1] ys + [4]                                                       
                                                         =  Cons(x,app(xs,ys))                                                          
        
                                           app(Nil(),ys) =  [1] ys + [3]                                                                
                                                         >= [1] ys + [0]                                                                
                                                         =  ys                                                                          
        
                                   part(x,Nil(),xs1,xs2) =  [1] xs1 + [1] xs2 + [4]                                                     
                                                         >= [1] xs1 + [1] xs2 + [0]                                                     
                                                         =  app(xs1,xs2)                                                                
        
                             part(x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         =  part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)                                    
        
         part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [1]                                            
                                                         =  part(x',xs,xs1,xs2)                                                         
        
          part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         =  part(x',xs,xs1,Cons(x,xs2))                                                 
        
                part[Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         =  part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)                             
        
                 part[Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [5]                                            
                                                         =  part(x',xs,Cons(x,xs1),xs2)                                                 
        
***** Step 1.b:6.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:part#(x,Nil(),xs1,xs2) -> c_5()
             
          
          2:W:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):6
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):5
          
          3:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5():1
          
          4:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5():1
          
          5:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):4
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):3
          
          6:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5():1
          
          7:W:qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):8
          
          8:W:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs)):7
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          8: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
          2: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          6: part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          4: part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          5: part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          3: part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          1: part#(x,Nil(),xs1,xs2) -> c_5()
***** Step 1.b:6.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part#(x,Nil(),xs1,xs2) -> c_5()
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):8
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):7
          
          2:S:qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):3
          
          3:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs)):2
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          4:W:part#(x,Nil(),xs1,xs2) -> c_5()
             
          
          5:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5():4
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          6:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5():4
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          7:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):6
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):5
          
          8:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x,Nil(),xs1,xs2) -> c_5():4
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: part#(x,Nil(),xs1,xs2) -> c_5()
**** Step 1.b:6.b:4.b:2: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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:
              part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          - Weak DPs:
              part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
              part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
              part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
              part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
              qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
              quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              >(0(),y) -> False()
              >(S(x),0()) -> True()
              >(S(x),S(y)) -> >(x,y)
              app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
              app(Nil(),ys) -> ys
              part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
              part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
              part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
              part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
              part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
              part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
          - Signature:
              {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
              ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
              ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
              ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
              ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
              quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
          - Weak DPs:
              part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
              part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
              part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
              part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
              part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              >(0(),y) -> False()
              >(S(x),0()) -> True()
              >(S(x),S(y)) -> >(x,y)
              app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
              app(Nil(),ys) -> ys
              part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
              part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
              part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
              part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
              part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
              part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
          - Signature:
              {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
              ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
              ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
              ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
              ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
***** Step 1.b:6.b:4.b:2.a:1: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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
          qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                ,part#(x,Cons(x',xs),Nil(),Nil()))
        and a lower component
          part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        Further, following extension rules are added to the lower component.
          qs#(x',Cons(x,xs)) -> quicksort#(xs)
          quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
          quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
****** Step 1.b:6.b:4.b:2.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
          
        Consider the set of all dependency pairs
          1: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
          2: qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
******* Step 1.b:6.b:4.b:2.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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_7) = {1},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {app,part,part[False][Ite],part[Ite],<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#,qs#
          ,quicksort#}
        TcT has computed the following interpretation:
                          p(0) = [1]                                    
                          p(<) = [1] x1 + [0]                           
                          p(>) = [0]                                    
                       p(Cons) = [1] x2 + [4]                           
                      p(False) = [0]                                    
                        p(Nil) = [0]                                    
                          p(S) = [1] x1 + [4]                           
                       p(True) = [0]                                    
                        p(app) = [1] x1 + [1] x2 + [0]                  
                   p(notEmpty) = [1] x1 + [0]                           
                       p(part) = [1] x2 + [1] x3 + [1] x4 + [4]         
           p(part[False][Ite]) = [1] x3 + [1] x4 + [1] x5 + [4]         
                  p(part[Ite]) = [1] x3 + [1] x4 + [1] x5 + [4]         
                         p(qs) = [0]                                    
                  p(quicksort) = [1]                                    
                         p(<#) = [1] x1 + [1] x2 + [1]                  
                         p(>#) = [1] x1 + [2]                           
                       p(app#) = [4] x1 + [1] x2 + [2]                  
                  p(notEmpty#) = [1] x1 + [1]                           
                      p(part#) = [2] x3 + [4]                           
          p(part[False][Ite]#) = [4] x4 + [1] x5 + [0]                  
                 p(part[Ite]#) = [1] x1 + [1] x2 + [2] x3 + [1] x5 + [1]
                        p(qs#) = [1] x2 + [0]                           
                 p(quicksort#) = [1] x1 + [1]                           
                        p(c_1) = [1] x1 + [0]                           
                        p(c_2) = [1]                                    
                        p(c_3) = [0]                                    
                        p(c_4) = [1]                                    
                        p(c_5) = [1]                                    
                        p(c_6) = [1] x1 + [2]                           
                        p(c_7) = [1] x1 + [3]                           
                        p(c_8) = [1] x1 + [0]                           
                        p(c_9) = [0]                                    
                       p(c_10) = [0]                                    
                       p(c_11) = [2]                                    
                       p(c_12) = [0]                                    
                       p(c_13) = [2] x1 + [0]                           
                       p(c_14) = [1]                                    
                       p(c_15) = [0]                                    
                       p(c_16) = [1] x1 + [0]                           
                       p(c_17) = [2] x1 + [4]                           
                       p(c_18) = [1] x1 + [0]                           
                       p(c_19) = [0]                                    
                       p(c_20) = [1]                                    
        
        Following rules are strictly oriented:
        quicksort#(Cons(x,Cons(x',xs))) = [1] xs + [9]                                                                
                                        > [1] xs + [8]                                                                
                                        = c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())),part#(x,Cons(x',xs),Nil(),Nil()))
        
        
        Following rules are (at-least) weakly oriented:
                                     qs#(x',Cons(x,xs)) =  [1] xs + [4]                                   
                                                        >= [1] xs + [4]                                   
                                                        =  c_7(quicksort#(xs))                            
        
                                     app(Cons(x,xs),ys) =  [1] xs + [1] ys + [4]                          
                                                        >= [1] xs + [1] ys + [4]                          
                                                        =  Cons(x,app(xs,ys))                             
        
                                          app(Nil(),ys) =  [1] ys + [0]                                   
                                                        >= [1] ys + [0]                                   
                                                        =  ys                                             
        
                                  part(x,Nil(),xs1,xs2) =  [1] xs1 + [1] xs2 + [4]                        
                                                        >= [1] xs1 + [1] xs2 + [0]                        
                                                        =  app(xs1,xs2)                                   
        
                            part(x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        =  part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)       
        
        part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [4]               
                                                        =  part(x',xs,xs1,xs2)                            
        
         part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        =  part(x',xs,xs1,Cons(x,xs2))                    
        
               part[Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        =  part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
        
                part[Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [8]               
                                                        =  part(x',xs,Cons(x,xs1),xs2)                    
        
******* Step 1.b:6.b:4.b:2.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******* Step 1.b:6.b:4.b:2.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):2
          
          2:W:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          2: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                   ,part#(x,Cons(x',xs),Nil(),Nil()))
******* Step 1.b:6.b:4.b:2.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

****** Step 1.b:6.b:4.b:2.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          
        Consider the set of all dependency pairs
          1: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          2: part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
          3: part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          4: part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          5: part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          6: qs#(x',Cons(x,xs)) -> quicksort#(xs)
          7: quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
          8: quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3,4,5}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
******* Step 1.b:6.b:4.b:2.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
        - Weak DPs:
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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},
          uargs(c_17) = {1},
          uargs(c_18) = {1},
          uargs(c_19) = {1},
          uargs(c_20) = {1}
        
        Following symbols are considered usable:
          {app,part,part[False][Ite],part[Ite],<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#,qs#
          ,quicksort#}
        TcT has computed the following interpretation:
                          p(0) = [0]                           
                          p(<) = [0]                           
                          p(>) = [0]                           
                       p(Cons) = [1] x2 + [1]                  
                      p(False) = [0]                           
                        p(Nil) = [0]                           
                          p(S) = [0]                           
                       p(True) = [0]                           
                        p(app) = [1] x1 + [1] x2 + [0]         
                   p(notEmpty) = [4] x1 + [2]                  
                       p(part) = [1] x2 + [1] x3 + [1] x4 + [0]
           p(part[False][Ite]) = [1] x3 + [1] x4 + [1] x5 + [0]
                  p(part[Ite]) = [1] x3 + [1] x4 + [1] x5 + [0]
                         p(qs) = [2] x2 + [1]                  
                  p(quicksort) = [1] x1 + [0]                  
                         p(<#) = [0]                           
                         p(>#) = [4] x1 + [1] x2 + [0]         
                       p(app#) = [1] x2 + [1]                  
                  p(notEmpty#) = [1]                           
                      p(part#) = [4] x2 + [3] x3 + [1] x4 + [5]
          p(part[False][Ite]#) = [4] x3 + [3] x4 + [1] x5 + [3]
                 p(part[Ite]#) = [4] x3 + [3] x4 + [1] x5 + [4]
                        p(qs#) = [6] x2 + [6]                  
                 p(quicksort#) = [6] x1 + [0]                  
                        p(c_1) = [2]                           
                        p(c_2) = [1]                           
                        p(c_3) = [0]                           
                        p(c_4) = [1]                           
                        p(c_5) = [0]                           
                        p(c_6) = [1] x1 + [0]                  
                        p(c_7) = [2]                           
                        p(c_8) = [1] x1 + [1] x2 + [0]         
                        p(c_9) = [2]                           
                       p(c_10) = [1]                           
                       p(c_11) = [0]                           
                       p(c_12) = [0]                           
                       p(c_13) = [1] x1 + [1]                  
                       p(c_14) = [1]                           
                       p(c_15) = [1]                           
                       p(c_16) = [4]                           
                       p(c_17) = [1] x1 + [2]                  
                       p(c_18) = [1] x1 + [0]                  
                       p(c_19) = [1] x1 + [0]                  
                       p(c_20) = [1] x1 + [0]                  
        
        Following rules are strictly oriented:
        part#(x',Cons(x,xs),xs1,xs2) = [4] xs + [3] xs1 + [1] xs2 + [9]              
                                     > [4] xs + [3] xs1 + [1] xs2 + [8]              
                                     = c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
        
        
        Following rules are (at-least) weakly oriented:
        part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) =  [4] xs + [3] xs1 + [1] xs2 + [7]                      
                                                         >= [4] xs + [3] xs1 + [1] xs2 + [7]                      
                                                         =  c_17(part#(x',xs,xs1,xs2))                            
        
         part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) =  [4] xs + [3] xs1 + [1] xs2 + [7]                      
                                                         >= [4] xs + [3] xs1 + [1] xs2 + [6]                      
                                                         =  c_18(part#(x',xs,xs1,Cons(x,xs2)))                    
        
               part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) =  [4] xs + [3] xs1 + [1] xs2 + [8]                      
                                                         >= [4] xs + [3] xs1 + [1] xs2 + [7]                      
                                                         =  c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
        
                part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) =  [4] xs + [3] xs1 + [1] xs2 + [8]                      
                                                         >= [4] xs + [3] xs1 + [1] xs2 + [8]                      
                                                         =  c_20(part#(x',xs,Cons(x,xs1),xs2))                    
        
                                      qs#(x',Cons(x,xs)) =  [6] xs + [12]                                         
                                                         >= [6] xs + [0]                                          
                                                         =  quicksort#(xs)                                        
        
                         quicksort#(Cons(x,Cons(x',xs))) =  [6] xs + [12]                                         
                                                         >= [4] xs + [9]                                          
                                                         =  part#(x,Cons(x',xs),Nil(),Nil())                      
        
                         quicksort#(Cons(x,Cons(x',xs))) =  [6] xs + [12]                                         
                                                         >= [6] xs + [12]                                         
                                                         =  qs#(x,part(x,Cons(x',xs),Nil(),Nil()))                
        
                                      app(Cons(x,xs),ys) =  [1] xs + [1] ys + [1]                                 
                                                         >= [1] xs + [1] ys + [1]                                 
                                                         =  Cons(x,app(xs,ys))                                    
        
                                           app(Nil(),ys) =  [1] ys + [0]                                          
                                                         >= [1] ys + [0]                                          
                                                         =  ys                                                    
        
                                   part(x,Nil(),xs1,xs2) =  [1] xs1 + [1] xs2 + [0]                               
                                                         >= [1] xs1 + [1] xs2 + [0]                               
                                                         =  app(xs1,xs2)                                          
        
                             part(x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         =  part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)              
        
         part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [0]                      
                                                         =  part(x',xs,xs1,xs2)                                   
        
          part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         =  part(x',xs,xs1,Cons(x,xs2))                           
        
                part[Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         =  part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)       
        
                 part[Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         >= [1] xs + [1] xs1 + [1] xs2 + [1]                      
                                                         =  part(x',xs,Cons(x,xs1),xs2)                           
        
******* Step 1.b:6.b:4.b:2.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******* Step 1.b:6.b:4.b:2.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
            qs#(x',Cons(x,xs)) -> quicksort#(xs)
            quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
            quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):5
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):4
          
          2:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          3:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          4:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):3
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):2
          
          5:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          6:W:qs#(x',Cons(x,xs)) -> quicksort#(xs)
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil())):8
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil()):7
          
          7:W:quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):1
          
          8:W:quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> quicksort#(xs):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: qs#(x',Cons(x,xs)) -> quicksort#(xs)
          8: quicksort#(Cons(x,Cons(x',xs))) -> qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
          7: quicksort#(Cons(x,Cons(x',xs))) -> part#(x,Cons(x',xs),Nil(),Nil())
          1: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          5: part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          3: part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          4: part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          2: part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
******* Step 1.b:6.b:4.b:2.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:6.b:4.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak DPs:
            part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
            part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
            part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
            part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
            part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):2
          
          2:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_2 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
             -->_1 qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs)):1
          
          3:W:part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2)):7
             -->_1 part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2)):6
          
          4:W:part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
          
          5:W:part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
          
          6:W:part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
             -->_1 part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2))):5
             -->_1 part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2)):4
          
          7:W:part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
             -->_1 part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: part#(x',Cons(x,xs),xs1,xs2) -> c_6(part[Ite]#(>(x',x),x',Cons(x,xs),xs1,xs2))
          7: part[Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_20(part#(x',xs,Cons(x,xs1),xs2))
          5: part[False][Ite]#(True(),x',Cons(x,xs),xs1,xs2) -> c_18(part#(x',xs,xs1,Cons(x,xs2)))
          6: part[Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_19(part[False][Ite]#(<(x',x),x',Cons(x,xs),xs1,xs2))
          4: part[False][Ite]#(False(),x',Cons(x,xs),xs1,xs2) -> c_17(part#(x',xs,xs1,xs2))
***** Step 1.b:6.b:4.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                  ,part#(x,Cons(x',xs),Nil(),Nil()))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/2,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                         ,part#(x,Cons(x',xs),Nil(),Nil())):2
          
          2:S:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))
                                                    ,part#(x,Cons(x',xs),Nil(),Nil()))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
***** Step 1.b:6.b:4.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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: qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          
        Consider the set of all dependency pairs
          1: qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          2: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:6.b:4.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,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_7) = {1},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {>,app,part,part[False][Ite],part[Ite],<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#,qs#
          ,quicksort#}
        TcT has computed the following interpretation:
                          p(0) = [4]                           
                          p(<) = [0]                           
                          p(>) = [0]                           
                       p(Cons) = [1] x2 + [4]                  
                      p(False) = [0]                           
                        p(Nil) = [3]                           
                          p(S) = [0]                           
                       p(True) = [0]                           
                        p(app) = [1] x1 + [1] x2 + [0]         
                   p(notEmpty) = [1]                           
                       p(part) = [1] x2 + [1] x3 + [1] x4 + [1]
           p(part[False][Ite]) = [1] x3 + [1] x4 + [1] x5 + [1]
                  p(part[Ite]) = [1] x3 + [1] x4 + [1] x5 + [1]
                         p(qs) = [1] x1 + [0]                  
                  p(quicksort) = [1]                           
                         p(<#) = [2] x2 + [0]                  
                         p(>#) = [1] x1 + [0]                  
                       p(app#) = [2] x2 + [1]                  
                  p(notEmpty#) = [2]                           
                      p(part#) = [2] x1 + [1] x4 + [0]         
          p(part[False][Ite]#) = [2] x2 + [2]                  
                 p(part[Ite]#) = [2] x1 + [1] x2 + [0]         
                        p(qs#) = [1] x2 + [0]                  
                 p(quicksort#) = [1] x1 + [3]                  
                        p(c_1) = [2]                           
                        p(c_2) = [4]                           
                        p(c_3) = [0]                           
                        p(c_4) = [0]                           
                        p(c_5) = [0]                           
                        p(c_6) = [4] x1 + [1]                  
                        p(c_7) = [1] x1 + [0]                  
                        p(c_8) = [1] x1 + [0]                  
                        p(c_9) = [1]                           
                       p(c_10) = [0]                           
                       p(c_11) = [1]                           
                       p(c_12) = [0]                           
                       p(c_13) = [1]                           
                       p(c_14) = [1]                           
                       p(c_15) = [0]                           
                       p(c_16) = [4]                           
                       p(c_17) = [4] x1 + [1]                  
                       p(c_18) = [2] x1 + [0]                  
                       p(c_19) = [1] x1 + [0]                  
                       p(c_20) = [4]                           
        
        Following rules are strictly oriented:
        qs#(x',Cons(x,xs)) = [1] xs + [4]       
                           > [1] xs + [3]       
                           = c_7(quicksort#(xs))
        
        
        Following rules are (at-least) weakly oriented:
                        quicksort#(Cons(x,Cons(x',xs))) =  [1] xs + [11]                                  
                                                        >= [1] xs + [11]                                  
                                                        =  c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))    
        
                                               >(0(),y) =  [0]                                            
                                                        >= [0]                                            
                                                        =  False()                                        
        
                                            >(S(x),0()) =  [0]                                            
                                                        >= [0]                                            
                                                        =  True()                                         
        
                                           >(S(x),S(y)) =  [0]                                            
                                                        >= [0]                                            
                                                        =  >(x,y)                                         
        
                                     app(Cons(x,xs),ys) =  [1] xs + [1] ys + [4]                          
                                                        >= [1] xs + [1] ys + [4]                          
                                                        =  Cons(x,app(xs,ys))                             
        
                                          app(Nil(),ys) =  [1] ys + [3]                                   
                                                        >= [1] ys + [0]                                   
                                                        =  ys                                             
        
                                  part(x,Nil(),xs1,xs2) =  [1] xs1 + [1] xs2 + [4]                        
                                                        >= [1] xs1 + [1] xs2 + [0]                        
                                                        =  app(xs1,xs2)                                   
        
                            part(x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        =  part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)       
        
        part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [1]               
                                                        =  part(x',xs,xs1,xs2)                            
        
         part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        =  part(x',xs,xs1,Cons(x,xs2))                    
        
               part[Ite](False(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        =  part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
        
                part[Ite](True(),x',Cons(x,xs),xs1,xs2) =  [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        >= [1] xs + [1] xs1 + [1] xs2 + [5]               
                                                        =  part(x',xs,Cons(x,xs1),xs2)                    
        
****** Step 1.b:6.b:4.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:6.b:4.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
            quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
             -->_1 quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil()))):2
          
          2:W:quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
             -->_1 qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: qs#(x',Cons(x,xs)) -> c_7(quicksort#(xs))
          2: quicksort#(Cons(x,Cons(x',xs))) -> c_8(qs#(x,part(x,Cons(x',xs),Nil(),Nil())))
****** Step 1.b:6.b:4.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            >(0(),y) -> False()
            >(S(x),0()) -> True()
            >(S(x),S(y)) -> >(x,y)
            app(Cons(x,xs),ys) -> Cons(x,app(xs,ys))
            app(Nil(),ys) -> ys
            part(x,Nil(),xs1,xs2) -> app(xs1,xs2)
            part(x',Cons(x,xs),xs1,xs2) -> part[Ite](>(x',x),x',Cons(x,xs),xs1,xs2)
            part[False][Ite](False(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,xs2)
            part[False][Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,xs1,Cons(x,xs2))
            part[Ite](False(),x',Cons(x,xs),xs1,xs2) -> part[False][Ite](<(x',x),x',Cons(x,xs),xs1,xs2)
            part[Ite](True(),x',Cons(x,xs),xs1,xs2) -> part(x',xs,Cons(x,xs1),xs2)
        - Signature:
            {/2,app/2,notEmpty/1,part/4,part[False][Ite]/5,part[Ite]/5,qs/2,quicksort/1,<#/2,>#/2,app#/2
            ,notEmpty#/1,part#/4,part[False][Ite]#/5,part[Ite]#/5,qs#/2,quicksort#/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0
            ,c_16/1,c_17/1,c_18/1,c_19/1,c_20/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,>#,app#,notEmpty#,part#,part[False][Ite]#,part[Ite]#
            ,qs#,quicksort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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